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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 187, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/187\hfil A priori estimates]
{A priori estimates and existence for quasilinear elliptic
 equations with nonlinear Neumann boundary conditions}

\author[Z. Hu, L. Wang, P. Zhao \hfil EJDE-2016/187\hfilneg]
{Zhe Hu, Li Wang, Peihao Zhao}

\address{Zhe Hu \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{huzhe@upc.edu.cn}

\address{Li Wang \newline
School of Mathematics and Statistics,
Lanzhou University, Lanzhou 730000, China}
\email{lwang10@lzu.edu.cn}

\address{Peihao Zhao \newline
School of Mathematics and Statistics,
 Lanzhou University, Lanzhou 730000,  China}
\email{zhaoph@lzu.edu.cn}

\thanks{Submitted March 19, 2016. Published July 12, 2016.}
\subjclass[2010]{26A33, 65M12, 65M06}
\keywords{m-Laplacian; nonlinear Neumann boundary conditions; 
\hfill\break\indent a priori estimates}

\begin{abstract}
 This article concerns the existence of positive  solutions for a
 nonlinear Neumann problem involving the m-Laplacian.
 The equation does not have a variational structure. We use a blow-up
 argument and a  Liouville-type theorem  to obtain a priori estimates
 and obtain the existence of positive solutions  by the Krasnoselskii
 fixed point theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction and statement of main results}

In this work we consider the  problem
\begin{equation}\label{eq1}
\begin{gathered}
\Delta_m u+B(z,u,\nabla  u)=0  \quad\text{in }  \Omega,\\
|\nabla u|^{m-2}\frac{\partial u}{\partial\nu}=g(z,u) \quad\text{on }
 \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded  domain  with smooth boundary in
 $\mathbb{R}^N (N\geq 2)$.   
$B(z,u,\mathbf{p}):\Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}$ 
is a continuous function. $\frac{\partial u}{\partial\nu}$ denotes the  
outward  normal derivative respect to $\partial \Omega$,
   $g(z,u):\partial\Omega \times \mathbb{R} \to \mathbb{R}$ is a continuous 
function.

 A function $u \in W^{1,m}(\Omega)\cap C(\bar{\Omega})$ is said to be a weak 
solution for \eqref{eq1}  if
$$
\int_\Omega |\nabla u|^{m-2}\nabla u\cdot\nabla\phi\,dz
-\int_{\partial\Omega} g(z,u)\phi\,d\sigma
=\int_\Omega B(z,u,\nabla u)\phi \,dz
$$
for any $\phi \in C^\infty(\bar \Omega)$. 

Similar problems have been studied in many articles, see e.g. 
\cite{BT}-\cite{WD}.
  When $B$ depends on $\nabla u$, variational methods are barely  used to
deal with equation ($\ref{eq1}$).  
 In this case, the question of the  existence of  solutions can be handled 
by  a priori estimates and topological methods.  Combining the blow-up 
(scaling) arguments with suitable Liouville-type theorems, we can derive 
a priori estimates.  The method was introduced  in \cite{GJ}, where 
Gidas and Spruck obtain a priori bounds for solutions of nonlinear elliptic 
boundary value problem with the nonlinearity depending on $x$ and $u$. 
Later, the method was used to systems in \cite{CMM}-\cite{AC} and more 
general cases   concerning a single equation were studied in 
\cite{DRuiz}-\cite{DMT}. Ruiz\cite{DRuiz} and Zou\cite{HHZ} consider 
nonlinear Dirichlet problem involving the $m$-Laplacian  with the nonlinearity 
depending on $x$, $u$ and $\nabla u$  under different conditions. 
In \cite{Yin Jingxue}, the power of growth of  $u$ and $\nabla u$ maybe 
critical or supercritical.
  In \cite{DGZ}, the authors obtain  similar results of generalized mean 
curvature equations.

 All articles mentioned before deal with the  Dirichlet problems.
We consider the m-Laplacian with nonlinear Neumann boundary conditions.
 Throughout this paper, we assume  $m \in (1,N)$, $p\in (m-1,m^*)$, 
where $m^*=\frac{Nm}{N-m}-1$. Let  $\alpha=\frac{p-(m-1)}{m}$  and 
$0<q<\frac{p+1}{m}(m-1)$.
First, we list some conditions to the nonlinear terms $B$ and $g$.

