\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 166, pp. 1--27.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/166\hfil Linear elliptic and parabolic PDEs]
{Linear elliptic and parabolic PDE\MakeLowercase{s} with nonlinear mixed boundary conditions
and spatial heterogeneities}

\author[S. Cano-Casanova \hfil EJDE-2018/166\hfilneg]
{Santiago Cano-Casanova}

\address{Santiago Cano-Casanova \newline
Grupo Din\'amica No Lineal,
Departamento de Matem\'atica Aplicada,
Universidad Pontificia Comillas,
Alberto Aguilera 25, 28015-Madrid, Spain}
\email{scano@comillas.edu}

\dedicatory{Communicated by Ratnasingham Shivaji}

\thanks{Submitted January 12, 2018. Published September 12, 2018.}
\subjclass[2010]{35J65, 35J25,35B09, 35B35, 35B40}
\keywords{Nonlinear mixed boundary conditions; positive solutions; 
\hfill\break\indent  spatial heterogeneities; nonlinear flux with arbitrary sign;
 blow up in finite time;
\hfill\break\indent elliptic and parabolic boundary value problems}

\begin{abstract}
 This article concerns the positive solutions of a boundary-value problem
 constituted by a linear elliptic partial differential equation, subject to
 nonlinear mixed boundary conditions containing spatial heterogeneities with
 arbitrary sign along the boundary. The results obtained in this work provide
 us the global bifurcation diagram of positive solutions, the pointing behavior
 of them when the parameters change and the dynamics of the positive solutions
 of the associated parabolic problem. The main contribution of this paper is to
 give general results about existence, uniqueness, stability and pointing
 behavior of positive solutions, for boundary-value problems with nonlinear
 boundary conditions of mixed type containing spatial heterogeneities.
 The main technical tools used to develop the mathematical analysis are local
 and global bifurcation, monotonicity techniques, the Characterization of the
 Strong Maximum Principle given by Amann and L\'opez-G\'omez \cite{ALG98}, %[5]
 blow up arguments and some of the techniques used in the previous works
 \cite{LGSL98,LG00, WW75,MY93}. % [19,20,33,34]
 The results obtained in this paper are the natural continuation of the previous
 ones in \cite{CC14}. %[11]
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{Section1}

In this article we consider the boundary-value problem with nonlinear mixed
boundary conditions and spatial heterogeneities given by
\begin{equation}\label{1}
\begin{gathered}
-\Delta u=\lambda u \quad\text{in } \Omega\,,\\
u=0 \quad\text{on } \Gamma_0\,,\\
\partial u+V(x)u=\gamma b(x)u^q \quad\text{on } \Gamma_1\,, q>1\,,
\end{gathered}
\end{equation}
where:
\begin{itemize}
\item[(i)] $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$ of class
${\mathcal C}^2$, with boundary $\partial \Omega=\Gamma_0 \cup \Gamma_1$,
where $\Gamma_0$ and $\Gamma_1$ are disjoint open and closed subsets of
$\partial \Omega$;

\item[(ii)] $-\Delta$ stands for the minus Laplacian operator in $\mathbb{R}^N$
and $\lambda \in \mathbb{R}$ is the bifurcation parameter;

\item[(iii)] the spatial heterogeneities on the boundary come given by the
potentials $V, b \in {\mathcal C}(\Gamma_1)$, where $b>0$ on $\Gamma_1$ and
$V$ possesses arbitrary sign in each point $x \in \Gamma_1$;

\item[(iv)] $\partial u (x)$ stands for the outer normal derivative of $u$ at
$x \in \Gamma_1$, and $\gamma \in \mathbb{R}$.
\end{itemize}

This work is devoted to analyzing the structure of the set of positive solutions
of \eqref{1} depending on the sign of the parameter $\gamma \in \mathbb{R}$
 on the nonlinear mixed boundary conditions, to ascertain the pointing
 behavior of positive solutions of \eqref{1} when $\gamma<0$ and $\lambda$
changes, and to obtain the dynamics of the positive solutions of the
associated parabolic problem to \eqref{1} depending on the values of
the bifurcation parameter $\lambda \in \mathbb{R}$ and on the sign of
the parameter $\gamma \in \mathbb{R}$.

By a \emph{positive solution} of \eqref{1} we will mean any couple
$(\lambda,u) \in \mathbb{R}\times W_p^2(\Omega)$ for some $p>N$, with $u>0$
in $\Omega$ satisfying \eqref{1}. It should be noted that
$W_p^2(\Omega) \subset {\mathcal C}^{2-\frac{N}{p}}(\bar{\Omega})$ and that
any function $u\in W_p^2(\Omega)$, $p>N$ is a.e. twice differentiable
 (cf.\ \cite[Theorem VIII.1]{St}). We will say that a positive solution
$(\lambda,u)$ of \eqref{1} is \emph{strongly positive} in $\Omega$,
 and we will denote it by $u\gg 0$, if $u(x)>0$ for all $x \in \Omega \cup \Gamma_1$
and $\partial u(x)<0$ for all $x \in \Gamma_0$.


In the particular case when $\gamma=0$, \eqref{1} becomes in a linear boundary-value
 problem which exhibits vertical bifurcation to positive solutions from the
trivial branch $(\lambda,u)=(\lambda,0)$ at a unique value of $\lambda$ which
will be denoted by $\sigma_1$. The results obtained along this work will
show that the sign of the parameter $\gamma$ plays a crucial role in our problem.
Indeed, we will see that although the partial differential equation of \eqref{1}
is linear, in the particular case when $\gamma<0$ the structure of the set
of positive solutions of $\eqref{1}$ is a typical structure of a sublinear problem,
whereas in the particular case when $\gamma>0$ it is the typical structure of
a superlinear problem. The same occurs in the study of the stability of the
positive solutions of \eqref{1}.

The main techniques used to carry out the mathematical analysis are monotonicity
techniques, local and global bifurcation, blow up arguments and some of the
techniques used in the previous works \cite{WW75,MY93,LGSL98,LG00}.

Hereinafter, for each $V\in {\mathcal C}(\Gamma_1)$, $\mathfrak{B}(V(x))$ will
stand for the boundary operator defined by
$$
 \mathfrak{B}(V(x))u:= \begin{cases}
 u& \text{on } \Gamma_0\,,\\
 \partial u+V(x)u& \text{on } \Gamma_1\,,
 \end{cases}
$$
 and $\mathfrak{D}$ the Dirichlet boundary operator on $\partial \Omega$.

From the results in \cite[Theorem 12.1]{Am83} and \cite[Theorem 2.2]{ALG98},
it is known that for any $K \in {\mathcal C}(\bar{\Omega})$ and
$V \in {\mathcal C}(\Gamma_1)$, the boundary eigenvalue problem
\begin{equation}\label{eip1}
\begin{gathered}
( -\Delta +K(x))\varphi=\sigma \varphi \quad \text{in } \Omega\,,\\
\mathfrak{B}(V(x))\varphi=\bar{0}\quad \text{on } \partial \Omega \,,
\end{gathered}
\end{equation}
admits a unique eigenvalue which possesses a positive eigenfunction,
unique up multiplicative constant, named \emph{principal eigenvalue} of \eqref{eip1}.
 Hereafter we will denote it by $\sigma_1^{\Omega}[-\Delta+K(x),\mathfrak{B}(V(x))]$.
Also the principal eigenvalue of \eqref{eip1} is simple and dominant in the sense
that any other eigenvalue $\lambda$ of \eqref{eip1} satisfies
$$
\mathfrak{R}e(\lambda)>\sigma_1^{\Omega}[-\Delta+K(x),\mathfrak{B}(V(x))]\,,
$$
where $\mathfrak{R}e(\lambda)$ stands for the real part of $\lambda$.
In addition, if $\varphi^*$ stands for the positive eigenfunction of \eqref{eip1}
associated to $\sigma_1^{\Omega}[-\Delta+K(x),\mathfrak{B}(V(x))]$,
unique up multiplicative constant, then
\begin{gather}\label{fin1}
\varphi^*\gg 0 \quad \text{in } \Omega, \\
\label{fin2}
\varphi^* \in W^2(\Omega):=\cap_{p>N}W_p^2(\Omega)\subset
{\mathcal C}^{1+\alpha}(\bar{\Omega}) \quad \text{for all} \quad \alpha \in (0,1)\,.
\end{gather}
Hereinafter we will denote
$$
\sigma_1:=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x))]\,,
$$
and by $\varphi_1>0$, the principal eigenfunction associated to the
principal eigenvalue $\sigma_1$, normalized so that
$\| \varphi_1 \|_{L^{\infty}(\Omega)}=1$. By \eqref{fin1} and \eqref{fin2}
we obtain
$$
\varphi_1 \gg 0 \quad \text{in } \Omega \quad \text{and} \quad
\varphi_1 \in {\mathcal C}^{1+\alpha}(\bar{\Omega}) \quad \forall \alpha \in (0,1)
$$
Also we will denote
$$
\sigma_0:=\sigma_1^{\Omega}[-\Delta, \mathfrak{D}]\,,
$$
that is, the principal eigenvalue of the $-\Delta$ operator in the domain
$\Omega$ under Dirichlet boundary conditions.
Owing to \cite[Proposition 3.1]{CCLG01} we know that
\begin{equation}\label{base}
\sigma_1<\sigma_0
\end{equation}
Finally, we will denote
$$
{\mathcal C}_{\Gamma_0}^1(\bar{\Omega}):= \left\{ u \in {\mathcal C}^1(\bar{\Omega})
: u|_{\Gamma_0}=0\right\}
$$
As it was mentioned, in the particular case when $\gamma=0$, \eqref{1} becomes
in the linear boundary-value problem
\begin{equation}\label{eigpro}
\begin{gathered}
-\Delta u=\lambda u \quad \text{in } \Omega\\
\mathfrak{B}(V(x))u=0\quad  \text{on } \partial \Omega
\end{gathered}
\end{equation}
and owing to \cite[Theorem 12.1]{Am83}, we know that \eqref{eigpro} possesses
positive solutions if, and only if $\lambda=\sigma_1$. In this case we obtain
vertical bifurcation to positive solutions from the trivial branch
$(\lambda,u)=(\lambda,0)$ at $\lambda=\sigma_1$, because all the positive
solutions of \eqref{eigpro} are positive multiple of $\varphi_1$,
being $\varphi_1$ the principal eigenfunction associated to the principal
eigenvalue $\sigma_1$ of \eqref{eigpro}, normalized so that
$\| \varphi_1 \|_{L^{\infty}(\Omega)}=1$.

There is a big amount of literature about the topics of existence,
uniqueness and stability of solutions of elliptic boundary-value problems with
nonlinear boundary conditions, and about the dynamics of the solutions of
parabolic problems with nonlinear boundary conditions, among others,
\cite{Am76NB,AMTR02,CC05,CCST,CC12,CC14,Chi91,GMRS08,LG91,LGMWPP,LGMW93, MoSu05,ARB01,
Rossi98, Rossi05, Um02,Um02BIS,Um04,WW95}.
The main contribution of this paper, together with the previous works
\cite{CC05,CCST,CC12,CC14,MoSu05}, lies in providing general results about
the structure of the set of positive solutions and about the stability of them,
for very general nonlinear boundary-value problems with nonlinear mixed boundary
conditions, containing spatial heterogeneities with arbitrary sign.
In particular, in this work, just as in \cite{CC14}, the outer normal derivative
of the solution $u$ depends in a nonlinear way of $u$, and may be positive,
negative and vanish in different regions of
$\Gamma_1$, depending on the sign of $\gamma b(x) u(x)^q-V(x)u(x)$ in each point
$x \in \Gamma_1$. In \cite{CC14} were analyzed the existence, uniqueness and
stability of the positive solutions of the semilinear boundary-value problem
 with nonlinear mixed boundary conditions
\begin{gather*}
-\Delta u=\lambda u-a(x)u^p\quad \text{in } \Omega\,,\; p>1\\
u=0\quad \text{on } \Gamma_0\,, \\
\partial u+V(x)u=b(x)u^q\quad \text{on } \Gamma_1\,,\; q>1
\end{gather*}
where the domain $\Omega$ and the potentials on the boundary
$V, b\in {\mathcal C}^1(\Gamma_1)$ possess similar properties to the considered
in this work, and the spatial heterogeneity $a \in {\mathcal C}(\bar{\Omega})$,
with $a>0$, satisfies that either
\begin{equation}\label{reco1}
\parbox{10cm}{$\Omega_0:=\operatorname{int}\{ x \in \Omega: a(x)=0\}$,
$\Omega_0 \in {\mathcal C}^2$, $\bar{\Omega}_0 \subset \Omega$, \
and $a(x)$ is bounded away from zero in any compact subsets of
$(\Omega \setminus \bar{\Omega}_0) \cup \Gamma_1$,}
\end{equation}
or
\begin{equation}\label{reco2}
a(x) \text{ is bounded away from zero in any compact subset } \Omega \cup \Gamma_1
\end{equation}
The results obtained in the current work are, in some sense, the natural
continuation of the results in \cite{CC14}, to cover the case therein
 when $\Omega_0=\Omega$, that is, when the potential $a=0$ in $\Omega$.
Clearly, the results obtained in this paper can not be obtained substituting
in \cite{CC14} $a=0$ in $\Omega$, because the case $\Omega_0=\Omega$ does not
satisfy the assumptions \eqref{reco1} neither \eqref{reco2} required in \cite{CC14}.

The organizations of this article is as follows:
Section \ref{Section1} is the Introduction.
Section \ref{Section2} contains results about the profile and regularity of
the positive solutions of \eqref{1} and the main results about local and
global bifurcation to positive solutions of \eqref{1} from the trivial
branch $(\lambda,u)=(\lambda,0)$. In Section \ref{Section3} is carried out
a very sharp analysis, in the particular case when $\gamma<0$, about the
global structure of the set of positive solutions of
\eqref{1}(Section \ref{Section3.1}), about the pointing behavior of the
positive solutions of \eqref{1} when $\lambda \uparrow \sigma_0$
(Section \ref{Section3.2}) and about the dynamics of the positive solutions
of the associated parabolic problem to \eqref{1} (Section \ref{Section3.3}).
Finally, in Section \ref{Section4} is analyzed the particular case when
$\gamma>0$, obtaining some results about the dynamics of the positive
solutions of the parabolic problem associated to \eqref{1}
(Section \ref{Section4.1}), and some results about the structure of the set
of positive solutions of \eqref{1} (Section \ref{Section4.2}).

\section{Regularity and bifurcation of positive solutions
to \eqref{1}}\label{Section2}

This section contains results about the profile and regularity of the positive
solutions of \eqref{1} and the main results about local and global bifurcation
to positive solutions of \eqref{1} from the trivial branch $(\lambda,u)=(\lambda,0)$.
The next result gives the regularity and profile of the positive solutions
of \eqref{1} and a necessary condition for the existence of them.

\begin{theorem}\label{regularity}
If $u_{\lambda}$ is a positive solution of \eqref{1} for the value $\lambda$
of the parameter, then
\begin{equation}\label{lambda}
\lambda=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x)-\gamma b(x)
u_{\lambda}^{q-1})]\,,
\end{equation}
$u_{\lambda} \in {\mathcal C}^{1+\alpha}(\bar{\Omega})$ for all
$\alpha \in (0,1)$, and $u_{\lambda}$ is strongly positive in $\Omega$.
\end{theorem}

\begin{proof}
 Let $u_{\lambda}$ be a positive solution of \eqref{1} for the value $\lambda$.
Then, $u_{\lambda}$ is a positive function in $\Omega$ satisfying the problem
\begin{gather*}
-\Delta u_{\lambda}= \lambda u_{\lambda} \quad\text{in } \Omega\,, \\
u_{\lambda}=0 \quad\text{on } \Gamma_0\,, \\
\big( \partial+V(x)-\gamma b(x)u_{\lambda}^{q-1}\big)u_{\lambda}=0 \quad\text{on }
 \Gamma_1\,,\; q>1\,;
\end{gather*}
that is, $\lambda$ is an eigenvalue of the problem
\begin{equation}\label{ip}
\begin{gathered}
-\Delta \theta= \lambda \theta \quad\text{in } \Omega\,,\\
\mathfrak{B}\big( V(x)-\gamma b(x)u_{\lambda}^{q-1}\big)\theta=0 \quad\text{on }
 \partial \Omega\,,\; q>1\,,
\end{gathered}
\end{equation}
and $\theta=u_{\lambda}$ is a positive eigenfunction of \eqref{ip} associated
to the eigenvalue $\lambda$. Then, owing to the uniqueness of the principal
eigenvalue of \eqref{ip} (cf.\ \eqref{eip1}, \cite{Am83}), we obtain \eqref{lambda}
 and that $u_{\lambda}$ is its principal eigenfunction.
The remaining assertions of the theorem follow from the structure and
regularity of the principal eigenfunction of \eqref{ip}
(cf.\ \eqref{fin1}, \eqref{fin2}, \cite{Am83}).
This completes the proof.
\end{proof}

Hereafter, by \emph{continuum} we will mean a closed and connected set.
The following theorem collects the main results about bifurcation of positive
 solutions of \eqref{1} from the trivial branch $(\lambda,u)=(\lambda,0)$.
It is \cite[Theorem 1.1]{CC12} for the particular case therein when $p=1$, $q>1$
and $a=0$ in $\Omega$.

\begin{theorem}\label{bf}
The following hold:

(i) The value $\lambda=\sigma_1$ is the unique bifurcation value to
 positive solutions of \eqref{1} from the trivial branch $(\lambda,u)=(\lambda,0)$.

