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

چکیده مقاله

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. © 2023, Allerton Press, Inc.