\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Variational and Topological Methods:
Theory, Applications, Numerical Simulations, and Open Problems (2012).
{\em Electronic Journal of Differential Equations},
Conference 21 (2014),  pp. 1--9.
ISSN: 1072-6691.  http://ejde.math.txstate.edu,
http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{1}
\title[\hfilneg EJDE-2014/Conf/21 \hfil Localization phenomena]
{Localization phenomena in a degenerate \\ logistic equation}

\author[J. M. Arrieta, R. Pardo, A. Rodr\'{\i}guez-Bernal \hfil EJDE-2014/Conf/21\hfilneg]
{Jos\'e M. Arrieta, Rosa Pardo, Anibal Rodr\'{\i}guez-Bernal}  % in alphabetical order

\address{Jos\'e M. Arrieta \newline
Departamento de Matem\'atica Aplicada, Universidad Complutense de Madrid,
28040--Madrid, Spain}
\email{arrieta@mat.ucm.es}

\address{Rosa Pardo \newline
Departamento de Matem\'atica Aplicada, Universidad Complutense de Madrid,
28040--Madrid, Spain}
\email{rpardo@mat.ucm.es}

\address{Anibal Rodr\'{\i}guez-Bernal \newline
Departamento de Matem\'atica Aplicada, Universidad Complutense de Madrid,
28040--Madrid, Spain. \newline
Instituto de Ciencias Matem\'aticas,CSIC-UAM-UC3M-UCM, 28049--Madrid, Spain}
\email{arober@mat.ucm.es}

\thanks{Published February 10, 2014.}
\subjclass[2000]{35B32, 35B35, 35B65, 35B40,  35B41, 35B44,  35J25}
\keywords{Logistic equation; positive solution;  bifurcation;localization; 
\hfill\break\indent  blow-up}

\begin{abstract}
 We analyze the behavior of positive solutions of elliptic equations
 with a degenerate logistic nonlinearity  and Dirichlet boundary
 conditions.  Our results concern existence and strong localization  in
 the spatial region in which the logistic nonlinearity cancels.
 This type of nonlinearity has applications in the nonlinear
 Schrodinger equation and the study of Bose-Einstein condensates. In
 this context, our analysis  explains the fact that the ground state
 presents a strong localization in the spatial region in which the
 nonlinearity cancels.
\end{abstract}

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

\section{Introduction}

In this paper we analyze the behavior of positive solutions of
elliptic  equations with  a degenerate logistic nonlinearity and
Dirichlet boundary conditions
  \begin{equation}  \label{eq:elliptic:problem}
\begin{gathered}
     - \Delta u  =   \lambda u -n(x) u^{\rho} \quad \text{in }  \Omega,\\
    u= 0 \quad  \text{on } \partial\Omega,
  \end{gathered}
\end{equation}
where  $\Omega \subset \mathbb{R}^N$, $N\geq 1$, is a bounded domain,
$\rho >1$, $\lambda \in \mathbb{R}$ and
$n(x) \geq 0$ in $\Omega$ and $n(x)$ is not identically zero.

We will also assume that $n(x)$
remains strictly positive near the boundary
of $\Omega$ and therefore
\begin{equation} \label{eq:vanishing_set}
K_0 =\{x\in \Omega : n(x) =0\}\subset\Omega\
\quad \text{and $K_0$ is a nonempty compact set}.
\end{equation}
Despite a large amount of mathematical literature in this kind of
logistic equations, see below, this  type of nonlinearity has applications 
in the nonlinear Schrodinger equation and the study of Bose-Einstein
condensates. In this context, assumption \eqref{eq:vanishing_set}  
implies  the fact that the
{\it ground state} presents a strong localization in the spatial
region $K_{0}$, see
\cite{perez-garcia09:_local_schroed} and references therein.

