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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 146, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2016/146\hfil Liouville-type theorems for elliptic inequalities]
{Liouville-type theorems for elliptic inequalities with
 power nonlinearities involving variable exponents for a fractional Grushin operator}

\author[M. Jleli, M. Kirane, B. Samet \hfil EJDE-2016/146\hfilneg]
{Mohamed Jleli, Mokhtar Kirane, Bessem Samet}

\address{Mohamed Jleli \newline
Department of Mathematics, College of Science,
 King Saud University, P.O. Box 2455, Riyadh 11451,
Saudi Arabia}
\email{jleli@ksu.edu.sa}

\address{Mokhtar Kirane \newline
 LaSIE, Facult\'e des Sciences et Technologies, Universit\'e de La Rochelle,
Avenue M. Cr\'epeau, 17042 La Rochelle, France}
\email{mokhtar.kirane@univ-lr.fr}

\address{Bessem Samet \newline
Department of Mathematics, College of Science,
King Saud University, P.O. Box 2455, Riyadh 11451,
Saudi Arabia}
\email{bsamet@ksu.edu.sa}

\thanks{Submitted March 1, 2016. Published June 14, 2016.}
\subjclass[2010]{35B53, 35R11, 35J70}
\keywords{Liouville-type theorem; elliptic inequalities; variable exponent;
\hfill\break\indent fractional Grushin operator}

\begin{abstract}
 We establish Liouville-type theorems for the elliptic inequality
 $$
 u\geq 0,\quad G_{\alpha,\beta,\theta}\big(u^{p(x,y)},u^{q(x,y)}\big)
 \geq u^{r(x,y)}, \quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
 $$
 where $G_{\alpha,\beta,\theta}$, $0<\alpha,\beta<2$, $\theta\geq 0$,
 is the fractional Grushin operator of mixed orders $\alpha,\beta$, defined by
 $$
 G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u
 +|x|^{2\theta} (-\Delta_y)^{\beta/2}v,
 $$
 where, $(-\Delta_x)^{\alpha/2}$ is the fractional Laplacian operator of
 order $\alpha$ with respect to the variable $x\in \mathbb{R}^{N_1}$,
 and $(-\Delta_y)^{\beta/2}$ is the fractional Laplacian operator of order
 $\beta$ with respect to the variable $y\in \mathbb{R}^{N_2}$.
 Here, $p,q,r: \mathbb{R}^{N_1}\times\mathbb{R}^{N_2}\to [1,\infty)$
 are measurable functions satisfying certain conditions.
\end{abstract}

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


\section{Introduction}

The standard Liouville theorem \cite{L} states that any bounded complex 
function which is harmonic (or holomorphic) on the entire space is constant.
 The first proof of this theorem was published by Cauchy \cite{C}. 
In the recent literature, Gidas and Spruck \cite{G} extended this result to 
the case of non-negative solutions of semilinear elliptic equations 
in the whole space $\mathbb{R}^N$ or in half-spaces. In the case of the
 whole  space $\mathbb{R}^N$, they established that if $1\leq r<\frac{N+2}{N-2}$,
 then the unique non-negative solution of
$$
-\Delta u=C u^r \quad \text{in } \mathbb{R}^N,
$$
where $C$ is a strictly positive constant, is the trivial solution. 
A simple proof based on the moving planes method was suggested by Chen 
and Li \cite{CL} in the whole range of $r$, i.e.,  $0< r<\frac{N+2}{N-2}$.  
This result is optimal in the sense that for any $r\geq \frac{N+2}{N-2}$, 
we have  infinitely many positive solutions.
The same result holds for the elliptic inequality
$$
-\Delta u\geq C u^r \quad \text{in } \mathbb{R}^N,
$$
see \cite{G2}. Berestycki \textit{et al.} \cite{B}, 
established Liouville-type theorems for semilinear elliptic inequalities 
of the form
$$
u\geq 0,\quad -\Delta u\geq h(x) u^r \quad \text{in } \Sigma,
$$
where $\Sigma$ is a cone in $\mathbb{R}^N$ and $h$ is a positive function.