We say $B(z,u, \mathbf{p})$ satisfies a growth-limit condition (G-L) if
    there exist positive constants $p$ and $K_i, i=1,2,3$, such that the
following:
\begin{enumerate}
\item There exists a bounded function
 $F:\mathbb{R}_+\to\mathbb{R}_+$ such that
 $$ 
|B(z,u,\mathbf{p} )| \leq K_1[1+u^p+F(|\mathbf{p}|)|\mathbf{p}|
 ^{\frac {mp}{p+1}}]
$$
 for all $(z,u,\mathbf{p} )\in \Omega \times \mathbb{R}_+ \times \mathbb{R}^N$, 
and $F(|\mathbf{p}|)\to0$ as $|\mathbf{p}|\to \infty$.

\item There exists a continuous function $b:\bar \Omega \to\mathbb{R}_+$ 
such that for any sequences
$\{(M_k,{\mathbf{p}}_k )\}\subset \mathbb{R}_+ \times \mathbb{R}^N$
 satisfying $M_k\to\infty$ and
${\mathbf{p}}_k=O(M^{1+\alpha}_k) $, it holds
$$
\lim_{k \to \infty}\frac{B(z,M_k,\mathbf{p}_k)}{M^p_k}=b(z)
$$
uniformly on $\Omega$.
\end{enumerate}

For the nonlinearity $g$ on the boundary  we assume the following  conditions:
\begin{itemize}

\item[(A1)] Assume that $g \in C(\partial\Omega\times \mathbb{R},\mathbb{R})$. 
There exist   constants $0<\mu_1,\mu_2 < 1$ and a nondecreasing continuous
function $\Gamma(t) :[0,\infty)\to (0,\infty) $ with
 $|\Gamma(t)|\leq K_2(1+t^q)$ such that
$$
|g(z,u)-g(y,v)|\leq \Gamma(\max\{|u|,|v|\})[|z-y|^{\mu_1}+|u-v|^{\mu_2}]  
$$
for all $ (z,u),(y,v)  \in \partial \Omega\times \mathbb{R}$.

\item[(A2)] $|g(z,u)|\leq K_3(1+|u|^q)$  for all 
$ (z,u)\in \partial\Omega\times \mathbb{R}$.

\item[(A3)] $g(z,u)\geq0$  for all $ (z,u)\in \partial\Omega\times \mathbb{R}_+$ 
and $g(z,0)=0 $ for all $z\in \partial\Omega$.

\end{itemize}

    The main ingredients of our arguments are  a priori estimates  on the pairs
 $(u,\lambda)$  solving the  problem
\begin{equation}\label{eq2}
   \begin{gathered}
   \Delta_m u+B(z,u,\nabla u)+\lambda= 0 \quad\text{in }  \Omega,\\
   |\nabla u|^{m-2}\frac{\partial u }{\partial \nu}=g(z,u) \quad\text{on }
  \partial \Omega.
   \end{gathered}
  \end{equation}
By the blow-up method, we first suppose by contradiction that there exists
 a sequence of unbounded solutions. Then by suitable scaling
 argument and taking advantage of the regularity results in \cite{fan2007global} 
(see also \cite{Lie}) we obtain a subsequence which  converges to a 
 nonnegative solution. That  contradicts   Liouville-type theorem on the entire 
space  $\mathbb{R}^N$ or on the half-space $\mathbb{R}^N_+$.
Our main results can be stated as follows.

\begin{theorem}\label{thm1.1}
Let $ \Omega$ be a  bounded smooth domain and assume that conditions 
{\rm(G-L), (A1)} and {\rm (A2)} hold.  Then there exists a positive
constant $C$ such that  $ \sup_{z\in\Omega}  u(z)+\lambda \leq C$
for all non-negative $ C^1 $solutions $u$ of \eqref{eq2}.
\end{theorem}

By this a priori estimates we can derive the existence of solutions for  
\eqref{eq1}. For this purpose, we need some further hypotheses.