Throughout this article we shall assume that the compact set $K_0$ 
and the function $n(x)$ satisfy the following  hypotheses:
\begin{itemize}
\item[(Hn)] $n(x)$ is a H\"older continuous function and
\[
n(x)  \geq  C\big(d_0(x)\big)^\gamma \quad \text{for some }\gamma > 0 .
\]
where $d_0(x) := \operatorname{dist}(x,K_{0})$,  and

\item[(HK)] $K_0=K_1\cup K_2\subset\Omega$,
where $K_1$ and $K_2$ are compact sets and
$K_1=\overline{\Omega}_0$ is the closure of a regular connected open set
$\Omega_0\neq \emptyset$,
$K_2$ has zero Lebesgue measure.
\end{itemize}
In some cases (HK) will be strengthened to
\begin{itemize}
\item[(HK')] $K_0$ satisfies  (HK) and
$K_2$ is a closed regular $d$-dimensional manifold, with
$d\leq N-1$.
\end{itemize}


When the set $K_{0}$ is empty, that is, if $n(x)$ is strictly
bounded away from zero, problem \eqref{eq:elliptic:problem} is
classical and well understood, see e.g. \cite{smoller} and references therein.
Also, when $K_{0}$ is
``smooth'' in the sense that in (HK) we have  $K_{0}= K_1=
\overline{\Omega}_{0}$ where $\Omega_{0}$ is a smooth
open set, and $K_2= \emptyset$, this problem has also been studied in
\cite{Ouyang,fraile96:_ellip, Gamez_97, garcia-melian98:_point,
  garcia-melian01:_uniquen, lopez-gomez98:_first}
and further developments in
\cite{gomez-renasco99_metasolutions,
  gomez-renasco-lopez-gomez02_metasolutions,
  lopez-gomez05_metasolutions}. Therefore here we focus on the effect
on the solutions of the presence of the part with  empty interior  $K_2$.

As a general notation, we will denote by $\lambda_1(U)$ the first
eigenvalue of the Laplace operator with Dirichlet boundary conditions
in the open and smooth set $U$.

As will be shown below,  by standard estimates on \eqref{eq:elliptic:problem}, if
the parameter $\lambda$ is below the value $\lambda_1(\Omega)$, the
unique non negative solution  is $u\equiv 0$. Moreover, as $\lambda$
crosses the value $\lambda_1(\Omega)$, a bifurcation phenomena takes
place and a unique  positive solution emanates from the trivial one.
This solution can be continued in
$\lambda$ up until it reaches some critical value, $\lambda_{c}$. By monotonicity
properties of the first eigenvalue (with respect to the domains and to
the potentials), it is an easy task to realize that the critical value
$\lambda_{c}$ is equal to $\lambda_1(\Omega_0)$, see Lemma
\ref{lem:implicitFThm}, part (i). Note that this is precisely the same
situation as when $K_{0}$ is ``smooth'', i.e. $K_2=\emptyset$. When
$K_{0}$ is empty, the picture is also as above, with $\lambda_{c}=
\infty$.

Our goal is then to  give a detailed description of the  behavior of
this branch of
solutions for $\lambda\in
(\lambda_1(\Omega), \lambda_1(\Omega_0))$ and specially as $\lambda\to
\lambda_1(\Omega_0)$. First we show that the  solutions blow up in
compact sets of $\Omega_0$
(see Lemma \ref{lem:grow_up_regularset} below). Also, we will show
that the solutions are
uniformly bounded in compact sets of $\Omega\setminus K_0$ (see
Proposition \ref{prop:boundedness_far_from_K0} below). Hence, it
remains to analyze the behavior of solutions in  $K_2$,  which  is not
so clear at all. In $K_2$ we have two competing mechanisms: on one hand the fact that
$n(x)\equiv 0$ in $K_2$ ``pushes'' the solution towards $+\infty$
while the fact that $K_2$ is not ``fat'' enough means that this effect
may not have enough room to force the solution to go  to infinity.

We will distinguish two situations for which we will be able to show
that the solutions remain bounded in $K_2$. In case $K_2\cap K_1=\emptyset$, then
any solution will be bounded in $K_2$, actually it will be so in a
neighborhood of $K_2$.  In the case $K_2\cap K_1\neq \emptyset$, it
will turn out that a balance between the geometry of $K_2$ and the strength of the
logistic term, given by the exponent $\rho$ and the behavior of the
function  $n(x)$ near $K_2$,  will determine the behavior of the
solution.
As a matter of fact we will be able to prove the following result.