Recently, several Liouville-type theorems were established  for various 
classes of degenerate elliptic equations.  Serrin and Zou \cite{S} 
generalized the standard Liouville theorem for $p$-harmonic functions on 
the whole space $\mathbb{R}^N$ and on exterior domains. 
In \cite{K,K2}, Liouville-type theorems for some linear degenerate 
elliptic operators such as  X-elliptic operators, Kohn-Laplacian 
and Ornstein-Uhlenbeck operators were proved. 
Dolcetta and Cutri \cite{D} established a Liouville-type theorem for an 
elliptic inequality involving the Grushin operator. More precisely, 
they considered the problem
\begin{equation}\label{PG}
u\geq 0,\quad G_\theta u \geq u^r \quad \text{in } 
\mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
\end{equation}
where $\theta>1$ and $G_\theta$ is the Grushin operator defined by
\begin{equation}\label{GO}
G_\theta u= (-\Delta_x) u+|x|^{2\theta} (-\Delta_y) u,\quad
 (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.
\end{equation}
They proved that if $1<r<\frac{Q}{Q-2}$, then the only solution of \eqref{PG} 
is the trivial solution. Here, $Q$ is the homogeneous dimension of the space, 
given by $Q=N_1+(\theta+1)N_2$. For other related results, we refer 
to \cite{A,M,M2,Y}.


Recently, a lot of attention  has been paid to the study of linear and nonlinear 
integral operators, involving the fractional Laplacian. 
In \cite{MA},  using the moving plane method, Ma and Chen  established
 a Liouville-type result for the system of equations
\begin{gather*}
(-\Delta)^{\mu/2} u = v^q,\\
(-\Delta)^{\mu/2} v = u^p,
\end{gather*}
where $\mu\in (0,2)$, $1<p,q\leq \frac{N+\mu}{N-\mu}$, and $N\geq 2$. 
Here, $(-\Delta)^{\mu/2}$ is the fractional Laplacian operator of order 
$\mu/2$. Using the test function method \cite{MP}, Dahmani \textit{et al.} \cite{DA} 
extended the result in \cite{MA} to various classes of systems involving  
fractional Laplacian operators with different orders. 
 Quaas and Xia \cite{Q} established Liouville-type results for a class of
fractional elliptic equations and systems in the half space.
 For other related works, we refer to \cite{F,F2,F3,GU,KI}, and the 
references therein.

This article is devoted to the study of nonexistence results of solutions 
for the elliptic inequality
\begin{equation}\label{PB}
u\geq 0,\quad  G_{\alpha,\beta,\theta}\big(u^{p(x,y)},u^{q(x,y)}\big) 
\geq u^{r(x,y)}, \quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
\end{equation}
where $G_{\alpha,\beta,\theta}$, $0<\alpha,\beta<2$, $\theta\geq 0$, 
is the fractional Grushin operator of mixed orders $\alpha,\beta$, defined by
$$
G_{\alpha,\beta,\theta}(u,v)
=(-\Delta_x)^{\alpha/2}u+|x|^{2\theta} (-\Delta_y)^{\beta/2}v,
$$
where, $(-\Delta_x)^{\alpha/2}$ is the fractional Laplacian operator of 
order $\alpha$ with respect to the variable $x\in \mathbb{R}^{N_1}$, 
and $(-\Delta_y)^{\beta/2}$ is the fractional Laplacian operator of 
order $\beta$ with respect to the variable $y\in \mathbb{R}^{N_2}$. 
Here, $p,q,r: \mathbb{R}^{N_1}\times\mathbb{R}^{N_2}\to [1,\infty)$ 
are supposed to be measurable functions satisfying certain conditions.
 Observe that the standard Grushin operator defined by \eqref{GO} can 
be written in the form
$$
G_\theta u= G_{2,2,\theta}(u,u).
$$
Up to our knowledge, there are not many works dealing with  
Liouville-type properties involving elliptic inequalities with variable 
exponents non-linearity. In this direction, we refer to the recent 
paper \cite{GG}.

Before stating and proving the main results of this work, let us present 
some basic definitions and some lemmas that will be used later.