(ii) A differentiable continuum $\mathfrak{C}$ of solutions of \eqref{1}
emanates from the bifurcation point $(\lambda,u)=(\sigma_1,0)$ and in a small
neighborhood ${\mathcal V}$ of $(\sigma_1,0)$ in
${\bf R} \times {\mathcal C}_{\Gamma_0}^1(\bar{\Omega})$, the unique non-trivial
solutions of \eqref{1} belong to $\mathfrak{C}$. In addition,
 \begin{equation}\label{---}
 \mathfrak{C} \cap {\mathcal V}=\{ (\lambda,u)=(\sigma_1+\mu_q(s),s(\varphi_1+v_q(s)))
: s\in (-\varepsilon,\varepsilon)\}
 \end{equation}
for $\varepsilon>0$ small enough, with
 \begin{equation}\label{+-+-}
 (\mu_q,v_q) \in {\mathcal C}^1((-\varepsilon,\varepsilon),{\bf R}
\times {\mathcal C}_{\Gamma_0}^1(\bar{\Omega}))\,, \quad (\mu_q(0),v_q(0))=(0,0)\,,
\end{equation}
and 
$\int_{\Omega}v_q(s)\varphi_1=0$ for all $s\in (-\varepsilon,\varepsilon)$.
 Furthermore,
 \begin{equation}\label{--}
 \lim_{s \to 0}\frac{\mu_q(s)}{s^{q-1}}={\mathcal D}(q,\gamma)\,,
 \end{equation}
 where
 \begin{equation}\label{-}
 {\mathcal D}(q,\gamma):=-\gamma \int_{\Gamma_1}b(x)\varphi_1^{q+1}\,.
 \end{equation}
\end{theorem}

\begin{remark}\label{remark1} \rm
It should be noted that owing to \eqref{---}, \eqref{+-+-}, \eqref{--} and \eqref{-} 
and since $b>0$ and $\varphi_1 \gg 0$ in $\Omega$, we obtain
the bifurcation to positive solutions from the trivial branch at $\lambda=\sigma_1$ 
produces for $s\in (0,\varepsilon)$ and it is supercritical if $\gamma<0$ and 
subcritical if $\gamma>0$.
\end{remark}

Hereinafter we will denote by $\mathfrak{C}^+$ the maximal subcontinuum
 of $\mathfrak{C}$ composed by the positive solutions of \eqref{1} emanating 
from the trivial branch at $\lambda =\sigma_1$.

\begin{remark}\label{remark2} \rm
Since $\lambda=\sigma_1$ is a simple eigenvalue of the linearization of \eqref{1} 
at $(\lambda,u)=(\sigma_1,0)$, and owing to the fact that $(\lambda,u)=(\sigma_1,0)$ 
is the unique bifurcation point to positive solutions of \eqref{1} from the trivial 
branch, it follows from the updated version of the Global Alternative of 
 Rabinowitz \cite[Theorem 1.27]{Rab71} given by L\'opez-G\'omez in
 \cite[Theorem 6.4.3]{LGli01}, that either
$\mathfrak{C}^+$ is unbounded in 
$\mathbb{R} \times {\mathcal C}_{\Gamma_0}^1(\bar{\Omega})$, 
or it contains a pair $(\tilde{\lambda},\tilde{u})$ with $\tilde{u} \gg 0$ 
in $\Omega$ satisfying
$$
\int_{\Omega}\tilde{u}\varphi_1=0\,,
$$
which is impossible since $\varphi_1 \gg 0$ in $\Omega$. 
Then, we obtain $\mathfrak{C}^+$ is unbounded in
 $\mathbb{R} \times {\mathcal C}_{\Gamma_0}^1(\bar{\Omega})$ and by the 
$L_p$-estimates, unbounded in $\mathbb{R}\times L^{\infty}(\Omega)$.
\end{remark}

\section{The case $\gamma<0$}\label{Section3}

In this section, in the particular case when $\gamma<0$, we will analyze 
the existence, uniqueness and stability of the positive solutions of \eqref{1}, 
and we will obtain the structure of the global bifurcation diagram of positive 
solutions of \eqref{1}, the pointing behavior of the positive solutions of \eqref{1}
 when $\lambda \uparrow \sigma_0$ and the dynamics of the positive solutions of 
the parabolic problem associated to \eqref{1}. Along this section we will denote
 by $\tilde{\gamma}:=-\gamma>0$, and hence, \eqref{1} will be written in the form
\begin{equation}\label{1-}
\begin{gathered}
-\Delta u=\lambda u \quad\text{in } \Omega\,, \\
u=0 \quad\text{on } \Gamma_0\,,\\
\partial u+V(x)u+\tilde{\gamma}b(x)u^q=0 \quad\text{on } \Gamma_1\,,\; q>1
\end{gathered}
\end{equation}

\subsection{Structure of the set of positive solutions to \eqref{1}} \label{Section3.1}

The following is the main result of this section, which gives the structure of 
the global bifurcation diagram of positive solutions of \eqref{1-}. 

\begin{theorem}\label{mt-1}
If
\begin{equation}\label{bsep}
b(x) \geq \underline{b}>0 \quad \text{for all }  x \in \Gamma_1\,,
\end{equation}
then the following hold:

(i) \eqref{1-} possesses a positive solution if, and only if
\begin{equation}\label{ex}
\sigma_1<\lambda<\sigma_0\,.
\end{equation}

(ii) The positive solution of \eqref{1-}, if it exists, is unique, 
strongly positive in $\Omega$, and linearly and globally asymptotically stable 
as steady-state of the parabolic problem associated to \eqref{1-}. 
Hereafter we will denote it by $u_{\lambda}$.

(iii) For any $\lambda \in (\sigma_1,\sigma_0)$, 
$\dot{u}_{\lambda}:=\frac{d\,u_{\lambda}}{d\, \lambda}$ is strongly positive in 
$\Omega$, that is,
\begin{equation}\label{pal1}
\dot{u}_{\lambda}(x)>0 \quad \forall  x\in \Omega \cup \Gamma_1 \quad \text{and}
\quad \partial \dot{u}_{\lambda}(x)<0 \quad \forall  x\in \Gamma_0
\end{equation}
In particular, for each $x \in \Omega \cup \Gamma_1$, the map
$(\sigma_1,\sigma_0) \to (0,\infty)$ defined by
\begin{equation}\label{pal3}
\lambda  \mapsto  u_{\lambda}(x)
\end{equation}
is strictly increasing.

(iv) There exist uniform $L^{\infty}(\Omega)$-bounds for the positive solutions 
of \eqref{1-} in compact intervals of $\lambda$ contained in $[\sigma_1,\sigma_0)$.

(v) The positive solutions of \eqref{1-} belong to a differentiable continuum 
$\mathfrak{C}^+$ of positive solutions, which emanates supercritically from 
the trivial branch at the unique bifurcation value to positive solutions of 
\eqref{1-} $\lambda=\sigma_1$, bifurcates from infinity at $\lambda=\sigma_0$ 
and it is increasing in $\| \cdot \|_{L^{\infty}(\Omega)}$ with the 
$\lambda-$parameter. In particular,
 \begin{gather}\label{pl}
 {\mathcal P}_{\lambda}(\mathfrak{C}^+)=[\sigma_1,\sigma_0), \\
 \lim_{\lambda \downarrow \sigma_1}\| u_{\lambda}\|_{L^{\infty}(\Omega)}=0\,, \quad
 \lim_{\lambda \uparrow \sigma_0}\| u_{\lambda}\|_{L^{\infty}(\Omega)}=\infty\,,
 \end{gather}
where ${\mathcal P}_{\lambda}(\mathfrak{C}^+)$ denotes the $\lambda$-projection 
of the continuum $\mathfrak{C}^+$ over the $\lambda$-axis.
\end{theorem}

To prove Theorem \ref{mt-1} we need some lemmas.
Next result gives a sufficient condition for the existence of a positive 
strict subsolution of \eqref{1-}.

\begin{lemma}\label{sub}
For each $\lambda > \sigma_1$, \eqref{1-} possesses a positive strict 
subsolution arbitrarily small, which is strongly positive in $\Omega$.
\end{lemma}

\begin{proof} 
Let $\lambda >\sigma_1$ be. Owing to the monotonicity and continuous dependence 
of the principal eigenvalue with respect to the potential on the boundary
 (cf.\ \cite[Proposition 3.5]{CCLG01}, \cite[Theorem 8.2, Remark 8.3]{CCLG01}), 
there exists $\varepsilon >0$ small enough such that
\begin{equation}\label{sub1}
\sigma_1:=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x))]
< \sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x)+\varepsilon)]<\lambda
\end{equation}
Let us fix $\varepsilon>0$ satisfying \eqref{sub1} and let us denote by
$$
\sigma_1^{\varepsilon}:=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x)+\varepsilon)]
$$
and by $\varphi_{\varepsilon}$ the principal eigenfunction associated to 
the principal eigenvalue $\sigma_{1}^{\varepsilon}$, normalized so that
\begin{equation}\label{sub2}
\| \varphi_{\varepsilon} \|_{L^{\infty}(\Omega)}=1
\end{equation}
By construction, $\varphi_{\varepsilon}$ is strongly positive in $\Omega$ and 
it satisfies the  problem
\begin{equation}\label{sub3}
\begin{gathered}
-\Delta \varphi_{\varepsilon}= \sigma_1^{\varepsilon} \varphi_{\varepsilon }
\quad \text{in } \Omega\,,\\
\varphi_{\varepsilon}=0 \quad\text{on } \quad \Gamma_0\,,\\
\big( \partial +V(x)+\varepsilon \big)\varphi_{\varepsilon}=0 \quad \text{on } 
\Gamma_1
\end{gathered}
\end{equation}
Now, let us consider the function
$\underline{u}_{\lambda}:=\alpha \varphi_{\varepsilon}$
for $\alpha>0$ satisfying
\begin{equation}\label{alphasmall}
0<\alpha<\Big( \frac{\varepsilon}{\tilde{\gamma}\| b \|_{L^{\infty}(\Gamma_1)}}
\Big)^{\frac{1}{q-1}}
\end{equation}
By construction, and thanks to \eqref{sub1}, \eqref{sub2}, \eqref{sub3} 
and \eqref{alphasmall}, it is easy to see that $\underline{u}_{\lambda}$ 
is a positive strict subsolution of \eqref{1-} for any fixed $\alpha>0$ 
satisfying \eqref{alphasmall}. Moreover, since $\varphi_{\varepsilon}$ is 
strongly positive in $\Omega$ and $\alpha>0$, we obtain
$ \underline{u}_{\lambda}$ is strongly positive in $\Omega$.
This completes the proof. 
\end{proof}

The next result gives a sufficient condition for the existence of a positive 
strict supersolution of \eqref{1-}.

\begin{lemma}\label{super}
If \eqref{bsep} holds, then for each
\begin{equation}\label{pv}
\lambda < \sigma_0
\end{equation}
Equation \eqref{1-} possesses a positive strict supersolution arbitrarily 
large and strongly positive in $\Omega$
\end{lemma}

\begin{proof} 
Let $\lambda <\sigma_0$ be. Owing to the dominance of the principal eigenvalue 
of the operator $-\Delta$ under Dirichlet boundary conditions 
(cf.\ \cite[Proposition 3.1, Corollary 9.2]{CCLG01}) and to the limiting behavior 
of the principal eigenvalue $\sigma_1^{\Omega}[-\Delta,\mathfrak{B}(n)]$ when 
$n \uparrow \infty$ (cf.\ \cite[Theorem 9.1]{CCLG01}), the following hold
\begin{equation}\label{super0}
\sigma_1^{\Omega}[-\Delta,{\mathfrak{B}(n)}]<\sigma_0\,, \quad \forall 
 n \in {\bf N}\,, \quad \lim_{n \uparrow \infty}\sigma_1^{\Omega}
[-\Delta,\mathfrak{B}(n)]=\sigma_0
\end{equation}
Then, owing to \eqref{pv} and \eqref{super0}, there exists $n \in {\bf N}$ 
large enough such that
\begin{equation}\label{super1}
\lambda <\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(n)]< \sigma_0
\end{equation}
Let us fix $n \in {\bf N}$ satisfying \eqref{super1} and let us denote by
$$
\sigma_1^n:=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(n)]
$$
and by $\varphi_1^n$ the principal eigenfunction associated to the principal 
eigenvalue $\sigma_{1}^{n}$, normalized so that
\begin{equation}\label{super2}
\| \varphi_1^n \|_{L^{\infty}(\Omega)}=1
\end{equation}
Set
\begin{equation}\label{mnl}
m_n:=\min_{x \in \Gamma_1}\varphi_1^n
\end{equation}
Since $\varphi_1^n$ is strongly positive in $\Omega$, we obtain
$\varphi_1^n(x)>0$ for all $x \in \Gamma_1$ and hence,
\begin{equation}\label{super22}
m_n>0
\end{equation}
By definition $\varphi_1^n$ satisfies the  problem
\begin{equation}\label{super3}
\begin{gathered}
-\Delta \varphi_1^n= \sigma_1^{n} \varphi_1^n \quad\text{in } \Omega\,,\\
\varphi_1^n=0 \quad\text{on } \Gamma_0\,,\\
( \partial +n )\varphi_1^n=0 \quad \text{on } \Gamma_1
\end{gathered}
\end{equation}
Now, let us consider the function
$\overline{u}_{\lambda}:=\kappa \varphi_1^n$
for $\kappa>0$ satisfying
\begin{equation}\label{kbig}
\kappa > \Big( \frac{\| V(x)-n\|_{L^{\infty}(\Gamma_1)}}
{\tilde{\gamma}\underline{b}m_n^{q-1}}\Big)^{\frac{1}{q-1}}
\end{equation}
By construction and owing to \eqref{super1}, \eqref{bsep}, \eqref{mnl}, 
\eqref{super22}, \eqref{super3} and \eqref{kbig}, it is easy to see that 
$\overline{u}_{\lambda}$ is a positive strict supersolution of \eqref{1-} 
for any $\kappa>0$ satisfying \eqref{kbig}. Moreover, since $\varphi_1^n$ 
is strongly positive in $\Omega$ and $\kappa>0$, we obtain
$\overline{u}_{\lambda}$ is strongly positive in $\Omega$.
This completes the proof. 
\end{proof}


\begin{proof}[Proof of Theorem \ref{mt-1}]
(i) To prove the necessary condition for the existence of positive solution 
of \eqref{1-}, let $u_{\lambda}$ be a positive solution of \eqref{1-} 
for the value $\lambda$ of the parameter. Then, owing to \eqref{lambda}, 
to the dominance of the principal eigenvalue of the operator $-\Delta$ 
in the domain $\Omega$ under Dirichlet boundary conditions 
(cf.\ \cite[Proposition 3.1]{CCLG01}), to the facts that $u_{\lambda}$ 
is strongly positive in $\Omega$, $b>0$ on $\Gamma_1$ and $\tilde{\gamma}>0$ 
and to the monotonicity of the principal eigenvalue with respect to the potential 
on the boundary conditions (cf.\ \cite[Proposition 3.5]{CCLG01}), we obtain
$$
\sigma_1<\lambda=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x)+\tilde{\gamma} 
b(x)u_{\lambda}^{q-1})]<\sigma_0\,,
$$
which proves \eqref{ex} and ends the proof of the necessary condition for the 
existence of positive solutions of \eqref{1-}.