\begin{theorem}\label{th:condition:bdd:0}
Assume $K_0$ satisfies {\rm (HK)} and $n(x)$ satisfies {\rm
  (Hn)}. Then for any $\lambda\in
(\lambda_1(\Omega),\lambda_1(\Omega_0))$ there exists a unique  positive solution of
\eqref{eq:elliptic:problem},  $\varphi_\lambda$, and we have
\begin{equation}\label{unbounded:1}
  \lim_{\lambda \to  \lambda_1(\Omega_0)} \varphi_\lambda(x) = \infty, \quad 
\text{for all $x\in\Omega_0$},
\end{equation}
and the limit is uniform in compact sets of $\Omega_0$.  Moreover,
we have the following two cases:
\begin{itemize}
\item[(i)] If $K_1\cap K_2=\emptyset$, then there exists a
$\delta>0$ and  $M>0$ such that
$$
|\varphi_\lambda(x)|\leq M,\quad \forall x,\, d(x,K_2)\leq \delta,
\quad \forall \, \lambda\in (\lambda_1(\Omega), \lambda_1(\Omega_0)).
$$

\item[(ii)] If $K_1\cap K_2\ne \emptyset$ and $K_{0}$ satisfies {\rm (HK')} 
and
\begin{equation}\label{eq:condition:bdd:1}
\gamma+2<(\rho -1)(N-d),
\end{equation}
then $\varphi_\lambda$ remains uniformly bounded on compact sets of
$\Omega\setminus K_1$. In particular it remains bounded  at each point
of  $K_2\setminus K_1$.
\end{itemize}
\end{theorem}


The proof of this result  relies on the following argument.
If we denote by $u$ a nonnegative solution of
\eqref{eq:elliptic:problem}, then we obtain first an upper bound of
$u$, independent of $\lambda$,   in compact sets of  
$\Omega\setminus K_0$. If $\bar
B(x_0,a)\subset \Omega\setminus K_0$, where $n(x)\geq n_0$ in this
ball, we may compare the solution $u$ with radial solutions of
singular Dirichlet problems, posed in $B(x_0,a)$,  going to  infinity
at the boundary, see \cite{garcia-melian98:_point, keller57:_delta,
  osserman57:_delta}.  By radial symmetry, the minimum of the singular
solution is attained at the center of the ball (that is in $x_0$), and
can be estimated in terms of $n_0$, $a$, $\rho$ and the dimension $N$.
Translating this result to our problem, we can move those balls for
points in $\Omega\setminus K_0$ next to the boundary of $K_0$,
and state some rate for the upper bounds in terms of some inverse
power of the distance to  the boundary of $K_0$.  This estimates
provide a rate at which the solution may diverge to infinity as we
approach $K_0$. See Lemma \ref{lem:singular_Dirichlet_pbm},
Proposition \ref{prop:boundedness_far_from_K0} and Lemma
\ref{lem:universal_upper_bound}.


Once this estimate is obtained we may consider a point $z\in
K_2\setminus K_1$ and consider for instance a small ball
$B(z,\delta)$, where in principle the solution $u$ may become
unbounded as $\lambda$ increases.
Nevertheless, the rate obtained with the argument
above may imply that the solution $u$ restricted to the sphere
$S(z,\delta)=\{ |x-z|=\delta\}$ is in $L^r(S(z,\delta))$ for some
$r\geq 1$, with a norm independent of $\lambda$.  Hence, $u$ will be a
solution of an elliptic problem in
$B(z,\delta)$ with an $L^r$ trace at the boundary. Elliptic regularity
will imply that the solution $u$ is bounded, independent of $\lambda$,
in compact sets of
$B(z,\delta)$ and in particular in a neighborhood of $z\in K_2$.
Therefore, we may obtain conditions on $\rho$, the dimensions $N$ and
$d$ and the rate $\gamma$ at which  $n(x)$ approaches to zero, see
(Hn), which may guarantee that the solution is bounded in
$K_2\setminus K_1$, see \eqref{eq:condition:bdd:1}.


This article is organized as follows. In Section
\ref{sec:behav-posit-equil} we have collected some relevant
results on the stationary solutions of logistic degenerated equations.
All those results are essentially well know in case $K_2=\emptyset$
and we now cover the case when $K_2\neq\emptyset$. 
In Section \ref{sec:bound-unbo-solut} we  state our main results.

\section{Existence of the positive equilibria}
\label{sec:behav-posit-equil}

Our main result in this Section states that for any $\lambda \in
\left(\lambda_1(\Omega) ,\lambda_1(\Omega_0)\right)$, there exists a
unique classical positive solution of \eqref{eq:elliptic:problem} and
their $L^\infty$-norms approach infinity as $\lambda \to \lambda_1(\Omega_0)$,
see Theorem \ref{th:implicitFThm}. As mentioned
before, this  result is already know in the particular case when
$n(x)$ is a smooth function, $K_2=\emptyset$, and $K_0=
K_1=\overline{\Omega_0}$,  an open set with regular boundary, see
\cite{Ouyang,Gamez_97,fraile96:_ellip}.


