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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 274, pp. 1--9.\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-2016/274\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions for nonlocal $p$-Laplacian problems}

\author[B. Emamizadeh, A. Farjudian \hfil EJDE-2016/274\hfilneg]
{Behrouz Emamizadeh, Amin Farjudian}

\address{Behrouz Emamizadeh (corresponding author)\newline
School of Mathematical Sciences,
The University of Nottingham Ningbo, \newline
Ningbo, 315100, China}
\email{Behrouz.Emamizadeh@nottingham.edu.cn}

\address{Amin Farjudian  \newline
Center for Research on Embedded Systems, 
Halmstad University, Halmstad, Sweden}
\email{Amin.Farjudian@gmail.com}

\thanks{Submitted June 12, 2016. Published October 12, 2016.}
\subjclass[2010]{35D05, 35J20, 35J92}
\keywords{Nonlocal problems; existence; uniqueness; 
\hfill\break\indent Schaefer's fixed point theorem}

\begin{abstract}
 We study the existence and uniqueness of positive solutions to
 a class of nonlocal boundary-value problems involving the $p$-Laplacian.
 Our main tools are a variant of the Schaefer's fixed point theorem,
 an inequality which suitably handles the $p$-Laplacian operator, and
 a Sobolev embedding which is applicable to the bounded domain.
\end{abstract}

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


\section{Introduction}

We study the boundary-value problem
\begin{equation} \label{mainbvp}
 \begin{gathered}
 -M(\|u\|^p)\Delta_pu=f(x,u) \quad \text{in } D,\\
 u=0 \quad\text{on } \partial D.
 \end{gathered}
\end{equation}
in which $\Delta_p$ denotes the $p$-Laplacian
 \begin{equation*}
 \Delta_p u=\nabla\cdot (|\nabla u|^{p-2}\nabla u)
 \end{equation*}
and $\|\cdot\|$ denotes the norm in $W^{1,p}_0(D)$,
 $\|u\|=\big(\int_D|\nabla u|^p \, dx\big)^{1/p}$. As for the
 functions $M: [0,\infty)\to [0,\infty)$ and
 $f:D\times\mathbb{R}\to\mathbb{R}$, we shall refer to the following assumptions:
 \begin{itemize}
 \item[(A1)] $M$ is continuous, and $M(t)\geq m_0>0$,
 where $m_0$ is a constant. Moreover, the function:
 \begin{equation*}
	\xi (t):=M(t)^{\frac{1}{p-1}}\,t^{1/p}
 \end{equation*}
 is invertible, and
 henceforth we let $q=\frac{p}{p-1}$.

 \item[(A2)] Let  $\hat M(t):=M(t^p)$. Then for a constant $\kappa$
 (to be defined by \eqref{estimate}), the function $\hat M$ is uniformly H\"{o}lder
 continuous with exponent $p-1$ in the interval $[0,\kappa)$. In
 other words
 \begin{equation*}
 L:=\sup_{t_1, t_2\in [0,\kappa),\,  t_1\neq t_2} 
\frac{|\hat M(t_1)-\hat M(t_2)|}{|t_1-t_2|^{p-1}} < \infty
 \end{equation*}
\end{itemize}
The principal eigenvalue of $-\Delta_p$ with Dirichlet boundary
conditions on $\partial D$ is defined as
\begin{equation}
\label{eq:lambda_defin}
\Lambda:=\inf_{u\in W^{1,p}_0(D),\, u\neq 0}
\frac{\int_D|\nabla u|^p \, dx}{\int_D|u|^p \, dx}.
\end{equation}
The eigenvalue $\Lambda$ is positive, isolated, and
simple \cite{Henrot:2006}. We impose the following minimum condition on $\Lambda$:
\begin{itemize}
\item[(A3)] $\hat m:=m_0-a/\Lambda >0$, in which $m_0$
 comes from (A1), the constant $a$ is introduced in (A5), and
 $\Lambda$ is the principal eigenvalue from~\eqref{eq:lambda_defin}.

