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\title[Nonstandard Dirichlet problems]{Nonstandard Dirichlet problems with \\ competing $(p,q)$-Laplacian, convection, \\ and convolution}
\author{Dumitru Motreanu}
\address{D{\'e}partement de Math{\'e}matiques \\
Universit{\'e} de Perpignan \\ 66860 Perpignan \\ France}
\email{motreanu@univ-perp.fr}
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\author{Viorica Venera Motreanu}
\address{Coll\`ege Jean Moulin \\ 14 rue Jean Moulin \\ 54510 Tomblaine \\ France}
\email{vmotreanu@gmail.com}
%
\subjclass{35J92 (primary); 47H30}
\keywords{competing $(p,q)$-Laplacian; Dirichlet problem;
convection; convolution; generalized solution; weak solution.}
\begin{abstract}
The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with
lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a
convection (i.e., it depends on the solution and its gradient)
composed with the convolution of the solution with an integrable
function. We prove the existence of a generalized solution through a
combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak  solution to the
respective elliptic problem.
\end{abstract}
\maketitle

\section{Introduction}\label{S1}

In this paper we consider the following quasilinear
problem with homogeneous Dirichlet boundary condition on a bounded
domain $\Omega\subset\mathbb{R}^N$ with the boundary
$\partial\Omega$, 

\begin{equation}\label{1}
\left\{
\begin{array}{ll}
-\Delta_p u+\mu\Delta_q u=f(x,\rho\ast u,\nabla(\rho\ast u)) & \qquad \hbox{in } \Omega,\\
u=0 & \qquad \hbox{on } \partial\Omega\\
\end{array}
\right.
\end{equation}
for $1<q<p<+\infty$, $\mu\in\mathbb{R}$, and $\rho\in L^1(\mathbb{R}^N)$. To ease the
exposition we assume $p<N$ mentioning that the complementary case
$p\geq N$ can be handled along the same lines.

\medskip

In order to simplify the notation, for any real number $r>1$, we set
$r'=r/(r-1)$ (the H\"older conjugate of $r$). In particular, we have
$p'=p/(p-1)<q'=q/(q-1)$. In the left-hand side of equation
\eqref{1} there are the negative $p$-Laplacian
$-\Delta_p:W_0^{1,p}(\Omega)\to W^{-1,p'}(\Omega)$ expressed as
$$
\langle-\Delta_p u,v\rangle=\int_\Omega |\nabla u(x)|^{p-2}\nabla
u(x)\cdot \nabla v(x)\,dx\ \hbox{ for all $u,v\in
W_0^{1,p}(\Omega)$}
$$
and the negative $q$-Laplacian $-\Delta_q:W_0^{1,q}(\Omega)\to
W^{-1,q'}(\Omega)$ expressed as
$$
\langle-\Delta_q u,v\rangle=\int_\Omega |\nabla u(x)|^{q-2}\nabla
u(x)\cdot \nabla v(x)\,dx\ \hbox{ for all $u,v\in
W_0^{1,q}(\Omega)$}.
$$
Hereafter, $|\cdot|$ stands for the Euclidean norm in $\mathbb{R}^N$. Since $1<q<p<+\infty$, it holds the continuous embedding
$W_0^{1,p}(\Omega)\hookrightarrow W_0^{1,q}(\Omega)$, so the
operator $-\Delta_p +\mu\Delta_q$ is well
defined on $W_0^{1,p}(\Omega)$. In the sequel, $p^*$ stands for the
Sobolev critical exponent $p^*=Np/(N-p)$ (recall that we assume
$p<N$). 

\medskip

The right-hand side of the equation in \eqref{1} is
described by means of a Carath\'{e}odory function $f:\Omega\times
\mathbb{R}\times\mathbb{R}^N\rightarrow \mathbb{R}$ (meaning that
$f(\cdot,s,\xi)$ is measurable on $\Omega$ for all $(s,\xi)\in
\mathbb{R}\times\mathbb{R}^N$ and $f(x,\cdot,\cdot)$ is continuous
for a.e. $x\in\Omega$) which is composed with the convolution
$$
\rho\ast u(x)=\int_{\mathbb{R^N}}\rho(x-y)u(y)\,dy\ \ \mbox{for a.e.
$x\in \mathbb{R}^N$}
$$
of some $\rho\in L^1(\mathbb{R}^N)$ and $u\in
W^{1,p}_0(\Omega)\subset W^{1,p}(\mathbb{R}^N)$. Notice that the
convolution $\rho\ast u$ is well defined.