We first state the following preliminary result. Assuming that for a
fixed value of the parameter $\lambda=\lambda_0$, there exists a
positive stationary solution of \eqref{eq:elliptic:problem}, then
$\lambda_0$ must lie inside a precise  open bounded interval.  Moreover, for this $\lambda_0$,
there is a small $\delta_0$ such that for each $\lambda\in
(\lambda_0-\delta, \lambda_0+\delta)$, there exists a unique positive
solution, which is smooth and increasing in the parameter. More
precisely, we have the following lemma.


\begin{lemma} \label{lem:implicitFThm}
Assume $n(x)$ is H\"older continuous and $K_{0}$ satisfies {\rm  (HK)}.
Assume that $\varphi_{0}$ is a nontrivial nonegative classical stationary 
solution of
\eqref{eq:elliptic:problem} for $\lambda= \lambda_{0}$. Then the following holds:
\begin{itemize}
\item[(i)]$\lambda_0 \in \left(\lambda_1(\Omega) ,\lambda_1(\Omega_0)\right)$

\item[(ii)] For each $\lambda$ in a neighborhood of $\lambda_{0}$ there exists a unique
nonnegative stationary solution of \eqref{eq:elliptic:problem},
$\varphi_{\lambda}$,  close to $\varphi_{0}$ which is moreover a smooth function of $\lambda$.
\item[(iii)] The equilibria $\varphi_{\lambda}$ is an increasing function of $\lambda$.
\end{itemize}
\end{lemma}

\begin{proof}
(i)  Assume that $(\lambda_0,\varphi_0)$ is a non  negative nontrivial 
stationary solution, then
\begin{equation} \label{first:eig}
  \lambda_0 = \lambda_1(-\Delta + n(x) \varphi_0^{\rho-1}, \Omega) ,
\end{equation}
that is, $\lambda_0$ is the first eigenvalue of the operator $-\Delta
+ n(x)\varphi_0^{\rho-1}$ in $\Omega$, with Dirichlet boundary
conditions.
This fact, together with the monotonicity of the first eigenvalue with
respect to the potential implies that, since $n(x) \varphi_0^{\rho-1}
\geq 0$,
\[
  \lambda_0 > \lambda_1(\Omega).
\]
On the other hand, the monotonicity with respect to the domain of this
eigenvalue gives
\[
 \lambda_0 <   \lambda_1(-\Delta + n(x) \varphi_0^{\rho-1}, \Omega_0).
\]
Also note that $n(x)=0$ on $\Omega_0$ and so
$$\lambda_1(-\Delta + n(x) \varphi_0^{\rho-1}, \Omega_0) =
\lambda_1(-\Delta, \Omega_0)=\lambda_1( \Omega_0) ,$$
and therefore, part (i) is already proved.


(ii) Since  $n$ is $C^{\alpha}$ H\"older continuous, we consider
  the map
\[
F: (\lambda ,u)\to -\Delta u-\lambda u+ n(x) u ^\rho
\]
from $\mathbb{R}\times C^{2,\alpha}_0(\overline{\Omega})\to
C^{\alpha}(\overline{\Omega})$ where
$C^{2,\alpha}_0(\overline{\Omega}):=\{u\in
C^{2,\alpha}(\overline{\Omega}): u=0,  \text{ on }\partial\Omega\}$. 
Then  $F$ is a continuously differentiable map, and we
apply the implicit function theorem at $(\lambda ,u)=(\lambda_0
,\varphi_0). $ By hypothesis $\varphi_0 $ is a nonnegative stationary
solution of \eqref{eq:elliptic:problem}, then $F(\lambda_0,\varphi_0)=0 $.

Moreover, the derivative with respect to $u$ at 
$(\lambda ,u)=(\lambda_0 ,\varphi_0)$ is 
$$
D_uF(\lambda_0 ,\varphi_0)=-\Delta -\lambda_0 + \rho n(x) \varphi_0 ^{\rho-1}.
$$
Since $\rho>1$ and taking into account the monotonicity of the first 
eigenvalue with respect to the potential and  \eqref{first:eig} we obtain
\[
   \lambda_1(-\Delta - \lambda_0 + \rho n(x) \varphi_0^{\rho-1}) >
   \lambda_1(-\Delta - \lambda_0 + n(x) \varphi_0^{\rho-1})=0.
\]
This implies that  the derivative $D_uF(\lambda_0 ,\varphi_0)$ is an isomorphism.