We say  $B$ satisfies a positivity condition:
\begin{itemize}
\item[(A4)]   There exists $L>0$ such that
$B(z,u,\mathbf{p})+L|u|^{m-1}\geq0 $for all 
$  (z,u,\mathbf{p})\in \Omega \times \mathbb{R}_+ \times \mathbb{R}^N$.
\end{itemize}
We call $B$ and $g$ ``super-linear" at the origin if
\begin{itemize}
\item[(A5)] There exists $L>0$ such that
$B(z,u,\mathbf{p})+L|u|^{m-1}=o(|u|^{m-1}+|\mathbf{p}|^{m-1})$, 
$ (z,u,\mathbf{p})\in \Omega \times \mathbb{R}_+ \times \mathbb{R}^N$,
$g(z,u)=o(|u|^{m-1})$, $ (z,u)\in \partial\Omega \times \mathbb{R}_+  $
as $(u,\mathbf{p})\to 0$ uniformly on $\overline{\Omega}$.
\end{itemize}

\begin{theorem}\label{thm1.2}
Let $ \Omega$ be a  bounded smooth domain and assume that $B$ and $g$ 
 satisfy conditions {\rm (G-L), (A1)--(A5)}.  Then   \eqref{eq1}
 has a positive solution.
\end{theorem}

 This paper is structured as follows. 
In section 2, we obtain the  a priori
 estimates for solutions of  \eqref{eq2}. 
In section 3, we obtain the existence result of \eqref{eq1}
 by the Krasnoselskii fixed-point theorem.

\section{A priori estimates}

 In this section, we prove Theorem \ref{thm1.1}, the main part of this article. 
The regularity for solutions and Liouville theorem play an important role 
in the  proof. We first list two lemmas which will be used later.

 \begin{lemma}[$ C^{1,\beta}$ Regularity \cite{fan2007global}]
\label{regularFan}
Let $ \Omega $ be a bounded domain in $ \mathbb{R}^N $ with smooth boundary, 
$ \beta, \mu_1,\mu_2\in(0,1)$.
 Suppose $ B:\Omega\times \mathbb{R}\times\mathbb{R}^N\to\mathbb{R}$ 
satisfy the condition
\begin{align}\label{regularfan5}
|B(x,u,\mathbf{p})|\leq\Lambda(|u|)(1+|\mathbf{p}|^{m}), \quad
\forall (x,u,\mathbf{p})\in \Omega\times \mathbb{R}\times\mathbb{R}^N
\end{align}
Suppose $g\in C(\partial\Omega\times\mathbb{R}, \mathbb{R})$  
satisfy the condition
\begin{equation*}\label{regulatitycon3}
  |g(x,\vartheta)-g(y,\omega)|\leq\Lambda(\max\{|\vartheta|,
|\omega|\})[|x-y|^{\mu_1}+|\vartheta-\omega|^{\mu_2}] ,\quad
  \forall x,y\in\partial\Omega, \forall\vartheta,\omega\in \mathbb{R} ,
\end{equation*}
where $ \Lambda:[0,\infty)\to(0,\infty) $ is a nondecreasing continuous function. \\
If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega) $ is a bounded generalized 
solution of the  boundary value problem
\begin{equation}\label{regular}
\begin{gathered}
\Delta_m u+B(z,u,\nabla  u)=0 ,\quad z\in  \Omega,\\
|\nabla u|^{m-2}\frac{\partial u}{\partial\nu}=g(z,u),\quad z\in  \partial\Omega,
\end{gathered}
\end{equation}
and satisfy $\sup_\Omega|u|\leq M_0$, then there is a positive constant 
\[
\beta=\beta(m,N,  \Lambda(M_0), M_0,\mu_1,\mu_2,
\sup|g(\partial\Omega\times[-M_0, M_0])|,\Omega) 
\]
 such that $ u $ 
is in $ C^{1,\beta}(\partial\Omega)$; moreover
\begin{equation}\label{regularbound}
  |u|_{C^{1,\beta}(\overline{\Omega})}
\leq C(m,N, \Lambda(M_0), M_0,\mu_1,\mu_2,\sup|g(\partial\Omega\times[-M_0, M_0])|,
\Omega)
  \end{equation}
\end{lemma}