\item[(A4)] The function $f$ is a Carath\'{e}odory
 function, and for some $ r \in ( 1, p^* - 1)$
\begin{equation} \label{f1}
 |f(x,s)|\leq A(x)|s|^r+B(x), \quad \forall x\in D,\;\forall s\in\mathbb{R},
 \end{equation}
 in which:
 \begin{itemize}
 \item $A\in L^\infty (D)$ is a non-negative function
 \item $B\in L^{1+1/r}(D)$
 \item $p^*=
 \begin{cases}
 \frac{np}{n-p} &\text{if } 1< p<n \\
 \infty &\text{if } p\geq n.
 \end{cases}$
\end{itemize}
 \item[(A5)] For some positive constants $a$ and $b$, 
$$
sf(x,s)\leq a|s|^p+b|s|,\quad \text{a.e. } x\in D, \; \forall s\in\mathbb{R}.
$$

 \item[(A6)] $f(x,s)\geq 0$ a.e. $x\in D$, for all
 $s\geq 0$ and $f(x,0)>0$, a.e. $x\in D$.

 \item[(A7)] For a positive constant $A$,
 \begin{equation*}
 (f(x,u)-f(x,v))(u-v)\leq A|u-v|^2, \quad \forall x\in D,\; u,v\in \mathbb{R}
 \end{equation*}
\end{itemize}

\begin{remark} \label{rmk1.1} \rm
 Note that when $p\geq 2$, the condition (A2) is satisfied when $M$ is a 
constant function, hence the boundary value problem \eqref{mainbvp} is 
no longer a nonlocal. Whence, even though the arguments to follow will hold 
for $p\geq 2$ but it is the case $1<p<2$ which is of interest.
 \end{remark}

 \begin{remark} \label{rmk1.2} \rm
 Let us mention that a function $M$ that satisfies the conditions (A1) and (A2) 
(these are the main conditions on $M$), for the case $p\in (1,2)$ is 
$M(t)=m_0+t^\beta$, where $\beta\geq\frac{1}{q}$. On the other hand, any 
function $f(x,s)$ which is bounded and $\frac{\partial}{\partial s} f(x,s)$ 
is uniformly bounded in $x$ satisfies (A4)--(A7).
 \end{remark}

The main results of this article are the following theorems.

\begin{theorem} \label{thm1}
 Under  assumptions {\rm (A1)--(A7)}, the
 boundary value problem \eqref{mainbvp} has a positive solution.
\end{theorem}

\begin{theorem} \label{thm2}
Suppose the conditions {\rm (A1)--(A7)} are satisfied. 
Then \eqref{mainbvp} has a unique positive solution,
provided that $L$ is sufficiently small and $m_0$ is sufficiently
large.
\end{theorem}

The special case of problem \eqref{mainbvp} when $p=2$ has been
considered in \cite{AlvesCorrea:Existence_Nonlinear:2001}, and 
\cite{Ma:Remarks_Kirchhoff:2005}. 
In the former, the authors impose conditions on the functions $M$ and $f$
so that it is possible to settle the issue of existence of solutions
via the Mountain Pass Theorem. However, in the latter the authors use
a different set of conditions, and apply the Galerkin method to obtain
their results (see also \cite{AlvesCorreaMa:Kirchhoff:2005}).

Our paper is motivated by \cite{Ma:Remarks_Kirchhoff:2005}. For the
result of Theorem \ref{thm1} regarding the existence of positive
solutions, we apply a variant of the Schaefer's fixed point theorem
coupled with a well known maximum principle. For the uniqueness result
of Theorem \ref{thm2}, we use the ideas of
\cite{Ma:Remarks_Kirchhoff:2005}. In proving both existence and
uniqueness of solutions we shall use an inequality which is particularly 
useful in dealing with the $p$-Laplace operator.
See inequality \eqref{inequality_p_power} in Lemma \ref{lemma:inequality_p_power}.

Nonlocal problems have been used in modeling various physical
phenomena, and the problem~\eqref{mainbvp} which we have considered in
this note is related to the steady state version of the Kirchhoff
equation \cite{Kirchhoff:Mechanik:1883}
\begin{equation}
\label{Kirchhoff}
u_{tt}-M\Big(\int_D|\nabla u|^2 \, dx\Big)\Delta u= f(x,t),
\end{equation}
where the coefficient of the diffusion term depends on the unknown
function $u(x,t)$ \emph{globally}. It was the paper
\cite{Lions:BVP_mathematical_physics:1978} by Lions that introduced an
abstract setting for \eqref{Kirchhoff}. Other relevant work are
\cite{Arosio_Panizzi:well_posedness_Kirchhoff:1996,
CousinFrotaLarkinMedeiros:Kirchhoff_Carrier:1997,
DAncona_Spagnolo:degenerate_Kirchhoff:1992}. Some
nonlocal problems in statistical mechanics are studied
in \cite{Biler_Hebisch_Nadzieja:Debye_system:1994,
Biler_Nadzieja:Nonlocal_statistical_mechanics:1997}.


