Infinitely Many Solutions for Schrödinger–Kirchhoff-Type Equations Involving the Fractional p(x, ·)-Laplacian

Abstract

The aim of this paper is to study the existence of infinitely many solutions for Schrödinger–Kirchhoff-type equations involving nonlocal $p(x, \cdot )$-fractional Laplacian $\left\{ {\begin{array}{*{20}{l}} {M({{\sigma }_{{p(x,y)}}}(u))\mathcal{L}_{K}^{{p(x, \cdot )}}(u) = \lambda {{{\left| u \right|}}^{{q(x) - 2}}}u + \mu {{{\left| u \right|}}^{{\gamma (x) - 2}}}u\;}&{{\text{in}}\;\Omega } \\ {u(x) = 0}&{{\text{in}}\;{{\mathbb{R}}^{N}}{\kern 1pt} \backslash {\kern 1pt} \Omega ,} \end{array}} \right.$ where ${{\sigma }_{{p(x,y)}}}(u) = \int_\mathcal{Q} \frac{{{{{\left| {u(x) - u(y)} \right|}}^{{p(x,y)}}}}}{{p(x,y)}}K(x,y)dxdy,$ $\mathcal{L}_{K}^{{p(x, \cdot )}}$ is a nonlocal operator with singular kernel $K$, $\Omega$ is a bounded domain in ${{\mathbb{R}}^{N}}$ with Lipschitz boundary $\partial \Omega$, $M:{{\mathbb{R}}^{ + }} \to \mathbb{R}$ is a continuous function, q, $\gamma \in C(\Omega )$ and $\lambda ,\mu$ are two parameters. Under some suitable assumptions, we show that the above problem admits infinitely many solutions by applying the Fountain Theorem and the Dual Fountain Theorem.