Existence of infinitely many solutions for the fractional Schr\"odinger- Maxwell equations

Abstract

In this paper, by using variational methods and critical point theory, we shall mainly study the existence of infinitely many solutions for the following fractional Schr\"odinger-Maxwell equations $$( -\Delta )^{\alpha} u+V(x)u+\phi u=f(x,u), \hbox{in } \mathbb{R}^3 ,$$ $$ (-\triangle)^{\alpha}\phi =K_{\alpha} u^2 \ \ \mathrm{in}\ \ \mathbb{R}^3 $$ where $\alpha \in (0,1],$ $K_{\alpha}=\dfrac{\pi^{-\alpha}\Gamma(\alpha)}{\pi^{-(3-2\alpha)/2}\Gamma((3-2\alpha)/2)},$ $( -\Delta )^{\alpha}$ stands for the fractional Laplacian. Under some more assumptions on $f,$ we get infinitely many solutions for the system.