\section{Preliminaries}

This section contains the basic material that we need 
for proving Theorems \ref{thm1} and \ref{thm2}. We begin with the
following definition.

\begin{definition} \label{def2.1} \rm
We say that $u\in W^{1,p}_0(D)$ is a solution of \eqref{mainbvp} 
if  the following integral equation is satisfied:
\begin{equation}\label{solution}
M(\|u\|^p) \int_D |\nabla u|^{p-2} \nabla u\cdot \nabla v \, dx
-\int_Dvf(x,u) \, dx=0,\quad \forall v\in W^{1,p}_0(D).
\end{equation}
\end{definition}
	
The convergence of the second integral in \eqref{solution} follows 
from the following general result regarding Nemytskii mappings.

\begin{lemma}\label{lem1}
Let $g:D\times\mathbb{R}\to\mathbb{R}$ be a Carath\'{e}odory function and suppose 
that there is a constant $c>0$, a function $l(x)\in L^\gamma(D)$ 
(where $1\leq\gamma\leq\infty$) and $\tau>0$ such that
\begin{equation*}
|g(x,s)|\leq c|s|^\tau +l(x), \quad \forall x\in D,\;\forall s\in\mathbb{R}.
\end{equation*}
Then $N_g:L^{\gamma\tau}(D)\to L^\gamma (D)$ defined by 
$N_g(u)(x)= g(x,u(x))$ is continuous and bounded, i.e.\ 
it maps bounded sets into bounded sets.
\end{lemma}

For a proof of the above lemma, see
\cite[Theorem 2.3]{de_Figueiredo:Lectures_Ekeland:1989}.
Let us review some basic facts regarding the  problem
\begin{equation} \label{bvp}
\begin{gathered}
 -\Delta_p u=h(x) \quad\text{in } D \\
 u=0  \quad \text{on }\partial D,
 \end{gathered}
\end{equation}
where $h(x)\in L^{1+1/r}(D)$. It is well known, see for 
example \cite{Stuwe:Partial Differential Equations:1990}, that \eqref{bvp} has a
unique solution $u\in W^{1,p}_0(D)$ which is the unique minimizer of
the strictly convex functional
\begin{equation*}
\Phi (w)=\frac{1}{p}\int_D|\nabla w|^p \, dx-\int_Dhw \, dx
\end{equation*}
relative to $w\in W^{1,p}_0(D)$. Therefore, the inverse mapping
\begin{equation*}
(-\Delta_p)^{-1}:L^{1+1/r}(D)\to W^{1,p}_0(D)
\end{equation*}
which takes every $h \in L^{1+1/r}(D)$ to the unique solution of
\eqref{bvp} is well-defined. It is straightforward to verify that
\begin{equation*}
 (-\Delta_p)^{-1}(\eta h)=\eta^{1/(p-1)}(-\Delta_p)^{-1}(h),
 \quad \forall h\in L^{1+1/r}(D), \eta > 0
\end{equation*}
and that the following inequality holds:
\begin{equation}
 \label{inequality1}
 \|(-\Delta_p)^{-1}(h)\|\leq C\|h\|^{1/(p-1)}_{1+1/r}, \quad \forall h\in L^{1+1/r}(D)
\end{equation}
where $C$ is a positive constant. Henceforth we shall use $C$ as a
generic symbol for the several constants which appear in various
places in this document, whose values could be different.