We now prove the sufficient condition \eqref{ex} for the existence of positive 
solution of \eqref{1-}. Indeed, owing to Lemma \ref{sub} and Lemma \ref{super}, 
for each $\lambda$ satisfying \eqref{ex}, there exist a positive strict 
subsolution $\underline{u}_{\lambda}$ of \eqref{1} arbitrarily small, 
and a positive strict supersolution $\overline{u}_{\lambda}$ of \eqref{1} 
arbitrarily large, both of them strongly positive in $\Omega$. 
Thanks to the fact that both of them are strongly positive in $\Omega$, 
taking $\alpha>0$ small enough in Lemma \ref{sub} or $\kappa>0$ large enough 
in Lemma \ref{super}, it is possible to obtain
$0<\underline{u}_{\lambda}<\overline{u}_{\lambda}$,
and hence, the sub-super solutions method (cf.\ \cite{Am76}) implies the 
existence of a positive solution $u_{\lambda}$ of \eqref{1-} with
$0<\underline{u}_{\lambda}<u_{\lambda}<\overline{u}_{\lambda}$,
for each $\lambda$ satisfying \eqref{ex}. This completes the proof of $i)$.


(ii) To prove the uniqueness of positive solution of \eqref{1-},
 when it exists, we will argue by contradiction. Let $\lambda$ be 
satisfying \eqref{ex} and suppose that $u_1$ and $u_2$ are two positive 
solutions of \eqref{1-} for the value $\lambda$ of the parameter with
\begin{equation}\label{u-1}
u_1 \neq u_2
\end{equation}
Owing to \eqref{lambda} we obtain
\begin{equation}\label{u0}
\lambda=\sigma_1^{\Omega}\big[-\Delta,\mathfrak{B}\big( V(x)+\tilde{\gamma} 
b(x) u_i^{q-1}\big)\big]\,, \quad i=1,2
\end{equation}
Arguing as in \cite[Theorem 4.1]{LGA14}, set
$$
J(t):=(tu_2+(1-t)u_1)^q\,, \quad t \in [0,1]
$$
By construction we obtain
$$
u_2^q-u_1^q=J(1)-J(0)=\int_0^1 J'(t)\, dt
=q (u_2-u_1) \int_0^1 (tu_2+(1-t)u_1)^{q-1}\,dt
$$
and hence,
\begin{equation}\label{pis1}
\frac{u_2^q-u_1^q}{u_2-u_1}=q\int_0^1 (tu_2+(1-t)u_1)^{q-1}\, dt
\end{equation}
Now, since $u_1 \gg 0$,
\begin{equation}\label{pis2}
(tu_2+(1-t)u_1)^{q-1} \gg t^{q-1}u_2^{q-1}\,, \quad 0\leq t<1\,,
\end{equation}
and hence, owing to \eqref{pis1} and \eqref{pis2} we obtain 
\begin{equation}\label{pis3}
\frac{u_2^q-u_1^q}{u_2-u_1}=q\int_0^1\, (tu_2+(1-t)u_1)^{q-1}\,dt
>q \int_0^1\, t^{q-1} u_2^{q-1}\,dt=u_2^{q-1}
\end{equation}
Now, let us consider the function $\Theta:=u_2-u_1$. By construction it satisfies
\begin{equation}\label{u1}
\begin{gathered}
(-\Delta-\lambda) \Theta=0 \quad\text{in } \Omega\,,\\
\Theta=0 \quad\text{on } \Gamma_0\,,\\
\partial \Theta+\Big( V(x)+\tilde{\gamma} b(x) \frac{u_2^q(x)-u_1^q(x)}{u_2(x)
-u_1(x)}\Big)\Theta=0 \quad \text{on } \Gamma_1
\end{gathered}
\end{equation}
Owing to \eqref{pis3} and the facts that $b>0$ and $\tilde{\gamma}>0$, 
it follows from the monotonicity of the principal eigenvalue with respect 
to the potential on the boundary (cf.\ \cite[Proposition 3.5]{CCLG01}) 
and \eqref{u0} that
$$
\sigma_1^{\Omega}\Big[ -\Delta, \mathfrak{B}\Big(V(x)+\tilde{\gamma}b(x)
 \frac{u_2^{q}-u_1^{q}}{u_2-u_1}\Big)\Big]
>\sigma_1^{\Omega}\big[ -\Delta, \mathfrak{B}\big(V(x)+\tilde{\gamma} b(x)u_2^{q-1}
\big)\big]=\lambda
$$
and hence,
\begin{equation}\label{u2}
\sigma_1^{\Omega}\Big[ -\Delta-\lambda, \mathfrak{B}\Big(V(x)+\tilde{\gamma}b(x)
 \frac{u_2^{q}-u_1^{q}}{u_2-u_1}\Big)\Big]>0
\end{equation}
Then, since 
$\sigma_1^{\Omega}[-\Delta-\lambda, \mathfrak{B}( V+\tilde{\gamma} b(x) 
\frac{u_2^q-u_1^q}{u_2-u_1} )]$ is the least eigenvalue of \eqref{u1} 
(cf.\ \cite[Theorem 12.1]{Am83}), and owing to \eqref{u2}, we obtain $0$
is not an eigenvalue of \eqref{u1} and therefore $\Theta=0$, which contradicts 
\eqref{u-1}. This completes the proof of the uniqueness of positive solution 
of \eqref{1-} when it exists.
The fact that $u_{\lambda}$ is strongly positive in $\Omega$ follows from 
Theorem \ref{regularity}.


 Now it will be proved that for each $\lambda$ satisfying \eqref{ex}, 
the unique positive solution $u_{\lambda}$ of \eqref{1-} is linearly 
asymptotically stable. Indeed, the linearization of \eqref{1-} in $u_{\lambda}$ 
is given by
\begin{equation}
\begin{gathered}
(-\Delta -\lambda) v=0\quad \text{in } \Omega\\
v=0\quad  \text{on } \Gamma_0\\
\Big( \partial+V(x)+\tilde{\gamma}qb(x)u_{\lambda}^{q-1}\Big)v=0 \quad
 \text{on } \Gamma_1
\end{gathered}
\end{equation}
Since by \eqref{lambda},
$$
\lambda=\sigma_1^{\Omega}[ -\Delta,\mathfrak{B}(V(x)+\tilde{\gamma}
b(x)u_{\lambda}^{q-1})]\,,
$$
and thanks to the facts that $b>0$, $q>1$ and $\tilde{\gamma}>0$, it follows 
from the monotonicity of the principal eigenvalue with respect to the potential 
on the boundary that
$$
\sigma_1^{\Omega}\big[ -\Delta - \lambda,\mathfrak{B}(V(x)
 +q\tilde{\gamma}b(x)u_{\lambda}^{q-1})\big]
> \sigma_1^{\Omega}\big[ -\Delta - \lambda,\mathfrak{B}(V(x)
 +\tilde{\gamma}b(x)u_{\lambda}^{q-1})\big]=0\,,
$$
which proves that $u_{\lambda}$ is linearly asymptotically stable. 
The proof of the fact that $u_{\lambda}$ is globally asymptotically stable 
as steady state of the parabolic problem associated to \eqref{1-} is given 
in Theorem \ref{dynamics}-ii).
This completes the proof of (ii).


(iii) Owing to (i), (ii) and \eqref{lambda}, for each
 $\lambda \in (\sigma_1,\sigma_0)$ there exists a unique positive solution 
$u_{\lambda}$ of \eqref{1-} and the following holds
\begin{equation}\label{ss0}
\lambda=\sigma_1^{\Omega}[-\Delta, \mathfrak{B}(V(x)+\tilde{\gamma}
 b(x)u_{\lambda}^{q-1})]
\end{equation}
Also, differentiating with respect to $\lambda$ in \eqref{1-}, we obtain
\begin{equation}\label{ss2}
\begin{gathered}
(-\Delta -\lambda) \dot{u}_{\lambda}=u_{\lambda}>0 \quad \text{in } \Omega\\
\dot{u}_{\lambda}=0\quad \text{on } \Gamma_0\\
(\partial +V(x)+\tilde{\gamma} b(x)qu_{\lambda}^{q-1})\dot{u}_{\lambda}=0\quad
 \text{on } \Gamma_1
\end{gathered}
\end{equation}
 Then, since $b>0$, $\tilde{\gamma}>0$ and $q>1$, and owing to \eqref{ss0} and 
to the monotonicity of the principal eigenvalue with respect to the potential 
on the boundary, the following hold
$$
\sigma_1^{\Omega}[-\Delta-\lambda,\mathfrak{B}(V(x)
+\tilde{\gamma}b(x)qu_{\lambda}^{q-1})]
>\sigma_1^{\Omega}[-\Delta-\lambda,\mathfrak{B}(V(x)+\tilde{\gamma} 
b(x)u_{\lambda}^{q-1})]=0
$$
Thus, the characterization of the strong maximum principle 
(cf.\ \cite[Theorem 2.4]{ALG98}) establishes that 
$(-\Delta-\lambda,\mathfrak{B}(V(x)+\tilde{\gamma}b(x)qu_{\lambda}^{q-1}),\Omega)$ 
satisfies the strong maximum principle and therefore, \eqref{ss2} implies that 
$\dot{u}_{\lambda}$ is strongly positive in $\Omega$.
This completes the proof of (iii).

(iv). It is a straightforward consequence of $iii)$, taking into account that 
owing to \eqref{ex} and \eqref{pal3} the map
$(\sigma_1,\sigma_0) \to \mathbb{R}^+$ given by
\[
\lambda \mapsto \| u_{\lambda}\|_{L^{\infty}(\Omega)}
\]
is increasing with $\lambda$ and $u_{\sigma_1}=0$. Then, if 
$[\alpha, \beta] \subset [\sigma_1,\sigma_0)$, we obtain 
$$
\| u_{\lambda} \|_{L^{\infty}(\Omega)}\leq \| u_{\beta} \|_{L^{\infty}(\Omega)} 
\quad \text{for all }  \lambda \in [\alpha,\beta]
$$
This completes the proof of (iv).

Now we are going to prove (v). The fact that $\lambda=\sigma_1$ is the unique 
bifurcation value to positive solutions of \eqref{1-} from the trivial branch 
$(\lambda,u)=(\lambda,0)$ was proved in Theorem \ref{bf}-i). 
The existence of the continuum $\mathfrak{C}^+$ of positive solutions of 
\eqref{1-} emanating supercritically from the point $(\lambda,u)=(\sigma_1,0)$ 
follows from Theorem \ref{bf} and Remark \ref{remark1}, taking into account that 
$\gamma:=-\tilde{\gamma}<0$ and $b>0$ on $\Gamma_1$. 
Denoting by ${\mathcal P}_{\lambda}(\mathfrak{C}^+)$ the $\lambda-$projection 
of $\mathfrak{C}^+$ on the $\lambda-$axis, it follows from \eqref{ex} that
\begin{equation}\label{pcc}
{\mathcal P}_{\lambda}(\mathfrak{C}^+)\subset [\sigma_1,\sigma_0)
\end{equation}
Owing to the fact that $(\sigma_1,0)$ is the unique bifurcation point of 
\eqref{1-} to positive solutions from the trivial branch $(\lambda,u)=(\lambda,0)$, 
it follows from the global bifurcation theory (cf.\ Remark \ref{remark2}) that 
the continuum $\mathfrak{C}^+$ is unbounded in $\mathbb{R} \times L^{\infty}(\Omega)$ 
and since \eqref{pcc} holds, we obtain $\mathfrak{C}^+$ is unbounded in
$L^{\infty}(\Omega)$, and therefore, $\mathfrak{C}^+$ must bifurcate to positive 
solutions from infinity at some value $\lambda^* \in [\sigma_1,\sigma_0]$.
 Now, the existence of uniform $L^{\infty}(\Omega)-$bounds for the positive
 solutions of \eqref{1-} in compact intervals of $\lambda$ contained in 
$[\sigma_1,\sigma_0)$, implies that $\mathfrak{C}^+$ must bifurcate from infinity 
in $L^{\infty}(\Omega)$ when $\lambda \uparrow \sigma_0$, and that 
$\lambda=\sigma_0$ is the unique bifurcation value to positive solutions 
of \eqref{1-} from infinity. Then, since $\mathfrak{C}^+$ bifurcates to positive 
solutions from the trivial branch at $\lambda=\sigma_1$ and from infinity 
at $\lambda=\sigma_0$, since \eqref{ex} holds and owing to the fact that 
$\mathfrak{C}^+$ is connected, we obtain
\begin{equation}\label{ec}
{\mathcal P}_{\lambda}(\mathfrak{C}^+)=[\sigma_1,\sigma_0)
\end{equation}
The fact that all the positive solutions of \eqref{1-} are contained in 
$\mathfrak{C}^+$ follows from \eqref{ex}, \eqref{ec}, from the fact that 
$\lambda=\sigma_1$ is the unique bifurcation value to positive solutions 
of \eqref{1-} from the trivial branch and from the uniqueness of positive 
solution of \eqref{1-} when it exists. Finally, the fact that $\mathfrak{C}^+$ 
is increasing in $L^{\infty}(\Omega)$ with $\lambda$, follows from (iii).
This completes the proof of (v).
\end{proof}

\subsection{Pointwise growth of positive solutions of \eqref{1-} when 
$\lambda \uparrow \sigma_0$}\label{Section3.2}

In this section we are going to analyze the pointing behavior of the positive 
solutions of \eqref{1-} when $\lambda \uparrow \sigma_0$. 
The original ideas given in the previous works \cite{LGSL98} and \cite{LG00} 
will play a crucial role to obtain the results of this section. 
Owing to Theorem \ref{mt-1} it is known that $\lambda=\sigma_0$ is the unique 
bifurcation value from infinity to positive solutions of \eqref{1-} and hence,
$$
\lim_{\lambda \uparrow \sigma_0}\| u_{\lambda} \|_{L^{\infty}(\Omega)}=\infty
$$
In this section we will prove that the growth to infinity of the positive 
solutions of \eqref{1-} when $\lambda \uparrow \sigma_0$ is not concentrated 
in some particular region of $\Omega \cup \Gamma_1$, but it occurs uniformly 
in any compact subset of $\Omega \cup \Gamma_1$. The following result 
establishes the uniform growth to infinity in compact subsets of $\Omega$ 
of the positive solutions of \eqref{1-} when $\lambda \uparrow \sigma_0$.

