\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 105, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/105\hfil Liouville-type theorems for an elliptic system] {Liouville-type theorems for an elliptic system involving fractional Laplacian operators with mixed order} \author[M. Jleli, B. Samet \hfil EJDE-2017/105\hfilneg] {Mohamed Jleli, 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{Bessem Samet \newline Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia} \email{bessem.samet@gmail.com} \dedicatory{Communicated by Mokhtar Kirane} \thanks{Submitted February 2, 2017. Published April 18, 2017.} \subjclass[2010]{35B53, 35R11} \keywords{Liouville-type theorem; nonexistence; fractional Grushin operator} \begin{abstract} We study the nonexistence of nontrivial solutions for the nonlinear elliptic system \begin{gather*} G_{\alpha,\beta,\theta}(u^{p},u^{q}) = v^{r}\\ G_{\lambda,\mu,\theta}(v^{s},v^{t}) = u^{m}\\ u,v\geq 0, \end{gather*} where $0<\alpha,\beta,\lambda,\mu\leq 2$, $\theta\geq 0$, $m>q\geq p\geq 1$, $r>t\geq s\geq 1$, and $G_{\alpha,\beta,\theta}$ is the fractional operator of mixed orders $\alpha,\beta$, defined by $$ G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u +|x|^{2\theta} (-\Delta_y)^{\beta/2}v, \quad \text{in }\mathbb{R}^{N_1} \times \mathbb{R}^{N_2}. $$ Here, $(-\Delta_x)^{\alpha/2}$, $0<\alpha<2$, is the fractional Laplacian operator of order $\alpha/2$ with respect to the variable $x\in \mathbb{R}^{N_1}$, and $(-\Delta_y)^{\beta/2}$, $0<\beta<2$, is the fractional Laplacian operator of order $\beta/2$ with respect to the variable $y\in \mathbb{R}^{N_2}$. Via a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters. \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} 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 is credited to Cauchy \cite{C}. In the recent literature, this result was extended to the case of non-negative solutions of semilinear elliptic equations in the whole space $\mathbb{R}^N$ or in half-spaces, by Gidas and Spruck \cite{G}. In the case of the whole space $\mathbb{R}^N$, they established that if $1\leq p<\frac{N+2}{N-2}$, then the unique non-negative solution of $$ -\Delta u=C u^p \quad \text{in } \mathbb{R}^N, $$ where $C$ is a stricly positive constant, is the trivial solution. Using the moving planes method, a simple proof was presented by Chen and Li \cite{CL} in the range $0< p<\frac{N+2}{N-2}$. This result is optimal in the sense that for any $p\geq \frac{N+2}{N-2}$, we have infinitely many positive solutions. Several Liouville-type results were proved for various classes of degenerate equations. In \cite{S}, Serrin and Zou generalized the standard Liouville theorem for p-harmonic functions on the whole space and on exterior domains. In \cite{K,K2}, Liouville-type properties for some degenerate elliptic operators such as X-elliptic