\begin{lemma} \label{lem2}
Assume that $u\in W^{1,p}_0(D)$ is a solution of \eqref{mainbvp}, and
that {\rm (A1)--(A6)} hold. Then
\begin{equation}
\label{estimate}
\|u\|\leq\Big(\frac{b  |D|^{1/q}}{\hat m  \Lambda^{1/p}}\Big)^{q/p}=:\kappa,
\end{equation}
where $q=\frac{p}{p-1}$. Here $|D|$ denotes the $n$-dimensional
Lebesgue measure of $D$.
 \end{lemma}

 \begin{proof}
 Setting $v=u$ in \eqref{solution}, assumption (A1) implies
\begin{equation} \label{eq1}
\begin{aligned}
 m_0\|u\|^p
&\leq M(\|u\|^p)\|u\|^p = \int_D f(x,u)u \, dx \\
&\leq  a\int_D|u|^p \, dx+b\int_D |u| \, dx \quad \text{(by assumption (A5))}  \\
&\leq a\frac{\|u\|^p}{\Lambda}+b|D|^{1/q}\|u\|_p \quad
 \text{(by \eqref{eq:lambda_defin} and H\"{o}lder)}\\
&\leq a\frac{\|u\|^p}{\Lambda}+\frac{b|D|^{1/q}\|u\|}{\Lambda^{1/p}} \quad
\text{(again by \eqref{eq:lambda_defin})}
\end{aligned}
\end{equation}
From these inequalities, we infer that
 \begin{equation*}
 \hat m\|u\|^p\leq \frac{b|D|^{1/q}\|u\|}{\Lambda^{1/p}},
 \end{equation*}
 which in turn implies the desired estimate \eqref{estimate}.
 \end{proof}

We also need the following variant of the Schaefer's fixed point
theorem, see for example \cite{Smart:Fixed Point Theorems:1980}, 
but we include the proof for the convenience of the reader.

\begin{lemma}  \label{lem3}
 Let $X$ be a Banach space and assume that:
 \begin{itemize}
 \item[(a)] $\mathcal{P}\subseteq X$ is a non-empty, closed, and convex subset
 of $X$.

 \item[(b)] $T:\mathcal{P}\to \mathcal{P}$ is a strongly continuous
 mapping, i.e.\ ~$T$ is continuous and for every bounded sequence
 $(u_n)\subseteq \mathcal{P}$, the image $(Tu_n)$ has a strongly
 convergent subsequence.

 \item[(c)] The set $S=\{x\in\mathcal{P} \mid x=\lambda Tx,\;\text{for some}\;\lambda\in [0,1]\}$ is bounded.
\end{itemize}
Then $T$ has a fixed point, i.e.\ there exists $x\in \mathcal{P}$ such that $Tx = x$.
 \end{lemma}

 \begin{proof}
 Consider the orthogonal projection $P:X\to\mathcal{P}$ of $X$
 on $\mathcal{P}$. This projection satisfies:
\begin{equation*}
\forall x\in X :\quad \|Px-x\|=\inf_{m\in\mathcal{P}}\|x-m\|,
 \end{equation*}
 The mapping $T\circ P:X\to \mathcal{P}\subseteq X$ is clearly
 strongly continuous. Define 
$S'=\{x\in X : x=\lambda (T\circ P)x,\text{ for some }
\lambda\in [0,1]\}$. Hence, $S'\subseteq  S$ and $S'$ is bounded.

 Now we can invoke the classical Schaefer's fixed point theorem,
 applied to $T\circ P$, and deduce that $T\circ P$ has a fixed point,
 say $x_0\in X$. Thus:
 \begin{align*}
 x_0 = T( P x_0) & \Rightarrow  x_0 \in \operatorname{range}(T) \\
 & \Rightarrow  x_0 \in \mathcal{P} \\
 & \Rightarrow  x_0 = P x_0 \\
 & \Rightarrow  x_0 = T x_0
 \end{align*}
Thus, $x_0$ is a desired fixed point of $T$.
\end{proof}

We shall also need the following result, see for example 
\cite{Damascelli:Comparison_Theorems:1998} and 
\cite{CorreaSantosJunior:Multiple:2014}

 \begin{lemma} \label{lemma:inequality_p_power}
 For any vectors $X, Y\in\mathbb{R}^n$, the following inequalities hold:
 \begin{equation}  \label{inequality_p_power}
 C_p \langle |X|^{p-2}X-|Y|^{p-2}Y,X-Y\rangle
\geq \begin{cases}
 {|X-Y|}^p &\text{if } p\geq 2 \\[4pt]
 \frac{|X-Y|^2}{(|X|+|Y|)^{2-p}} & \text{if }1\leq p\leq 2,
 \end{cases}
 \end{equation}
in which $\langle\cdot,\cdot\rangle$ denotes the usual dot product in
 $\mathbb{R}^n$, and $C_p$ is a constant depending on $p$.
 \end{lemma}

