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} (-Δ)_{p(\cdot)}^{s} u&=λ~ a(x)| u|^{q(x)-2}u+\frac{α(x)}{α(x)+β(x)}c(x)| u|^{α(x)-2}u| v| ^{β(x)},\hspace{2mm} x\in Ω; \\ (-Δ)_{p(\cdot)}^{s} v&=μ~ b(x)| v|^{q(x)-2}v+\frac{α(x)}{α(x)+β(x)}c(x)| v|^{α(x)-2}v| u| ^{β(x)},\hspace{2.5mm} x\in Ω; \\ u=v&=0 ,\hspace{1cm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Omega \subset {\mathbb{R}}^N,N\ge 2$ 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 $q,\alpha ,\beta$.