The nonlocal operator $(-\Delta)^s$, $0<s<1$, is defined for any function 
$h$ in the Schwartz class through the Fourier transform
$$
(-\Delta)^sh(x)=\mathcal{F}^{-1}\left(|\xi|^{2s}\mathcal{F}(h)(\xi)\right)(x),
$$
where $\mathcal{F}$ stands for the Fourier transform and $\mathcal{F}^{-1}$ 
for its inverse. It can be also defined via the Riesz potential
$$
(-\Delta)^sh(x)=c_{N,s} \text{ PV } 
\int_{\mathbb{R}^N} \frac{h(x)-h(\overline{x})}{|x-\overline{x}|^{N+2s}}
\,d\overline{x},
$$
where $c_{N,s}$ is a normalisation constant and PV  is the Cauchy principal 
value (see \cite{LA,ST}).

\begin{lemma}[\cite{J}] \label{L1}
Suppose that $\delta\in (0,2)$, $\beta+1\geq 0$, and 
$\psi\in C_0^\infty(\mathbb{R}^N)$, $\psi\geq 0$. Then the
following point-wise inequality holds:
$$
(-\Delta)^{\delta/2}\psi^{\beta+2}(x)\leq (\beta+2)
\psi^{\beta+1}(x)(-\Delta)^{\delta/2}\psi(x).
$$
\end{lemma}

\begin{lemma}[$\varepsilon$-Young's inequality] \label{L2} 
Let $1<p,q<\infty$, and $\frac{1}{p}+\frac{1}{q}=1$. Then
$$
ab\leq \varepsilon a^p +C(\varepsilon) b^q, \quad (a,b>0, \varepsilon>0),
$$
where $C(\varepsilon)=(\varepsilon p)^{-q/p}q^{-1}$.
\end{lemma}

For a measurable function $p: \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}\to [1,\infty)$, 
we denote by
$L^{p(\cdot,\cdot)}(\mathbb{R}^{N_1}\times \mathbb{R}^{N_2})$
the Lebesgue space with variable exponent, defined by
\begin{align*}
&L^{p(\cdot,\cdot)}(\mathbb{R}^{N_1}\times \mathbb{R}^{N_2})\\
&=\Big\{u:\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}\to \mathbb{R}:
 u \text{ measurable, } \int_{ \mathbb{R}^{N_1}
\times \mathbb{R}^{N_2}} |u|^{p(x,y)}\,dx\,dy<\infty\big\}.
\end{align*}
We denote by $L_{\rm loc}^{p(\cdot,\cdot)}(\mathbb{R}^{N_1}\times \mathbb{R}^{N_2})$
 the set defined by
\begin{align*}
&L_{\rm loc}^{p(\cdot,\cdot)}(\mathbb{R}^{N_1}\times \mathbb{R}^{N_2})\\
&=\big\{u:\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}\to \mathbb{R}:
 u \text{ measurable, } \int_{K} |u|^{p(x,y)}\,dx\,dy<\infty,\,
 K \text{ compact}\big\}.
\end{align*}
For more details on Lebesgue spaces with variable exponents,
 we refer to \cite{AN}.

\section{Main results}

We consider the elliptic inequality \eqref{PB} under the assumptions:
\begin{itemize}
\item[ ] $\theta\geq 0$, $0<\alpha,\beta<2$,
\item[ ] $p,q,r\in L^\infty(\mathbb{R}^N)$, $N=N_1+N_2$,
\item[ ] $r(x,y)>\max\{p(x,y),q(x,y)\}\geq 1$,
\item[ ] $\lambda:=\inf_{(x,y)\in\mathbb{R}^{N_1}\times
\mathbb{R}^{N_2}} \left\{r(x,y)-p(x,y)\right\}>0$,
\item[ ] $\mu:=\inf_{(x,y)\in\mathbb{R}^{N_1}\times
\mathbb{R}^{N_2}} \left\{r(x,y)-q(x,y)\right\}>0$.
\end{itemize}
The definition of solutions we adopt for \eqref{PB} is the following.

\begin{definition} \label{def2.1} \rm
We say that $u$ is a weak solution of \eqref{PB}, if 
$u\in L_{\rm loc}^{i(\cdot,\cdot)}(\mathbb{R}^{N_1}\times \mathbb{R}^{N_2})$, 
$i\in\{p,q,r\}$, $u\geq 0$,  and
\begin{align*}
&\int_{\mathbb{R}^{N}} u^{p(x,y)} (-\Delta_x)^{\alpha/2} \varphi \,dx\,dy +
\int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} (-\Delta_y)^{\beta/2} 
\varphi \,dx\,dy \\
&\geq  \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi\,dx\,dy,
\end{align*}
for all $\varphi\in C_0^\infty(\mathbb{R}^N)$, $\varphi\geq 0$.
\end{definition}