 The following lemma is elementary, so we omit its proof.

 \begin{lemma} \label{lem5}
 Let $t_n\stackrel{\mathbb{R}}{\to}t$ and $f_n\stackrel{L^p}{\to} f$. Then
 $t_nf_n\stackrel{L^p}{\to} tf$.
 \end{lemma}

 \begin{lemma}  \label{lem6}
 If $f_n\stackrel{L^{1+1/r}}{\to} f$, then 
$(-\Delta_p)^{-1}(f_n)\stackrel{W^{1,p}_0}{\to} (-\Delta_p)^{-1}(f)$.
 \end{lemma}

 \begin{proof}
 Set $v_n=(-\Delta_p)^{-1}(f_n)$ and $v=(-\Delta_p)^{-1}(f)$. Thus:
\begin{gather*}
 -\Delta_p v_n=f_n \quad \text{in } D \\
 v_n=0 \quad \text{on } \partial D
\end{gather*}
and
\begin{gather*} 
-\Delta_p v=f \quad \text{in } D \\
 v=0 \quad \text{on } \partial D
\end{gather*}
The derivation of the following equation is then straightforward.
 \begin{equation}\label{eq10}
\begin{aligned}
&\int_D(|\nabla v_n|^{p-2}\nabla v_n-|\nabla v|^{p-2}\nabla v)\cdot
 (\nabla v_n-\nabla v) \, dx\\
& = \int_D(f_n-f)(v_n-v) \, dx.
\end{aligned}
 \end{equation}
 Applying the H\"{o}lder's inequality and the embedding
 $W^{1,p}_0(D)\to L^{1+1/r}(D)$, a bound on the integral on the right
 hand side of \eqref{eq10} is obtained as follows:
 \begin{equation}  \label{eq11}
\begin{aligned}
 \int_D(f_n-f)(v_n-v) \, dx 
& \leq  \|f_n-f\|_{1+1/r}\|v_n-v\|_{r+1}  \\
& \leq  C\|f_n-f\|_{1+1/r}\|v_n-v\|.
\end{aligned}
 \end{equation}
Now we consider two cases:
\smallskip

\noindent\textbf{Case $p\geq 2$.}
 From Lemma \ref{lemma:inequality_p_power} (setting
 $X=\nabla v_n$, $Y=\nabla v$), \eqref{eq10} and \eqref{eq11}, we
 obtain
 \begin{equation*}
 \|\nabla v_n-\nabla v\|_p^p\leq C\|f_n-f\|_{1+1/r}\|v_n-v\|,
 \end{equation*}
 hence $\|v_n-v\|\leq C\|f_n-f\|_{1+1/r}^{1/(p-1)}$. Thus,
 $v_n\to v$ in $W^{1,p}_0(D)$.
\smallskip

\noindent\textbf{Case $1\leq p\leq 2$.}
 This case requires more work. We  begin with the observation
\begin{equation}\label{eq12}
\begin{aligned}
&\|\nabla v_n-\nabla v\|_p^p\\
& = \int_D\frac{|\nabla v_n-\nabla
 v|^p}{(|\nabla v_n|+|\nabla v|)^{\frac{p(2-p)}{2}}}\; (|\nabla
v|+|\nabla v|)^{\frac{p(2-p)}{2}} \, dx\\
&\leq \Big(\int_D\frac{|\nabla v_n-\nabla v|^2}{(|\nabla v_n|+|\nabla
 v|)^{2-p}} \, dx\Big)^{p/2}\Big(\int_D(|\nabla v_n|+|\nabla
 v|)^p \, dx\Big)^{(2-p)/2}
\end{aligned}
\end{equation}
This follows from  H\"{o}lder's inequality which is applicable
since $\frac{2}{p}\geq 1$. Applying inequality \eqref{inequality1}, we
obtain $\|v_n\|\leq C\|f_n\|_{1+1/r}^{1/(p-1)}$ and 
$\|v\|\leq C\|f\|_{1+1/r}^{1/(p-1)}$. Since $(f_n)$ is bounded in 
$L^{1+1/r}(D)$,
we infer that $\max  (\|v_n\|, \|v\|)\leq C$, for all 
$n\in \mathbb{N}$. Thus, from \eqref{eq12} we obtain
\begin{equation} \label{eq13}
\|\nabla v_n-\nabla v\|^p_p
\leq C\Big(\int_D\frac{|\nabla v_n-\nabla v|^2}{(|\nabla v_n|+|\nabla v|)^{2-p}}
\Big)^{p/2}
\end{equation}
Now, by setting $X=\nabla v_n$ and $Y=\nabla v$ in Lemma
\ref{lemma:inequality_p_power}, together with \eqref{eq10}, \eqref{eq11}, and
\eqref{eq13} we find that
\begin{equation*}
\|v_n-v\|^p=\|\nabla v_n-\nabla\|_p^p\leq C\|f_n-f\|_{1+1/r}\|v_n-v\|,
\end{equation*}
This implies that $\|v_n-v\|\leq C\|f_n-f\|_{1+1/r}^{1/(p-1)}$. So,
$v_n\to v$ in $W^{1,p}_0(D)$, as desired.
The proof is complete.
\end{proof}