operators, Kohn-Laplacian and Ornstein-Uhlenbeck operators were presented. In \cite{D}, Dolcetta and Cutri considered an elliptic inequality involving the Grushin operator. More precisely, they studied the problem \begin{equation}\label{PG} u\geq 0,\quad G_\theta u \geq u^p \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 $11$, under the initial boundary conditions \begin{equation}\label{PK2} u(x,0)=u_0(x)\geq 0,\quad v(x,0)=v_0(x)\geq 0,\quad x\in \mathbb{R}^N, \end{equation} where $$ \Delta_i=\sum_{j=1}^{N_i} \frac{\partial^2}{\partial x_j^2},\quad i=1,2,\quad x_j\in R_i,\quad N_i=\dim(R_i)\leq N, $$ $R_i$ is a subspace of $\mathbb{R}^N$, and the algebraic sum $R_1+R_2=\mathbb{R}^N$. In the case of $R_1\neq R_2$, they proved that any solution to \eqref{PK}-\eqref{PK2} blows up in finite time if $$ \max\big\{\alpha_1-\frac{N_1}{2}-\frac{n_2}{2q},\alpha_2-\frac{N_2}{2} -\frac{n_1}{2p}\big\}>0, $$ where $\alpha_1=\frac{p+1}{pq-1}$, $\alpha_2=\frac{q+1}{pq-1}$, and $n_i=N_i-\dim(R_1\cap R_2)$, $i=1,2$. For other results in this directions, we refer to \cite{A,LE,M,M2,Y}. Recently, a lot of attention has been paid to the study of Liouville-type properties for elliptic equations and inequalities involving fractional operators. In \cite{MA}, via the moving plane method, Ma and Chen obtained a Liouville-type result for the system of equations \begin{gather*} (-\Delta)^{\mu/2} u = v^q\\ (-\Delta)^{\mu/2} v = u^p \\ u,v\geq 0, \end{gather*} where $\mu\in (0,2)$, $1q\geq p\geq 1$, $r>t\geq s\geq 1$, and $G_{\alpha,\beta,\theta}$ is the fractional operator of mixed orders $\alpha,\beta$, defined by defined by $$ G_{\alpha,\beta,\theta}(u,v)=(-\Delta_x)^{\alpha/2}u+|x|^{2\theta} (-\Delta_y)^{\beta/2}v, \quad \text{ in }\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, $$ where, $(-\Delta_x)^{\alpha/2}$, $0<\alpha<2$, is the fractional Laplacian operator of order $\alpha/2$ with respect to the variable $x\in \mathbb{R}^{N_1}$, and $(-\Delta_y)^{\beta/2}$, $0<\beta<2$, is the fractional Laplacian operator of order $\beta/2$ with respect to the variable $y\in \mathbb{R}^{N_2}$. Observe that the standard Grushin operator defined by \eqref{GO} can be written in the form $$ G_\theta u= G_{2,2,\theta}(u,u). $$ Via a weak formulation approach, we provide sufficient conditions for the nonexistence of nontrivial solutions to system \eqref{S1} in terms of space dimension and system parameters. 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 $(-\Delta)^s$, $0 \max\{\alpha_2,\theta_2, \alpha_2\theta_2\}$. Suppose that \begin{gather*} X^{\alpha_1} \leq A_1 Y+B_1 Y^{\theta_2},\\ Y^{\theta_1} \leq A_2 X+B_2 X^{\alpha_2}. \end{gather*} Then there is some constant $C>0$ such that \begin{align*} Y^{\alpha_1\theta_1} &\leq C\big[ \left(A_2^{\alpha_1}A_1 \right)^{\frac{\alpha_1\theta_1}{\alpha_1\theta_1-1}} +\left(A_2^{\alpha_1}B_1\right)^{\frac{\alpha_1\theta_1}{\alpha_1\theta_1-\theta_2}}\\ &\quad +\left(B_2^{\alpha_1}A_1^{\alpha_2}\right)^{\frac{\alpha_1\theta_1}{\alpha_1\theta_1 -\alpha_2}} +\left(B_2^{\alpha_1}B_1^{\alpha_2} \right)^{\frac{\alpha_1\theta_1}{\alpha_1\theta_1-\alpha_2\theta_2}}\big]. \end{align*} \end{lemma} \section{Main results} In this section, we state an prove the main results in this paper. We consider the elliptic system \eqref{S1} under the assumptions \begin{equation}\label{ass} 0<\alpha,\beta,\lambda,\mu\leq 2,\quad \theta\geq 0,\,\, m>q\geq p\geq 1,\quad r>t\geq s\geq 1. \end{equation} We adopt the following definition of solutions for \eqref{S1}. \begin{definition} \label{def2.1} \rm We say that the pair $(u,v)$ is a weak solution of \eqref{S1} if, $u\geq 0$, $v\geq 0$, $(u,v)\in L^m_{\rm loc}(\mathbb{R^N})\times L^r_{\rm loc} (\mathbb{R^N})$, $N=N_1+N_2$, and \begin{gather*} \int_{\mathbb{R}^N} v^r\varphi\,dx\,dy = \int_{\mathbb{R}^N} u^p (-\Delta_x)^{\alpha/2}\varphi\,dx\,dy +\int_{\mathbb{R}^N} |x|^{2\theta} u^q (-\Delta_y)^{\beta/2}\varphi\,dx\,dy,\\ \int_{\mathbb{R}^N} u^m\varphi\,dx\,dy = \int_{\mathbb{R}^N} v^s (-\Delta_x)^{\lambda/2}\varphi\,dx\,dy +\int_{\mathbb{R}^N} |x|^{2\theta} v^t (-\Delta_y)^{\mu/2}\varphi\,dx\,dy, \end{gather*} for every $\varphi\in C_0^\infty(\mathbb{R}^N)$, $\varphi\geq 0$. \end{definition} Let us introduce the following parameters: \begin{gather*} Q_1=\frac{m}{mr-ps}\,(\alpha s+\lambda r), \quad \overline{Q}_1=\frac{r}{mr-ps}\,(\lambda p+\alpha m),\\ Q_2=\frac{m}{mr-qs}\,\big(\lambda r-(2\theta-\beta(\theta+1))s\big), \\ \overline{Q}_2=\frac{r}{mr-tp}\,\big(\alpha m-(2\theta-\mu(\theta+1))p\big),\\ Q_3=\frac{m}{mr-pt}\,\big(\alpha t-(2\theta-\mu(\theta+1))r\big),\\ \overline{Q}_3=\frac{r}{mr-sq}\,\big(\lambda q-(2\theta-\beta(\theta+1))m\big),\\ Q_4=\frac{m}{mr-qt}\,\Big((\mu(\theta+1)-2\theta)r+(\beta(\theta+1)-2\theta)t\Big),\\ \overline{Q}_4=\frac{r}{mr-qt}\,\Big((\beta(\theta+1)-2\theta)m+(\mu(\theta+1)-2\theta)q\Big). \end{gather*} Our main result in this article is the following Liouville-type theorem. \begin{theorem}\label{T1} Let $(u,v)$ be a weak solution of system \eqref{S1}. Under assumptions \eqref{ass}, if \begin{equation}\label{eq1} Q<\max\{\Lambda_1,\Lambda_2\}, \end{equation} where \[ Q=N_1+N_2(\theta+1),\quad \Lambda_1 = \min\{Q_1,Q_2,Q_3,Q_4\},\quad \Lambda_2 = \min\{\overline{Q}_1,\overline{Q}_2,\overline{Q}_3,\overline{Q}_4\}, \] then the solution $(u,v)$ is trivial. \end{theorem} \begin{proof} Suppose that $(u,v)$ is a weak solution of \eqref{S1} such that $(u,v)\not\equiv (0,0)$. Let $\omega$ be a real number such that \begin{equation}\label{choix} \omega> \max\big\{\frac{m}{m-q},\frac{r}{r-t}\big\}. \end{equation} By the weak formulation of \eqref{S1}, for all $\varphi\in C_0^\infty(\mathbb{R}^N)$, $\varphi\geq 0$, we have \begin{equation}\label{W1} \int_{\mathbb{R}^N} v^r\varphi^\omega\,dx\,dy = \int_{\mathbb{R}^N} u^p (-\Delta_x)^{\alpha/2}\varphi^\omega\,dx\,dy +\int_{\mathbb{R}^N} |x|^{2\theta} u^q (-\Delta_y)^{\beta/2}\varphi^\omega\,dx\,dy \end{equation} and \begin{equation}\label{W2} \int_{\mathbb{R}^N} u^m\varphi^\omega\,dx\,dy = \int_{\mathbb{R}^N} v^s (-\Delta_x)^{\lambda/2}\varphi^\omega\,dx\,dy +\int_{\mathbb{R}^N} |x|^{2\theta} v^t (-\Delta_y)^{\mu/2}\varphi^\omega\,dx\,dy. \end{equation} Using Lemma \ref{L1} and H\"older's inequality with parameters $\frac{m}{p}$ and $\frac{m}{m-p}$, we obtain \begin{align*} &\int_{\mathbb{R}^N} u^p (-\Delta_x)^{\alpha/2}\varphi^\omega\,dx\,dy\\ &\leq \omega \int_{\mathbb{R}^N} u^p \varphi^{\omega-1} |(-\Delta_x)^{\alpha/2}\varphi|\,dx\,dy\\ &=\omega \int_{\mathbb{R}^N} u^p \varphi^{\frac{\omega p}{m}} \varphi^{(\omega-1-\frac{\omega p}{m})} |(-\Delta_x)^{\alpha/2}\varphi|\,dx\,dy\\ &\leq \omega\Big(\int_{\mathbb{R}^N} u^m\varphi^\omega\,dx\,dy\big)^{p/m} \Big(\int_{\mathbb{R}^N} \varphi^{(\omega-1-\frac{\omega p}{m})\frac{m}{m-p}}| (-\Delta_x)^{\alpha/2}\varphi|^{\frac{m}{m-p}}\,dx\,dy\Big)^{\frac{m-p}{m}}\\ &=\omega\Big(\int_{\mathbb{R}^N} u^m\varphi^\omega\,dx\,dy\Big)^{p/m} \Big(\int_{\mathbb{R}^N} \varphi^{\omega-\frac{m}{m-p}}|(-\Delta_x)^{\alpha/2}\varphi|^{\frac{m}{m-p}} \,dx\,dy\Big)^{\frac{m-p}{m}}. \end{align*} Note that thanks to the choice \eqref{choix} of the parameter $\omega$, we have $$ \int_{\mathbb{R}^N} \varphi^{\omega-\frac{m}{m-p}}|(-\Delta_x)^{\alpha/2}\varphi|^{\frac{m}{m-p}}\,dx\,dy <\infty. $$ Therefore, we have the estimate \begin{equation} \label{es1} \begin{aligned} &\int_{\mathbb{R}^N} u^p (-\Delta_x)^{\alpha/2}\varphi^\omega\,dx\,dy\\ &\leq \omega\Big(\int_{\mathbb{R}^N} u^m\varphi^\omega\,dx\,dy\Big)^{p/m} \Big(\int_{\mathbb{R}^N} \varphi^{\omega-\frac{m}{m-p}}|(-\Delta_x)^{\alpha/2}\varphi|^{\frac{m}{m-p}}\,dx\,dy \Big)^{\frac{m-p}{m}}. \end{aligned} \end{equation} Again, using Lemma \ref{L1} and H\"older's inequality with parameters $\frac{m}{q}$ and $\frac{m}{m-q}$, we obtain \begin{align*} & \int_{\mathbb{R}^N} |x|^{2\theta} u^q (-\Delta_y)^{\beta/2}\varphi^\omega\,dx\,dy\\ &\leq \omega \int_{\mathbb{R}^N} u^q |x|^{2\theta}\varphi^{\omega-1} |(-\Delta_y)^{\beta/2}\varphi|\,dx\,dy \\ &= \omega \int_{\mathbb{R}^N} u^q \varphi^{\frac{\omega q}{m}} |x|^{2\theta}\varphi^{(\omega-1-\frac{\omega q}{m})} |(-\Delta_y)^{\beta/2}\varphi|\,dx\,dy \\ &\leq \omega \Big(\int_{\mathbb{R}^N} u^m \varphi^{\omega} \,dx\,dy\Big)^{q/m} \Big(\int_{\mathbb{R}^N} |x|^{\frac{2\theta m}{m-q}} \varphi^{\left(\omega-1-\frac{\omega q}{m} \right)\frac{m}{m-q}} |(-\Delta_y)^{\beta/2}\varphi|^{\frac{m}{m-q}}\,dx\,dy \Big)^{\frac{m-q}{m}}\\ &=\omega \Big(\int_{\mathbb{R}^N} u^m \varphi^{\omega} \,dx\,dy\Big)^{q/m} \Big( \int_{\mathbb{R}^N} |x|^{\frac{2\theta