Given $R>0$, we denote by $\Omega_{R,\theta}$ the subset of $\mathbb{R}^{N_1}\times
\mathbb{R}^{N_2}$ defined by
$$
\Omega_{R,\theta}=\Big\{(x,y)\in \mathbb{R}^{N_1}\times
\mathbb{R}^{N_2}:
 1\leq \frac{|x|^2}{R^2}+\frac{|y|^2}{R^{2(\theta+1)}}\leq 2\Big\}.
$$
We have the following Liouville-type theorem for the elliptic inequality \eqref{PB}.

\begin{theorem}\label{T1}
Suppose that
\begin{equation}\label{AS}
\lim_{R\to \infty} \Big( \int_{\Omega_{R,\theta}} 
R^{\frac{-\alpha r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy
+\int_{\Omega_{R,\theta}} R^{\frac{[2\theta-\beta(\theta+1)]
 r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy\Big)=0.
\end{equation}
Then inequality \eqref{PB} has  no nontrivial weak solution.
\end{theorem}


\begin{proof}
Suppose that $u$ is a nontrivial weak solution of \eqref{PB}. 
Let $\omega$ be a real number such that
\begin{equation}\label{com}
\omega> \max\Big\{\frac{\|r\|_{L^\infty(\mathbb{R}^N)}}{\lambda},
\frac{\|r\|_{L^\infty(\mathbb{R}^N)}}{\mu},1\Big\} .
\end{equation}
By the weak formulation of \eqref{PB}, we have
\begin{equation}\label{eq1}
\begin{aligned}
&\int_{\mathbb{R}^{N}} u^{p(x,y)} (-\Delta_x)^{\alpha/2} \varphi^\omega \,dx\,dy 
+ \int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} (-\Delta_y)^{\beta/2} 
 \varphi^\omega \,dx\,dy\\
&\geq  \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^\omega\,dx\,dy,
\end{aligned}
\end{equation}
for all $\varphi\in C_0^\infty(\mathbb{R}^N)$, $\varphi\geq 0$. 
By Lemma \ref{L1}, we have
$$
\int_{\mathbb{R}^{N}} u^{p(x,y)} (-\Delta_x)^{\alpha/2} \varphi^\omega \,dx\,dy 
\leq \omega \int_{\mathbb{R}^{N}} u^{p(x,y)} \varphi^{\omega-1} 
|(-\Delta_x)^{\alpha/2} \varphi|\,dx\,dy.
$$
Using the $\varepsilon$-Young inequality (see Lemma \ref{L2}) with parameters 
$s(x,y)=\frac{r(x,y)}{p(x,y)}$ and
$s'(x,y)=\frac{r(x,y)}{r(x,y)-p(x,y)}$, for all $\varepsilon>0$, we obtain
\begin{align*}
&\int_{\mathbb{R}^{N}} u^{p(x,y)} \varphi^{\omega-1} |
 (-\Delta_x)^{\alpha/2} \varphi|\,dx\,dy\\
&=\int_{\mathbb{R}^{N}} u^{p(x,y)} \varphi^{\frac{\omega}{s(x,y)}}
 \varphi^{\omega-1-\frac{\omega}{s(x,y)}} |(-\Delta_x)^{\alpha/2} \varphi|\,dx\,dy\\
&\leq  \varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy\\
&\quad +\int_{\mathbb{R}^{N}} C_1(x,y,\varepsilon) \varphi^{[\omega-1-\frac{\omega}{s(x,y)}]s'(x,y)}|(-\Delta_x)^{\alpha/2} \varphi|^{s'(x,y)}\,dx\,dy,
\end{align*}
where
\[
C_1(x,y,\varepsilon)=\Big(\frac{\varepsilon r(x,y)}{p(x,y)}
\Big)^{\frac{-p(x,y)}{r(x,y)-p(x,y)}}\Big(\frac{r(x,y)-p(x,y)}{r(x,y)}\Big),
\]
$(x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}$, and $\varepsilon>0$.
Observe that for all $\varepsilon>0$, we have 
$C_1(\cdot,\cdot,\varepsilon)\in L^\infty(\mathbb{R}^N)$. 
In fact, under the considered assumptions, we have
$$
C_1(x,y,\varepsilon)
\leq \varepsilon^{\frac{\|p\|_{L^\infty(\mathbb{R}^N)}}{\lambda}},\quad 
(x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.
$$
Let $C_1(\varepsilon)=\|C_1(\cdot,\cdot,\varepsilon)\|_{L^\infty(\mathbb{R}^N)}$. 
Therefore,
\begin{align*}