\begin{theorem}\label{ws1}
Assume \eqref{bsep}. Then
\begin{equation}\label{ro0}
\lim_{\lambda \uparrow \sigma_0}u_{\lambda}=\infty \quad \text{and} \quad
\lim_{\lambda \uparrow \sigma_0}\dot{u}_{\lambda}=\infty
\end{equation}
uniformly in compact subsets of $\Omega$, where 
$\dot{u_{\lambda}}:=\frac{du_{\lambda}}{d \lambda}$
\end{theorem}

\begin{proof} 
Let $\varphi_0$ be the principal eigenfunction of $-\Delta$ in the domain 
$\Omega$ under Dirichlet boundary conditions, normalized so that 
$\| \varphi_0 \|_{L^{\infty}(\Omega)}=1$, let us fix 
$\lambda_1 \in (\sigma_1,\sigma_0)$ and let $u_{\lambda_1}$ be the unique 
positive solution of \eqref{1-} for such a value $\lambda=\lambda_1$. 
Since $\varphi_0$ and $u_{\lambda_1}$ are strongly positive in $\Omega$, 
there exists $\alpha>0$ such that
$$
u_{\lambda_1} \gg \alpha \varphi_0 \quad \text{in } \Omega\,,
$$
and since the branch of positive solutions
$\mathfrak{C}^+=\{ u_{\lambda}: \lambda \in (\sigma_1,\sigma_0)\}$
is increasing with $\lambda$ (cf.\ Theorem \ref{mt-1}-iii)), we obtain 
\begin{equation}\label{ro3}
u_{\lambda}>u_{\lambda_1}>\alpha \varphi_0 \quad \text{in } \Omega\,, \;
\lambda \in (\lambda_1,\sigma_0)\,.
\end{equation}
Also, we obtain 
\begin{equation}\label{ro4}
\sigma_1^{\Omega}[-\Delta -\lambda, {\mathcal D}]=\sigma_0-\lambda>0\,, \quad 
\forall  \lambda \in (\lambda_1,\sigma_0)\,.
\end{equation}
Now, differentiating \eqref{1-} with respect to $\lambda$ gives
\begin{equation}\label{ro1}
\begin{gathered}
(-\Delta -\lambda)\dot{u}_{\lambda}=u_{\lambda} \quad \text{in } \Omega\\
\dot{u}_{\lambda}=0 \quad\text{on } \Gamma_0\\
(\partial+V(x)+q\tilde{\gamma}b(x)u_{\lambda}^{q-1})\dot{u}_{\lambda}=0
\quad \text{on } \Gamma_1
\end{gathered}
\end{equation}
and taking into account \eqref{pal1} and \eqref{ro3}, we obtain for
$\lambda \in (\lambda_1,\sigma_0)$,
\begin{equation}\label{ro5}
\begin{gathered}
( -\Delta -\lambda )\dot{u}_{\lambda}=u_{\lambda}>\alpha \varphi_0 \quad
 \text{in } \Omega\\
\dot{u}_{\lambda}=0\quad \text{on } \Gamma_0\\
\dot{u}_{\lambda}>0\quad \text{on } \Gamma_1
\end{gathered}
\end{equation}
Now, since by definition,
\[
\varphi_0= \begin{cases}
\frac{1}{\sigma_0-\lambda}(-\Delta-\lambda)\varphi_0 & \text{in } \Omega, \\
0 \quad &\text{on } \partial \Omega=\Gamma_0 \cup \Gamma_1\,,
\end{cases}
\]
Equation \eqref{ro5} becomes
\begin{gather*}
( -\Delta -\lambda )\big(\dot{u}_{\lambda}-\frac{\alpha \varphi_0}{\sigma_0-\lambda} 
\big)>0 \quad \text{in } \Omega\\
\dot{u}_{\lambda}-\frac{\alpha \varphi_0}{\sigma_0-\lambda}=0\quad\text{on } \Gamma_0\\
\dot{u}_{\lambda}-\frac{\alpha \varphi_0}{\sigma_0-\lambda}>0\quad\text{on } \Gamma_1
\end{gather*}
and hence,
\begin{equation}\label{ro555}
\begin{gathered}
( -\Delta -\lambda )\big(\dot{u}_{\lambda}-\frac{\alpha \varphi_0}{\sigma_0-\lambda}
 \big)>0 \quad \text{in } \Omega\\
\dot{u}_{\lambda}-\frac{\alpha \varphi_0}{\sigma_0-\lambda}>0\quad\text{on }
 \partial \Omega
\end{gathered}
\end{equation}
Owing to the characterization of the strong maximum principle 
(cf.\ \cite[Theorem 2.4]{ALG98}) it follows from \eqref{ro4} that the problem 
$(-\Delta-\lambda,\Omega,{\mathcal D})$ satisfies the strong maximum principle 
for each $\lambda \in (\lambda_1,\sigma_0)$ and hence, \eqref{ro555} implies that
\begin{equation}\label{ro666}
\dot{u}_{\lambda}>\frac{\alpha \varphi_0}{\sigma_0-\lambda}\,, \quad 
x \in \Omega\,,\quad \lambda \in (\lambda_1,\sigma_0)
\end{equation}
Now, let $K \subset \Omega$ be a compact subset in $\Omega$ and let us denote
$m_K:=\min_{x \in K}\varphi_0>0$.
Owing to \eqref{ro666} we obtain for $\lambda \in (\lambda_1,\sigma_0)$,
\begin{equation}\label{ro6}
\dot{u}_{\lambda}>\frac{\alpha \varphi_0}{\sigma_0-\lambda}
\geq \frac{\alpha m_K}{\sigma_0-\lambda}\,, \quad x \in K\,,
\end{equation}
and taking limits in \eqref{ro6} when $\lambda \uparrow \sigma_0$ it is 
obtained that
$\lim_{\lambda \uparrow \sigma_0}\dot{u}_{\lambda}=\infty$
uniformly in $K$.
Finally, integrating \eqref{ro6} in $[\lambda_1, \lambda]$ gives
\begin{equation}
u_{\lambda} \geq u_{\lambda_1}+\alpha m_K \ln 
\big( \frac{\sigma_0-\lambda_1}{\sigma_0-\lambda} \big)\,, \quad x \in K
\end{equation}
and therefore,
$\lim_{\lambda \uparrow \sigma_0}u_{\lambda}=\infty$ uniformly in $K$.
This completes the proof. 
\end{proof}

To prove that the positive solutions of \eqref{1-} also grow to infinity uniformly 
on $\Gamma_1$, we need the following lemmas.
The following lemma gives a comparison result, and it may be proved following 
similar arguments to the used in the proof of \cite[Proposition 3.2]{CC14}.

\begin{lemma}\label{comparison}
Let $u_{\lambda}$ and $\theta_{\lambda}$ be a positive solution and a positive 
strict subsolution of \eqref{1-}, respectively, for the value $\lambda$ of 
the parameter. Then
\begin{equation}\label{comparacions}
\theta_{\lambda} \ll u_{\lambda} \quad \text{in } \Omega
\end{equation}
\end{lemma}

The following lemma will be proved adapting to our current framework some of 
the original ideas given in \cite{LGSL98}, assuming for it that the component
 $\Gamma_1$ of $\partial \Omega$ is of class ${\mathcal C}^3$ in $\mathbb{R}^N$.

\begin{lemma}\label{tech}
Assume that the component $\Gamma_1$ of $\partial \Omega$ is of class 
${\mathcal C}^3$ in $\mathbb{R}^N$, and let be the domain
$$
\Omega_{\delta}:=\Omega \cup \{ x\in \mathbb{R}^N \setminus \Omega: 
\operatorname{dist}(x,\Gamma_1)<\delta \}\,,
$$
for $\delta>0$ small enough, and $\varphi_0$ and $\varphi_{\delta}$ 
the principal eigenfunctions associated to $-\Delta$ operator in the 
domains $\Omega$ and $\Omega_{\delta}$, respectively, under Dirichlet 
boundary conditions, normalized so that
$$
\| \varphi_0 \|_{L^{\infty}(\Omega)}=1\,, \quad
\| \varphi_{\delta} \|_{L^{\infty}(\Omega_{\delta})}=1
$$
Then, there exists $\bar{z}, \underline{z} \in \Gamma_1$ such that
\begin{gather}\label{marr1}
\|\varphi_{\delta}\|_{L^{\infty}(\Gamma_1)}
=-\partial \varphi_0(\bar{z}) \delta+o(\delta), \\
\label{mabb1}
\min_{x \in \Gamma_1}\varphi_{\delta}
=-\partial \varphi_0(\underline{z}) \delta+o(\delta)
\end{gather}
\end{lemma}

\begin{proof} Since $\Gamma_1$ is a compact surface of class ${\mathcal C}^3$ 
in $\mathbb{R}^N$, let $n:=n(x) \in {\mathcal C}^2(\Gamma_1;\mathbb{R}^N)$ 
be the ${\mathcal C}^2$ outward unit normal field to $\Gamma_1$ and for 
$\delta>0$ small enough, let 
$$
A_{\delta}:=\{ x\in \mathbb{R}^N: \operatorname{dist}(x,\Gamma_1)<\delta\}
$$
be a tubular $\delta-$neighborhood of $\Gamma_1$, and
$$
\Gamma_{\delta}:=\left\{ x\in \mathbb{R}^N \setminus \Omega :
 \operatorname{dist}(x,\Gamma_1)=\delta \right\}
$$
Then, for every $x \in A_{\delta}$, there exists a unique $y \in \Gamma_1$ 
and $\tau \in (-\delta,\delta)$ such that
\begin{equation}\label{tub1}
x=y-\tau n(y)
\end{equation}
By restricting $\delta>0$ if it is necessary, the implicit function theorem 
gives the existence of two unique mappings 
$\tau \in {\mathcal C}^2(A_{\delta};\mathbb{R})$ and 
$\pi \in {\mathcal C}^2({\mathcal A}_{\delta};\Gamma_1)$ such that
\begin{equation}\label{tub2}
x=\pi(x)-\tau(x)n(\pi(x))\,, \quad x\in A_{\delta}
\end{equation}
Let $\hat{\tau} \in {\mathcal C}(\bar{\Omega}; \mathbb{R})$ be the extension to 
$\bar{\Omega}$ by $\hat{\tau}(x)=\delta$ if 
$\operatorname{dist}(x,\Gamma_1) \geq \delta$, 
$\hat{n}\in {\mathcal C}^2(\bar{\Omega}; \mathbb{R}^N)$ any regular extension 
of the vector field $n(\pi(x))$ to $\bar{\Omega}$ and let us consider any 
function $\xi \in {\mathcal C}^3([0,\infty);[0,\infty))$ satisfying
$\xi(0)=1$, $\xi(\tau) \xi'(\tau)<0$ for $\tau \in [0,\frac{\delta}{2})$ and 
$\xi(\tau)=0$ for $\tau \geq \frac{\delta}{2}$. Now, let us consider the mapping
$$
H(x):=\xi(\hat{\tau}(x))\hat{n}(x)\,, \quad x\in \bar{\Omega}
$$
This map is of class ${\mathcal C}^2$ and it satisfies 
$$
H(x)= \begin{cases}
0 &\text{if } \operatorname{dist}(x,\Gamma_1) \geq \delta/2\\
\xi(\tau(x))n(\pi(x)) & \text{if } 0<\operatorname{dist}(x,\Gamma_1)<\delta/2\\
n(x)& \text{if } x \in \Gamma_1
\end{cases}
$$
Let us consider the mapping $T_{\delta}:\bar{\Omega} \,\mapsto\, \mathbb{R}^N$ 
defined by
$$
T_{\delta}:=I+\delta H\,,
$$
where $I$ stands for the identity map in $\mathbb{R}^N$. 
Owing to \cite[Theorem 3.1]{LGSL98}, 
$T_{\delta} \in {\mathcal C}^2(\bar{\Omega}; \mathbb{R}^N)$ and 
$T_{\delta}: \bar{\Omega}\mapsto \bar{\Omega}_{\delta}$ is a bijection and a 
real holomorphic family in $\delta \cong 0$(cf.\ \cite[Section 2]{LGSL98}). 
Now, set
$$
y=T_{\delta}(x) \in \bar{\Omega}_{\delta}\,, \quad 
\psi_{\delta}(x)=\varphi_{\delta}(y)=\varphi_{\delta}(T_{\delta}(x))\,, 
\quad x\in \bar{\Omega}
$$
By construction and definition we obtain
$T_{\delta}(\Gamma_1)=\Gamma_{\delta}$ and
\begin{equation}\label{rest1}
\psi_{\delta}|_{\Gamma_1}=(\varphi_{\delta} \circ T_{\delta})
|_{\Gamma_1}=\varphi_{\delta}|_{T_{\delta}(\Gamma_1)}
=\varphi_{\delta}|_{\Gamma_{\delta}}=0
\end{equation}
To prove \eqref{marr1}, for each $\delta>0$ small enough, let 
$\bar{y}_{\delta} \in \Gamma_1 \subset \Omega_{\delta}$ be such that
\begin{equation}\label{f4}
\| \varphi_{\delta}\|_{L^{\infty}(\Gamma_1)}=\varphi_{\delta}(\bar{y}_{\delta})\,,
\end{equation}
and $x_{\delta} \in \Omega$ such that $T_{\delta}(x_{\delta})=\bar{y}_{\delta}$. 
Let $z_{\delta} \in \Gamma_1$ be such that $\pi(x_{\delta})=z_{\delta}$.
 We have that $x_{\delta}=z_{\delta}-\delta n(z_{\delta})$.
Since $z_{\delta}\in \Gamma_1$, it follows from \eqref{rest1} that
$\psi_{\delta}(z_{\delta})=0$ and since
$$
\varphi_{\delta}(\bar{y}_{\delta})=\varphi_{\delta}(T_{\delta}(x_{\delta}))
=\psi_{\delta}(x_{\delta})=\psi_{\delta}(z_{\delta}-\delta n(z_{\delta}))\,,
$$
arguing as in \cite[Theorem 4.3]{LGSL98} we find that
\begin{equation}\label{marr4}
\begin{aligned}
\frac{\varphi_{\delta}(\bar{y}_{\delta})}{\delta}
&=\frac{\psi_{\delta}(x_{\delta})}{\delta} 
=\frac{\psi_{\delta}(z_{\delta}-\delta n(z_{\delta}))
 -\psi_{\delta}(z_{\delta})}{\delta}\\
&=-\int_0^1 \langle \nabla \psi_{\delta}(z_{\delta}-t \delta n(z_{\delta})),
 n(z_{\delta})\rangle\, dt
\end{aligned}
\end{equation}
Since
$\{ z_{\delta} : \delta>0\} \subset \Gamma_1$
and $\Gamma_1$ is compact, taking limits when $\delta \to 0$, module some 
subsequence, we obtain there exists $\bar{z} \in \Gamma_1$ such that
$\lim_{\delta \downarrow 0}z_{\delta}=\bar{z} \in \Gamma_1$.
Then, taking limits in \eqref{marr4} when $\delta \to 0$ we find that
\begin{equation}\label{f6}
\lim_{\delta \to 0}\frac{\varphi_{\delta}(\bar{y}_{\delta})}{\delta}
=-\lim_{\delta \to 0}\int_0^1 \langle \nabla \psi_{\delta}(z_{\delta}-t \delta n(z_{\delta})),n(z_{\delta})\rangle\, dt=-\partial \varphi_0 (\bar{z})\,,
\end{equation}
and therefore, \eqref{f4} and \eqref{f6} imply
$$
\| \varphi_{\delta} \|_{L^{\infty}(\Gamma_1)}
=\varphi_{\delta}(\bar{y}_{\delta})=-\partial \varphi_0(\bar{z})\delta+o(\delta)
$$
This completes the proof of \eqref{marr1}. 