\medskip

There are two noticeable aspects related to the right-hand side of
the equation in \eqref{1}. The first one is the fact
that it exhibits dependence not only with respect to the solution
$u$ but also with respect to its gradient $\nabla u$. Such a term is
usually called convection and its presence prevents us to make use
of variational methods. A systematic study of problems with
convection can be found in \cite{M}. A second significant feature
related to the right-hand side of the equation in \eqref{1} is the fact that the convection is composed with a
convolution which is nonlocal operator. The study of the problems
involving the composition of convection and convolution has been
started in \cite{MM}, specifically for problem \eqref{1} with $\mu\leq 0$. This study incorporates the case where the operator is the $p$-Laplacian $-\Delta_p$ (for $\mu=0$) and the ordinary $(p,q)$-Laplacian $-\Delta_p-\Delta_q$ (for $\mu=-1$). The investigation of a (nonsmooth) version of problem
\eqref{1} for an arbitrary $\mu\in\mathbb{R}$, but without convection and convolution, was initiated in
\cite {LLMZ}. Problem \eqref{1} with the ``competing'' $(p,q)$-Laplacian
$-\Delta_p +\Delta_q$ (i.e., in the case where $\mu=1$) and convection but without convolution was
addressed in \cite{M1}.

\medskip

Let $\lambda_{1,p}>0$ denote the first eigenvalue of the negative $p$-Laplacian on $W^{1,p}_0(\Omega)$, which is given by the following variational characterization (see, e.g., \cite[\S9.2]{MMP}),
\begin{equation}
\label{lambda1}
\lambda_{1,p}=\min\Big\{\frac{\|\nabla u\|_{L^p(\Omega,\mathbb{R}^N)}^p}{\|u\|_{L^p(\Omega)}^p}:u\in W^{1,p}_0(\Omega)\setminus\{0\}\Big\}.
\end{equation}
We assume that the following growth condition for $f(x,s,\xi)$ is
satisfied.

\begin{assumption}
\label{H}
There holds
\begin{equation}\label{6}
|f(x,s,\xi)|\leq \sigma(x)+a_1|s|^{p-1}+a_2|\xi|^{p-1}
\end{equation}
for a.e. $x\in\Omega$, all $s\in\mathbb{R}$, and
$\xi\in\mathbb{R}^N$, with a function $\sigma\in L^{r'}(\Omega)$
where $r\in[1,p^*)$ and constants $a_1,a_2\geq 0$ satisfying
\begin{equation}\label{7}
\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}(a_1\lambda_{1,p}^{-1} +a_2N^{p-1}\lambda_{1,p}^{-\frac{1}{p}})<1.
\end{equation}
\end{assumption}

\begin{remark}
The condition \eqref{7} in Assumption \ref{H} can be expressed by saying that the parameter $\rho\in L^1(\mathbb{R}^N)$ in problem \eqref{1} is small enough with respect to its $L^1$ norm.
\end{remark}

\begin{remark}
{\rm (a)} If the Carath\'eodory function $f$ satisfies the growth condition
$$
|f(x,s,\xi)|\leq \sigma(x)+a_1|s|^{p-1}+a_2|\xi|^{\beta}
$$
as in \eqref{6} except that the exponent of $|\xi|$ is some $\beta\in[0,p-1)$, then Assumption \ref{H} is fulfilled provided that 
$$
a_1\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}<\lambda_{1,p}.
$$

\noindent {\rm (b)} If $f$ satisfies the stronger growth condition
$$
|f(x,s,\xi)|\leq \sigma(x)+a_1|s|^{\alpha}+a_2|\xi|^{\beta}
$$
with $\alpha,\beta\in[0,p-1)$, then Assumption \ref{H} is fulfilled.
\end{remark}