m}{m-q}} \varphi^{\omega-\frac{m}{m-q}} |(-\Delta_y)^{\beta/2}\varphi|^{\frac{m}{m-q}}\,dx\,dy \Big)^{\frac{m-q}{m}}. \end{align*} From the choice \eqref{choix} of the parameter $\omega$, we have $$ \int_{\mathbb{R}^N} |x|^{\frac{2\theta m}{m-q}} \varphi^{\omega-\frac{m}{m-q}} |(-\Delta_y)^{\beta/2}\varphi|^{\frac{m}{m-q}}\,dx\,dy<\infty. $$ Therefore, we have the estimate % \label{es2} \begin{align*} &\int_{\mathbb{R}^N} |x|^{2\theta} u^q (-\Delta_y)^{\beta/2}\varphi^\omega\,dx\,dy\\ &\leq \omega \Big(\int_{\mathbb{R}^N} u^m \varphi^{\omega} \,dx\,dy\Big)^{q/m} \Big( \int_{\mathbb{R}^N} |x|^{\frac{2\theta m}{m-q}} \varphi^{\omega-\frac{m}{m-q}} |(-\Delta_y)^{\beta/2}\varphi|^{\frac{m}{m-q}}\,dx\,dy \Big)^{\frac{m-q}{m}}. \end{align*} Combining this with \eqref{W1} and \eqref{es1}, we obtain \begin{equation}\label{EES1} \int_{\mathbb{R}^N} v^r \varphi^\omega\,dx\,dy \leq A_\varphi \Big(\int_{\mathbb{R}^N} u^m \varphi^{\omega} \,dx\,dy\Big)^{p/m} +B_\varphi \Big(\int_{\mathbb{R}^N} u^m \varphi^{\omega} \,dx\,dy\Big)^{q/m}, \end{equation} where \begin{gather*} A_\varphi = \omega \Big(\int_{\mathbb{R}^N} \varphi^{\omega-\frac{m}{m-p}}|(-\Delta_x)^{\alpha/2}\varphi|^{\frac{m}{m-p}} \,dx\,dy\Big)^{\frac{m-p}{m}},\\ B_\varphi = \omega \Big( \int_{\mathbb{R}^N} |x|^{\frac{2\theta m}{m-q}} \varphi^{\omega-\frac{m}{m-q}} |(-\Delta_y)^{\beta/2}\varphi|^{\frac{m}{m-q}}\,dx\,dy \Big)^{\frac{m-q}{m}}. \end{gather*} Similarly, using H\"older's inequality with parameters $\frac{r}{s}$ and $\frac{r}{r-s}$, we obtain \begin{equation} \label{es11} \begin{aligned} &\int_{\mathbb{R}^N} v^s (-\Delta_x)^{\lambda/2}\varphi^\omega\,dx\,dy\\ &\leq \omega\Big(\int_{\mathbb{R}^N} v^r\varphi^\omega\,dx\,dy\Big)^{s/r} \Big(\int_{\mathbb{R}^N} \varphi^{\omega-\frac{r}{r-s}}|(-\Delta_x)^{\lambda/2}\varphi|^{\frac{r}{r-s}} \,dx\,dy\Big)^{\frac{r-s}{r}}. \end{aligned} \end{equation} Again, H\"older's inequality with parameters $\frac{r}{t}$ and $\frac{r}{r-t}$ yields \begin{equation} \label{es22} \begin{aligned} &\int_{\mathbb{R}^N} |x|^{2\theta} v^t (-\Delta_y)^{\mu/2}\varphi^\omega\,dx\,dy\\ &\leq \omega \Big(\int_{\mathbb{R}^N} v^r \varphi^{\omega} \,dx\,dy\Big)^{t/r} \Big( \int_{\mathbb{R}^N} |x|^{\frac{2\theta r}{r-t}} \varphi^{\omega-\frac{r}{r-t}} |(-\Delta_y)^{\mu/2}\varphi|^{\frac{r}{r-t}}\,dx\,dy \Big)^{\frac{r-t}{r}}. \end{aligned} \end{equation} Combining \eqref{W2} with the estimates \eqref{es11} and \eqref{es22}, we obtain \begin{equation}\label{EES2} \int_{\mathbb{R}^N} u^m \varphi^\omega\,dx\,dy \leq C_\varphi \Big(\int_{\mathbb{R}^N} v^r \varphi^{\omega} \,dx\,dy \Big)^{s/r} +D_\varphi \Big(\int_{\mathbb{R}^N} v^r \varphi^{\omega} \,dx\,dy\Big)^{t/r}, \end{equation} where \begin{gather*} C_\varphi = \omega \Big(\int_{\mathbb{R}^N} \varphi^{\omega-\frac{r}{r-s}}|(-\Delta_x)^{\lambda/2}\varphi|^{\frac{r}{r-s}} \,dx\,dy\Big)^{\frac{r-s}{r}},\\ D_\varphi = \omega \Big( \int_{\mathbb{R}^N} |x|^{\frac{2\theta r}{r-t}} \varphi^{\omega-\frac{r}{r-t}} |(-\Delta_y)^{\mu/2}\varphi|^{\frac{r}{r-t}}\,dx\,dy \Big)^{\frac{r-t}{r}}. \end{gather*} Let \[ X=\Big(\int_{\mathbb{R}^N} u^m \varphi^\omega\,dx\,dy\Big)^{p/m}, \quad Y=\Big(\int_{\mathbb{R}^N} v^r \varphi^\omega\,dx\,dy\Big)^{s/r}. \] Combining the estimates \eqref{EES1} and \eqref{EES2}, we obtain the system of inequalities \begin{gather*} X^{m/p} \leq C_\varphi Y+D_\varphi Y^{\frac{t}{s}}, \\ Y^{r/s} \leq A_\varphi X+B_\varphi X^{\frac{q}{p}}. \end{gather*} Using Lemma \ref{L2}, we obtain \begin{equation}\label{ESTY} \begin{aligned} Y^{\frac{mr}{ps}} &\leq C \Big( \Big(A_\varphi^{m/p}C_\varphi\Big)^{\frac{mr}{mr-ps}} +\Big(A_\varphi^{m/p}D_\varphi\Big)^{\frac{mr}{mr-pt}}\\ &\quad +\Big(B_\varphi^{m/p}C_\varphi^{\frac{q}{p}}\Big)^{\frac{mr}{mr-qs}} +\Big(B_\varphi^{m/p}D_\varphi^{\frac{q}{p}}\Big)^{\frac{mr}{mr-qt}}\Big). \end{aligned} \end{equation} Similarly, we obtain \begin{equation}\label{ESTX} \begin{aligned} X^{\frac{mr}{ps}} &\leq C \Big( \Big(C_\varphi^{r/s}A_\varphi\Big)^{\frac{mr}{mr-ps}} +\Big(C_\varphi^{r/s}B_\varphi\Big)^{\frac{mr}{mr-qs}} \\ &\quad +\Big(D_\varphi^{r/s}A_\varphi^{\frac{t}{s}}\Big)^{\frac{mr}{mr-pt}} +\Big(D_\varphi^{r/s}B_\varphi^{\frac{t}{s}}\Big)^{\frac{mr}{mr-qt}}\Big). \end{aligned} \end{equation} Now, 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 $\varphi_0$ is the classical cutoff function, that is, $\varphi_0\in C_0^\infty(0,\infty)$ is a smooth decreasing function such that \begin{gather*} 0\leq \varphi_0\leq 1, \quad |\varphi_0'(\eta)|\leq C\eta^{-1}, \\ \varphi_0(\eta)=\begin{cases} 1 &\text{if } 0<\eta\leq 1,\\ 0 &\text{if } \eta\geq 2. \end{cases} \end{gather*} We use the change of variables $$ x=Rz\quad\text{and}\quad y=R^{\theta+1}w. $$ In this case, we have $$ \eta:=\frac{|x|^2}{R^2}+\frac{|y|^2}{R^{2(\theta+1)}}=|z|^2+|w|^2,\quad (z,w)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}. $$ Let $\Omega$ be the subset of $\mathbb{R}^{N_1}\times \mathbb{R}^{N_2}$ defined by $$ \Omega=\{(z,w)\in \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}: 1\leq |z|^2+|w|^2\leq 2\}. $$ We have the following estimates. \smallskip \noindent$\bullet$ Estimate of $A_\varphi$. Using the above change of variables, we obtain $$ A_\varphi=\omega R^{\frac{Q(m-p)-\alpha m}{m}} \Big(\int_{\Omega} [\varphi_0(\eta)]^{\omega-\frac{m}{m-p}} |(-\Delta_z)^{\alpha/2} \varphi_0(\eta)|^{\frac{m}{m-p}}\,dz\,dw\Big)^{\frac{m-p}{m}}. $$ Observe that $$ \int_{\Omega} [\varphi_0(\eta)]^{\omega-\frac{m}{m-p}} |(-\Delta_z)^{\alpha/2} \varphi_0(\eta)|^{\frac{m}{m-p}}\,dz\,dw $$ is a real number independent on $R$. Therefore, we have \begin{equation}\label{ESA} A_\varphi=C R^{\frac{Q(m-p)-\alpha m}{m}}, \end{equation} where $C$ is a positive constant independent on $R$. \smallskip \noindent $\bullet$ Estimate of $B_\varphi$. Using the same change of variable as above, we obtain \begin{align*} B_\varphi&=\omega R^{\frac{(2\theta-\beta(\theta+1))m+Q(m-q)}{m}} \\ &\quad\times \Big(\int_{\Omega} |z|^{\frac{2\theta m}{m-q}} [\varphi_0(\eta)]^{\omega-\frac{m}{m-q}} |(-\Delta_w)^{\beta/2} \varphi_0(\eta)|^{\frac{m}{m-q}}\,dz\,dw\Big)^{\frac{m-q}{m}}. \end{align*} Since $$ \int_{\Omega} |z|^{\frac{2\theta m}{m-q}} [\varphi_0(\eta)]^{\omega-\frac{m}{m-q}} |(-\Delta_w)^{\beta/2} \varphi_0(\eta)|^{\frac{m}{m-q}}\,dz\,dw $$ is a real number independent on $R$, we have \begin{equation}\label{ESB} B_\varphi=C R^{\frac{\left(2\theta-\beta(\theta+1)\right)m+Q(m-q)}{m}}. \end{equation} \noindent $\bullet$ Estimate of $C_\varphi$. We argue as previously, to obtain \begin{equation}\label{ESC} C_\varphi=C R^{\frac{Q(r-s)-\lambda r}{r}}. \end{equation} \noindent $\bullet$ Estimate of $D_\varphi$. We have the estimate \begin{equation}\label{ESD} D_\varphi=C R^{\frac{\left(2\theta-\mu(\theta+1)\right)r+Q(r-t)}{r}}. \end{equation} Using the estimates \eqref{ESTX}, \eqref{ESA}, \eqref{ESB}, \eqref{ESC} and \eqref{ESD}, we obtain \begin{equation}\label{FEX} X^{\frac{mr}{ps}} \leq C\left(R^{\tau_1}+R^{\tau_2}+R^{\tau_3}+R^{\tau_4}\right), \end{equation} where \begin{gather*} \tau_1 = \big(\frac{rm}{rm-ps}\big) \Big(\frac{Q(mr-ps)-m(\lambda r+\alpha s)}{ms} \Big),\\ \tau_2 = \big(\frac{rm}{rm-qs}\big) \Big(\frac{Q(mr-sq)+m(s(2\theta-\beta(\theta+1)) -\lambda r)}{ms}\Big),\\ \tau_3 = \big(\frac{rm}{rm-pt}\big) \Big(\frac{Q(mr-pt)+m(r(2\theta-\mu(\theta+1)) -\alpha t)}{ms}\Big),\\ \tau_4 = \big(\frac{rm}{rm-qt}\big) \Big(\frac{Q(mr-qt)+m(r(2\theta-\mu(\theta+1)) +(2\theta-\beta(\theta+1)))}{ms}\Big). \end{gather*} Similarly, using the estimates \eqref{ESTY}, \eqref{ESA}, \eqref{ESB}, \eqref{ESC} and \eqref{ESD}, we obtain \begin{equation}\label{FEY} Y^{\frac{mr}{ps}} \leq C\left(R^{\kappa_1}+R^{\kappa_2}+R^{\kappa_3}+R^{\kappa_4}\right), \end{equation} where \begin{gather*} \kappa_1 = \big(\frac{rm}{rm-ps}\big) \Big(\frac{Q(mr-ps)-r(\alpha m+\lambda p)}{rp} \Big),\\ \kappa_2 = \big(\frac{rm}{rm-tp}\big) \Big(\frac{Q(mr-pt) +r(p(2\theta-\mu(\theta+1))-\alpha m)}{rp}\Big),\\ \kappa_3 = \big(\frac{rm}{rm-sq}\big) \Big(\frac{Q(mr-sq) +r(m(2\theta-\beta(\theta+1))-\lambda q)}{rp}\Big),\\ \kappa_4 = \big(\frac{rm}{rm-tq}\big) \Big(\frac{Q(mr-qt) +r(m(2\theta-\beta(\theta+1))+(2\theta-\mu(\theta+1)))}{rp}\Big). \end{gather*} Now, using \eqref{eq1}, we can see that $$ \max\{\tau_i: i=,1,2,3,4\}<0 $$ or $$ \max\{\kappa_i: i=,1,2,3,4\}<0. $$ \smallskip \noindent\textbf{Case 1.} If $\max\{\tau_i: i=,1,2,3,4\}<0$. In this case, passing to the limit as $R\to \infty$ in \eqref{FEX}, and using the monotone convergence theorem, we obtain $$ \lim_{R\to \infty} \Big(\int_{\mathbb{R}^N} u^m \Big[\varphi_0\Big(\frac{|x|^2}{R^2} +\frac{|y|^2}{R^{2(\theta+1)}}\Big)\Big]^\omega \,dx\,dy\Big)^{r/s} =\Big(\int_{\mathbb{R}^N} u^m \,dx\,dy\Big)^{r/s}=0, $$ which yields $(u,v)\equiv (0,0)$, that is a contradiction with the fact that $(u,v)$ is a nontrivial solution. \smallskip \noindent\textbf{Case 2.