 \begin{lemma}\label{lem3}
   Let $b>0$ be a constant. Then
the problem
\begin{gather*}
\Delta_m u+b u^p = 0 \quad \text{in } \mathbb{R}^N_+\\ 
|\nabla u| ^{m-2} \frac{\partial u}{\partial \nu}=0 \quad \text{on } 
\partial\mathbb{R}^N_+,
\end{gather*}
does not admit any non-negative non-trivial solutions  when $p\in (m-1,m^*)$.
\end{lemma}

 We sketch a proof of Lemma \ref{lem3}, our approach is similar to the one
 used in  \cite{CP}. 
Assume that the equation has a non-negative non-trivial solution $\omega$. 
By reflection with respect to the hyperplane $z_N=0$,  we obtain  
$\tilde{\omega}$ which is a non-negative non-trivial solution of corresponding 
equation on entire space, as the reader can see in \cite{HHZ}. 
That is a contradiction and we prove Lemma \ref{lem3}.


\begin{proof}[Proof of Theorem \ref{thm1.1}]
We argue by contradiction and suppose that the conclusion is not true. 
Then there exists a sequence of positive solutions $\{u_k,\lambda_k\}$ 
of \eqref{eq2} such that
\begin{equation}\label{j3.1}
\lim_{k\to\infty}(\|u_k\|_{L^\infty(\Omega)}+\lambda_k)=\infty.
\end{equation}
For $u_k \in C(\bar{\Omega})$, there exists $\xi^k\in\Omega$, such that
$M_k=\max _ {z\in\Omega}u_k(z)=u_k(\xi^k)$, $k=1,2,\dots$
We introduce the transform
\begin{equation}\label{j3.2}
w_k(y)=N^{-1}_k u_k(z) , \quad  y=(z-\zeta^k)N^\alpha_k
\end{equation}
where $N_k, \zeta^k$ will be determined later.
Denote $\Omega_k=\{y\in \mathbb{R}^N|z=N^{-\alpha}_ky+\zeta^k\in\Omega \}$ 
being the image of $\Omega$
after the transform \eqref{j3.2}.
By direct calculations, $w_k$ satisfies
\begin{equation}\label{j3.3}
\begin{gathered}
\Delta_mw_k+N^{-(1+\alpha)(m-1)-\alpha}_k [ B(N_k^{-\alpha} y
+\zeta^k,N_kw_k,N_k^{1+\alpha}\nabla w_k)+\lambda_k]=0  \quad \text{in}  \Omega_k,\\
|\nabla w_k|^{m-2}\frac{\partial w_k}{\partial\nu}
=N^{-(1+\alpha)(m-1)}_k  g(N_k^{-\alpha}y+\zeta^k,N_kw_k) \quad 
\text{on} \partial \Omega_k.
\end{gathered}
\end{equation}
 For convenience, we denote
\begin{gather*}
\theta_k (y,w_k,\nabla w_k)=N^{-(1+\alpha)(m-1)-\alpha}_k [B(N_k^{-\alpha} y
+\zeta^k,N_kw_k,N_k^{1+\alpha} \nabla w_k )+\lambda_k]   \quad \text{in } \Omega_k, \\
\sigma_k (y,w_k)=N^{-(1+\alpha)(m-1)}_kg(N_k^{-\alpha} y+\zeta^k,N_kw_k)   \quad
  \text{on }  \partial \Omega_k.
\end{gather*}
 We divide the proof into two cases.

