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 $$ \left\{\begin{array}{ll} M \, \left(\displaystyle\int_{\Omega\times \Omega} \ \displaystyle{\frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y) \ |x-y|^{N+p(x,y)s}}} \ dx \, dy\right)(-\Delta)^{s}_{p(x,\cdot)}u = f(x,u), \quad x\in \Omega, \\ \\ u= 0, \quad x\in \partial\Omega, \end{array}\right.$$ where $\Omega\subset\mathbb{R}^{N}$, $N\geq 2$ is a bounded smooth domain, $s\in(0,1),$ $p: \overline{\Omega}\times \overline{\Omega} \rightarrow (1, \infty)$, $(-\Delta)^{s}_{p(x,\cdot)}$ 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.