\section{Proofs of main theorems}

To prove Theorem \ref{thm1} we shall apply Lemma \ref{lem3}. 
To this end, we set $\mathcal{P}=L^{r+1}_+(D)$. Note that by
Lemma~\ref{lem1} we have $\forall u\in \mathcal{P}: N_f(u)\in
L^{1+1/r}(D)$. From assumption $(A6)$ we infer that $N_f(u)$ is
non-negative. For every $u \in \mathcal{P}$, we define:
\begin{equation*}
Tu=t^{1/p}\frac{v}{\|v\|},
\end{equation*}
in which $v:=(-\Delta_p)^{-1}(N_f(u))$ and
$t:=\xi^{-1}(\|v\|)$. Observe that $w=Tu$ satisfies
\begin{equation}  \label{eq20}
 \begin{gathered}
 -M(\|w\|^p)\Delta_p w=f(x,u) \quad \text{in } D\\
 w=0 \quad \text{on }\partial D.
 \end{gathered}
\end{equation}
Since $Tu\in W^{1,p}_0(D)$, the embedding 
$W^{1,p}_0(D)\to L^{r+1}(D)$ implies $Tu\in L^{r+1}(D)$. 
Thus, by applying a classical maximum principle (see for example
\cite{Vazquez:Strong_Maximum_Principle_quasilinear:1984}) to
\eqref{eq20}, we deduce that $w$ is positive, i.e.\ $Tu\in
L^{r+1}_+(D)$.

The above discussion ensures that the mapping $T: \mathcal{P}\to
\mathcal{P}$ is well defined. Note that if $u$ is a fixed point of $T$,
then $u$ will be a solution of \eqref{mainbvp}. The existence of such
a fixed point will confirm the assertion of Theorem \ref{thm1}.


\subsection{Proof of Theorem \ref{thm1}}
We just need to verify that
the mapping $T$ satisfies the hypotheses of Lemma \ref{lem3}.

\subsection*{Continuity}
 Let $(u_n)\subseteq \mathcal{P}$ be a sequence such
 that $u_n\to u$ in $L^{r+1}(D)$. Note that since $\mathcal{P}$
 is closed then $u$ must be non-negative. We need to show that
 $Tu_n\to Tu$ in $L^{r+1}(D)$. In view of the embedding
 $W^{1,p}_0(D)\to L^{r+1}(D)$, it suffices to show
 $Tu_n\to Tu$ in $W^{1,p}_0(D)$. To this end, we first recall
 Lemma \ref{lem1} which ensures that $N_f(u_n)\to N_f(u)$ in
 $L^{1+1/r}(D)$. Whence, by Lemma \ref{lem6}:
 \begin{equation*}
 (-\Delta_p)^{-1}(N_f(u_n))\to (-\Delta_p)^{-1}(N_f(u)) \quad
 \text{in } W^{1,p}_0(D).
 \end{equation*}
By the continuity of the norm we also have
\begin{equation*}
\|(-\Delta_p)^{-1}(N_f(u_n))\|\to
\|(-\Delta_p)^{-1}(N_f(u))\|.
\end{equation*}
On the other hand,
\begin{equation*}
Tu_n=t_n^{1/p}\frac{(-\Delta_p)^{-1}(N_f(u_n))}{\|(-\Delta_p)^{-1}(N_f(u_n))\|},
\end{equation*}
in which $t_n=\xi^{-1}(\|(-\Delta_p)^{-1}(N_f(u_n))\|)$.
 Since $\xi$ is continuous, we obtain
\begin{equation*}
t_n\to t:=\xi^{-1}(\|(-\Delta_p)^{-1}(N_f(u))\|).
\end{equation*}
Now we apply Lemma \ref{lem5} to conclude that $Tu_n\to Tu$ in 
$W^{1,p}_0(D)$, as desired.