By a {\em generalized solution} to problem \eqref{1} we mean any function
$u\in W_0^{1,p}(\Omega)$ for which there exists a sequence
$\{u_n\}_{n\geq 1}$ in $W_0^{1,p}(\Omega)$ such that
\begin{itemize}
\item[\rm (a)]\ $u_n\rightharpoonup u$ in $W^{1,p}_0(\Omega)$ as
$n\to\infty$;
\item[\rm (b)]\ $-\Delta_pu_n+\mu\Delta_qu_n-f(\cdot,\rho\ast u_n(\cdot), \nabla(\rho\ast
u_n)(\cdot))\rightharpoonup 0$ in $W^{-1,p'}(\Omega)$ as
$n\to\infty$;
\item[\rm (c)]
$\displaystyle \lim_{n\to\infty}\langle -\Delta_pu_n+\mu\Delta_qu_n,u_n-u\rangle=0$.
\end{itemize}

\medskip

The essential point in our work is that the driving operator
$-\Delta_p +\mu\Delta_q$ in problem \eqref{1} has a fundamentally different behavior depending on whether $\mu\leq 0$ or $\mu>0$.
Indeed, in the latter case, the operator
 lacks the ellipticity: notice for instance that, for a nonzero $u_0\in W_0^{1,p}(\Omega)$ and
a number $\lambda>0$, the quantity
$$
\langle -\Delta_p(\lambda u_0)+\mu\Delta_q(\lambda u_0),\lambda
u_0\rangle=\lambda^p\|\nabla u_0\|_{L^p(\Omega,\mathbb{R}^N)}^p-\lambda^q\mu\|\nabla u_0\|_{L^q(\Omega,\mathbb{R}^N)}^q
$$
does not keep a constant sign if $\mu>0$. It is positive for $\lambda>0$
sufficiently large and it is negative for $\lambda>0$ sufficiently
small. In view of this, in \cite{LLMZ}, the operator $-\Delta_p +\mu\Delta_q$ for $\mu>0$ was
called a competing $(p,q)$-Laplacian. Due to the
lack of ellipticity there is no available method to handle problem
\eqref{1} for arbitrary $\mu$. In order to bypass this drawback, the notion of
generalized solution was introduced in \cite{LLMZ} for a counterpart
of problem \eqref{1} without convolution. Note that, in the case where $\mu\leq 0$, the notions of generalized solution and weak solution coincide (see Lemma \ref{L3.3}).
In Theorem \ref{T1}, we prove the
existence of a generalized solution to problem \eqref{1} for arbitrary $\mu$. Our approach relies on a fixed-point theorem and approximation process.
Our treatment of problem \eqref{1} is unified in the sense that it does not distinguish according to the sign of $\mu$.
 

\section{Preliminaries}\label{S2}

In the sequel, the space $W^{1,p}_0(\Omega)$ is considered endowed with the
norm $\|\nabla(\cdot)\|_{L^p(\Omega,\mathbb{R}^N)}$. 

\subsection{Galerkin basis} \label{S2.1}

Due to the density of
$C^{\infty}_0(\Omega)$ in $W^{1,p}_0(\Omega)$, the Banach space
$W^{1,p}_0(\Omega)$ with $1<p<+\infty$ is separable. Therefore, there
exists a Galerkin basis of $W^{1,p}_0(\Omega)$, that is a sequence
$\{X_n\}_{n\geq 1}$ of vector subspaces of $W^{1,p}_0(\Omega)$
satisfying
\begin{itemize}
\item[\rm (i)] $\dim X_n<\infty,\quad \forall n\geq 1$;
\item[\rm (ii)] $X_n\subset X_{n+1}, \quad \forall n\geq 1$;
\item[\rm (iii)]
$\displaystyle \overline{\bigcup_{n\geq 1} X_n}=W^{1,p}_0(\Omega).$
\end{itemize}
For the rest of the paper we fix a Galerkin basis $\{X_n\}_{n\geq
1}$ of $W^{1,p}_0(\Omega)$.