The proof of \eqref{mabb1} follows exactly the same steps than in the proof
 of \eqref{marr1}, changing in \eqref{f4} the existence of 
$\bar{y}_{\delta}\in \Gamma_1 \subset \Omega_{\delta}$ satisfying \eqref{f4}, 
by the existence of $\tilde{y}_{\delta}\in \Gamma_1 \subset \Omega_{\delta}$
such that
$$
\min_{x \in \Gamma_1}\varphi_{\delta}=\varphi_{\delta}(\tilde{y}_{\delta})>0\,,
$$
where later, module some subsequence, we will get that
$$
\lim_{\delta \downarrow 0}\tilde{z}_{\delta}=\underline{z}\in \Gamma_1\,,
$$
being $\tilde{x}_{\delta} \in \Omega$, 
$T_{\delta}(\tilde{x}_{\delta})=\tilde{y}_{\delta}$ and 
$\pi(\tilde{x}_{\delta})=\tilde{z}_{\delta}$.
This completes the proof of \eqref{mabb1} and of the result. 
\end{proof}

The following result establishes the uniform growth to infinity on $\Gamma_1$
 of the positive solutions of \eqref{1-} when $\lambda \uparrow \sigma_0$. 
Part of its proof is based in some of the original ideas given in \cite{LGSL98}.

\begin{theorem}\label{ws2} 
Assume \eqref{bsep} and that the component $\Gamma_1$ of $\partial \Omega$ 
is of class ${\mathcal C}^3$ in $\mathbb{R}^N$. Then
\begin{equation}\label{bl}
\lim_{\lambda \uparrow \sigma_0}u_{\lambda}(x)
=\infty \quad \text{uniformly on } \Gamma_1
\end{equation}
\end{theorem}

\begin{proof} Set, for $\delta>0$ small enough, the domain
$$
\Omega_{\delta}:=\Omega \cup \{ x\in \mathbb{R}^N \setminus \Omega: 
\operatorname{dist}(x,\Gamma_1)<\delta\}\,,
$$
and let $(\sigma_0,\varphi_0)$ and $(\sigma^{\delta},\varphi_{\delta})$
 be the principal eigen-pairs associated to $-\Delta$ operator in the domains 
$\Omega$ and $\Omega_{\delta}$, respectively, under Dirichlet boundary conditions, 
with the eigenfunctions normalized so that
$$
\| \varphi_0\|_{L^{\infty}(\Omega)}=1\,, \quad 
\| \varphi_{\delta}\|_{L^{\infty}(\Omega_{\delta})}=1
$$
By definition and since $\varphi_0$ and $\varphi_{\delta}$ are strongly positive 
in $\Omega$ and $\Omega_{\delta}$, respectively, we obtain
$$
\varphi_0(x)=0\,, \quad \partial \varphi_0(x)<0 \quad 
\forall x \in \partial \Omega=\Gamma_0 \cup \Gamma_1\,,
$$
and since $\Gamma_1 \subset \Omega_{\delta}$ for all $\delta>0$,
\begin{equation}\label{positivo}
\varphi_{\delta}(x)>0 \quad \forall x \in \Gamma_1
\end{equation}
Let us denote 
\begin{gather}\label{defa0}
\alpha_0(x):=-\partial \varphi_0(x)>0 \quad \forall  x \in \partial \Omega\,, \\
\label{defm0}
\underline{\alpha}_0:=\min_{x \in \Gamma_1} \alpha_0(x)>0\,, \quad 
\bar{\alpha}_0:=\| \alpha_0(x)\|_{L^{\infty}(\Gamma_1)}\,, \\
\label{ib}
\bar{b}:=\| b(x)\|_{L^{\infty}(\Gamma_1)}>0\,, \quad 
\bar{V}:=\| V(x)\|_{L^{\infty}(\Gamma_1)} \geq 0
\end{gather}
Let us fix $\varepsilon >0$ such that
\begin{equation}\label{epsilon}
0<\varepsilon< \frac{\underline{\alpha}_0}{1+\bar{V}}
\end{equation}
and $k_0>0$ large enough such that
\begin{equation}\label{km}
0<\frac{1}{k_0}<\underline{\alpha}_0-\varepsilon (1+\bar{V})
\end{equation}
Thanks to the monotonicity of the principal eigenvalue with respect to the 
domain (cf.\ \cite{LG96}, \cite[Proposition 3.2]{CCLG01}), it is known that
\begin{equation}\label{dn}
\sigma^{\delta}<\sigma_0 \quad \text{for all } \delta>0,
\end{equation}
and by the continuous dependence of the principal eigenvalue with respect
 to perturbations of the domain around its Dirichlet boundary
 (cf.\ \cite[Theorem 4.2]{LG96}, \cite[Theorems 7.1 and 7.4]{CCLG01}), we obtain
\begin{equation}\label{cff}
\lim_{\delta \downarrow 0}\sigma^{\delta}=\sigma_0\,.
\end{equation}
Then, by \eqref{base}, \eqref{dn} and \eqref{cff}, there exists $\delta_0>0$ 
such that
\begin{equation}\label{dn1}
\sigma_1<\sigma^{\delta}<\sigma_0 \quad \forall  \delta \in (0,\delta_0]
\end{equation}
Also, by construction, owing to the regularity of the principal eigenfunctions 
$\varphi_{\delta}$ and $\varphi_0$ and to the results in
\cite{LGSL98}, there exists $\delta_1 \in (0,\delta_0]$ such that
\begin{gather}\label{conv}
\partial \varphi_{\delta}(x)\leq \partial \varphi_0(x)+\varepsilon\,, \quad 
\forall \delta \in (0,\delta_1]\,, \; \forall x \in \Gamma_1, \\
\label{conver}
\varphi_{\delta}(x)\leq \varphi_0(x)+\varepsilon 
=\varepsilon\,, \quad \forall \delta \in (0,\delta_1]\,, \; \forall x \in \Gamma_1
\end{gather}
Let $\bar{y}_{\delta}\in \Gamma_1$ be such that
\begin{equation}\label{m}
\varphi_{\delta}(\bar{y}_{\delta})=\|\varphi_{\delta}\|_{L^{\infty}(\Gamma_1)}
\end{equation}
(cf.\ \eqref{f4}). Owing to \eqref{marr1} we obtain the existence of 
$\bar{z} \in \Gamma_1$ such that
\begin{equation}\label{o}
\varphi_{\delta}(\bar{y}_{\delta})
=\| \varphi_{\delta}\|_{L^{\infty}(\Gamma_1)}
=\alpha_0(\bar{z})\delta+o(\delta)\,.
\end{equation}
Now, let us consider the function defined in $\Omega_{\delta}$ by
\begin{equation}
v_{\delta}:= C(\delta)\varphi_{\delta}\,,
\end{equation}
where 
\begin{equation}\label{defc}
C=C(\delta):=\frac{1}{(k_0 \bar{b} \tilde{\gamma})^{\frac{1}{q-1}}
(\bar{\alpha}_0\delta)^{\frac{q}{q-1}}}>0\,,
\end{equation}
being $\bar{\alpha}_0$, $\bar{b}$ and $k_0$ defined by \eqref{defm0}, \eqref{ib} 
and \eqref{km}, respectively. We are going to prove that there exists 
$\delta_2 \in (0,\delta_1]$, such that for each $\delta \in (0,\delta_2]$ and 
$\lambda \in (\sigma^{\delta},\sigma_0)$, the function
$$
u_{\delta}:=v_{\delta}|_{\bar{\Omega}}=C(\delta) \varphi_{\delta}|_{\bar{\Omega}}
$$
is a positive strict subsolution of \eqref{1-}. Indeed, by construction the 
following holds in $\Omega$ for each $\delta \in (0,\delta_1]$ and 
$\lambda \in (\sigma^{\delta},\sigma_0)$
\begin{equation}\label{bc0}
(-\Delta -\lambda)u_{\delta}=C(\delta)(\sigma^{\delta}-\lambda)\varphi_{\delta}<0
\end{equation}
In regard to the boundary conditions on $\Gamma_1$, by \eqref{defa0}, \eqref{defm0}, 
\eqref{ib}, \eqref{conv}, \eqref{conver}, \eqref{m}, \eqref{o} and \eqref{defc}, 
the following hold on $\Gamma_1$:
\begin{align*}
&\left(\partial +V(x) \right) u_{\delta}(x)+\tilde{\gamma}b(x) u_{\delta}^q(x)\\
&=C \left( \partial \varphi_{\delta}(x)
 +V(x)\varphi_{\delta}(x)+\tilde{\gamma}b(x)C^{q-1}\varphi_{\delta}^q(x) \right)\\
&\leq  C \left( \partial \varphi_{0}(x)+\varepsilon (1+\bar{V})
 +\tilde{\gamma}\bar{b}C^{q-1}\varphi_{\delta}^q(y_{\delta})\right)\\
&=  C \left( -\alpha_0(x)+\varepsilon (1+\bar{V})
 +\tilde{\gamma}\bar{b}C^{q-1}(\alpha_0(z_0)\delta+o(\delta))^q\right)\\
&\leq  C(-\underline{\alpha}_0+\varepsilon (1+\bar{V})
 +\tilde{\gamma}\bar{b}C^{q-1}(\bar{\alpha}_0 \delta +o(\delta))^q)\\
&=C(-\underline{\alpha}_0+\varepsilon (1+\bar{V})
 +\tilde{\gamma}\bar{b}C^{q-1} \left( \bar{\alpha}_0^q \delta^q +o(\delta^q) \right))\\
&=C\Big( -\underline{\alpha}_0+\varepsilon(1+\bar{V})
 +\frac{1}{k_0}+\frac{1}{k_0 \bar{\alpha}_0^q}\frac{o(\delta^q)}{\delta^q}\Big)\,.
\end{align*}
Then, by \eqref{km}, there exists $\delta_2 \in (0,\delta_1]$ such that for each 
$\delta \in (0,\delta_2]$ and $x \in \Gamma_1$,
$$
\left( \partial +V(x) \right) u_{\delta}(x)+\tilde{\gamma}b(x)u_{\delta}^q(x) 
\leq C \Big( -\underline{\alpha}_0+\varepsilon (1+\bar{V})
 +\frac{1}{k_0}+\frac{1}{k_0 \bar{\alpha}_0^q}\frac{o(\delta^q)}{\delta^q}\Big)<0
$$
and hence,
\begin{equation}\label{bc2}
\partial u_{\delta}+V(x)u_{\delta}+\tilde{\gamma} b(x)u_{\delta}^q<0 \quad 
\text{on } \Gamma_1\,, \;\delta \in (0,\delta_2]
\end{equation}
Also, by construction
\begin{equation}\label{bc3}
u_{\delta}\big|_{\Gamma_0}=C(\delta)\varphi_{\delta}|_{\Gamma_0}=0
\end{equation}
Then, \eqref{bc0}, \eqref{bc2} and \eqref{bc3} give that for 
$\delta \in (0,\delta_2]$ and $\lambda \in (\sigma^{\delta},\sigma_0)$
\begin{equation}
\begin{gathered}
(-\Delta -\lambda) u_{\delta}<0 \quad \text{in } \Omega\\
u_{\delta}=0\quad \text{on } \Gamma_0\\
\partial u_{\delta}+V(x)u_{\delta}+\tilde{\gamma} b(x)u_{\delta}^q<0\quad
 \text{on } \Gamma_1
\end{gathered}
\end{equation}
and therefore, $u_{\delta}$ is a positive strict subsolution of \eqref{1-} 
for $\delta>0$ small enough and $\lambda \in (\sigma^{\delta},\sigma_0)$.

Now we  prove \eqref{bl}. From \eqref{positivo}, for each $\delta \in (0,\delta_2]$ 
there exists $\tilde{y}_{\delta} \in \Gamma_1$ such that
\begin{equation}\label{s}
\min_{x \in \Gamma_1}\varphi_{\delta}(x)=\varphi_{\delta}(\tilde{y}_{\delta})>0\,,
\end{equation}
and from \eqref{mabb1}, there exists $\underline{z} \in \Gamma_1$ such that
\begin{equation}\label{ss}
\varphi_{\delta}(\tilde{y}_{\delta})=\alpha_0(\underline{z})\delta+o(\delta)
\end{equation}
Thus, from \eqref{s}, \eqref{ss} and the definition of the constants 
$C=C(\delta)>0$ and $\underline{\alpha}_0$ (cf.\ \eqref{defc}, \eqref{defm0}), 
for each $x \in \Gamma_1$ and $\delta \in (0,\delta_2]$ we obtain
$$
u_{\delta}(x)=C\varphi_{\delta}(x)\geq C\varphi_{\delta}(\tilde{y}_{\delta})
=\frac{\alpha_0(\underline{z})\delta+o(\delta)}
{\left( k_0 \bar{b} \tilde{\gamma}\right)^{\frac{1}{q-1}} 
\big(\bar{\alpha}_0 \delta \big)^{\frac{q}{q-1}}}
\geq \frac{\underline{\alpha}_0\delta+o(\delta)}
{\big( k_0 \bar{b} \tilde{\gamma}\big)^{\frac{1}{q-1}} 
\big(\bar{\alpha}_0 \delta \big)^{\frac{q}{q-1}}}\,,
$$
and therefore, since $q>1$, it holds
\begin{equation}\label{rl}
\begin{aligned}
\liminf_{\delta \downarrow 0}u_{\delta}(x) 
&\geq \lim_{\delta \downarrow 0}\frac{\underline{\alpha}_0 \delta 
 +o(\delta)}{k_0^{\frac{1}{q-1}}\tilde{\gamma}^{\frac{1}{q-1}}
 \bar{b}^{\frac{1}{q-1}} \bar{\alpha}_0^{\frac{q}{q-1}}\delta^{\frac{q}{q-1}}} \\
&= \lim_{\delta \downarrow 0}\frac{\underline{\alpha}_0}{k_0^{\frac{1}{q-1}} 
 \tilde{\gamma}^{\frac{1}{q-1}}\bar{b}^{\frac{1}{q-1}} 
 \bar{\alpha}_0^{\frac{q}{q-1}}\delta^{\frac{1}{q-1}}}=\infty
\end{aligned}
\end{equation}
uniformly on $\Gamma_1$. Now, due to the fact that $u_{\delta}(x)$ is a positive 
strict subsolution of \eqref{1-} for any $\delta \in (0,\delta_2]$ and 
$\lambda \in (\sigma^{\delta},\sigma_0)$, it follows from Lemma \ref{comparison} 
and from the existence and uniqueness of positive solution $u_{\lambda}$ 
of \eqref{1-} for each $\lambda \in (\sigma^{\delta},\sigma_0)$ (cf.\ \eqref{dn1} 
and Theorem \ref{mt-1}) that
\begin{equation}\label{rk}
u_{\delta}< u_{\lambda} \text{ in } \Omega, \quad 0<\delta \leq \delta_2\,,
 \quad \sigma^{\delta}<\lambda<\sigma_0\,,
\end{equation}
and therefore, \eqref{rl}, \eqref{rk} and \eqref{cff} imply \eqref{bl}.
This completes the proof. 
\end{proof}

\subsection{Dynamics of the positive solutions of the parabolic problem for 
$\gamma<0$}\label{Section3.3}

In this section we will analyze, in the particular case when $\gamma<0$, 
depending on the values of the parameter $\lambda$, the longtime behavior of 
the positive solutions of the parabolic problem associated to \eqref{1-}, 
given by
\begin{equation}\label{pp1}
\begin{gathered}
w_t-\Delta w=\lambda w\quad \text{in } \Omega \times (0,\infty)\\
w=0\quad \text{on }  \Gamma_0 \times (0,\infty)\\
\partial w+V(x)w=\gamma b(x) w^q\quad  \text{on } \Gamma_1 \times (0,\infty)\\
w(x,0)=u_0>0\quad  \text{in } \Omega\,.
\end{gathered}
\end{equation}
In this section $\Theta_{\lambda}(x,t;u_0)$ stands for the solution of \eqref{pp1} 
for the value $\lambda$ of the parameter, $u_{\lambda}$ will stand for the unique 
positive solution of \eqref{1-} when it exists, that is, when 
$\lambda \in (\sigma_1,\sigma_0)$ and
$T(t)$ will stand for the $L_p$-evolution operator associated with 
$\Delta+\lambda$ under the linear homogeneous mixed boundary conditions given
 by the boundary operator $\mathfrak{B}(V(x))$. The solution 
$\Theta_{\lambda}(x,t;u_0)$ is globally defined in time, since it satisfies
\begin{gather*}
(\Theta_{\lambda}(\cdot,t;u_0))_t-(\Delta + \lambda) \Theta_{\lambda}(\cdot,t;u_0)=0
\quad \text{in } \Omega \times (0,\infty)\\
\Theta_{\lambda}(\cdot,t;u_0)=0 \quad \text{on }  \Gamma_0 \times (0,\infty)\\
(\partial +V(x))\Theta_{\lambda}(\cdot,t;u_0)<0 \quad \text{on }
 \Gamma_1 \times (0,\infty)\\
\Theta_{\lambda}(\cdot,0;u_0)=u_0>0\quad \text{in } \Omega\,;
\end{gather*}
that is, it is a positive strict subsolution of the linear heat equation in 
the domain $\Omega$ under the linear homogeneous mixed boundary conditions
$$\mathfrak{B}(V(x))\Theta_{\lambda}=0\,,
$$
and therefore, by the parabolic maximum principle we obtain
$$
0 \ll \Theta_{\lambda}(x,t;u_0) \ll T(t)u_0\,.
$$
Hereafter we will say that a non-negative steady-state $\tilde{u}_{\lambda}$ 
of \eqref{1-} for the value $\lambda$ of the parameter is globally asymptotically 
stable, if
$$
\lim_{t \uparrow \infty} \| \Theta_{\lambda}(\cdot,t;u_0)
-\tilde{u}_{\lambda}(\cdot)\|_{L^{\infty}(\Omega)}=0\,,
$$
for any initial data $u_0>0$.

Arguing as in \cite[Theorem 2.2]{LG00} and by Lemma \ref{sub}, Lemma \ref{super} 
and Theorem \ref{mt-1}, we obtain the following result, which gives the
 dynamics of the solution $\Theta_{\lambda} (\cdot,t;u_0)$ of the parabolic 
problem \eqref{pp1}, with $u_0>0$, depending on the value $\lambda$ of the parameter.

\begin{theorem}\label{dynamics}
Under the assumptions of Theorem \ref{mt-1} the following hold:
\begin{itemize}
\item[(i)] If $\lambda \leq \sigma_1$, then $u=0$ is globally asymptotically 
stable for \eqref{pp1}.

