Existence of infinitely many small solutions for sublinear fractional Kirchhoff-Schrodinger-Poisson systems

Abstract

We study the Kirchhoff-Schrodinger-Poisson system $$\displaylines{ m([u]_{\alpha}^2)(-\Delta)^\alpha u+V(x)u+k(x)\phi u = f(x,u), \quad x\in\mathbb{R}^3,\cr (-\Delta)^\beta \phi = k(x)u^2, \quad x\in\mathbb{R}^3, }$$ where $[\cdot]_{\alpha}$ denotes the Gagliardo semi-norm, $(-\Delta)^{\alpha}$ denotes the fractional Laplacian operator with $\alpha,\beta\in (0,1]$, $4\alpha+2\beta\geq 3$ and $m:[0,+\infty)\to[0,+\infty)$ is a Kirchhoff function satisfying suitable assumptions. The functions V(x) and k(x) are nonnegative and the nonlinear term f(x,s) satisfies certain local conditions. By using a variational approach, we use a Kajikiya's version of the symmetric mountain pass lemma and Moser iteration method to prove the existence of infinitely many small solutions.

Keywords

• Kirchhoff-Schrodinger-Poisson equation
• fractional Laplacian
• variational method