\subsection{Rellich-Kondrachov theorem}
For $1<p<N$, as known from the Rellich-Kondrachov theorem, the
Sobolev space $W^{1,p}_0(\Omega)$ is compactly embedded into
$L^{\theta}(\Omega)$ if $1\leq \theta < p^\ast(=\frac{Np}{N-p})$ and
continuously embedded if $\theta=p^\ast$. For every
$\theta\in[1,p^\ast]$ we denote by $S_\theta>0$ the best constant for this embedding, hence
\begin{eqnarray}\label{8}
\|u\|_{L^\theta(\Omega)}\leq S_\theta\, \|\nabla u\|_{L^p(\Omega,\mathbb{R}^N)}, \quad
\forall u\in W^{1,p}_0(\Omega).
\end{eqnarray}
For $\theta=p$, we have that $S_p=\lambda_{1,p}^{-\frac{1}{p}}$ (see \eqref{lambda1}).

\subsection{Convolution}

For easy reference we list a few useful properties of the
convolution $\rho\ast u$ of $\rho\in L^1(\mathbb{R}^N)$ and $u\in
W^{1,p}_0(\Omega)$; we refer to \cite[\S4.4, \S9.1]{B} for details.
In order to have well defined the convolution $\rho\ast u$ of
$\rho\in L^1(\mathbb{R}^N)$ with $u\in W^{1,p}_0(\Omega)$, it is
convenient to consider the Sobolev space $W^{1,p}_0(\Omega)$
embedded in $W^{1,p}(\mathbb{R}^N)$ by identifying every $u\in
W^{1,p}_0(\Omega)$ with its extension equal to zero outside
$\Omega$. The convolution $\rho\ast u$ is defined by
$$
\rho\ast u(x)=\int_{\mathbb{R^N}}\rho(x-y)u(y)\,dy\ \ \mbox{for a.e.
$x\in \mathbb{R}^N$}.
$$
The weak partial
derivatives of the convolution $\rho\ast u$ are expressed by
$$
\frac{\partial}{\partial x_i}(\rho\ast u)=\rho\ast \frac{\partial
u}{\partial x_i}\in L^{p}(\mathbb{R}^N),\ \ \forall i=1,\ldots,N.
$$
There hold the estimates
\begin{equation}\label{2}
\|\rho\ast u\|_{L^r(\mathbb{R}^N)}\leq
\|\rho\|_{L^1(\mathbb{R}^N)}\|u\|_{L^r(\Omega)}
\end{equation}
whenever $r\in[1,p^*]$ and
\begin{equation}\label{3}
\Big\|\rho\ast \frac{\partial u}{\partial
x_i}\Big\|_{L^p(\mathbb{R}^N)}\leq
\|\rho\|_{L^1(\mathbb{R}^N)}\Big\|\frac{\partial u}{\partial
x_i}\Big\|_{L^p(\Omega)},\ \ \forall i=1,\ldots,N.
\end{equation}
Using the convexity of the function
$t\mapsto t^p$ on $(0,+\infty)$ and \eqref{3}, we derive that
\begin{eqnarray}
&&\|\nabla(\rho\ast
u)\|_{L^p(\mathbb{R}^N,\mathbb{R}^N)}^p=\int_{\mathbb{R}^N}|\nabla(\rho\ast
u)|^p\,dx
=\int_{\mathbb{R}^N}\left(\sum_{i=1}^N\Big(\rho\ast\frac{\partial
u}{\partial x_i}\Big)^2\right)^{\frac{p}{2}}dx \nonumber \\
&&\leq
\int_{\mathbb{R}^N}\left(\sum_{i=1}^N\Big|\rho\ast\frac{\partial
u}{\partial x_i}\Big|\right)^pdx\leq
N^{p-1}\sum_{i=1}^N\Big\|\rho\ast\frac{\partial u}{\partial
x_i}\Big\|_{L^p(\mathbb{R}^N)}^p\nonumber \\
&&\leq
N^{p-1}\|\rho\|_{L^1(\mathbb{R}^N)}^p\sum_{i=1}^N\Big\|\frac{\partial
u}{\partial x_i}\Big\|_{L^p(\Omega)}^p\leq
N^p\|\rho\|_{L^1(\mathbb{R}^N)}^p\|\nabla
u\|_{L^p(\Omega,\mathbb{R}^N)}^p. \label{4}
\end{eqnarray}