\item[(ii)] If $\sigma_1<\lambda<\sigma_0$, then $u_{\lambda}$ is globally 
asymptotically stable for \eqref{pp1}.

\item[(iii)] If $\lambda \geq \sigma_0$, then for any $u_0>0$,
\begin{equation}\label{mhj}
\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0)=\infty
\end{equation}
uniformly in compact subsets $K \subset \Omega$. If in addition 
$\Gamma_1$ is of class ${\mathcal C}^3$, then \eqref{mhj} holds uniformly in 
compact subsets $K \subset \Omega \cup \Gamma_1$.
\end{itemize}
\end{theorem}

\begin{proof}
(i) Let $u_0>0$ be, fix $t_1>0$ and let us consider $\tilde{u}_0=u(\cdot,t_1;u_0)$. 
Owing to the parabolic maximum principle we know that
\begin{equation}\label{pmp}
\tilde{u}_0 \gg 0 \quad \text{in } \Omega
\end{equation}
 Since $\lambda \leq \sigma_1<\sigma_0$, it follows from Lemma \ref{super} 
the existence of a positive strict supersolution $\bar{u}_{\lambda}$ of \eqref{1-}, 
strongly positive in $\Omega$, such that
$$
\tilde{u}_0<\bar{u}_{\lambda}\,,
$$
and owing to the parabolic maximum principle, we obtain
\begin{equation}\label{const1}
0 \ll \Theta_{\lambda}(\cdot,t;u_0)
=\Theta_{\lambda}(\cdot,t-t_1;\tilde{u}_0)
\leq \Theta_{\lambda}(\cdot,t-t_1;\bar{u}_{\lambda})
\end{equation}
Now, thanks to the results in \cite{Sa73}, we know that 
$\Theta_{\lambda}(\cdot,t-t_1;\bar{u}_{\lambda})$ is a decreasing function in 
$t>t_1$ which converges to a non-negative solution of \eqref{1-}, and due to 
the fact that $u=0$ is the unique nonnegative solution of \eqref{1-} for 
$\lambda \leq \sigma_1$ (cf.\ Theorem \ref{mt-1}), taking limits in \eqref{const1}
 when $t \uparrow \infty$ we obtain
$$
\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0)
=\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t-t_1;\bar{u}_{\lambda})=0\,,
$$
which completes the proof of (i).

(ii) Let $u_0>0$ be, fix $t_1>0$ and let us consider 
$\tilde{u}_0=u(\cdot,t_1;u_0) \gg 0$ (cf.\ \eqref{pmp}). Since 
$\lambda \in (\sigma_1,\sigma_0)$, owing to Lemma \ref{sub} and Lemma \ref{super}, 
we obtain the existence of a positive strict subsolution $\underline{u}_{\lambda}$ 
of \eqref{1-} and a positive strict supersolution $\bar{u}_{\lambda}$ of \eqref{1-}, 
both of them strongly positive in $\Omega$, such that
$$
0<\underline{u}_{\lambda}<\tilde{u}_0<\bar{u}_{\lambda} \quad \text{in } \Omega
$$
Then, by the parabolic maximum principle, we obtain
\begin{equation}\label{const2}
0 \ll \Theta_{\lambda}(\cdot,t-t_1;\underline{u}_{\lambda})
\leq \Theta_{\lambda}(\cdot,t;u_0)=\Theta_{\lambda}(\cdot,t-t_1;\tilde{u}_0)
\leq \Theta_{\lambda}(\cdot,t-t_1;\bar{u}_{\lambda})
\end{equation}
On the other hand, thanks to the results in \cite{Sa73}, we know that 
$\Theta_{\lambda}(\cdot,t-t_1;\underline{u}_{\lambda})$ and 
$\Theta_{\lambda}(x,t-t_1;\bar{u}_{\lambda})$ are an increasing and a decreasing 
function in $t>t_1$, respectively, converging to a nonnegative solution 
of \eqref{1-}. Then, since for $\lambda \in (\sigma_1,\sigma_0)$ there exists 
a unique positive solution $u_{\lambda}$ of \eqref{1-} (cf.\ Theorem \ref{mt-1}), 
we obtain
$$
\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t-t_1,\bar{u}_{\lambda})
=u_{\lambda}\,, \quad
\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t-t_1,\underline{u}_{\lambda})
=u_{\lambda}\,,
$$
and therefore, taking limits in \eqref{const2} when $t \uparrow \infty$, 
we have
$$
\lim_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0)=u_{\lambda}\,,
$$
which completes the proof of (ii).

(iii) Let $u_0>0$ be and let us consider the global solution 
$\Theta_{\lambda}:=\Theta_{\lambda}(\cdot,t;u_0)>0$ of the parabolic problem 
\eqref{pp1} for the value $\lambda$ of the parameter with initial data $u_0>0$.
 Since $\lambda \geq \sigma_0$, for all $\varepsilon>0$ small enough, 
$\lambda\geq \sigma_0 >\sigma_0-\varepsilon > \sigma_1$. Then
$$
\left(\Theta_{\lambda}\right)_t-\Delta \Theta_{\lambda}=
\lambda \Theta_{\lambda}>(\sigma_0-\varepsilon)\Theta_{\lambda}\,, \quad \text{in } 
\Omega \times (0,\infty)\,,
$$
and hence, $\Theta_{\lambda}$ is a positive strict supersolution of the problem
\begin{equation}\label{const3}
\begin{gathered}
w_t-\Delta w=(\sigma_0-\varepsilon)w\quad \text{in } \Omega \times (0,\infty)\\
w=0 \quad \text{on } \Gamma_0 \times (0,\infty)\\
\partial w+V(x)w=\gamma b(x) w^q \quad \text{on } \Gamma_1 \times (0,\infty)\\
w(\cdot,0)=u_0>0 \quad \text{in } \Omega
\end{gathered}
\end{equation}
Then, by the parabolic maximum principle,
\begin{equation}\label{const9}
\Theta_{\lambda}(\cdot,t;u_0) > \Theta_{\sigma_0-\varepsilon}(\cdot,t;u_0)\,,
\end{equation}
where $\Theta_{\sigma_0-\varepsilon}(\cdot,t,u_0)$ stands for the solution 
of \eqref{const3}. Now, since $\sigma_0-\varepsilon \in (\sigma_1,\sigma_0)$, 
owing to $ii)$ we obtain $u_{\sigma_0-\varepsilon}$ is globally asymptotically 
stable for \eqref{pp1}. Then
\begin{equation}\label{const4}
\lim_{t \uparrow \infty}\| \Theta_{\sigma_0-\varepsilon}(\cdot,t;u_0)
- u_{\sigma_0-\varepsilon}(\cdot)\|_{L^{\infty}(\Omega)}=0\,,
\end{equation}
and hence, \eqref{const9} and \eqref{const4} imply 
\begin{equation}\label{const10}
\liminf_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0) 
\geq u_{\sigma_0-\varepsilon}(\cdot)\,.
\end{equation}
Now, taking limits in \eqref{const10} when $\varepsilon \downarrow 0$ we obtain
\begin{equation}\label{kkjjll}
\liminf_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0) 
\geq \lim_{\varepsilon \downarrow 0}u_{\sigma_0-\varepsilon}(\cdot)\,,
\end{equation}
and since by Theorem \ref{ws1}
\begin{equation}\label{star1}
\lim_{\varepsilon \downarrow 0}u_{\sigma_0-\varepsilon}(\cdot)=\infty
\end{equation}
uniformly in compact subsets $K \subset \Omega$, we obtain that
\begin{equation}\label{star2}
\liminf_{t \uparrow \infty}\Theta_{\lambda}(\cdot,t;u_0) =\infty
\end{equation}
uniformly in compact subsets $K \subset \Omega$. 
If in addition $\Gamma_1$ is of class ${\mathcal C}^3$ in $\mathbb{R}^N$, 
then owing to Theorem \ref{ws2} and \eqref{kkjjll}, \eqref{star1} holds 
uniformly on $\Gamma_1$ and therefore \eqref{star2} holds uniformly 
in compact subsets of $\Omega \cup \Gamma_1$.
This completes the proof. 
\end{proof}

\section{The case $\gamma>0$}\label{Section4}

In this section we obtain, in the particular case when $\gamma>0$, 
some results about existence and non existence of positive solutions of
 \eqref{1}, about the stability of them and about the structure of the 
global bifurcation diagram of positive solutions of \eqref{1}. 
Also, we will obtain some results about the dynamics of the positive solutions 
of the parabolic problem associated to \eqref{1}, given by
\begin{equation}\label{parproblem}
\begin{gathered}
v_t-\Delta v=\lambda v \quad \text{in } \Omega \times (0,T)\,,\\
v=0\quad \text{on } \Gamma_0 \times (0,T)\,,\\
\partial v+V(x)v=\gamma b(x)v^q\quad \text{on } \Gamma_1 \times (0,T)\,,\quad q>1\\
v(\cdot,0)=u_0>0\quad \text{in } \Omega\,,
\end{gathered}
\end{equation}
where $T>0$ is its time of existence.

\subsection{Dynamics of  positive solutions of the parabolic problem for $\gamma>0$}
\label{Section4.1}
In this section we analyze the longtime behavior or blow up in finite time of 
the positive solutions of \eqref{parproblem}. In this section 
$\Theta_{\lambda}(x,t;u_0)$ will stand for the solution of \eqref{parproblem} 
for the value $\lambda$ of the parameter.
The main result of this section is the following.

\begin{theorem}\label{parabolico}
Assume $\gamma>0$. Then, the following hold:
\begin{itemize}
\item[(i)] For any $\lambda$ sufficiently negative,
 $\Theta_{\lambda}(x,t;u_0)$ is globally defined in time and
\begin{equation}\label{ri}
\| \Theta_{\lambda}(\cdot,t;u_0)\|_{L^{\infty}(\Omega)} \to 0 \quad \text{as } t\to \infty
\end{equation}

\item[(ii)] If \eqref{bsep} holds, $\lambda \geq \sigma_1$ and $u_0$ is large enough, 
then $\Theta_{\lambda}(x,t;u_0)$ blows up in finite time uniformly in compact 
subsets $Q \subset \Omega \cup \Gamma_1$.
\end{itemize}
\end{theorem}

\begin{proof}
(i) Let us take some $p>2q-1$, let us fix some $\alpha>0$ satisfying
\begin{equation}\label{ca}
0<\alpha <\big( \gamma \| b\|_{L^{\infty}(\Gamma_1)}\big)^{-\frac{1}{q-1}}\,,
\end{equation}
let $\tilde{V}$ be any extension of $V$ from $\Gamma_1$ to
$\partial \Omega=\Gamma_0 \cup \Gamma_1$ with 
$\tilde{V} \in {\mathcal C}^1(\partial \Omega)$, and let us consider the 
parabolic problem
\begin{equation}\label{par1}
\begin{gathered}
w_t-\Delta w=-w^p \quad \text{in } \Omega \times (0,T)\\
\partial w+\tilde{V}(x)w=w^q\quad \text{on } \partial \Omega \times (0,T)\\
w(x,0)=\frac{1}{\alpha}u_0>0\quad \text{in } \Omega\,.
\end{gathered}
\end{equation}
By \cite[Lemma 4.5]{CC14}, obtained adapting to our framework the original ideas 
given in \cite[Lemma 5.1]{AMTR02} and \cite[Theorem 2.3]{AMTR02} for $m=1$ 
therein (also cf.\ \cite{Chi91},\cite{ARB01}), the solution $w(x,t)$ 
of \eqref{par1} is globally defined in time ($T=\infty$) and globally bounded 
in $(x,t) \in \bar{\Omega} \times [0,\infty)$. Now, let us consider the function
$$
\bar{v}(x,t):=\alpha e^{-t}w(x,t)
$$
Since $w(x,t)$ is globally bounded in $\bar{\Omega} \times [0,\infty)$, we obtain
\begin{equation}\label{lim1}
\lim_{t \uparrow \infty}\| \bar{v}(x,t) \|_{L^{\infty}(\Omega)}=0
\end{equation}
Now we prove that for $\alpha>0$ satisfying \eqref{ca} and $\lambda$ sufficiently 
negative, the function $\bar{v}$ is a positive strict supersolution 
of \eqref{parproblem}. Indeed, since $w>0$ is globally bounded in 
$\bar{\Omega}\times [0,\infty)$, by construction the following hold in 
$\Omega \times [0,\infty)$ for any $\alpha>0$ and $\lambda$ negative enough
\begin{equation}\label{d1}
\begin{aligned}
\bar{v}_t-\Delta \bar{v}-\lambda \bar{v}
&= \alpha e^{-t}w \big(-(1+\lambda)-w^{p-1} \big)\\
&\geq  \alpha e^{-t}w \big( -(1+\lambda)
 -\| w \|_{L^{\infty}(\Omega \times [0,\infty))}^{p-1} \big)>0
\end{aligned}
\end{equation}
On the other hand, as for the boundary conditions, owing to \eqref{ca} and 
since $q>1$, $\gamma>0$ and $b>0$, the following hold on $\Gamma_1$
\begin{equation}\label{d2}
\begin{aligned}
\partial \bar{v}+V(x)\bar{v}-\gamma b(x) \bar{v}^q
&=  \alpha e^{-t}w^q \big( 1-\gamma b(x) \alpha^{q-1}e^{-(q-1)t} \big)\\
&\geq  \alpha e^{-t}w^q \left( 1-\gamma \| b \|_{L^{\infty}(\Gamma_1)}\alpha^{q-1} \right)>0
\end{aligned}
\end{equation}
Also, since $w>0$ in $\bar{\Omega}$, we obtain
\begin{equation}\label{d3}
\bar{v}\geq 0 \quad \text{on } \Gamma_0
\end{equation}
Finally,
\begin{equation}\label{333}
\bar{v}(x,0)=\alpha w(x,0)=u_0(x)
\end{equation}
Thus \eqref{d1}, \eqref{d2}, \eqref{d3} and \eqref{333} show that for any
 $\lambda$ negative enough, $\bar{v}$ is a positive strict supersolution 
of \eqref{parproblem} and hence, owing to the parabolic maximum principle we obtain
\begin{equation}\label{rf}
0<\Theta_{\lambda}(x,t;u_0) \leq \bar{v}(x,t)\,, \quad 
(x,t) \in \Omega \times [0,\infty)\,,
\end{equation}
where $\Theta_{\lambda}(x,t;u_0)$ stands for the solution of \eqref{parproblem}.
Now, \eqref{rf} and \eqref{lim1} give \eqref{ri} and complete the proof of (i).


(ii) Let us denote
$\tilde{d}(x):=\operatorname{dist}(x,\Gamma_0)$,
and for each $\delta>0$ set
$$
\Gamma_{0}^{\delta}:=\{ x\in \bar{\Omega} : \tilde{d}(x) < \delta \}
$$
Since $\partial \Omega =\Gamma_0 \cup \Gamma_1 \in {\mathcal C}^2$, there exists 
$\delta >0$ small enough such that $\tilde{d} \in {\mathcal C}^2(\Gamma_0^{\delta})$ 
and for each $x \in \Gamma_0^{\delta}$, there exists a unique $y(x) \in \Gamma_0$ 
such that $\tilde{d}(x)=| y(x)-x|$ and if $n=n(x)$ denotes the unit (outward) 
normal to $\Gamma_0$ at $y(x)$, then
$$
| \nabla \tilde{d} |^2=\Big(\frac{\partial \tilde{d}}{\partial n} \Big)^2=1\,, 
\quad \| \Delta \tilde{d} \|_{L^{\infty}(\Gamma_0^{\delta})} \leq C\,,
$$
for some $C>0$ depending only on the curvature of $\partial \Omega$
 (cf.\ \cite[Lemma 3.1]{PQ93}, \cite[Lemmas 14.16 and 14.17]{GT83}).
Let us take any smooth extension $d \in {\mathcal C}^2(\bar{\Omega})$ of 
$\tilde{d}$, from $\Gamma_0^{\delta}$ to $\bar{\Omega}$, satisfying
\begin{equation}\label{propd1}
| \nabla d | \geq \mu>0 \quad \text{for all }  x\in \Omega\,,
\end{equation}
for some $\mu>0$, and
\begin{equation}\label{propd2}
d(x)>0 \quad \text{for all }  x\in \Omega \cup \Gamma_1 \quad \text{and} \quad 
\| \Delta d\|_{L^{\infty}(\Omega)}\leq \tilde{C}\,,
\end{equation}
for some $\tilde{C}>0$.