\subsection*{Case 1}\label{case1}
For a subsequence, but still indexed by $k$, it holds
$$
\lim_ {k\to\infty}\frac{\lambda_k}{M_k^p}=0,
$$
which implies that $M_k\to\infty$  as  $k\to\infty$.
In the transform \eqref{j3.2}, take
$N_k=M_k,\zeta^k=\xi^k$, then 
$$
\lim_{k\to\infty}\frac{\lambda_k}{N_k^{p}}
=\lim_{k\to\infty}\frac{\lambda_k}{M_k^{p}}=0
$$
and
\begin{equation}\label{j3.6}
0<w_k(y)\leq \frac{M_k}{N_k}=1, \quad y\in\Omega_k; \quad  w_k(0)=1.
 \end{equation}

Using the  Part 1 of growth-limit condition (G-L)  we obtain that (for $k$ large enough)
\begin{equation}\label{j3.7}
|\theta_k ( y,w_k,\nabla w_k)|\leq  K_1(3+|\nabla w_k|^{m})+1,
 \end{equation}
 In condition (A1), constant $\mu_2 \in (0, 1)$   can be replaced 
by $\mu_3\in (0,min\{\mu_2,\frac{p+1}{m}(m-1)-q\}]$    such that
   \begin{equation}\label{j3.8}
 |g(z,u)-g(y,v)|\leq \tilde{K_2}[1+(\max\{|u|,|v|\})^q] 
(|z-y|^{\mu_1}+|u-v|^{\mu_3})
\end{equation}
for all $(z,u),(y,v) \in \partial \Omega\times \mathbb{R}$.
 by  assumptions (A1) and (A2). Then  we have
 \begin{equation}\label{j3.9}
 \begin{split}
&|\sigma_k(x,\omega)-\sigma_k (y,\vartheta)| \\
&\leq     \tilde{K} _2M_k^{-\frac{p+1}{m}(m-1)}
[1+M_k^q(\max\{| \omega|,|\vartheta|\})^q](M_k^{-\alpha\mu_1}|x-y|^{\mu_1}
    +M_k^{\mu_3}|\omega-\vartheta|^{\mu_3})\\
 & \leq \tilde{K} _2[1+(\max\{| \omega|,|\vartheta|\})^q] (|x-y|^{\mu_1}+|\omega-\vartheta|^{\mu_3}) ,\\
 \end{split}
 \end{equation}
By condition (A2), we have
\begin{equation}\label{ineq}
  M_k^{-\frac{p+1}{m}(m-1)}g( M_k^{-\frac{p-(m-1)}{m}}y+\xi^k,M_kw_k)
\leq K_3 M_k^{-\frac{p+1}{m}(m-1)}(1+|M_kw_k|^q).
\end{equation}
The transform \eqref{j3.2} flatten the boundary $\partial\Omega$, 
then for $k$ large enough,
$ \|\partial\Omega_k\|_{1,\beta_0}\leq\|\partial\Omega\|_{1,\beta_0}$.
Now we use a $C^{1,\beta}$ regularity result Lemma \ref{regularFan}.
From \cite{fan2007global} (see also  \cite{Lie}) and \eqref{j3.7}, \eqref{j3.9} 
to conclude that there exist positive constants $\beta=\beta(K_1,N,m)\in(0,\beta_0)$ 
and $C=C(K_1,K_2,K_3,N,m,\Gamma(1), \sup _{\partial\Omega\times[-1,1]}
 g(z,t),\Omega)>0$  such that
\begin{equation}\label{j3.10}
\|w_k\|_{C^{1,\beta}(\overline{\Omega_k})}\leq C,
 \end{equation}
where $C$ is a constant independent of $k$.

Set $d_k=\operatorname{dist}(\xi^k, \partial \Omega)$, then 
$\operatorname{dist}(O, \partial \Omega_k)=M_k^{\alpha}d_k$.
Next we consider two subcases:

 \subsubsection*{Unbounded $\{M_k^{\alpha}d_k \}$} \label{subcase1}
The sequence $\{M_k^{\alpha}d_k\}$ is unbounded, we assume there exists a 
subsequence  $\{M_k^{\alpha}d_k\} \to \infty$ as $k\to \infty$.
  With the aid of \eqref{j3.6} and  \eqref{j3.10}, we can apply the 
Arzela-Ascoli theorem and the diagonal line argument  to infer that  
 there exists $w\in C^{1}(\mathbb{R}^N) $, such that
\begin{equation}\label{j3.11}
\lim_{k\to\infty}w_k(y)=w(y)\geq0,  w(0)=1,
\end{equation}
uniformly on any compact subset of $\mathbb{R}^N$ in $C^{1}-$topology.