\subsection{Fixed point theorem}
An essential tool in our approach will be the following consequence
of Brouwer's fixed point theorem (see \cite[page 37]{S}).

\begin{lemma}\label{L1}
Let $X$ be a finite-dimensional space endowed with the norm
$\|\cdot\|_X$ and let $A:X\to X^*$ be a continuous mapping. Assume
that there is a constant $R>0$ such that
$$
\langle A(v),v\rangle\geq 0\ \hbox{for all $v\in X$ with
$\|v\|_X=R$}.
$$
Then there exists $u\in X$ with $\|u\|_X\leq R$ satisfying $A(u)=0$.
\end{lemma}

\section{Main result}\label{S3}

In this section we provide our main result regarding the existence of solutions to problem \eqref{1}.

\subsection{Nonlinear operator associated to problem \eqref{1}}
Hereafter we consider the operator $A:W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$ given by
\begin{equation}
\label{A}
\langle A(u),v\rangle=\langle-\Delta_p u+\mu\Delta_q
u,v\rangle-\int_{\Omega}f(x,\rho\ast u(x),\nabla(\rho\ast
u)(x))v(x)\,dx
\end{equation}
which arises from problem \eqref{1}.

\begin{lemma}
\label{L3.1}
Suppose that \eqref{6} in Assumption \ref{H} is fulfilled. Then, the operator
$A:W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$ defined in \eqref{A} is continuous.
\end{lemma}

\begin{proof}  Relations \eqref{2} and \eqref{4} imply that the operator $T:W^{1,p}_0(\Omega)\to L^p(\Omega)\times L^p(\Omega)^N$ given by $T(u)=(\rho*u|_\Omega,\nabla(\rho*u)|_\Omega)$ is linear and continuous. The growth condition in \eqref{6} allows to apply the Krasnoselskii theorem \cite{Krasnoselskii} which implies that the Nemytskii operator
$$
N_f:L^p(\Omega)\times L^p(\Omega)^N\to L^{p'}(\Omega),\ (v,w)\mapsto f(\cdot,v(\cdot),w(\cdot))
$$
is well defined and continuous. We infer that the operator
\begin{equation}
\label{10-new}
W^{1,p}_0(\Omega)\to L^{p'}(\Omega),\ u\mapsto f(\cdot,\rho*u(\cdot),\nabla(\rho*u)(\cdot))
\end{equation}
is continuous as the composition of continuous operators.
Note also that $L^{p'}(\Omega)$ is continuously embedded in $W^{-1,p'}(\Omega)$. 

The operators $-\Delta_p:W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$ and $-\Delta_q:W^{1,q}_0(\Omega)\to W^{-1,q'}(\Omega)$ are continuous. Since $q<p$ and $\Omega$ is bounded, we have that $W^{1,p}_0(\Omega)$ is continuously embedded in $W^{1,q}_0(\Omega)$ and $W^{-1,q'}(\Omega)$ is continuously embedded in $W^{-1,p'}(\Omega)$. Therefore, $-\Delta_p+\mu\Delta_q:W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$ is continuous.

Altogether, this shows that the operator $A$ is continuous.
\end{proof}

\subsection{Finite-dimensional approximations}

Given a Galerkin basis $\{X_n\}_{n\geq 1}$ of $W^{1,p}_0(\Omega)$,
we construct a corresponding sequence of approximate solutions
related to problem \eqref{1}.