} If $\max\{\kappa_i: i=,1,2,3,4\}<0$. As in the previous case, passing to the limit as $R\to \infty$ in \eqref{FEY}, and using the monotone convergence theorem, we obtain $$ \lim_{R\to \infty} \Big(\int_{\mathbb{R}^N} v^r \Big[\varphi_0 \Big(\frac{|x|^2}{R^2}+\frac{|y|^2}{R^{2(\theta+1)}}\Big)\Big]^\omega \,dx\,dy \Big)^{m/p}=\Big(\int_{\mathbb{R}^N} v^r \,dx\,dy\Big)^{m/p}=0, $$ which yields $(u,v)\equiv (0,0)$, that is a contradiction. In both cases, we get a contradiction. As consequence, we infer that the only weak solution to system \eqref{S1} is the trivial solution. \end{proof} The following Liouville-type results follow from Theorem \ref{T1}. Taking $\alpha=\lambda$, $\beta=\mu=2$ and $p=s=q=t=1$ in Theorem \ref{T1}, we obtain the following Liouville-type property. \begin{corollary}\label{CR1} Let $(u,v)$ be a weak solution of the elliptic system \begin{gather*} (-\Delta_x)^{\alpha/2} u+|x|^{2\theta} (-\Delta_y) u = v^r\\ (-\Delta_x)^{\alpha/2} v+|x|^{2\theta} (-\Delta_y) v = u^m, u,v\geq 0, \end{gather*} where $0<\alpha\leq 2$, $\theta\geq 0$, $m>1$ and $r>1$. If $$ Q< \frac{\alpha}{mr-1}\max\left\{m(r+1),r(m+1)\right\}, $$ then $(u,v)$ is trivial. \end{corollary} Taking $\alpha=2$ in Corollary \ref{CR1}, we obtain the following Liouville-type property for an elliptic system involving the standard Grushin operator. \begin{corollary}\label{CR2} Let $(u,v)$ be a weak solution of the elliptic system \begin{gather*} (-\Delta_x) u+|x|^{2\theta} (-\Delta_y) u = v^r\\ (-\Delta_x) v+|x|^{2\theta} (-\Delta_y) v = u^m \\ u,v\geq 0, \end{gather*} where $\theta\geq 0$, $m>1$ and $r>1$. If $$ Q< \frac{2}{mr-1}\max\left\{m(r+1),r(m+1)\right\}, $$ then the solution $(u,v)$ is trivial. \end{corollary} Taking $u=v$ and $m=r$ in Corollary \ref{CR1}, we obtain the following result. \begin{corollary}\label{CR33} Let $u$ be a weak solution of the elliptic equation $$ (-\Delta_x)^{\alpha/2} u+|x|^{2\theta} (-\Delta_y) u = u^r,\quad u\geq 0, $$ where $0<\alpha\leq 2$, $\theta\geq 0$. If \begin{equation}\label{KAR} 11$ and $r>1$. If $$ Q< \frac{\beta(\theta+1)-2\theta}{mr-1}\max\left\{m(r+1),r(m+1)\right\}, $$ then the solution $(u,v)$ is trivial. \end{corollary} \begin{remark}\rm Taking $\beta=2$ in Corollary \ref{CR44}, we obtain the Liouville-type property given by Corollary \ref{CR2}. \end{remark} Taking $u=v$ and $m=r$ in Corollary \ref{CR44}, we obtain the following result. \begin{corollary}\label{CR331} Let $u$ be a weak solution of the elliptic equation $$ (-\Delta_x) u+|x|^{2\theta} (-\Delta_y)^{\beta/2} u = u^r,\quad u\geq 0, $$ where $0<\beta\leq 2$, $\theta\geq 0$. If $$ 1