Now, let $\varphi$ be the solution of the initial-value problem
\begin{equation}\label{fii}
\begin{gathered}
\varphi'(t)=\varphi^q(t)\,, \quad t>0\\
\varphi(0)=\alpha >0\,,
\end{gathered}
\end{equation}
for some fixed $\alpha$ satisfying
\begin{equation}\label{alpha}
\alpha>\Big(\frac{1+| \sigma_1 | \| d \|_{L^{\infty}(\Omega)}
+\| \Delta d \|_{L^{\infty}(\Omega)}}{q \mu^2}\Big)^{\frac{1}{q-1}}\,.
\end{equation}
The solution is
$$
\varphi(t):=\frac{\alpha}{\left( 1-(q-1)\alpha^{q-1}t\right)^{\frac{1}{q-1}}}\,.
$$
This function  blows up in finite time, at
\begin{equation}\label{blowt}
T^*=T^*(\alpha,q):=\frac{1}{(q-1)\alpha^{q-1}}\,,
\end{equation}
and since $\varphi(t)>0$ for all $t \in [0,T^*)$ and $\varphi'=\varphi^q>0$, 
we obtain $\varphi$ is increasing and the following hold
\begin{equation}\label{desifi}
\varphi(t)\geq \varphi(0)=\alpha>0\,, \quad 
\varphi'=\varphi^q \geq \alpha^q>0\,, \quad 
\varphi''=q \varphi^{q-1}\varphi'\geq q\alpha^{q-1}\varphi'>0
\end{equation}
Hereafter we will denote 
$$
s=s(x,t):=t+d(x)\,.
$$
By construction we obtain if $x \in \Omega \cup \Gamma_1$, then $s>t$, and
 if $x \in \Gamma_0$, then $s=t$. Also, set
$$
d_1:=\min_{x \in \Gamma_1}d(x)>0\,.
$$
Since $\varphi $ is increasing,  for each $x \in \Gamma_1$ the 
following holds
\begin{equation}\label{wdf}
\varphi(d(x))\geq \varphi(d_1)>\varphi(0)\,.
\end{equation}
Now, let us consider for each $x \in \Gamma_1$ the function
$$
g(x,t):=\frac{\varphi(t)}{\varphi(t+d(x))}=\frac{\varphi(t)}{\varphi(s)}\,.
$$
Since $s>t$ on $\Gamma_1$ and $\varphi$ is increasing and positive, we obtain
$$
g_t(x,t)=\frac{\partial g}{\partial t}(x,t)
=\frac{\varphi(t)}{\varphi(s)}\left( \varphi^{q-1}(t)-\varphi^{q-1}(s)\right)<0\,, 
\quad x\in \Gamma_1
$$
Hence, for each fixed $x \in \Gamma_1$, $g(x,t)$ is decreasing in $t$ and owing 
to \eqref{wdf} we obtain
\begin{equation}\label{gesi}
g(x,t)<g(x,0)=\frac{\varphi(0)}{\varphi(d(x))}
\leq \frac{\varphi(0)}{\varphi(d_1)}:=\varepsilon<1
\end{equation}
Also, for each $x \in \Gamma_1$ we obtain
\begin{equation}\label{wdf2}
\frac{\left( \varphi(s)-\varphi(t)\right)^q}{\varphi'(s)}=
\frac{\left( \varphi(s)-\varphi(t)\right)^q}{\varphi^q(s)}=
\Big( 1-\frac{\varphi(t)}{\varphi(s)}\Big)^q=
(1-g(x,t))^q
\end{equation}
Now, let us consider the function
\begin{equation}\label{subp}
\tilde{u}(x,t)=k\left( \varphi(t+d(x))-\varphi(t)\right)\,,
\end{equation}
where
\begin{equation}\label{k}
k>\Big( \frac{\big(\| \partial d \|_{L^{\infty}(\Gamma_1)}
 +\| V \|_{L^{\infty}(\Gamma_1)}\| d \|_{L^{\infty}(\Omega)}\big) }
{\gamma \underline{b}(1-\varepsilon)^q} \Big)^{\frac{1}{q-1}}>0\,,
\end{equation}
being $\varepsilon \in (0,1)$ and $\mu>0$ defined by \eqref{gesi} and 
\eqref{propd1}, respectively. It must be pointed out that for each 
$x \in \Omega \cup \Gamma_1$, the function $\tilde{u}(x,t)$ blows up in finite 
time $t^*(x)=T^*-d(x)$, where $T^*$ is the time defined by \eqref{blowt}.

To prove the result we are going to prove that if $u_0>0$ is large enough, 
then $\tilde{u}(x,t)$ is a positive subsolution of \eqref{parproblem} which 
blows up in finite time. The construction of the subsolution $\tilde{u}(x,t)$
 made to prove the result, is strongly motivated, up the necessary and 
involved changes and modifications to adapt it to our mixed boundary conditions 
with spatial heterogeneities, for the previous constructions made in \cite{WW75}
 and \cite{MY93} for other kind of problems.
Indeed, by construction
\begin{gather}\label{derivadas}
\tilde{u}_t=k(\varphi'(s)-\varphi'(t))\,, \quad 
\Delta \tilde{u}=k(\varphi''(s)| \nabla d |^2+\varphi'(s) \Delta d), \\
\label{derivadas1}
\partial \tilde{u}=k\varphi'(s)\partial d\,, \quad x\in \Gamma_1\,.
\end{gather}
Owing to \eqref{desifi}, if $x \in \Omega \cup \Gamma_1$ there exists 
$\xi_1(x,t) \in (t,s)$ such that
\begin{equation}\label{desi3}
\varphi'(t)d(x)\leq \varphi(s)-\varphi(t)=\varphi'(\xi_1(x,t))d(x)
\leq \varphi'(s)d(x)
\end{equation}
Also, for each $\lambda \geq \sigma_1$ and $x \in \Omega$ we obtain
\begin{equation}\label{lifi}
(\lambda-\sigma_1)(\varphi(s)-\varphi(t))\geq 0
\end{equation}
Then, owing to \eqref{subp}, \eqref{derivadas}, \eqref{desifi}, \eqref{propd1}, 
\eqref{lifi}, \eqref{desi3} and \eqref{alpha}, the following hold in $\Omega$,
\begin{align*}
&\frac{1}{k}(\tilde{u}_t-\Delta \tilde{u}-\lambda \tilde{u}) \\
&=\varphi'(s)-\varphi'(t)-\varphi''(s)| \nabla d |^2 -\varphi'(s)\Delta d -\lambda (\varphi(s)-\varphi(t))\\
&\leq \varphi'(s)-\varphi'(t)-q\alpha^{q-1}\varphi'(s)| \nabla d |^2 -\varphi'(s)\Delta d -\lambda (\varphi(s)-\varphi(t))\\
&\leq\varphi'(s) \left[ 1-q \alpha^{q-1} \mu^2 -\Delta d \right] -\varphi'(t)-(\lambda-\sigma_1)(\varphi(s)-\varphi(t))-\sigma_1 (\varphi(s)-\varphi(t))\\
&\leq \varphi'(s) \left[ 1-q \alpha^{q-1} \mu^2 -\Delta d \right]+| \sigma_1 | (\varphi(s)-\varphi(t))\\
&\leq \varphi'(s) \left[ 1-q \alpha^{q-1} \mu^2 -\Delta d \right]+| \sigma_1 | \varphi'(s) \| d \|_{L^{\infty}(\Omega)}\\
&\leq \varphi'(s)\left[ 1-q \alpha^{q-1}\mu^2 
 +\| \Delta d \|_{L^{\infty}(\Omega)}
 +| \sigma_1 | \| d \|_{L^{\infty}(\Omega)} \right]
<0\,,
\end{align*}
and therefore,
\begin{equation}\label{omg}
\tilde{u}_t-\Delta \tilde{u}-\lambda \tilde{u}<0 \quad \text{if} \quad
 (x,t) \in \Omega \times (0,t^*(x))
\end{equation}

As for the boundary conditions, owing to \eqref{subp}, \eqref{derivadas1},
 \eqref{desi3}, \eqref{wdf2}, \eqref{gesi}, \eqref{bsep} and \eqref{k}, 
the following hold on $\Gamma_1$,
\begin{align*}
&\partial \tilde{u} \quad +V(x)\tilde{u}-\gamma b(x) \tilde{u}^q \\
&= k(\varphi'(s) \partial d +V(x)(\varphi(s)-\varphi(t))
 -\gamma b(x)k^q(\varphi(s)-\varphi(t))^q\\
&= k\varphi'(s)\big[\partial d+V(x)\frac{\varphi(s)
 -\varphi(t)}{\varphi'(s)}-\gamma b(x) k^{q-1}
 \frac{(\varphi(s)-\varphi(t))^q}{\varphi'(s)}\big]\\
&\leq k\varphi'(s)\left[\| \partial d \|_{L^{\infty}
 (\Gamma_1)}+\| V \|_{L^{\infty}(\Gamma_1)} \| d \|_{L^{\infty}(\Omega)}-\gamma b(x) 
 k^{q-1} \left( 1-g(x,t)\right)^q\right]\\
&\leq  k\varphi'(s)\left[\| \partial d \|_{L^{\infty}(\Gamma_1)}
 +\| V\|_{L^{\infty}(\Gamma_1)}\| d \|_{L^{\infty}(\Omega)}
 -\gamma \underline{b}k^{q-1}(1-\varepsilon)^q\right]
<0\,;
\end{align*}
therefore
\begin{equation}\label{bond1}
\partial \tilde{u}+V(x)\tilde{u}-\gamma b(x) \tilde{u}^q < 0 \quad \text{if} \quad 
(x,t)\in \Gamma_1 \times (0,t^*(x))
\end{equation}

Also, since $d(x)=0$ for all $x \in \Gamma_0$, we obtain
\begin{equation}\label{bond0}
\tilde{u}(x,t)=0 \quad \text{for all } x\in \Gamma_0, \; t \in [0,T^*)
\end{equation}
Finally, if the initial data $u_0$ is large enough to satisfy
\begin{equation}\label{incond}
\tilde{u}(x,0)=k(\varphi(d(x))-\varphi(0))=k(\varphi(d(x))-\alpha)\leq u_0(x)\,,
\end{equation}
then, by \eqref{omg}, \eqref{bond1}, \eqref{bond0} and \eqref{incond}, 
the following holds
\begin{gather*}
\tilde{u}_t-\Delta \tilde{u}-\lambda \tilde{u}< 0\quad \text{if }
 (x,t) \in \Omega \times (0,t^*(x))\,,\\
\tilde{u}=0\quad  \text{if } (x,t) \in \Gamma_0 \times (0,t^*(x))\,,\\
\partial \tilde{u}+V(x) \tilde{u}< \gamma b(x)\tilde{u}^q \quad \text{if }
 (x,t) \in \Gamma_1 \times (0,t^*(x))\,,\\
\tilde{u}(x,0) \leq u_0(x) \quad \text{if }  x \in \Omega\,;
\end{gather*}
that is, $\tilde{u}(x,t)$ is a positive strict subsolution of \eqref{parproblem} 
for each initial data $u_0>0$ satisfying \eqref{incond}.
Then, owing to the parabolic maximum principle we obtain the solution 
$\Theta_{\lambda}(x,t;u_0)$ of \eqref{parproblem} holds
\begin{equation}\label{wdf3}
\Theta_{\lambda}(x,t;u_0)\geq \tilde{u}(x,t)\,, \quad 
(x,t) \in \Omega \times [0,t^*(x))
\end{equation}
Now, let us consider a compact set $Q \subset \Omega \cup \Gamma_1$. 
Since $d(x)>0$ for all $x \in Q$, we obtain there exists $d_Q>0$ such that
\begin{equation}\label{dq}
d(x) \geq d_Q>0 \quad \text{for all }  x\in Q.
\end{equation}
Thus, from \eqref{wdf3}, \eqref{desi3} and \eqref{dq}, we obtain that
if $(x,t) \in Q \times [0,T^*)$ then
\begin{equation}\label{wdf6}
\Theta_{\lambda}(x,t;u_0)\geq \tilde{u}(x,t)=k(\varphi(s)-\varphi(t))
\geq k \varphi'(t) d(x)= k \varphi^q(t)d(x)\geq kd_Q \varphi^q(t)\,,
\end{equation}
and therefore, since $\varphi$ blows up in finite time $t=T^*$, 
owing to \eqref{wdf6}, so do $\tilde{u}(x,t)$ and $\Theta_{\lambda}(x,t;u_0)$ 
uniformly in $Q$.
This completes the proof of (ii), and the result. 
\end{proof}

\begin{remark} \rm
If the domain $\Omega$ is in some sense a nice domain, then
$$
d(x)=\tilde{d}(x)=\operatorname{dist}(x,\Gamma_0) \quad \text{for all } 
 x \in \bar{\Omega}\,,
$$
that is, in the proof of Theorem \ref{parabolico}-ii) is not necessary to 
take an smooth extension $d \in {\mathcal C}^2(\bar{\Omega})$ of $\tilde{d}$, 
from $\Gamma_0^{\delta}$ to $\bar{\Omega}$, different from $\tilde{d}(x)$, 
because $\tilde{d}$ already satisfies \eqref{propd1} and \eqref{propd2}. 
For instance, if $\Omega$ is the annulus of $\mathbb{R}^N$,\,$N>1$ given by
$$
\Omega:=\{ x \in \mathbb{R}^N: R_1<| x |<R_2\}
$$
with
$$
\Gamma_0:=\{ x \in \mathbb{R}^N: | x| =R_1\}\,, \quad
\Gamma_1:=\{ x \in \mathbb{R}^N: | x| =R_2\}\,,
$$
for some $0<R_1<R_2$, then for all $x \in \bar{\Omega}$ we obtain
$$
\tilde{d}(x)=\operatorname{dist}(x,\Gamma_0)=| x |-R_1\,, \quad 
| \nabla d(x)|=1\,, \quad |\Delta d(x) |=\frac{N-1}{| x |} \leq \frac{N-1}{R_1}\,,
$$
and \eqref{propd1} and \eqref{propd2} hold for
$$
d(x)=\tilde{d}(x)\,, \quad \mu=1\,, \quad \tilde{C}=\frac{N-1}{R_1}>0.
$$
\end{remark}

\begin{remark} \rm
In the particular case when $\gamma>0$, $\lambda \geq \sigma_1$ and \eqref{bsep}
 holds, taking into account in the proof of Theorem \ref{parabolico}(ii)
 the definition and properties of the functions $\varphi(t)$ and $d(x)$ 
(cf.\ \eqref{fii}, \eqref{propd1}, \eqref{propd2}) and the constant $k$ 
(cf.\ \eqref{k}), we obtain \eqref{incond} gives a structure for the initial
 data $u_0>0$, in terms of $d(x)$, to ensure that the positive solution 
$\Theta_{\lambda}(x,t;u_0)$ of \eqref{parproblem} blows up in finite time
\end{remark}

\begin{corollary}
Under the general assumptions, let us consider the superlinear parabolic problem
\begin{equation}\label{superparproblem}
\begin{gathered}
w_t-\Delta w=\lambda w +a(x)w^p\quad \text{in } \Omega \times (0,T)\,, \; p>1\\
w=0 \quad \text{on } \Gamma_0 \times (0,T)\\
\partial w+V(x)w=\gamma b(x)w^q \quad \text{on } \Gamma_1 \times (0,T)\,, \quad q>1\\
w(\cdot,0)=u_0>0\quad \text{in } \Omega\,,
\end{gathered}
\end{equation}
where the potential $a \in {\mathcal C}(\bar{\Omega})$, $a>0$ in $\Omega$, 
$\gamma>0$, $\lambda \geq \sigma_1$ and \eqref{bsep} holds. 
Then, if $u_0$ is large enough, the solution $w_{\lambda}(x,t;u_0)$ of 
\eqref{superparproblem} blows up in finite time uniformly in compact subsets 
$Q \subset \Omega \cup \Gamma_1$.
\end{corollary}

\begin{proof}
The result follows from the parabolic maximum principle and 
Theorem \ref{parabolico}(ii), taking into account that the solution 
$\Theta_{\lambda}(x,t;u_0)$ of \eqref{parproblem} is a positive strict subsolution 
of \eqref{superparproblem}.
\end{proof}

\subsection{Structure of the set of positive solutions of \eqref{1}}
\label{Section4.2}

In this section we  analyze, in the particular case when $\gamma>0$, 
the structure of the set of positive solutions of \eqref{1} and the linear 
stability of them. The main result of this section is the following.

\begin{theorem}\label{gamma+}
Assume $\gamma >0$. Then, the following hold:
\begin{itemize}
\item[(i)] If $u_{\lambda}$ is a positive solution of \eqref{1} for the value 
$\lambda$ of the parameter, then
\begin{equation}\label{exis++}
\lambda<\sigma_1
\end{equation}

\item[(ii)] There exists $\lambda^*<0$, such that \eqref{1} does not admit a 
positive solution for $\lambda <\lambda^*$.