\subsection*{Compactness}
 Consider a bounded sequence $(u_n)\subseteq \mathcal{P}$. Setting $w_n=Tu_n$,
 we will have
\begin{equation}  \label{eq30}
 \begin{gathered}
 -M(\|w_n\|^p)\Delta_pw_n=f(x,u_n) \quad\text{in } D\\
 w_n=0 \quad \text{on }\partial D.
 \end{gathered}
\end{equation}
From \eqref{eq30} we obtain
\begin{equation*}
M(\|w_n\|^p)  \|w_n\|^p=\int_D f(x,u_n)w_n \, dx.
\end{equation*}
An application of H\"{o}lder's inequality then gives
\begin{equation} \label{eq31}
M(\|w_n\|^p) \, \|w_n\|^p\leq \|N_f(u_n)\|_{1+1/r}\|w_n\|_{r+1}.
\end{equation}
%
\noindent
The inequality \eqref{eq31}, the embedding $W^{1,p}_0(D)\to
L^{r+1}(D)$, and the assumption (A1) together lead to
\begin{equation*}
m_0\|w_n\|^p\leq C\|N_f(u_n)\|_{1+1/r}\|w_n\|,
\end{equation*}
Hence, we get $\|w_n\|\leq C \| N_f(u_n)\|^{\frac{1}{p-1}}_{1+1/r}$. 
This, coupled with the boundedness of the operator $N_f$ (see Lemma
\ref{lem1}), implies that $(w_n)$ is bounded in $W^{1,p}_0(D)$. So,
there exists a subsequence $(w_{n_j})\subseteq (w_n)$ such that
$w_{n_j}\rightharpoonup w$ in $W^{1,p}_0(D)$, for some 
$w\in W^{1,p}_0(D)$. Since the embedding $W^{1,p}_0(D)\to L^{r+1}(D)$ is compact,
 we deduce that $w_{n_j}\to w$ in $L^{r+1}(D)$. 
This means that $(Tu_{n_j})$ is strongly convergent in
$L^{r+1}(D)$ and as a result $T:\mathcal{P}\to \mathcal{P}$ is compact.

\subsection*{Boundedness of $S$}
The final step is to prove the  boundedness of the set
\begin{equation*}
S=\{u\in\mathcal{P} : u=\lambda Tu,\text{ for some }\lambda\in [0,1]\}.
\end{equation*}
To that end, let us fix a $u\in S$ and assume that $u=\lambda Tu$ for
some $\lambda\in [0,1]$. Thus, we must have
\begin{equation*}
 u=\lambda t^{1/p}\frac{(-\Delta_p)^{-1}(N_f(u))}{\|(-\Delta_p)^{-1}(N_f(u))\|},
\end{equation*}
where $t=\xi^{-1}(\|(-\Delta_p)^{-1}(N_f(u))\|)$. Since 
$\|u\|=\lambda t^{1/p}$ and assuming that $\lambda\neq 0$, then 
$t=\|u\|^p/\lambda^p$ and $M(\frac{\|u\|^p}{\lambda^p})=M(t)$. So, we obtain
\begin{equation}  \label{eq40}
\begin{aligned}
-M\Big(\frac{\|u\|^p}{\lambda^p}\Big)\Delta_p u 
& = \frac{M(t)\lambda^{p-1}t^{1/q}}{\|(-\Delta_p)^{-1}(N_f(u))\|^{p-1}}\;
f(x,u) \\
& = \lambda^{p-1} f(x,u),
\end{aligned}
\end{equation}
where $q=\frac{p}{p-1}$. Since $u\in W^{1,p}_0(D)$, from \eqref{eq40},
(A1), and (A5) one gets
\begin{equation} \label{eq41}
\begin{aligned}
m_0\|u\|^p
&\leq  M\Big(\frac{\|u\|^p}{\lambda^p}\Big)\|u\|^p
 =\lambda^{p-1}\int_D f(x,u) u \, dx  \\
&\leq \lambda^{p-1}\Big(a\int_D|u|^p \, dx+b\int_D|u| \, dx\Big)  \\
&\leq  a\frac{\|u\|^p}{\Lambda}+\frac{b|D|^{1/q}\|u\|}{\Lambda^{1/p}}.
\end{aligned}
\end{equation}
From \eqref{eq41} and (A3) we obtain $\|u\|\leq\kappa$ (which was
defined in \eqref{estimate}). Note that in case $\lambda=0$, this last
inequality trivially holds.

