Infinitely many solutions for Schr\"odinger-Kirchhoff-type equations involving the fractional $p(x,\cdot)$-Laplacian

M. Mirzapour

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2023), pp. 23-34

Résumé

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 $$ \begin{array}{ll} M \big(\sigma_{p(x,y)}(u)\big)\mathcal{L}_K^{p(x,\cdot)} (u) =\lambda \vert u\vert^{q(x)-2}u+\mu \vert u \vert^{\gamma(x)-2}u \text{ in } \Omega,\\ u(x)=0 \textrm{ in } \mathbb{R}^{N}\backslash \Omega, \end{array} $$ where $$ \sigma_{p(x,y)}(u)=\int _{\mathcal{Q}} \frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)}K(x,y) dx dy, $$ $\mathcal{L}_{K}^{p(x,\cdot)}$ is a nonlocal operator with singular kernel $K$, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary $\partial \Omega$, $M:\mathbb{R}^+ \rightarrow \mathbb{R}$ 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.

Détail

Keywords: fractional $p(x,\cdot)$-Laplacian, Schrödinger-Kirchhoff-type problem, variational methods.