&\int_{\mathbb{R}^{N}} u^{p(x,y)} \varphi^{\omega-1} |(-\Delta_x)^{\alpha/2}
  \varphi|\,dx\,dy\\
&\leq  \varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy
 +C_1(\varepsilon) \int_{\mathbb{R}^{N}}  
\varphi^{[\omega-1-\frac{\omega}{s(x,y)}]s'(x,y)}|(-\Delta_x)^{\alpha/2} 
\varphi|^{s'(x,y)}\,dx\,dy.
\end{align*}
Observe that thanks to \eqref{com}, we have
$$
\int_{\mathbb{R}^{N}}  \varphi^{[\omega-1-\frac{\omega}{s(x,y)}]s'(x,y)}|
(-\Delta_x)^{\alpha/2} \varphi|^{s'(x,y)}\,dx\,dy<\infty.
$$
Indeed, we have
\begin{align*}
&\int_{\mathbb{R}^{N}}  \varphi^{[\omega-1-\frac{\omega}{s(x,y)}]s'(x,y)}|
(-\Delta_x)^{\alpha/2} \varphi|^{s'(x,y)}\,dx\,dy\\
&=\int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-p(x,y)}}|
(-\Delta_x)^{\alpha/2} \varphi|^{\frac{r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy.
\end{align*}
On the other hand,  from \eqref{com}, we have
$$
\frac{r(x,y)}{r(x,y)-p(x,y)}
 \leq \frac{\|r\|_{L^\infty(\mathbb{R}^N)}}{\lambda}<\omega, 
\quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.
$$
As consequence, we have the  estimate
\begin{equation} \label{es1}
\begin{aligned}
& \int_{\mathbb{R}^{N}} u^{p(x,y)} (-\Delta_x)^{\alpha/2} \varphi^\omega \,dx\,dy\\
&  \leq   \omega
\varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy\\
&\quad +C_1(\varepsilon) \omega \int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-p(x,y)}}|(-\Delta_x)^{\alpha/2} \varphi|^{\frac{r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy.
\end{aligned}
\end{equation}
Again, using Lemma \ref{L1}, we obtain
$$
\int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} (-\Delta_y)^{\beta/2}
\varphi^\omega \,dx\,dy
\leq \omega \int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)}
\varphi^{\omega-1} |(-\Delta_y)^{\beta/2} \varphi|\,dx\,dy.
$$
Using the $\varepsilon$-Young inequality  with parameters
$k(x,y)=\frac{r(x,y)}{q(x,y)}$ and
$k'(x,y)=\frac{r(x,y)}{r(x,y)-q(x,y)}$, for all $\varepsilon>0$, we obtain
\begin{align*}
&\int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} \varphi^{\omega-1}
|(-\Delta_y)^{\beta/2} \varphi|\,dx\,dy\\
&= \int_{\mathbb{R}^{N}} u^{q(x,y)} \varphi^{\frac{\omega}{k(x,y)}}
\varphi^{\omega-1-\frac{\omega}{k(x,y)}} |x|^{2\theta}
|(-\Delta_y)^{\beta/2} \varphi|\,dx\,dy\\
&\leq   \varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy\\
&\quad +\int_{\mathbb{R}^{N}} C_2(x,y,\varepsilon)
\varphi^{[\omega-1-\frac{\omega}{k(x,y)}]k'(x,y)} |x|^{2\theta k'(x,y)}
|(-\Delta_y)^{\beta/2} \varphi|^{k'(x,y)}\,dx\,dy,
\end{align*}
where
$$
C_2(x,y,\varepsilon)=\Big(\frac{\varepsilon r(x,y)}{q(x,y)}
\Big)^{\frac{-q(x,y)}{r(x,y)-q(x,y)}}
\Big(\frac{r(x,y)-q(x,y)}{r(x,y)}\Big),\quad
(x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},\; \varepsilon>0.
$$
As previously, under the considered assumptions, we have
\[
C_2(x,y,\varepsilon)\leq \varepsilon^{\frac{\|q\|_{L^\infty(\mathbb{R}^N)}}{\mu}},
\]
$(x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}$,
which implies that  $C_2(\cdot,\cdot,\varepsilon)\in L^\infty(\mathbb{R}^N)$,