So, for each $\lambda$ in a neighborhood of $\lambda_0$ there is a unique 
solution $\varphi_\lambda$ of \eqref{eq:elliptic:problem} in a neighborhood 
of $\varphi_0$ and the map $\lambda \to \varphi_\lambda $ is continuously 
differentiable with $\varphi_{\lambda_0}=\varphi_0$,
ending this part of the proof.

(iii) Let
$$
v:=\frac{d\varphi_\lambda}{d\lambda},
$$
taking derivatives with respect to $\lambda$ in  \eqref{eq:elliptic:problem} 
we obtain
\begin{gather*}
    - \Delta v  =   \lambda v +\varphi -\rho n(x) \varphi^{\rho-1} v\quad
 \text{in }  \Omega\\
    v= 0 \quad  \text{on }  \partial\Omega .
  \end{gather*}
We shall reason as before. Since
$  \lambda = \lambda_1(-\Delta + n(x) \varphi_\lambda^{\rho-1})$
 and
\[
   \lambda_1(-\Delta - \lambda + \rho n(x) \varphi_\lambda^{\rho-1})
   > \lambda_1(-\Delta - \lambda + n(x) \varphi_\lambda^{\rho-1}) = 0,
\]
the  maximum principle gives
$$
v:=\frac{d\varphi_\lambda}{d\lambda}>0,
$$
therefore $\varphi_\lambda$ is an increasing function  of $\lambda$.
\end{proof}


The next result gives some ``spectral'' property of the set $K_{0}$
that will be used below.

\begin{lemma}\label{convergence-first-eigenvalue}
Assume $K_{0}$ satisfies {\rm   (HK)}. If we denote by 
$U_\delta=\{x\in \Omega: d(x,K_0)<\delta\}$, then
\begin{equation}\label{convergence-eigenvalues}
\lambda_1(U_\delta)\nearrow \lambda_1(\Omega_0),\quad \hbox{ as }
\delta \to 0 .
\end{equation}
\end{lemma}

\begin{proof}
Observe that the family $U_\delta$ is decreasing in $\delta$ and we
have $\Omega_0\subset K_0\subset U_\delta$. Therefore,
$\lambda_1(U_\delta)$ is an increasing sequence in $\delta$
with $\lambda_1(U_\delta)<\lambda_1(\Omega_0)$.  Nevertheless,
$U_\delta$ does not converge in the Haussdorf distances to $\Omega_0$
so the convergence stated in \eqref{convergence-eigenvalues} is not
obvious at all.

Notice first that if $K_1\cap K_2=\emptyset$ then for
$\delta<\frac{1}{2}d(K_1,K_2)$, we have $U_\delta=U_\delta^1\cup
U_\delta^2$, where $U_\delta^i=\{x\in \Omega: d(x,K_i)<\delta\}$ for
$i=1,2$ and $U_\delta^1\cap U_\delta^2=\emptyset$.   This implies that
$\lambda_1(U_\delta)=$min$\{ \lambda_1(U_\delta^1),
\lambda_1(U_\delta^2)\}$.  But since
$|K_2|=0$ then $|U_\delta^2|\to 0$ and therefore
$\lambda_1(U_\delta^2)\to +\infty$.  To see this, we just use
Faber-Krahn inequality, see for instance \cite{henrot}.  This implies
that $\lambda_1(U_\delta)=\lambda_1(U_\delta^1)$ and since $\Omega_0$
is a smooth open set, then $\lambda_1(U_\delta^2)\to
\lambda_1(\Omega_0)$, see  \cite{C-H, B-V}.

If $K_1\cap K_2\ne\emptyset$, then the argument is not so
straightforward.  Nevertheless, since $|K_2|=0$,  we have that for
each fixed ball $B\subset \mathbb{R}^N\setminus \bar\Omega_0$ we have
$|B\cap U_\delta|\to 0$ as $\delta \to 0$ and this implies, see
\cite{A, D} that $\lambda_1(U_\delta)\to \lambda_1(\Omega_0)$.
\end{proof}


Next, we  state the following result. For each parameter inside the
interval determined in Lemma \ref{lem:implicitFThm}, part (i), there
exists a unique positive solution. Moreover, the
$L^\infty$ norm of the solutions grows to infinity as the parameter
$\lambda$ approaches $\lambda_1(\Omega_0)$.

\begin{theorem} \label{th:implicitFThm}
Assume $K_{0}$ satisfies {\rm   (HK)}. Then the  following holds:
\begin{itemize}
\item[(i)] For any
$\lambda \in \left(\lambda_1(\Omega) ,\lambda_1(\Omega_0)\right)$
there exists a unique strictly positive classical solution
$\varphi_{\lambda}\in C^{2}_0(\overline{\Omega})$  of
\eqref{eq:elliptic:problem}.