\begin{proposition}\label{P1}
Suppose that Assumption \ref{H} is fulfilled. Then, for every $n\geq 1$, there exists $u_n\in
X_n$ such that
\begin{equation}\label{5}
\langle-\Delta_p u_n+\mu\Delta_q u_n,v\rangle=\int_{\Omega}f(x,\rho\ast
u_n(x),\nabla(\rho\ast u_n)(x))v(x)\,dx
\end{equation}
for all $v\in X_n$.
Moreover, the sequence $\{u_n\}_{n\geq 1}$ so obtained is bounded in $W^{1,p}_0(\Omega)$.
\end{proposition}

\begin{proof}
On each finite-dimensional space $X_n$ we consider the mapping
$A_n:X_n\to X_n^*$ defined by
$$
\langle A_n(u),v\rangle=\langle-\Delta_p u+\mu\Delta_q
u,v\rangle-\int_{\Omega}f(x,\rho\ast u(x),\nabla(\rho\ast
u)(x))v(x)\,dx
$$
for all $u,v\in X_n$. Note that $A_n$ is continuous (see Lemma \ref{L3.1}). Our goal is to apply Lemma \ref{L1} to the operator
$A_n$. To this end, we note from \eqref{6} in Assumption \ref{H} and
H\"older's inequality that
\begin{eqnarray*}
&&\langle A_n(v),v\rangle=\int_{\Omega}(|\nabla v|^p-\mu|\nabla
v|^q-f(x,\rho\ast v,\nabla(\rho\ast v))v)\,dx\\
&&\geq \|\nabla v\|_{L^p(\Omega,\mathbb{R}^N)}^p-\mu|\Omega|^{\frac{p-q}{p}}\|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^q-\|\sigma\|_{L^{r'}(\Omega)}\|v\|_{L^r(\Omega)}\\
&&-a_1\|\rho\ast v\|_{L^p(\mathbb{R}^N)}^{p-1}\|v\|_{L^p(\Omega)}
-a_2\|\nabla(\rho\ast
v)\|_{L^p(\mathbb{R}^N,\mathbb{R}^N)}^{p-1}\|v\|_{L^p(\Omega)}
\end{eqnarray*}
for all $v\in X_n$. Hereafter, we denote by $|\Omega|$ the Lebesgue measure of $\Omega$. Then \eqref{2}, \eqref{4}, and \eqref{8} lead to
the estimate
\begin{eqnarray}
 && \langle A_n(v),v\rangle \geq \|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^p-\mu|\Omega|^{\frac{p-q}{p}}\|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^q \nonumber \\
&&-\|\sigma\|_{L^{r'}(\Omega)}\|v\|_{L^r(\Omega)}-a_1\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}\| v\|_{L^p(\Omega)}^p \nonumber \\
&&-a_2N^{p-1}\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}\|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^{p-1}\|v\|_{L^p(\Omega)} \nonumber \\
&&\geq \|\nabla v\|_{L^p(\Omega,\mathbb{R}^N)}^p-\mu|\Omega|^{\frac{p-q}{p}}\|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^q -S_r\|\sigma\|_{L^{r'}(\Omega)}\|\nabla v\|_{L^p(\Omega,\mathbb{R}^N)} \nonumber \\
&&-(a_1S_p^p\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}
+a_2S_pN^{p-1}\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1})\|\nabla
v\|_{L^p(\Omega,\mathbb{R}^N)}^p \label{10new}
\end{eqnarray}
for all $v\in X_n$. Taking into account \eqref{7} (recall that $S_p=\lambda_{1,p}^{-\frac{1}{p}}$) and that $p>q>1$,
the following estimate is true
$$
\langle A_n(v),v\rangle\geq 0\ \hbox{whenever $v\in X_n$ with
$\|\nabla v\|_{L^p(\Omega,\mathbb{R}^N)}=R$}
$$
provided $R>0$ is sufficiently large. Then Lemma \ref{L1} yields the existence of $u_n\in X_n$ satisfying
$A_n(u_n)=0$, that is, \eqref{5}.