Multiplying \eqref{j3.3} by a test function 
$\phi\in C^\infty(\mathbb{R}^N) $ and integrating by parts on  
$\Omega_k$, we obtain
\begin{align*}
&\int_{\Omega_k}|\nabla w_k|^{m-2}\nabla w_k\cdot\nabla\phi  dy 
 -\int_{\partial\Omega_k} M_k^{-\frac{p+1}{m}(m-1)}
 g( M_k^{-\frac{p-(m-1)}{m}}y+\xi^k,M_kw_k) \phi  ds \\
&= M_k^{-p}\int_{\partial\Omega_k}[ B(M^{-\frac{p-(m-1)}{m}}_k y
 +\xi^k,M_kw_k,M^{\frac{p+1}{m}}_k\nabla w_k)+\lambda_k]  \phi  dy.
\end{align*}

On account of \eqref{ineq}, we have 
$\lim_{k\to\infty} M_k^{-\frac{p+1}{m}(m-1)}
g( M_k^{-\frac{p-(m-1)}{m}}y+\xi^k,M_kw_k)=0$.
Combining the condition (G-L) part 2 with the above equality, we obtain
$$
\Delta_m w+b(\xi^0)w^p=0  \text{in} \mathbb{R}^N,
$$
where $\xi^0=\lim_{k\to\infty} \xi^k\in \bar{\Omega}$, but $w(0)=1$. 
This contradicts the Liouville-type theorem on entire space 
$\mathbb{R}^N$ \cite[Therorem 1.1 ]{HHZ}.

\subsubsection*{Bounded $\{M_k^{\alpha}d_k\}$}
 The sequence $\{M_k^{\alpha}d_k\}$ is bounded
 as $k \to \infty$. So there exists a subsequence such that
 $\{M_k^{\alpha}d_k\}\to \varepsilon\geq 0$.  Denote 
$z=(z',z_N)=(z_1,\dots,z_{N-1},z_N)$ for any $z\in \mathbb{R}^{N}$.
 With proper translation and rotation, one may assume
$\xi^k=(0',|\xi^k|),  d_k=\operatorname{dist}(O,\xi^k )=|\xi^k|$,
where $O=(0',0)\in\partial \Omega$ is the origin in $\mathbb{R}^{N}$  
and $\xi^k$ is the positive $z_N$-direction.
By the transform \eqref{j3.2}  for any $y \in \Omega_k$, we have 
$y_N> -\varepsilon$ and
the sequence of the domains $\Omega_k$ converges to the half-space, namely
$\lim_{k\to\infty}\Omega_k
=\mathbb{R}^N_\varepsilon:=\{y\in\mathbb{R}^n|y_N>-\varepsilon\}$.

By  similar arguments as in 2.1.1, we deduce  from 
 \eqref{j3.2}-\eqref{j3.10} that there exists  
$w \in C^{1}(\overline{\mathbb{R}^N_\varepsilon})$  such that
\begin{equation}\label{j3.13}
\lim_{k\to\infty}w_k(y)=w(y)\geq 0, \quad w(0)=1,
\end{equation}
uniformly on any compact subset of $\overline{\mathbb{R}^N_\varepsilon}$ 
in $C^{1}-$topology.
By the same approach in \emph{2.1.1}, we have
\begin{gather*}
\Delta_m w+b(\xi^0)w^p=0 \quad     \text{in } \mathbb{R}^N_\varepsilon, \\
|\nabla w|^{m-2}\frac{\partial w}{\partial\nu}=0  \quad      \text{on } 
\partial\mathbb{R}^N_\varepsilon,
\end{gather*}
where $\xi^0=\lim_{k\to\infty} \xi^k\in \overline{\Omega}$.
 On account of  the Liouville-type theorem on half space
 $\mathbb{R}^N_\varepsilon$ in Lemma \ref{lem3}, this is a contradiction.