\item[(ii)]  furthermore, as $\lambda \to  \lambda_1(\Omega_0)$, we have
\begin{equation}\label{unbounded}
  \quad  \|\varphi_{\lambda}\|_{L^\infty (\Omega)} \to \infty .
\end{equation}
\end{itemize}
\end{theorem}


\begin{proof}
(i) If $\lambda \in (\lambda_1(\Omega), \lambda_1(\Omega_0))$, by
sub-supersolutions method, we will prove that there is a bounded
solution of \eqref{eq:elliptic:problem}.
Specifically, observe that  $\underline{u}:=\varepsilon\Phi_1$ is a
subsolution  choosing $\varepsilon$ small enough, in particular
for any $\varepsilon\leq
\big(\frac{\lambda-\lambda_1(\Omega)}{\|n\|_\infty}\big)^{1/(p-1)}$.


On the other hand,  from Lemma \ref{convergence-first-eigenvalue},  we
can choose regular domains $\Omega_1$, $\Omega_2$, with
$$
\Omega_0\subset K_0 \Subset \Omega_1\Subset
\Omega_2\subset \Omega
$$
such that $\lambda < \lambda_1(\Omega_2)< \lambda_1(\Omega_1)< \lambda_1(\Omega_0)$.
Set $w\in C^2(\overline{\Omega})$ a function strictly positive such that
$$
w(x):=\begin{cases} 1 & \text{for } x\in\Omega\setminus\Omega_2\\
\Phi_1(\Omega_2) & \text{for } x\in\Omega_1
\end{cases}
$$
where  $\Phi_1(\Omega_2)>0$ is the first eigenfunction corresponding
to the eigenvalue problem in $\Omega_2$ with Dirichlet boundary
conditions.

Then a  supersolution can be chose in the following way
$\overline{u}:=Mw$ for $M$ big enough, \cite{Gamez_97}.
Thus existence of a pair of ordered positive solutions $\varphi_1 \leq
\varphi_2$,  follows from \cite{Am76}.

To prove uniqueness, observe that if $\varphi_1 \leq
\varphi_2$ are not the same, then we would have
\[
    \lambda = \lambda_1(-\Delta + n(x) \varphi_2^{\rho-1}) >
    \lambda_1(-\Delta + n(x) \varphi_1^{\rho-1}) = \lambda ,
\]
which is absurd.

(ii) From the monotonicity in $\lambda$, see Lemma
\ref{lem:implicitFThm}, there exists the monotone pointwise limit
\[
  \varphi^{*} (x) = \lim_{\lambda \to \lambda_1 (\Omega_0)}
 \varphi_{\lambda}(x).
\]
We next prove \eqref{unbounded}. In fact, otherwise, we get   
$\varphi^{*} \in L^\infty (\Omega)$ and by   elliptic regularity we would have
$\|\varphi_{\lambda}\|_{W^{2,p}(\Omega)}\leq C$, for all $\lambda \in
\left(\lambda_1(\Omega) ,\lambda_1(\Omega_0)\right)$ and any 
$1<p<\infty$.

Sobolev's compact imbedding  Theorem implies then that  at least for a
subsequence, $\varphi_{\lambda}\to \varphi^{*}$ in
$W^{1,p}(\Omega)\hookrightarrow C(\bar{\Omega})$, for $p>N$ and  therefore
$\varphi^{*}$ is a weak solution of
\begin{gather*}
    - \Delta \varphi^{*}  =   \lambda_1 (\Omega_0) \varphi^{*}  -
    n(x) (\varphi^{*})^{\rho} \quad \text{in }  \Omega\\
    \varphi^{*}= 0 \quad \text{on }  \partial\Omega .
  \end{gather*}
Moreover,  $\varphi^{*}$ is bounded and therefore, by a bootstrap argument
$\varphi^{*}$ will be a classical solution of
\eqref{eq:elliptic:problem} with $\lambda =\lambda_1
(\Omega_0)$, which contradicts part (i) of Lemma
\ref{lem:implicitFThm}, which  ends the proof.
\end{proof}

\begin{remark}
It can be shown that  $\varphi_{\lambda}$ is globally  asymptotically stable for nonnegative nontrivial
solutions of \eqref{eq:elliptic:problem}; see \cite{A-P-RB12}.