for all $\varepsilon>0$.
Let $C_2(\varepsilon)=\|C_2(\cdot,\cdot,\varepsilon)\|_{L^\infty(\mathbb{R}^N)}$.
Therefore, we have
\begin{align*}
&\int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} \varphi^{\omega-1} |(-\Delta_y)^{\beta/2} \varphi|\,dx\,dy\\
&\leq  \varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy\\
&\quad+C_2(\varepsilon) \int_{\mathbb{R}^{N}}  \varphi^{[\omega-1-\frac{\omega}{k(x,y)}]k'(x,y)} |x|^{2\theta k'(x,y)} |(-\Delta_y)^{\beta/2} \varphi|^{k'(x,y)}\,dx\,dy.
\end{align*}
On the other hand, we have
\begin{align*}
&\int_{\mathbb{R}^{N}}  \varphi^{[\omega-1-\frac{\omega}{k(x,y)}]k'(x,y)}
  |x|^{2\theta k'(x,y)} |(-\Delta_y)^{\beta/2} \varphi|^{k'(x,y)}\,dx\,dy\\
&=\int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-q(x,y)}}
  |x|^{\frac{2\theta r(x,y)}{r(x,y)-q(x,y)} } |(-\Delta_y)^{\beta/2}
  \varphi|^{\frac{r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{align*}
From \eqref{com}, we have
$$
\frac{r(x,y)}{r(x,y)-q(x,y)} \leq \frac{\|r\|_{L^\infty(\mathbb{R}^N)}}{\mu}<\omega,
 \quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2};
$$
then
$$
\int_{\mathbb{R}^{N}}  \varphi^{[\omega-1-\frac{\omega}{k(x,y)}]k'(x,y)}
|x|^{2\theta k'(x,y)} |(-\Delta_y)^{\beta/2} \varphi|^{k'(x,y)}\,dx\,dy<\infty.
$$
As consequence, we have the  estimate
\begin{equation} \label{es2}
\begin{aligned}
 &\int_{\mathbb{R}^{N}} |x|^{2\theta} u^{q(x,y)} (-\Delta_y)^{\beta/2} \varphi^\omega \,dx\,dy\\
  &\leq   \omega \varepsilon \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^{\omega}\,dx\,dy\\
 &\quad +C_2(\varepsilon) \omega \int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-q(x,y)}} |x|^{\frac{2\theta r(x,y)}{r(x,y)-q(x,y)} } |(-\Delta_y)^{\beta/2} \varphi|^{\frac{r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{aligned}
\end{equation}
Now, combining \eqref{eq1}, \eqref{es1} and \eqref{es2}, we obtain
\begin{align*}
&(1-2\omega\varepsilon) \int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^\omega\,dx\,dy\\
&\leq  C_1(\varepsilon) \omega \int_{\mathbb{R}^{N}}
 \varphi^{\omega-\frac{r(x,y)}{r(x,y)-p(x,y)}}|(-\Delta_x)^{\alpha/2}
  \varphi|^{\frac{r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy\\
&\quad +C_2(\varepsilon) \omega \int_{\mathbb{R}^{N}}
 \varphi^{\omega-\frac{r(x,y)}{r(x,y)-q(x,y)}}
 |x|^{\frac{2\theta r(x,y)}{r(x,y)-q(x,y)} } |
 (-\Delta_y)^{\beta/2} \varphi|^{\frac{r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{align*}
Taking $\varepsilon=(4\omega)^{-1}$, we obtain
\begin{equation}\label{FE}
\int_{\mathbb{R}^{N}} u^{r(x,y)} \varphi^\omega\,dx\,dy
\leq C \left(A(\varphi)+B(\varphi)\right),
\end{equation}
where
\begin{align*}
A(\varphi)= \int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-p(x,y)}}|
 (-\Delta_x)^{\alpha/2} \varphi|^{\frac{r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy,\\
B(\varphi)= \int_{\mathbb{R}^{N}}  \varphi^{\omega-\frac{r(x,y)}{r(x,y)-q(x,y)}}
 |x|^{\frac{2\theta r(x,y)}{r(x,y)-q(x,y)} } |(-\Delta_y)^{\beta/2}
 \varphi|^{\frac{r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{align*}