Finally,  by invoking the embedding $W^{1,p}_0(D)\to
L^{r+1}(D)$, we infer that $\|u\|_{r+1}\leq C$. Whence, $S$ is
bounded, as desired. This completes the proof.

\subsection{Proof of Theorem \ref{thm2}}
The existence of solutions is guaranteed by Theorem \ref{thm1}. 
We prove uniqueness by contradiction. Let us assume that $u_1$ and $u_2$ 
are two solutions of \eqref{mainbvp}, satisfying
\begin{equation} \label{eq50}
 \begin{gathered} 
-M(\|u_i\|^p)\Delta_p u_i=f(x,u_i) \quad\text{in } D \\
 u_i=0 \quad \text{on }\partial D,
\end{gathered}
\end{equation}
for $i=1,2$.
From \eqref{eq50} we obtain
\begin{equation} \label{eq51}
\begin{aligned}
&\int_D(M(\|u_1\|^p)\|\nabla u_1\|^{p-2}\nabla u_1-M(\|u_2\|^p)\|\nabla
u_2\|^{p-2}\nabla u_2)\cdot\nabla w \, dx\\
& = \int_D(f(x,u_1)-f(x,u_2))w \, dx.
\end{aligned}
\end{equation}
By rearranging  terms, we obtain
\begin{equation}\label{eq52}
\begin{aligned}
&M(\|u_2\|^p)\int_D(|\nabla u_1|^{p-2}\nabla u_1 - |\nabla
u_2|^{p-2}\nabla u_2)\cdot\nabla w \, dx \\
&=(M(\|u_2\|^p)-M(\|u_1\|^p))\int_D|\nabla u_1|^{p-2}\nabla
u_1\cdot\nabla w \, dx \\
&\quad + \int_D(f(x,u_1)-f(x,u_2))w \, dx \\
&\leq L \big|\,\|u_2\|-\|u_1\|\,\big|^{p-1}\|u_1\|^{p-1}\|w\|+A\|w\|_p^p,
\end{aligned}
\end{equation}
where we have used (A2) and (A7) in the last inequality. Note that
\begin{equation} \label{eq53}
L|\,\|u_2\|-\|u_1\|\,|^{p-1}\;\|u_1\|^{p-1}\|w\|+A\|w\|_p^p
\leq \Big(L\kappa^{p-1}+\frac{A}{\Lambda}\Big)\|w\|^p.
\end{equation}
On the other hand, using similar arguments as in the proof of 
Lemma \ref{lem6}, we obtain the  estimate
\begin{equation} \label{eq54}
\int_D(|\nabla u_1|^{p-2}\nabla u_1-|\nabla u_2|^{p-2}\nabla u_2)\cdot\nabla w \, dx
\geq C\|w\|^p,
\end{equation}
in which the constant $C$ depends on $\kappa$ if $p<2$, otherwise it
does not. From \eqref{eq52}, \eqref{eq53}, and \eqref{eq54} we obtain
\begin{equation} \label{eq55}
m_0C\|w\|^p\leq\Big(L\kappa^{p-1}+\frac{A}{\Lambda}\Big)\|w\|^p.
\end{equation}
Since $u_1\neq u_2$, \eqref{eq55} implies
\begin{equation}\label{eq56}
m_0C- A \Lambda^{-p} \leq L\kappa^{p-1}.
\end{equation}
Now, if $m_0$ is large enough, and $L$ is small enough as
\begin{equation*}
 m_0 > A C^{-1} \Lambda^{-p} \quad \text{and} \quad 
L<\frac{m_0C-A\Lambda^{-p}}{\kappa^{p-1}}.
\end{equation*}
then we obtain the desired contradiction, and the proof is
 complete. 


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