\end{remark}

\section{Boundedness and unboundedness of solutions}
\label{sec:bound-unbo-solut}

The questions are now: What happens as $\lambda \to  \lambda_1(\Omega_0)$?
Where and how solutions become unbounded?

The first that we can say is that the blow-up is a complete blow-up at
every point in $\Omega_{0}$.
For the  for the proofs of the  following   results, we refer to \cite{A-P-RB12}.

\begin{lemma} \label{lem:grow_up_regularset}
Assume $K_{0}$ satisfies {\rm   (HK)} and let  $\{\varphi_\lambda\}$ for
$\lambda\in(\lambda_1(\Omega),\lambda_1(\Omega_0))$
denote the family of positive solutions of \eqref{eq:elliptic:problem}.
Then
\[
  \lim_{\lambda \to \lambda_1(\Omega_{0})} \varphi_\lambda(x) = \infty, \quad 
\text{for all $x\in \Omega_{0}$} .
\]
\end{lemma}


To obtain upper bounds on the solutions outside $\Omega_{0}$ we will 
use the following Lemma, see \cite{garcia-melian98:_point}.
This Lemma analyzes the minimum of a radially symmetric solution of a
singular logistic equation with constant coefficients and going to
infinity at the boundary, see  \cite{keller57:_delta,  osserman57:_delta}.


\begin{lemma} \label{lem:singular_Dirichlet_pbm}
Assume $\rho>1$ and $\lambda, \beta >0$ and consider a ball in
$\mathbb{R}^{N}$ of radius $a>0$ and the following singular Dirichlet
problem
\begin{gather*}
- \Delta z = \lambda z -\beta z^{\rho} \quad \text{in } B(0,a) \\
z= \infty \quad \text{on }\partial B(0,a).
\end{gather*}
Then, there exists a unique positive radial solution, $z_{a}(x)$.
Moreover, the solution satisfies
$$
\Big( {\lambda \over \beta} \Big)^{1/(\rho -1)} \leq
z_{a}(0)= \inf_{B(0,a)} z_{a}(x) 
\leq \Big(   {\lambda (\rho +1) \over 2\beta} +
\frac{B}{\beta a^{2}}
\Big)^{1/(\rho -1)}
$$
for some constant $B=B(\rho,N)>0$, $B$ independent of $\lambda$.
\end{lemma}

The above Lemma gives a local upper bound.

\begin{proposition} \label{prop:boundedness_far_from_K0}
Let $x_{0} \in \Omega \setminus K_{0}$ and let $\varphi > 0$ be a
stationary solution of \eqref{eq:elliptic:problem} for some
$\lambda < \lambda_1(\Omega_0)$. Then there exists $a>0$
and $M>0$  independent of $\lambda$, such that
\[
  0\leq \varphi (x) \leq M, \quad \forall x \in B(x_{0}, a).
\]
\end{proposition}

\begin{proof}
Let $x_{0} \in \Omega\setminus K_{0}$ and let $a>0$ be such that
$B(x_{0}, 3a) \subset    \Omega\setminus K_{0}$. Denote
$$
\beta =\inf\{ n(x), \; x \in B(x_{0}, 2a)\} >0.
$$
For each $y\in B(x_0,a)$, consider $z(x)$ the
translation to  $B(y, a)$ of the function in Lemma
\ref{lem:singular_Dirichlet_pbm}, with $\lambda=\lambda_1(\Omega_0)$.
Hence $z(x)$ is a
supersolution for $\varphi(x)$ and then
\[
  \varphi(x) \leq z(x), \quad x\in B(y, a).
\]
In particular,  taking $x=y$, we have
$$
\varphi(y)\leq  \Big(   {\lambda_1(\Omega_0)(\rho +1) \over 2\beta} +
\frac{B}{\beta a^{2}}\Big)^{1/(\rho -1)}, \quad \forall y\in B(x_0,a),
$$
which proves the result with 
$M=\left(   {\lambda_1(\Omega_0)(\rho +1) \over 2\beta} +\frac{B}{\beta a^{2}}
\right)^{1/(\rho -1)}$.
\end{proof}