Let $\varphi_0$  be the standard cutoff function; that is,
$\varphi_0\in C_0^\infty(0,\infty)$  is a smooth decreasing function such that
$$
0\leq \varphi_0\leq 1,\quad  |\varphi_0'(\sigma)| \leq \frac{C}{\sigma}, 
\quad  \varphi_0(\sigma)=\begin{cases}
1 &\text{ if } 0<\sigma \leq 1,\\ 
0 &\text{ if } \sigma \geq 2.
\end{cases}
$$
As a test function, we take
$$
\varphi(x,y)=\varphi_0\Big(\frac{|x|^2}{R^2}+\frac{|y|^2}{R^{2(\theta+1)}}\Big), 
\quad (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
$$
where $R>0$ is a real number (large enough). Let $\Omega$ be the subset 
of $\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}$ defined by
$$
\Omega=\big\{(z,w)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}: 
1\leq |z|^2+|w|^2\leq 2\big\}.
$$
Let
$$
\eta(z,w)=|z|^2+|w|^2,\quad (z,w)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}.
$$
Using the change of variables
$$
z=\frac{x}{R},\quad  w=\frac{y}{R^{2(\theta+1)}},
$$
we obtain
\begin{align*}
 A(\varphi)
&= \int_{\Omega} [\varphi_0(\eta)]^{\omega-s'(Rz,R^{\theta+1}w)} 
 |(-\Delta_z)^{\alpha/2} \varphi_0(\eta)|^{s'(Rz,R^{\theta+1}w)}\\
&\quad\times  R^{N_1+N_2(\theta+1)-\alpha s'(Rz,R^{\theta+1}w)}\,dz\,dw\\
&\leq C \int_{\Omega} R^{N_1+N_2(\theta+1)-\alpha s'(Rz,R^{\theta+1}w)}\,dz\,dw\\
&= C\int_{\Omega_R} R^{\frac{-\alpha r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy.
\end{align*}
Therefore, we have the estimate
\begin{equation}\label{ees1}
A(\varphi)\leq C\int_{\Omega_R} R^{\frac{-\alpha r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy.
\end{equation}
Under the same change of variables, we obtain
\begin{align*}
B(\varphi) &\leq  C \int_{\Omega} R^{N_1+N_2(\theta+1)
 +[2\theta-\beta(\theta+1)]k'(Rz,R^{\theta+1}w)}\,dz\,dw\\
&= C \int_{\Omega_R} R^{\frac{[2\theta-\beta(\theta+1)]r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{align*}
Therefore, we have the estimate
\begin{equation}\label{ees2}
B(\varphi)\leq C\int_{\Omega_R}
 R^{\frac{[2\theta-\beta(\theta+1)]r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy.
\end{equation}
Combining \eqref{FE}, \eqref{ees1} and \eqref{ees2}, we obtain
\begin{align*}
&\int_{\mathbb{R}^N} u^{r(x,y)} \varphi_0^\omega\Big(\frac{|x|^2}{R^2}
 +\frac{|y|^2}{R^{2(\theta+1)}}\Big)\,dx\,dy\\
&\leq C\Big(\int_{\Omega_R} R^{\frac{-\alpha r(x,y)}{r(x,y)-p(x,y)}}\,dx\,dy
+\int_{\Omega_R} R^{\frac{[2\theta-\beta(\theta+1)]r(x,y)}{r(x,y)-q(x,y)}}\,dx\,dy
\Big).
\end{align*}
Passing to the limit as $R\to \infty$ in the above inequality, 
 using the monotone convergence theorem and \eqref{AS},  we obtain
$$
\int_{\mathbb{R}^N} u^{r(x,y)} \,dx\,dy=0,
$$
which is a contradiction with the fact that $u$ is a nontrivial solution.
\end{proof}