\subsection*{Case 2}
 There exists $c_0>0$ such that
$$
\liminf_{k\to\infty} \frac{\lambda_k^{1/p}}{M_k}=c_0,
$$
which implies that $\lambda_k\to\infty$  as  $k\to\infty$.
Fix any $x_0\in\Omega$ and   take
$N_k=\lambda_k^{1/p}$, $\zeta^k=x_0$  in \eqref{j3.2}, then we have 
\begin{gather*}
\lim_{k\to\infty}\frac{\lambda_k}{N_k^{p}}=1, \\
0<w_k(y)\leq \frac{M_k}{N_k}=\frac{1}{c_0},  y\in\Omega_k.
\end{gather*}
Now since $\operatorname{dist}(O,\partial\Omega_k)
=N^\alpha_k\operatorname{dist}(x_0,\partial\Omega)\to\infty$ 
as $k\to\infty$, then $\Omega_k$ converges to the entire space $\mathbb{R}^N$.
 By similar  procedure in $\emph{2.1}$, we  obtain that there exists 
$w\in C^{1}(\mathbb{R}^N) $ such that
$$
\lim_{k\to\infty} w_k(y)=w(y)\geq0
$$
uniformly on any compact subset of $\mathbb{R}^N$ in $C^{1}$-topology 
and $w$ satisfies
\[
\Delta_m w+b(x_0)w^p+1=0 \quad \text{in } \mathbb{R}^N.
\]
This contradicts the Liouville-type theorem on entire space
 $\mathbb{R}^N$ \cite[Lemma 2.8 Part 1]{HHZ}.

In conclusion, the hypothesis  \eqref{j3.1} is invalid. 
We completed the proof.
\end{proof}

\section{Existence}
In this section, we prove the existence of a positive solution for \eqref{eq1}.
 We use a version of a fixed point theorem of Krasnoselskii \cite{DRuiz}.
 In this procedure,  the a priori estimates  Theorem \ref{thm1.1} are crucial.

\begin{lemma} \label{lem4}
Let $\mathcal{C}$ be a cone in  a Banach $X$ space and
 $\Lambda :\mathcal{C}\to \mathcal{C}$ a compact operator such that 
$\Lambda(0)=0$. Assume that there exists $r>0$, satisfying:
\begin{itemize}
\item[(1)]  $u\neq t\Lambda(u)$ for all $\|u\|=r, t\in[0,1]$.
\end{itemize}
Assume also that there exists a compact homotopy 
$H:[0,1]\times \mathcal{C}\to \mathcal{C}$, and $R>r$ such that
\begin{itemize}
\item[(2)]  $\Lambda(u)=H(0,u) $ for  all $ u \in \mathcal{C}$;

\item[(3)]  $H(t,u)\neq u  $for  any $ \|u\|=R , t\in[0,1]$;

\item[(4)]  $H(1,u)\neq u  $for  any  $\|u\|\leq R $.
\end{itemize}
Let $D=\{u \in \mathcal{C}:r<\|u\|<R\}$, then $\Lambda$ has a fixed point in $D$.
\end{lemma}


\begin{proof}[Proof of Theorem \ref{thm1.2}] 
We use Lemma \ref{lem4}.
For each $f\in C(\overline{\Omega}),h\in C^\gamma(\partial\Omega)$, 
we denote by $K(f,h) \in C^{1,\beta}(\overline{\Omega})$ the unique weak 
solution of the problem
\begin{gather*} %\label{eq11}
  -\Delta_m u+L|u|^{m-2}u= f \quad \text{in }  \Omega,\\
   |\nabla u|^{m-2}\frac{\partial u }{\partial \nu}=h \quad \text{on }  
\partial \Omega.
   \end{gather*}
 The operator
  $K:C(\overline{\Omega})\times C^{\gamma}(\partial\Omega) 
\to C^{1,\beta}(\overline{\Omega})$
   is bounded, continuous and positive, that is 
$K(f,h)\geq0$ provided $f,h\geq0$  \cite[Proposition  2.7(3),(4)]{FD}.
Define $T:C^1(\overline{\Omega})\to C(\overline{\Omega})
\times C^\gamma(\partial\Omega)$,   
$T(u)=(B(z,u,\nabla u)+L|u|^{m-2}u, g(x,u))$. 
$T$ is bounded   and  continuous.
Define  $\Lambda=K\circ T:C^1(\overline{\Omega})
\to C^{1,\beta}(\overline{\Omega})\hookrightarrow C^1(\overline{\Omega})$.

