\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 09, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2016/09\hfil Non-local elliptic systems] {Non-local elliptic systems on the \\ Heisenberg group} \author[N. Al-Salti, S. Kerbal \hfil EJDE-2016/09\hfilneg] {Nasser Al-Salti, Sebti Kerbal} \address{Nasser Al-Salti \newline Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khodh 123, Muscat, Oman} \email{nalsalti@squ.edu.om} \address{Sebti Kerbal \newline Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khodh 123, Muscat, Oman} \email{skerbal@squ.edu.om} \thanks{Submitted September 13, 2015. Published January 6, 2016.} \subjclass[2010]{35R03, 35J60, 35D02} \keywords{Fractional diffusion operator; nonlinear elliptic systems; \hfill\break\indent convexity inequality} \begin{abstract} We present Liouville type results for certain systems of nonlinear elliptic equations containing fractional powers of the Laplacian on the Heisenberg group. Our method of proof is based on the test function method and a recent inequality proved by Alsaedi, Ahmad, and Kirane, leading to the derivation of sufficient conditions in terms of space dimension and systems parameters. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} \label{intro} This article concerns Liouville type results for two nonlinear systems of elliptic equations with nonlocal diffusions posed on the Heisenberg group. We start with the system \begin{equation}\label{System1} \begin{gathered} (-\Delta_{\mathbb{H}})^{\mu/2} u = | v |^q ,\quad q>1, \\ (-\Delta_{\mathbb{H}})^{\nu/2} v = | u |^p ,\quad p>1, \end{gathered} \end{equation} posed in ${\mathbb{R}}^{2N+1}$, and where the fractional power of the Laplacian on the Heisenberg group $(- \Delta_{\mathbb{H}})^{\delta/2}$ ($0 < \delta < 2$) accounts for anomalous diffusion and is to be defined later. Using the test function method and a variant of Cordoba-Cordoba's inequality \cite{CC} for the Heisenberg group proved in \cite{AAK}, we find a relation relating $N, \mu, \nu, p$ and $q$ leading to Liouville type results. Let us point out that we overcome a difficulty raised by the test function by using the inequality proved in \cite{AAK} for $(- \Delta_{\mathbb{H}})^{\frac{\mu}{2}}$. Then we consider the system \begin{equation}\label{System3} \begin{gathered} (- \Delta_{\mathbb{H}})^{\mu_1/2} |u| + (- \Delta_{\mathbb{H}})^{\mu_2/2} |v| = |v|^q, \quad q>1, \\ (- \Delta_{\mathbb{H}})^{\nu_1/2} |v| + (- \Delta_{\mathbb{H}})^{\nu_2/2} |u| = |u|^p, \quad p>1, \end{gathered} \end{equation} where $0<\mu_{i},\nu_{i}\leq 2$ ($i=1,2$) are constants. Here the positivity condition on the solutions is omitted and replaced by the absolute value of $u$ and $v$. \section{Preliminaries} For the reader's convenience, let us briefly recall the definition and basic properties of the Heisenberg group and the inequality in \cite{AAK}. \subsection{Heisenberg group} The Heisenberg group $\mathbb{H}$, whose points will be denoted by $\eta = (x, y, \tau)$, is the Lie group $(\mathbb{R}^{2N+1}, \circ)$ with the non-commutative group operation $ \circ$ defined by \[ \eta \circ \tilde \eta =( x + \tilde x, y + \tilde y, \tau + \tilde \tau + 2(x\cdot\tilde y - \tilde x\cdot y)), \] where ``$\cdot$'' is the usual inner product in $\mathbb{R}^{N}$. The Laplacian $\Delta_{\mathbb{H}}$ over $\mathbb{H}$ is obtained from the vector fields $X_{i}= \frac{\partial}{\partial x_{i}} + 2 y_{i} \frac{\partial}{\partial \tau}$ and $Y_{i}= \frac{\partial}{\partial y_{i}} - 2 x_{i} \frac{\partial}{\partial \tau} $, by \begin{equation}\label{Lapformula} \Delta_{\mathbb{H}} = \sum_{i=1}^{N}(X_{i}^{2} + Y_{i}^{2}). \end{equation} Explicit computation gives the expression \begin{equation}\label{explicitexpression} \Delta_{\mathbb{H}}= \sum_{i=1}^{N} \Big(\frac{\partial^{2}}{\partial x_{i}^{2}} + \frac{\partial^{2}}{\partial y_{i}^{2}} +4 y_{i}\frac{\partial^{2}}{\partial x_{i}\partial \tau} - 4 x_{i}\frac{\partial^{2}}{\partial y_{i}\partial \tau} + 4 (x_{i}^{2}+ y_{i}^{2}) \frac{\partial^{2}}{\partial \tau^{2}} \Big). \end{equation} A natural group of dilations on $\mathbb{H}$ is given by \[ \delta_{\lambda}(\eta)=(\lambda x,\lambda y,\lambda ^{2} \tau), \quad \lambda >0, \] whose Jacobian determinant is $\lambda^{Q}$, where \begin{equation}\label{dimenssion} Q=2N+2 \end{equation} is the homogeneous dimension of $\mathbb{H}$. The operator $\Delta_{\mathbb{H}}$ is a degenerate elliptic operator. It is invariant with respect to the left translation of $\mathbb{H}$ and homogeneous with respect to the dilations $\delta_{\lambda}$. More precisely, we have \begin{equation}\label{dilation} \begin{gathered} \Delta_{\mathbb{H}}(u ( \eta \circ \tilde \eta)) = (\Delta_{\mathbb{H}}u)(\eta \circ \tilde \eta),\\ \Delta_{\mathbb{H}}(u \circ \delta_{\lambda}) = \lambda^{2}(\Delta_{\mathbb{H}}u ) \circ \delta_{\lambda}, \quad \eta, \tilde \eta \in \mathbb{H}. \end{gathered} \end{equation} The natural distance from $\eta $ to the origin is \begin{equation}\label{distance} | \eta |_{\mathbb{H}}= \Big( \tau^{2} + \Big(\sum_{i=1}^{N} (x_{i}^{2}+ y_{i}^{2})\Big)^{2} \Big)^{1/4}. \end{equation} \subsection{Fractional powers of sub-elliptic Laplacians} The representation of the fractional power of $ (- \Delta_{\mathbb{H}})^{s} $ is given by the following theorem. \begin{theorem} \label{thm2.1} The operator $\Delta_{\mathbb{H}}$ is a positive self-adjoint operator with domain $W_{\mathbb{H}}^{2,2}(\mathbb{H})$. Denote now by $\{E(\lambda)\}$ the spectral resolution of $\Delta_{\mathbb{H}}$ in $L^{2}(\mathbb{H})$. If $\alpha>0$, then $$ (-\Delta_{\mathbb{H}})^{\alpha/2} =\int_0^{+\infty}\lambda^{\alpha/2}\,dE(\lambda), $$ with domain $$ W_{\mathbb{H}}^{\alpha,2}(\mathbb{H}):=\{v\in L^{2}(\mathbb{H}); \int_0^{+\infty}\lambda^{\alpha}\,d\langle E(\lambda)v,v\rangle <\infty\},$$ endowed with graph norm. \end{theorem} \begin{proposition}[\cite{AAK}] \label{Prop1} Assume that the function $\varphi \in C^{\infty}_0({\mathbb{R}}^{2N+1})$. Then \begin{equation}\label{AAKinequality} \sigma \varphi^{\sigma-1} (- \Delta_{\mathbb{H}})^{\sigma/2}\varphi \geq (- \Delta_{\mathbb{H}})^{\sigma/2}\varphi^{\sigma} \end{equation} holds point-wise. \end{proposition} A proof of the above proposition can be found in \cite{AAK}. \section{Main results} The definition of solutions we adopt for system \eqref{System1} is as follows. \begin{definition} \label{DefSol} \rm We say that the pair $(u,v)$ is a weak solution of \eqref{System1}, if \begin{gather} (u,v)\in L^p_{\rm loc}({\mathbb{R}}^{2N+1}) \times L^q_{\rm loc}({\mathbb{R}}^{2N+1}), \nonumber \\ \label{weaksoleqn1} \int_{{\mathbb{R}}^{2N+1}} u {(-\Delta)^{\mu/2} \psi}\,d x = \int_{{\mathbb{R}}^{2N+1}} | v |^q \psi \,d x , \\ \label{weaksoleqn2} \int_{{\mathbb{R}}^{2N+1}} v {(-\Delta)^{\nu/2} \psi} \,d x = \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi \,d x , \end{gather} for any nonnegative test function $ \psi \in \mathcal{C}^{\infty}_0 ({\mathbb{R}}^{2N+1})$. \end{definition} Before we present our results, let us mention some important works on Liouville type theorems for the classical nonlinear elliptic equations/systems on the Heisenberg group. V\'{e}ron and Pohozaev \cite{VP} improved the study of Birindelli, Capuzzo Dolcetta and Cutri \cite{BirCapCut} concerning the equation \begin{equation}\label{BirCapCutinequality} \Delta_{\mathbb{H}}(a u) + | u|^p\leq 0 \end{equation} with a bounded function $a$ and $1

p' $ (${(\beta-1)p'-\beta \frac{p'}{p}>0})$ we have $$ \int_{{\mathbb{R}}^{2N+1}} \psi^{(\beta-1)p'-\beta \frac{p'}{p}} |{(-\Delta_{\mathbb{H}})^{\delta/2} \psi}|^{p'} \,d \eta = \int_{K} \psi^{(\beta-1)p'-\beta \frac{p'}{p}} |{(-\Delta_{\mathbb{H}})^{\delta/2} \psi}|^{p'} \,d \eta <\infty,\\ $$ where $K:= \operatorname{supp}(\psi) $ stands for support of $\psi$, and $p+p'=pp'$. For the proof of our main results, we consider a cut-off function $\varphi\in \mathcal{C}^{\infty}_0 ({\mathbb{R}})$ such that $0 \leq \varphi \leq 1$, $|\varphi'(r)| \leq \frac{C}{r}$, and for any $r > 0 $, \[ \varphi(r)=\begin{cases} 1 & \text{if } r\leq 1,\\ 0 & \text{if } r\geq 2. \end{cases} \] \begin{proof}[Proof of Theorem \ref{tabs0}] From \eqref{weaksoleqn1} and \eqref{weaksoleqn2} we have \begin{gather*} \int_{{\mathbb{R}}^{2N+1}} u {(-\Delta_{\mathbb{H}})^{\mu/2} \psi^{\beta}} d \eta = \int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d \eta , \\ \int_{{\mathbb{R}}^{2N+1}} v {(-\Delta_{\mathbb{H}})^{\nu/2} \psi^{\beta}} d \eta = \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta , \end{gather*} for any nonnegative test function $ \psi^{\beta} \in \mathcal{C}^{\infty}_0 ({\mathbb{R}}^{N})$ with $\beta >\max{(p',q')}$. Using the convexity inequality in Proposition \ref{Prop1} and the H\"older inequality, we estimate the first integral over $K$ as follows, \begin{align*} &\int_{{\mathbb{R}}^{2N+1}} u (-\Delta_{\mathbb{H}})^{\mu/2} \psi^{\beta} \,d \eta \\ &\leq \beta \int_{K} u \psi^{\beta/p} \psi^{-\beta/p} \psi^{\beta-1} {(-\Delta_{\mathbb{H}})^{\mu/2} \psi} \,d \eta\\ & \leq \beta \Big(\int_{K}| u |^p \psi^{\beta} \,d \eta \Big)^{1/p} \Big(\int_{K} \psi^{(\beta-1)p'-\beta \frac{p'}{p}} |{(-\Delta_{\mathbb{H}})^{\mu/2} \psi}|^{p'} \,d \eta \Big)^{1/p'}, \end{align*} where $K:= \operatorname{supp}( \psi)$ and ${{p+p'=pp'}}$. Similarly, we obtain the estimate for the second integral \begin{align*} &\int_{{\mathbb{R}}^{2N+1}} v {(-\Delta_{\mathbb{H}})^{\nu/2} \psi^{\beta}} \,d \eta \\ &\leq \beta \int_{K} v \psi^{\beta/q} \psi^{-\beta/q} \psi^{\beta-1} {(-\Delta_{\mathbb{H}})^{\nu/2} \psi} \,d \eta \\ &\leq \beta \Big(\int_{K}| v |^q \psi^{\beta} \,d \eta \Big)^{1/q} \Big(\int_{K} \psi^{(\beta-1)q'-\beta \frac{q'}{q}} |{(-\Delta_{\mathbb{H}})^{\nu/2} \psi}|^{q'} \,d \eta \Big)^{1/q'}, \end{align*} where $q+q'=qq'$. If we set $$ \mathcal{A}( r, \delta) := \Big(\int_{K} \psi^{(\beta-1)r'-\beta \frac{r'}{r}} |{(-\Delta_{\mathbb{H}}) ^{\delta/2} \psi}|^{r'} \,d \eta \Big)^{1/r'}, $$ then we can write \begin{gather}\label{Ineg1} \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} d \eta \leq \beta \mathcal{A}( q, \nu) \Big( \int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} d\eta \Big)^{1/q}, \\ \label{Ineg2} \int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} d\eta \leq \beta\mathcal{A}( p, \mu) \Big( \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} d\eta \Big)^{1/p}. \end{gather} Therefore, \begin{equation}\label{Ineg3} \Big(\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d \eta \Big)^{1/q} \leq \beta^{1/q}\Big( \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta \Big)^{1/pq} \Big( \mathcal{A}( p, \mu) \Big)^{1/q}. \end{equation} Using \eqref{Ineg1} and \eqref{Ineg3}, we obtain $$ \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta \leq \beta^{1+ 1/q}\Big(\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta \Big)^{1/pq} \Big( \mathcal{A}( q, \nu) \Big) \Big( \mathcal{A} ( p, \mu) \Big)^{1/q}, $$ and consequently, $$ \Big(\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta \Big)^{1- 1/(pq)} \leq \beta^{1+ 1/q} \Big( \mathcal{A}( q, \nu) \Big) \Big( \mathcal{A}( p, \mu) \Big)^{1/q}. $$ Similarly, we obtain $$ \Big(\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta}\,d \eta \Big)^{1- 1/(pq)} \leq \beta^{1+ 1/p}\Big( \mathcal{A}( p, \mu) \Big) \Big( \mathcal{A}( q, \nu) \Big)^{1/p}. $$ Now, we take \[ \psi(\eta)=\varphi\Big( \frac{\tau^{2}+|x|^{4}+|y|^{4}}{R^{4}}\Big), \] and change variables from $\eta=(x,y,\tau)$ to $\tilde \eta=( \tilde x , \tilde y , \tilde\tau) $ as follows: $$ \tau=R^{2}\tilde \tau,\quad x=R\tilde x, \quad y=R\tilde y. $$ Using $$ {|(-\Delta_{\mathbb{H}})^{\nu/2} \psi|}^{p'} =R^{-p'\mu} {|(-\Delta_{\mathbb{H}})^{\nu/2}\varphi( \tilde{\eta})|}^{p'} $$ and $d\eta=R^{Q} d\tilde\eta$, we obtain \begin{equation}\label{Ineg3A} \mathcal{A}( p, \mu)\leq C R^{-\mu +\frac{Q}{p'}} \end{equation} where \begin{gather*} C= \beta^{1+ 1/p} \Big(\int_{\Omega} \varphi^{(\beta-1)p'-\beta \frac{p'}{p}} |{(-\Delta_{\mathbb{H}})^{\mu/2} \varphi}|^{p'} \,d \tilde{\eta} \Big)^{1/p'}, \\ \Omega=\Big\{(\tilde x,\tilde y,\tilde\tau)\in{\mathbb{R}}^{2N+1}: {\tilde\tau}^{2}+|\tilde