It remains to show that the sequence $\{u_n\}_{n\geq 1}$ is bounded in $W^{1,p}_0(\Omega)$.
By inserting $v=u_n\in X_n$ in \eqref{10new}, we find that
\begin{eqnarray*}
&&\big(1-\|\rho\|_{L^1(\mathbb{R}^N)}^{p-1}(a_1S_p^{p} +a_2S_pN^{p-1})\big)\|\nabla u_n\|_{L^p(\Omega,\mathbb{R}^N)}^p\\
&&\leq \mu|\Omega|^{\frac{p-q}{p}}\|\nabla
u_n\|_{L^p(\Omega,\mathbb{R}^N)}^q+S_r\|\sigma\|_{L^{r'}(\Omega)}\|\nabla
u_n\|_{L^p(\Omega,\mathbb{R}^N)}.
\end{eqnarray*}
The desired conclusion is readily obtained from assumption
\eqref{7} and the fact that $p>q>1$.
\end{proof}

\subsection{Main result on the existence of a solution to problem \eqref{1}} \label{S4}

First, we show that the notions of generalized solution and weak solution coincide for problem \eqref{1} in the case where $\mu\leq 0$.

\begin{lemma}
\label{L3.3}
Suppose that $\mu\leq 0$. For every $u\in W^{1,p}_0(\Omega)$, the following conditions are equivalent:
\begin{itemize}
\item[\rm (i)] $u$ is a weak solution to problem \eqref{1}, that is, $u$ satisfies
$$
\langle-\Delta_p u+\mu\Delta_q u,v\rangle=\int_{\Omega}f(x,\rho\ast
u(x),\nabla(\rho\ast u)(x))v(x)\,dx
$$
for all $v\in W_0^{1,p}(\Omega)$;
\item[\rm (ii)] $u$ is a generalized solution to problem \eqref{1}. 
\end{itemize}
\end{lemma}

\begin{proof}
The implication {\rm (i)}$\Rightarrow${\rm (ii)} is immediate (take $u_n=u$) and actually does not require the condition that $\mu\leq 0$. Conversely, assume that $u$ is a generalized solution to problem \eqref{1}, and let $\{u_n\}_{n\geq 1}$ be a sequence satisfying conditions (a)--(c) of the definition of generalized solution with respect to $u$.
Using the monotonicity of the operator $-\Delta_q$ we note that
\begin{eqnarray*}
\langle -\Delta_pu_n,u_n-u\rangle & \leq & \langle
-\Delta_pu_n,u_n-u\rangle
-\mu\langle-\Delta_qu_n+\Delta_qu,u_n-u\rangle\\
 & = & \langle -\Delta_pu_n+\mu\Delta_qu_n,u_n-u\rangle -\mu\langle
\Delta_qu,u_n-u\rangle.
\end{eqnarray*}
By  {\rm (a)} and {\rm (c)}, this leads to
$$
\limsup_{n\to\infty}\langle -\Delta_pu_n,u_n-u\rangle\leq 0.
$$
Then we are able to conclude the strong convergence $u_n\to u$ in
$W^{1,p}(\Omega)$ (see, e.g., \cite[Proposition 2.72]{MMP}). By Lemma \ref{L3.1}, this implies that $A(u_n)\to A(u)$ in $W^{-1,p'}(\Omega)$, where $A:W^{1,p}_0(\Omega)\to W^{-1,p'}(\Omega)$ is the operator defined in \eqref{A}. In view of condition (b) of the definition of generalized solution, this yields $A(u)=0$, which precisely means that $u$ is a weak solution to problem~\eqref{1}.
\end{proof}

We can now state our main result.

\begin{theorem}\label{T1}
Suppose that Assumption \ref{H} holds. Then there exists a generalized
solution to problem \eqref{1}. In particular, if $\mu\leq 0$, there exists a weak solution to problem \eqref{1}.
\end{theorem}

