Existence and nonexistence of solutions to nonlinear gradient elliptic systems involving \((p(x),q(x))\)-Laplacian operators (Q2877650)

In this paper the following \((p(x),q(x))\)-gradient elliptic system is studied NEWLINE\[NEWLINE \begin{cases} -\Delta_{p(x)}u=c(x)u|u|^{\alpha-1}|v|^{\beta+1} & \text{ in }\Omega, \\ -\Delta_{q(x)}v=c(x)v|v|^{\beta-1}|u|^{\alpha+1} & \text{ in }\Omega, \\ u=v=0 & \text{ on } \partial \Omega.\end{cases} NEWLINE\]NEWLINE Here the \(p(x)\)-Laplacian is defined by \(\displaystyle \Delta_{p(x)}u=\text{div}(|\nabla u|^{p(x)-2}\nabla u)\), \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(p,q: \Omega\rightarrow (1,+\infty)\) and \(c: \Omega\rightarrow \mathbb{R}\) are measurable functions.NEWLINENEWLINEThe authors first prove a Pohozaev-type identity for the above system and then use this identity to prove a non-existence result for the system under a set of conditions on \(p(x)\), \(q(x)\) and \(c(x)\). Then the authors use the so-called fibering method introduced by Pohozaev to establish the existence of nontrivial solutions in the generalized Sobolev space \(W_0^{1,p(x)}(\Omega)\times W_0^{1,q(x)}(\Omega)\) under appropriate conditions.