x|^{4}+|\tilde y|^{4}\leq 2 \Big\}. \end{gather*} So, we have \begin{gather*} \Big(\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta \Big)^{1- 1/(pq)} \leq \; CR^{{\theta}_{1}}, \\ \Big(\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d \eta \Big)^{1- 1/(pq)} \leq \; C R^{{\theta}_{2}}, \end{gather*} where \begin{gather*} {\theta}_{1} =(-\mu p' +Q)\frac{1}{p'q} +(-\nu q'+Q)\frac{1}{q'}, \\ {\theta}_{2} =(-\nu q' +Q)\frac{1}{pq'} +(-\mu p'+Q)\frac{1}{p'} . \end{gather*} Now, using \eqref{hyp1}, we can see that if \[ {\theta}_{1} < 0\quad \Longleftrightarrow\quad Q< (\frac{pq}{pq-1})(\frac{\mu}{q}+\nu) \] or \[ {\theta}_{2} < 0\quad \Longleftrightarrow\quad Q< (\frac{pq}{pq-1})(\frac{\nu}{p}+\mu); \] that is, \[ Q < (\frac{pq}{pq-1})\max \Big\{\frac{\nu}{p}+\mu , \frac{\mu}{q}+\nu \Big\}, \] then, we have $$ \lim_{R \to \infty}\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d \eta =\int_{{\mathbb{R}}^{2N+1}} | u |^p\,d \eta=0 $$ or $$ \lim_{R \to \infty}\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d \eta =\int_{{\mathbb{R}}^{2N+1}} | v |^q\,d \eta=0; $$ therefore $ (u,v) \equiv (0,0)$. This completes the proof. \end{proof} In the case of a single equation $$ { (-\Delta_{\mathbb{H}})^{\mu /2} u = | u |^p, \quad u \geq 0 \quad \text{in } \mathbb{R}}^{N} $$ using the scaled variables as in the proof of Theorem \ref{tabs0}, one can verify that if $ 1< p< \frac{Q}{Q-\mu}$ then the solution is trivial. \begin{proof}[Proof of Theorem \ref{tabsS30}] Let $(u,v)$ be a weak solution of system \eqref{System3}. Following the same method as in the proof of Theorem \ref{tabs0} for system \eqref{System1}, one obtains $$ \int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d\eta \leq \beta \mathcal{A}( q, \nu_1) \Big( \int_{K} | v |^q \psi^{\beta} \,d\eta \Big)^{1/q} +\beta \mathcal{A}( p, \nu_2) \Big( \int_{K} | u |^p \psi^{\beta} \,d\eta \Big)^{1/p}, $$ and $$ \int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d\eta \leq \beta\mathcal{A}( p, \mu_1) \Big( \int_{K} | u |^p \psi^{\beta} \,d\eta \Big)^{1/p}+ \beta\mathcal{A}( q, \mu_2) \Big( \int_{K} | v |^q \psi^{\beta} \,d\eta\Big)^{1/q}. $$ Similarly, we have \begin{align*} \Big(\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,d\eta\Big)^{pq} & \leq C \Big\{\Big(\mathcal{A}( p, \nu_2)\Big)^{\frac{pq}{p-1}} + \Big(\mathcal{A}( q, \nu_1)\Big)^q \Big(\mathcal{A}( q, \mu_2)\Big)^{\frac{q}{q-1}}\\ &\quad +\Big(\big( \mathcal{A}( q, \nu_1) \big)^q \mathcal{A}_{{\beta}}( p, \mu_1)\Big)^{\frac{pq}{pq-1}}\Big\}, \end{align*} and \begin{align*} \Big(\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,dx\Big)^{pq} & \leq C \Big\{\Big(\mathcal{A}( q, \mu_2)\Big)^{\frac{pq}{q-1}} + \Big(\mathcal{A}( p, \mu_1)\Big)^p \Big(\mathcal{A}( p, \nu_2)\Big)^{\frac{p}{p-1}}\\ &\quad + \Big(\big( \mathcal{A}( p, \mu_1)\big)^p \mathcal{A}( q, \nu_1)\Big)^{\frac{pq}{pq-1}} \Big\}. \end{align*} Also, using the arguments of the previous theorem, we obtain $$ \Big(\int_{{\mathbb{R}}^{2N+1}} | u |^p \psi^{\beta} \,dx\Big)^{pq} \leq