\begin{proof}
Consider the sequence $\{u_n\}_{n\geq 1}\subset W^{1,p}_0(\Omega)$
constructed in Proposition \ref{P1}. As asserted therein, this sequence is bounded in $W^{1,p}_0(\Omega)$. In view of the
reflexivity of the space $W^{1,p}_0(\Omega)$, we can pass to a
subsequence still denoted by $\{u_n\}_{n\geq 1}$ such that
\begin{equation}\label{10}
u_n\rightharpoonup u\ \hbox{in $W^{1,p}_0(\Omega)$}
\end{equation}
with some $u\in W^{1,p}_0(\Omega)$. 
Moreover, since the sequence $\{u_n\}_{n\geq 1}$ is bounded in $W^{1,p}_0(\Omega)$, invoking the continuity of the operator in \eqref{10-new}, we have that
\begin{equation}
\label{14-new}
\mbox{the sequence $\{f(\cdot,\rho\ast u_n,\nabla(\rho\ast
u_n))\}_{n\geq 1}$ is bounded in $L^{p'}(\Omega)$.}
\end{equation}
On the basis
of the reflexivity of $W^{-1,p'}(\Omega)$, we can assume that
\begin{equation}\label{13}
-\Delta_pu_n+\mu\Delta_qu_n-f(\cdot,\rho\ast u_n,\nabla(\rho\ast
u_n))\rightharpoonup \eta\ \hbox{in $W^{-1,p'}(\Omega)$}
\end{equation}
with some $\eta\in W^{-1,p'}(\Omega)$.

Now let $v\in \bigcup_{n\geq 1}X_n$. Fix an integer $m\geq 1$ such
that $v\in X_m$. Proposition \ref{P1} provides that \eqref{5} holds
for all $n\geq m$. Letting $n\to\infty$ in \eqref{5}, by means of
\eqref{13} we get
$$
\langle \eta,v\rangle=0\ \hbox{for all $v\in \bigcup\limits_{n\geq
1}X_n$}.
$$
By the density of $\bigcup_{n\geq 1}X_n$ in $W_0^{1,p}(\Omega)$ (see
{\rm (iii)} in the definition of Galerkin basis in Section \ref{S2.1}), it
turns out that $\eta=0$. Therefore, \eqref{13} renders
\begin{equation}\label{14}
-\Delta_pu_n+\mu\Delta_qu_n-f(\cdot,\rho\ast u_n,\nabla(\rho\ast
u_n))\rightharpoonup 0\ \hbox{in $W^{-1,p'}(\Omega)$}.
\end{equation}

Next, setting $v=u_n$ in \eqref{5}, we obtain
\begin{equation}\label{15}
\langle -\Delta_pu_n+\mu\Delta_qu_n,u_n\rangle-\int_{\Omega}f(x,\rho\ast u_n,\nabla(\rho\ast u_n))u_n\,dx=0
\end{equation}
for all $n\geq 1$,
while \eqref{14} gives
\begin{equation}\label{16}
\langle -\Delta_pu_n+\mu\Delta_qu_n,u\rangle-\int_{\Omega}f(x,\rho\ast
u_n,\nabla(\rho\ast u_n))u\,dx \to 0
\end{equation}
as $n\to\infty$.
Altogether, \eqref{15} and \eqref{16} yield
\begin{equation}\label{17}
\langle-\Delta_pu_n+\mu\Delta_qu_n,u_n-u\rangle-\int_{\Omega}f(x,\rho\ast
u_n,\nabla(\rho\ast u_n))(u_n-u)\,dx\to 0
\end{equation}
as $n\to\infty$.
Moreover,
from \eqref{10}, Rellich-Kondrachov compact embedding theorem
which ensures that $u_n\to u$ strongly in
$L^p(\Omega)$, and \eqref{14-new}, we derive that
\begin{equation}\label{18}
\lim_{n\to\infty}\int_{\Omega}f(x,\rho\ast u_n,\nabla(\rho\ast
u_n))(u_n-u)\,dx=0.
\end{equation}
Inserting \eqref{18} into \eqref{17} enables us to assert
\begin{equation}\label{19}
\lim_{n\to\infty}\langle -\Delta_pu_n+\mu\Delta_qu_n,u_n-u\rangle=0.
\end{equation}

At this point we can notice that \eqref{10}, \eqref{14}, and
\eqref{19} are just the conditions {\rm (a)}, {\rm (b)}, and {\rm (c)} expressing
that $u\in W_0^{1,p}(\Omega)$ is a generalized solution to problem
\eqref{1}, which proves the first assertion in the theorem. The last assertion in the theorem is a consequence of Lemma \ref{L3.3}.
\end{proof}

\subsubsection*{Acknowledgement} The authors are grateful to Dr. Lucas Fresse for important remarks.

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\end{document}