It is clear that the fixed-point of operator $\Lambda$ is a solution of  
\eqref{eq1}.
 The operator $\Lambda$ is continuous and compact since $K\circ T$ is 
continuous and bounded and the embedding 
$C^{1,\beta}(\overline{\Omega})\hookrightarrow C^1(\overline{\Omega})$ is compact.

Let $X:=C^1(\overline{\Omega})$, $\mathcal{C}$$=\{u \in X|u\geq 0\}$ is 
a cone in $X$. In the sequel, $\|\cdot\|$ denotes the supremum $C^1$-norm 
on $\overline{\Omega}$.
$\Lambda(0)=0$ since $K(0,0)=0$.  By the weak comparison principle  
for the m-Laplace operator with Neumann boundary condition 
(by the  Maximum principle in \cite{MAX}) and conditions (A4) and (A3),
 we have $\Lambda:\mathcal{C}\to \mathcal{C}$.

First we verify condition (1) of Lemma \ref{lem4}.
Consider $u=\lambda \Lambda(u)$ in $\mathcal{C}\backslash\{0\}$ 
for certain $\lambda \in [0,1]$, that is, $u$ satisfies the following equation
\begin{equation}
\begin{gathered}
-\Delta_m u+L|u|^{m-2}u=\lambda^{m-1}[B(z,u,\nabla u)+L|u|^{m-2}u]  \quad \text{in}    \Omega,\\
|\nabla u|^{m-2}\frac{\partial u}{\partial \nu}=\lambda^{m-1}g(x,u)  \quad \text{on}    \partial\Omega,
\end{gathered}
\end{equation}

By taking $u$ as a test function and using the condition (A5), we have
\begin{align*}
&\int_\Omega |\nabla u|^{m} { d}z+\int_\Omega L|u|^{m}{ d}z \\
&=\lambda ^{m-1}\int_\Omega (B(z,u,\nabla u)u+L|u|^{m}) { d}z
+\lambda ^{m-1}\int_{\partial\Omega} g(z,u)u{ d}s  \\
& =  \int_\Omega o(|u|^{m}+|\nabla u|^{m}) { d}z
 +\int_{\partial\Omega} o(|u|^{m}){ d}s
\end{align*}
as $\|u\|\to 0$. Hence we can choose $r>0$ small enough such that  
equation $u=\lambda \Lambda(u)$ has no positive solutions in 
$B_{r}(0)\backslash\{0\}$ for all $\lambda \in [0,1]$.

Now we verify (2)--(4) of Lemma \ref{lem4}. 
By Theorem \ref{thm1.1}, there exists a positive constant $\lambda_0$, 
such that problem \eqref{eq2} has no solution.
Define $H :[0,1]\times \mathcal{C}\to \mathcal{C}$ as
 $H(t,u)=K\circ(T(u)+t(\lambda_0,0))$. Clearly, $u=H(t,u)$ is
 equivalent to
\begin{equation}\label{eq4}
\begin{gathered}
\Delta_m u+B(z,u,\nabla  u)+t\lambda_0=0  \quad\text{in }  \Omega,\\
|\nabla u|^{m-2}\frac{\partial u}{\partial\nu}=g(z,u) \quad\text{on } \partial\Omega.
\end{gathered}
\end{equation}
Obviously $H(0,u)=\Lambda (u)$ for any $u\in \mathcal{C}$,
namely, (2) holds.
 By Theorem \ref{thm1.1}  solutions of  \eqref{eq4} are a priori bounded in the 
uniform norm. There exists a constant $R>r$, such that each solution 
of \eqref{eq4} satisfies $\|u\|_{C^1(\overline{\Omega})}<R$, and then 
(3) holds. When $t=1$, \eqref{eq4} has no solution in view of the choice 
of the number $\lambda_0$, this implies (4) holds. Therefore the mapping
 $\Lambda$ has a fixed point $u\in \mathcal{C}$ and $r< \|u\|<R$, 
which is a non-negative solution of  \eqref{eq1}. 
The proof is complete.
\end{proof}

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