C ( R^{\gamma'_{1}} +R^{\gamma'_{2}}+R^{\gamma'_{3}}), $$ where \begin{gather*} { \gamma'_1 } = { \big(-\nu_2 + \frac{Q}{p'} \big) \frac{pq}{p-1}}, \\ { \gamma'_2 } = { \big(-\nu_1 + \frac{Q}{q'}\big)q +\big( -\mu_2 + \frac{Q}{q'}\big)\frac{q}{q-1}}, \\ { \gamma'_3 } = { \Big(\big(-\nu_1 + \frac{Q}{q'}\big)q + \big( -\mu_1 + \frac{Q}{p'}\big)\Big)\frac{pq}{pq-1}}, \\ \Big(\int_{{\mathbb{R}}^{2N+1}} | v |^q \psi^{\beta} \,d\eta\Big)^{pq} \leq C ( R^{\theta'_{1}} +R^{\theta'_{2}}+R^{\theta'_{3}}), \end{gather*} where \begin{gather*} {\theta'_1} = { (-\mu_2 + \frac{Q}{q'} ) \frac{pq}{q-1}}, \\ {\theta'_2} = {(-\mu_1 + \frac{Q}{p'})p +( -\nu_2 + \frac{Q}{p'})\frac{p}{p-1}}, \\ {\theta'_3} = \Big((-\mu_1 + \frac{Q}{p'})p +( -\nu_1 + \frac{Q}{q'})\Big)\frac{pq}{pq-1}. \end{gather*} Taking either $\max(\gamma'_{1}, \gamma'_{2}, \gamma'_{3}) < 0$ or $ \max( \theta'_{1}, \theta'_{2}, \theta'_{3}) <0$, and using the same arguments as in the previous proofs one can show that $u =v=0$. \end{proof} \subsection*{Acknowledgments} Authors acknowledge financial support from The Research Council (TRC), Oman. This work is funded by TRC under the research agreement no. ORG/SQU/CBS/13/030. \begin{thebibliography}{10} \bibitem{AAK} A. Alsaedi, B. Ahmad, M. Kirane; \emph{Nonexistence of global solutions of nonlinear space-fractional equations on the Heisenberg group}, Electronic Journal of Differential Equations, 2015, 227,2015, 1-10. \bibitem{BirCapCut} I. Birindelli, I. Capuzzo Dolcetta, A. Cutri; \emph{Liouville theorems for semilinear equations on the Heisenberg group}, Annales de l'Institut Henri Poincare, 14, 1997, 295-308. \bibitem{CC} A. C\'{o}rdoba, D. C\'{o}rdoba; \emph{A maximum principle applied to quasi-geostrophic equations}, Comunn. Math. Phys. 249, 2004, 511-528. \bibitem{DaKaKe}Z. Dahmani, F. Karami, S. Kerbal; \emph{Nonexistence of positive solutions to nonlinear nonlocal elliptic systems}, J. Math. Anal. Appl., 346, 2008, 22-29. \bibitem{GarLan} N. Garofalo, E. Lanconelli; \emph{Existence and nonexistence results for semilinear equations on the Heisenberg group}, Indiana Univ. Math. J., 41, 1992, 71-98. \bibitem{KiQua} M. Kirane, M. Qafsaoui; \emph{Global nonexistence for the Cauchy problem of some nonlinear reaction-diffusion systems}, J. Math. Anal. Appl., 268, 2002, 217-243. \bibitem{QuaXia} A. Quaas, A. Xia; \emph{Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space}, Calc. Var. Partial Differential Equations, 526, 2014, 1-19. \bibitem{Ugu} F. Uguzzoni; \emph{A non-existence theorem for a semilinear Dirichlet problem involving critical exponent on halfspaces of the Heisenberg group}, Nonlinear differ. equ. appl., 6, 1999, 191-206. \bibitem{VP} L. V\'{e}ron, S. I. Pohozaev; \emph{Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group}, Manuscripta Math., 102, 2000, 85-99. \bibitem{Xu} L. Xu; \emph{Semi-linear Liouville theorems in the Heisenberg group via vector field methods}, Journal of Differential Equations, 247, 2009, 2799-2820. \end{thebibliography} \end{document}