Assume now that the two parts $K_1$ and $K_2$ of $K_{0}$ are
disjoint. The following result shows that, for
$\lambda\to\lambda_1(\Omega_0)$, all solutions of \eqref{eq:elliptic:problem}
remain bounded in $K_2$,  while they start to grow up in $K_1$.

\begin{theorem} \label{thr:progressive_grow_up}
Assume $K_{0}$ satisfies {\rm   (HK)} and
$ K_1 \cap K_2 =\emptyset$.
Then the following holds
\begin{itemize}
\item[(i)] There exists a $\delta>0$ and  $M>0$ such that
$$
|\varphi_\lambda(x)|\leq M,\quad \forall x:\, d(x,K_2)\leq \delta,
\quad \forall \, \lambda\in (\lambda_1(\Omega),
\lambda_1(\Omega_0)) .
$$

\item[(ii)] For  $\lambda \to\lambda_1(\Omega_0)$ all solution of
\eqref{eq:elliptic:problem} are bounded on $K_2$.

\item[(iii)]  If $\lambda \to \lambda_1(\Omega_0)$  then the pointwise limit 
of the solutions of \eqref{eq:elliptic:problem} is unbounded on  $K_1$.
\end{itemize}
\end{theorem}

Now we turn to the case in which $K_1$ and $K_2$ are  glued
together. First using Lemma \ref{lem:singular_Dirichlet_pbm}  we prove the
following universal bounds for solutions of
\eqref{eq:elliptic:problem}.


\begin{lemma} \label{lem:universal_upper_bound}
Assume that $n(x)$ satisfies {\rm (Hn)}.
Then there exists a constant $A$, independent of $\lambda$ such that  for
any solution of \eqref{eq:elliptic:problem}
 we have
\[
0 \leq   \varphi (x) \leq h(x) = \Big( \frac{A }{d_0(x)}
   %\displaystyle\inf_{B\left(x,\frac12 d_0(x)\right)} n}
  \Big)^{\frac{\gamma +2}{\rho -1}}
\]
with $d_0(x) = \operatorname{dist}(x,K_{0})$.
\end{lemma}

The following result will be used further below and gives a criteria
to check whether a function that is infinity on a smooth compact set of
measure zero, is integrable. As shown below, this criteria depends on
the  dimension  of the set and the rate at which the function diverges on
it.

\begin{lemma} \label{lem:integral_on_fractal}
Assume $K\subset \mathbb{R}^{N}$ is a closed regular $d-$dimensional
manifold with $d\leq N-1,$ and consider a function defined on a bounded  
neighborhood $\Omega$ of $K$ of the form
\[
  f(x) = \big(dist(x,K)\big)^{-\alpha}\quad \text{for } \alpha >0.
\]
If $r\geq 1$ satisfies
$r\alpha < N-d$,
then $f\in L^{r}(\Omega)$.
\end{lemma}

With all these  we can state the following result.

\begin{theorem}\label{th:condition:bdd}
Assume $K_0$ satisfies {\rm (HK')} and
$$
K_1 \cap K_2 \neq
\emptyset.
$$
Assume $n(x)$ satisfies {\rm (Hn)}.
Assume also that
\[
\gamma+2<(\rho -1)(N-d) .
\]
Then, the positive solutions of \eqref{eq:elliptic:problem}
remain  bounded on compact sets of
$\Omega\setminus K_1$. In particular they  remain bounded  at each
point of  $K_2\setminus K_1$.
\end{theorem}

\begin{remark} \rm
It is an interesting open problem to determine whether we always obtain 
that the solution of \eqref{eq:elliptic:problem}
are bounded in compact sets of $\Omega\setminus K_1$ or, in the contrary, 
that we have cases in which $\varphi_\lambda$ becomes infinity in $K_2$ 
as $\lambda\to \lambda_1(\Omega_0)$.
\end{remark}

\begin{remark} \rm
This work is still in progress, and we refer to \cite{A-P-RB12} for details and
more general results, including more general configurations for the
set $K_{0}$ and the analysis of the solutions of the parabolic problem
associated to \eqref{eq:elliptic:problem}.
\end{remark}

\subsection*{Acknowledgments}
This research was supported by Projects MTM2009-07540,
MTM2012-31298 and GR35/10-A,  Grupo 920894 BSCH-UCM, Grupo de
Investigaci\'on CADEDIF, Spain.


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