Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities

Abstract

In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \begin{equation*} \;\;\; \begin{array}{rl} (-\Delta)_{p(\cdot)}^{s} u&=\lambda~ a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)},\hspace{2mm} x\in \Omega; \\ (-\Delta)_{p(\cdot)}^{s} v&=\mu~ b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x)+\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\hspace{2.5mm} x\in \Omega; \\ u=v&=0 ,\hspace{1cm} x\in \Omega^c:=\mathbb R^N\setminus\Omega, \end{array} \end{equation*} where $\Omega\subset\mathbb R^N,~N\geq2$ is a smooth bounded domain, $\lambda,\mu>0$ are the parameters, $s\in(0,1),$ $p\in C(\mathbb R^N\times \mathbb R^N,(1,\infty))$ and $q,\alpha,\beta\in C(\overline{\Omega},(1,\infty))$ are the variable exponents and $a,b,c\in C(\overline{\Omega},[0,\infty))$ are the non-negative weight functions. We show that there exists $\Lambda>0$ such that for all $\lambda+\mu<\Lambda$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,\alpha,\beta$.