In the case of constant exponents, we have the following Liouville-type theorem.

\begin{theorem}\label{T2}
Let $u$ be a non-negative weak solution of the elliptic inequality
$$
(-\Delta_x)^{\alpha/2} u^p+|x|^{2\theta} (-\Delta_y)^{\beta/2} u^q \geq u^r,\quad
 (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
$$
where $0<\alpha,\beta<2$, $\theta\geq 0$, and $N=N_1+N_2\geq 2$. Suppose that
\begin{equation}\label{CDC}
1\leq \max\{p,q\}<r<Q\min\Big\{\frac{p}{Q-\alpha}, 
\frac{q}{\theta(2-\beta)+Q-\beta}\Big\},
\end{equation}
where $Q=N_1+N_2(\theta+1)$. Then $u$ is trivial.
\end{theorem}

\begin{proof}
Following the proof of Theorem \ref{T1} and taking
$$
(p(x,y),q(x,y),r(x,y))=(p,q,r),\quad (x,y)\in \mathbb{R}^{N_1}\times 
\mathbb{R}^{N_2},
$$
we obtain
$$
A(\varphi)\leq C |\Omega| R^{N_1+N_2(\theta+1)-\frac{\alpha r}{r-p}},\quad
 B(\varphi)\leq C|\Omega| R^{N_1+N_2(\theta+1)
 +\frac{[2\theta-\beta(\theta+1)]r}{r-q}}.
$$
Using \eqref{FE}, we obtain
\begin{equation} \label{Good}
\begin{aligned}
& \int_{\mathbb{R}^N} u^{r(x,y)} \varphi_0^\omega
 \Big(\frac{|x|^2}{R^2}+\frac{|y|^2}{R^{2(\theta+1)}}\Big)\,dx\,dy\\
&\leq C\Big(R^{N_1+N_2(\theta+1)-\frac{\alpha r}{r-p}}+R^{N_1+N_2(\theta+1)
 +\frac{[2\theta-\beta(\theta+1)]r}{r-q}}\Big).
\end{aligned}
\end{equation}
Now, we impose the conditions
\begin{gather*}
N_1+N_2(\theta+1)-\frac{\alpha r}{r-p}<0,\\
N_1+N_2(\theta+1)+\frac{[2\theta-\beta(\theta+1)]r}{r-q}<0,
\end{gather*}
which are equivalent to
$$
r<Q\min\big\{\frac{p}{Q-\alpha}, \frac{q}{\theta(2-\beta)+Q-\beta}\big\}.
$$
Therefore, under the condition \eqref{CDC}, passing to the limit as
$R\to \infty$ in \eqref{Good}, we obtain
$$
\int_{\mathbb{R}^N} u^{r} \,dx\,dy=0,
$$
which proves that $u$ is trivial.
\end{proof}

For the limit cases $\alpha\to 2^-$ and $\beta\to 2^-$, we obtain
 the following Liouville-type theorem.

\begin{corollary}\label{T3}
Let $u$ be a non-negative weak solution of the elliptic inequality
$$
(-\Delta_x) u^p+|x|^{2\theta} (-\Delta_y) u^q \geq u^r,\quad 
(x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
$$
where  $\theta\geq 0$ and $N=N_1+N_2\geq 2$. Suppose that
$$
1\leq \max\{p,q\}<r<\frac{Q\min\{p,q\}}{Q-2}.
$$
Then $u$ is trivial.
\end{corollary}

The above corollary follows by taking $\alpha=\beta=2$ in Theorem \ref{T2}, 
The following Liouville-type result which was established by Dolcetta 
and Cutri \cite{D} is an immediate consequence of Corollary \ref{T3}.

\begin{corollary}\label{T4}
Let $u$ be a non-negative weak solution of the elliptic inequality
$$
(-\Delta_x)u+|x|^{2\theta} (-\Delta_y) u \geq u^r,\quad
 (x,y)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2},
$$
where  $\theta\geq 0$ and $N=N_1+N_2\geq 2$. Suppose that
$$
1<r<\frac{Q}{Q-2}.
$$
Then $u$ is trivial.
\end{corollary}

The above corollary follows by taking
$p=q=1$ in Corollary \ref{T3}.

\begin{remark} \label{rmk2.6} \rm
The obtained results in this paper can be extended to
various classes of systems of elliptic inequalities including the system
\begin{gather*}
(-\Delta_x)^{\alpha/2} u^{p(x,y)}+|x|^{2\theta} 
(-\Delta_y)^{\beta/2}u^{q(x,y)} \geq v^{r(x,y)},\\
(-\Delta_x)^{\gamma/2} v^{\mu(x,y)}+|x|^{2\lambda} 
 (-\Delta_y)^{\tau/2}v^{\sigma(x,y)} \geq  u^{\xi(x,y)},
\end{gather*}
with appropriate functional parameters.
\end{remark}

\subsection*{Acknowledgements}
The third author extends his appreciation to Distinguished Scientist
Fellowship Program (DSFP) at King Saud University (Saudi Arabia).



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\end{document}