Nehari manifold approach for fractional $p(\cdot)$-Laplacian system involving concave-convex nonlinearities

Abstract

In this article, using Nehari manifold method we study the multiplicity of solutions of the nonlocal elliptic system involving variable exponents and concave-convex nonlinearities, $$\displaylines{ (-\Delta)_{p(\cdot)}^{s} u=\lambda a(x)| u|^{q(x)-2}u+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| u|^{\alpha(x)-2}u| v| ^{\beta(x)}, \quad x\in \Omega; \cr (-\Delta)_{p(\cdot)}^{s} v=\mu b(x)| v|^{q(x)-2}v+\frac{\alpha(x)}{\alpha(x) +\beta(x)}c(x)| v|^{\alpha(x)-2}v| u| ^{\beta(x)},\quad x\in \Omega; \cr u=v=0,\quad x\in \Omega^c:=\mathbb R^N\setminus\Omega, }$$ where $\Omega\subset\mathbb R^N$, $N\geq2$ is a smooth bounded domain, $\lambda,\mu>0$ are parameters, and $s\in(0,1)$. We show that there exists $\Lambda>0$ such that for all $\lambda+\mu<\Lambda$, this system admits at least two non-trivial and non-negative solutions under some assumptions on $q,\alpha,\beta,a,b,c$.

Keywords

• nonlocal problem with variable exponents
• elliptic system
• nehari manifold
• fibering map
• concave-convex nonlinearities