Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition

Abstract

We are interested in the existence of solutions for the following fractional $p(x,\cdot )$-Kirchhoff type problem $$\{\begin{array}{l}M\left({\int }_{\Omega \times \Omega }\frac{|u(x)-u(y){|}^{p(x,y)}}{p(x,y)|x-y{|}^{N+p(x,y)s}}dxdy\right)(-\Delta {)}_{p(x,\cdot )}^{s}u=f(x,u),x\in \Omega ,\\ \\ u=0,x\in \partial \Omega ,\end{array}$$ where $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$ is a bounded smooth domain, $s\in (0,1),$ $p:\overline{\Omega }\times \overline{\Omega }\to (1,\infty )$, $(-\Delta {)}_{p(x,\cdot )}^s$ denotes the $p(x,\cdot )$-fractional Laplace operator, $M:[0,\infty )\to [0,\infty ),$ and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ are continuous functions. Using variational methods, especially the symmetric mountain pass theorem due to Bartolo-Benci-Fortunato (Nonlinear Anal. 7:9 (1983), 981-1012), we establish the existence of infinitely many solutions for this problem without assuming the Ambrosetti-Rabinowitz condition. Our main result in several directions extends previous ones which have recently appeared in the literature.

References in corpus (1)

• Existence of solutions for systems arising in electromagnetism