\item[(iii)] The value $\lambda=\sigma_1$ is the unique bifurcation value to 
positive solutions from the trivial branch $(\lambda,u)=(\lambda,0)$. 
From it emanates subcritically a continuum $\mathfrak{C}^+$ of positive 
solutions of \eqref{1}, which also bifurcates to positive solutions from 
infinity at least in some value $\lambda_{\infty} \in [\lambda^*,\sigma_1]$. 
In particular, $\mathfrak{C}^+$ is unbounded in $L^{\infty}(\Omega)$ and
 \begin{equation}\label{lp}
 {\mathcal P}_{\lambda}(\mathfrak{C}^+ \setminus 
\{ (\sigma_1,0)\} ) \subset [\lambda^*,\sigma_1)\,.
 \end{equation}

\item[(iv)] All the positive solutions of \eqref{1} are unstable.
\end{itemize}
\end{theorem}

\begin{proof}
(i) Let $u_{\lambda}>0$ be a positive solution of \eqref{1} for the value 
$\lambda$ of the parameter. Then, since $b>0$, $\gamma>0$ and owing to 
\eqref{lambda} and to the monotonicity of the principal eigenvalue with 
respect to the potential on the boundary conditions, we obtain
$$
\lambda=\sigma_1^{\Omega}[-\Delta,\mathfrak{B}(V(x)-\gamma b(x)u_{\lambda}^{q-1})]
<\sigma_1^{\Omega}[-\Delta,\mathfrak{B}(V(x))]=\sigma_1\,,
$$
which proves \eqref{exis++}.

(ii) Arguing in a similar way to \cite[Lemma 4.4]{GMRS08} and 
\cite[Theorem 4.6]{CC14}, the result follows from Theorem \ref{parabolico}-i).

(iii) The existence of the continuum $\mathfrak{C}^+$ of positive solutions 
of \eqref{1} emanating subcritically from the trivial branch at the unique 
bifurcation value to positive solutions $\lambda=\sigma_1$, follows from 
Theorem \ref{bf} and Remark \ref{remark1}, taking into account that $b>0$ 
and $\gamma>0$. On the other hand, \eqref{lp} follows from $i)$ and $ii)$. 
Now, since $\mathfrak{C}^+$ is unbounded in $\mathbb{R} \times L^{\infty}(\Omega)$ 
(cf.\ Remark \ref{remark2}) and since by \eqref{lp} 
${\mathcal P}_{\lambda}(\mathfrak{C}^+)$ is bounded in $\mathbb{R}$, we obtain 
$\mathfrak{C}^+$ is unbounded in $L^{\infty}(\Omega)$ and therefore, it must
 bifurcate from infinity at least in some value 
$\lambda_{\infty} \in [\lambda^*,\sigma_1]$.

(iv) To prove the result we will prove that the principal eigenvalue of 
the linearization of \eqref{1} at any positive solution $u_{\lambda}$ of it is 
negative. Indeed, let $u_{\lambda}>0$ be a positive solution of \eqref{1} for 
some value $\lambda$ of the parameter with $\lambda < \sigma_1$. 
The linearization of \eqref{1} at $u_{\lambda}$ is given by
\begin{equation}
\begin{gathered}
(-\Delta -\lambda)w=0 \quad \text{in } \Omega\\
\mathfrak{B}\big( V(x)-\gamma q b(x)u_{\lambda}^{q-1} \big)w=0 \quad
\text{on } \partial \Omega
\end{gathered}
\end{equation}
Then, taking into account that $\gamma>0$, $b>0$, $u_{\lambda}>0$ and $q>1$, 
it follows from \eqref{lambda} and from the monotonicity of the principal 
eigenvalue with respect to the potential on the boundary condition that
$$
\sigma_1^{\Omega}\big[-\Delta-\lambda, \mathfrak{B}
\big( V(x)-\gamma q b(x)u_{\lambda}^{q-1}\big)\big]
<\sigma_1^{\Omega}\big[-\Delta-\lambda, \mathfrak{B}
\big( V(x)-\gamma b(x)u_{\lambda}^{q-1}\big)\big]=0
$$
which completes the proof of (iv), and of the theorem.
\end{proof}

\begin{remark} \rm
In the particular case when $\gamma>0$, if we denote 
\begin{gather*}
\Lambda :=\{ \lambda \in \mathbb{R}: \eqref{1} \text{ possesses positive solution } 
\} \subset [\lambda^*,\sigma_1), \\
\tilde{\lambda}:=\inf \Lambda \geq \lambda^*\,, \quad
\hat{\lambda}:= \inf \{ \, \lambda : (\lambda,u) \in \mathfrak{C}^+\,\}\,,
\end{gather*}
it might occur that $\tilde{\lambda}<\hat{\lambda}$, by the existence of an 
isola $\mathfrak{G}$ of positive solutions of \eqref{1} such that
$$
\lambda^* \leq \tilde{\lambda}\leq \inf 
\{ \, \lambda: (\lambda,u) \in \mathfrak{G}\}<\hat{\lambda}
$$
In fact, $\Lambda$ might be unconnected, since it might occur that
$$
\tilde{\lambda}\leq \inf\{\lambda: (\lambda,u)\in \mathfrak{G}\}
<\bar{\lambda}:= \sup\{\,\lambda: (\lambda,u)\in \mathfrak{G}\}<\hat{\lambda}
$$
and that \eqref{1} does not possess positive solutions for 
$\lambda \in (\bar{\lambda},\hat{\lambda})$. In this kind of problems, if $u_1$ 
is a positive solution of \eqref{1} for $\lambda=\lambda_1$, then small 
positive multiples of $u_1$ are not positive subsolutions of \eqref{1} 
for $\lambda >\lambda_1$, and large positive multiples of $u_1$ are not 
positive supersolutions of \eqref{1} for $\lambda< \lambda_1$.
\end{remark}



\subsection*{Acknowledgments}
This research was supported by the Ministry of Economy and Competivity under 
grant MTM2015-65899-P.
The author wants to thank the anonymous referees for their useful comments
and remarks which have contributed to improve the earlier version of this paper.



\begin{thebibliography}{99}

\bibitem{LGA14} Aleja, D.; L\'opez-G\'omez, J.; 
\emph{Some paradoxical effects of the advection on a class of diffusive equation 
in Ecology}, Discrete and Continuous Dynamical Systems B,\textbf{19} (2014) 
10, 3031-3056.

\bibitem{Am76} Amann, H.; 
\emph{Fixed point equations and nonlinear eigenvalue problems in ordered Banach 
spaces}, SIAM Rev, \textbf{18} (1976), 620--709.

\bibitem{Am76NB} Amann, H.; 
\emph{Nonlinear elliptic equations with nonlinear boundary conditions},
in New Developments in Differential Equations (ed. W. Eckhaus), Math Studies, 
\textbf{21}, North-Holland, Amsterdam (1976), 43--63.

\bibitem{Am83} Amann, H.; 
\emph{Dual semigroups and second order linear elliptic boundary-value problems}, 
Israel Journal of Mathematics, \textbf{45} (1983), 225--254.

\bibitem{ALG98} Amann, H.; L\'opez-G\'omez, J.; 
\emph{A priori bounds and multiple solutions for superlinear indefinite 
elliptic problems}, J. Differential Equations, \textbf{146} (1998), 336-374.

\bibitem{AMTR02} Andreu, F.; Maz\'on, J. M.; Toledo, J.; Rossi, J.D.; 
\emph{Porous medium equation with absorption and a nonlinear boundary condition},
 Nonlinear Analysis, \textbf{49} (2002), 541-563.

\bibitem{CCLG01} Cano-Casanova, S.; L\'opez-G\'omez, J.; 
\emph{Properties of the principal eigenvalues of a general class of non-classical
 mixed boundary-value problems}, Journal of Differential Equations, 
\textbf{178} (2002), 123--211.

\bibitem{CC05} Cano-Casanova, S.; 
\emph{On the existence and uniqueness of positive solutions of the Logistic 
elliptic BVP with nonlinear mixed boundary conditions}, Nonlinear Analysis, 
(2005), e1399-e1407.

\bibitem{CCST} Cano-Casanova, S.; 
\emph{On the positive solutions of the Logistic weighted elliptic BVP with 
sublinear mixed boundary conditions}, in Spectral Theory and Nonlinear Analysis 
with Applications to Spatial Ecology, World Scientific Ed., 2005, 1--15.

\bibitem{CC12} Cano-Casanova, S.; 
\emph{Bifurcation to positive solutions in BVPs of Logistic type with nonlinear 
indefinite mixed boundary conditions}, Discrete and Continuous Dynamical Systems, 
Suplement 2013, (2013), 95--104.

\bibitem{CC14} Cano-Casanova, S.; 
\emph{Nonlinear mixed boundary conditions in BVPs of Logistic type with spatial 
heterogeneities and a nonlinear flux on the boundary with arbitrary sign. 
The case $p>2q-1$}, Journal of Differential Equations, \textbf{256} (2014), 82-107.

\bibitem{Chi91} Chipot, M.; Fila, M.; Quittner, P.; 
\emph{Stationary solutions, blow up and convergence to stationary solutions 
for semilinear parabolic equations with nonlinear boundary conditions}, 
Acta Math. Univ. Comenianae, \textbf{60} (1991), 35--103.

\bibitem{GMRS08} Garcia-Meli\'an, J.; Morales-Rodrigo, C.; Rossi, J. D.; Su\'arez, A.;
 \emph{Nonnegative solutions to an elliptic problem with nonlinear absorption and
 a nonlinear incoming flux on the boundary}, Ann. Mat. Pura Appl., 
\textbf{187} (2008), 459-486.

\bibitem{GT83} Gilbarg, D; Trudinger, N. S.; 
\emph{Elliptic partial differential equations of second order}, second edition, 
Springer, Berlin-Heidelberg-New York-Tokyo, 1983.

\bibitem{LG91} L\'opez-G\'omez, J.; Marquez, V.; Wolanski, N.;
\emph{Blow up results and localization of blow up points for the heat equations
 with a nonlinear boundary conditions}, J. Differential Equations, 
\textbf{92} (1991), 384--401.

\bibitem{LGMWPP} L\'opez-G\'omez, J.; M\'arquez, V.; Wolanski, N.; 
\emph{Global behavior of positive solutions to a semilinear equation with a 
nonlinear flux condition}, IMA Preprint Series \textbf{810}, May 1991. 
Institute for Mathematics and Its Applications. University of Minnesota.

\bibitem{LGMW93} L\'opez-G\'omez, J.; M\'arquez, V.; Wolanski, N.; 
\emph{Dynamic behavior of positive solutions to reaction-diffusion problems 
with nonlinear absorption through the boundary}, 
Revista de la Uni\'on Matem\'atica Argentina, \textbf{38} (1993), 196--209.

\bibitem{LG96} L\'opez-G\'omez, J.; 
\emph{The maximum principle and the existence of principal eigenvalues for 
some linear weighted boundary value problems}, Journal of Differential Equations,
 \textbf{127} (1996), 263--294.

\bibitem{LGSL98} L\'opez-G\'omez, J.; Sabina de Lis, J.C.; 
\emph{First variations of principal eigenvalues with respect to the domain and 
point-wise growth of positive solutions for problems where the bifurcations 
from infinity occurs}, Journal of Differential Equations \textbf{148} (1998), 
47--64.

\bibitem{LG00} L\'opez-G\'omez, J.; 
\emph{Large solutions, metasolutions, and asymptotic behaviour of the regular 
positive solutions of sublinear parabolic problems}, 
Electron. J. Diff. Equations, Conf. 05, (2000), 135--171.

\bibitem{LGli01} L\'opez-G\'omez, J.; 
\emph{Spectral Theory and Nonlinear Functional Analysis}, 
in Research Notes in Mathematics, vol. 426, Chapman and Hall/CRC,
 Boca Raton, Florida, 2001.

\bibitem{MoSu05} Morales-Rodrigo, C.; Su\'arez, A.; 
\emph{Some elliptic problems with nonlinear boundary conditions}, in
Spectral Theory and Nonlinear Analysis with Applications to Spatial Ecology, 
World Scientific Ed., 2005, 175--199.

\bibitem{PQ93} Quittner, P.;
\emph{On global existence and stationary solutions for two classes of semilinear 
parabolic problems}, Comment. Math. Univ. Carolin., \textbf{34,1} (1993), 105--124.

\bibitem{Rab71} Rabinowitz, P. H.; 
\emph{Some global results for nonlinear eigenvalue problems}, 
Journal of Functional Analysis, \text{7} (1971), 487-513.

\bibitem{ARB01} Rodriguez-Bernal, A.; Tajdine, A.; 
\emph{Nonlinear Balance for Reaction-Diffusion Equations under Nonlinear 
Boundary Conditions: Dissipativity and Blow-up}, 
Journal of Differential Equations, \textbf{169} (2001), 332-372.

\bibitem{Rossi98} Rossi, J. D.; 
\emph{The blow-up rate for a semilinear parabolic equation with a nonlinear 
boundary condition}, Acta Math. Univ. Comenianae, \textbf{67} (1998), 343-350.

\bibitem{Rossi05} Rossi, J.D.;
\emph{Elliptic problems with nonlinear boundary conditions and the Sobolev 
trace theorem}, Handbook of Differential Equations: Stationary Partial Differential 
Equations, Vol. 2, Chapter 5. Edited by M. Chipot and P. Quittner. Elsevier (2005), 
311-406.

\bibitem{Sa73} Sattinger, D.; 
\emph{Topics in Stability and Bifurcation Theory}, Lectures Notes in Mathematics, 
309, Springer, Berlin/New York, 1973.

\bibitem{St} Stein, E. M.; \emph{Singular Integrals and Differentiability Properties 
of Functions}, Princeton Univ. Press, Princeton, NJ, 1970.

\bibitem{Um02} Umezu, K.; 
\emph{Behavior and stability of positive solutions of nonlinear elliptic boundary 
value problems arising in population dynamics}, Nonlinear Analysis,
 \textbf{49} (2002), 817--840.

\bibitem{Um02BIS} Umezu, K.; 
\emph{Bifurcation from infinity for asymptotically linear elliptic eigenvalue 
problems}, Journal of Mathematical Analysis and Applications, \textbf{267} (2002), 
651-664.

\bibitem{Um04} Umezu, K.; 
\emph{Local bifurcation analysis and stability of steady-state solutions for
 diffusive equations with nonlinear boundary conditions}, 
Comm. Appl. Anal. \textbf{8} (2004), 533-547.

\bibitem{WW75} Walter, W.; 
\emph{On existence and nonexistence in the large of solutions of parabolic 
differential equations with a nonlinear boundary condition}, 
SIAM J. Math. Anal., \textbf{6}, No.1, (1975), 85-90.

\bibitem{MY93} Wang, M.; Wu, Y.; 
\emph{Global existence and blow-up problems for quasilinear parabolic equations 
with nonlinear boundary conditions}, SIAM J. Math. Anal., \textbf{24}, 
No. 6, (1993), 1515-1521.

\bibitem{WW95} Wu, Y.; Wang, M.; 
\emph{Existence and nonexistence of global solution of nonlinear parabolic 
equation with nonlinear boundary condition}, Chin. Ann. Math., 
\textbf{16B} (1995), 371-378.

\end{thebibliography}

\end{document}