Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency

Abstract

In this article we study the fractional Schrodinger-Poisson system $$\displaylines{ \epsilon^{2s}(-\Delta)^s u+V(x)u=\phi |u|^{2^*_s-3}u,\quad x\in \mathbb{R}^3, \cr (-\Delta)^s\phi=|u|^{2^*_s-1}, \quad x\in \mathbb{R}^3, }$$ where \(s\in(1/2,1)\), \(\epsilon>0\) is a parameter, \(2^*_s=6/(3-2s)\) is the critical Sobolev exponent, \(V\in L^{\frac{3} {2s}}(\mathbb{R}^3)\) is a nonnegative function which may be zero in some region of \(\mathbb{R}^3\). By means of variational methods, we present the number of high energy bound states with the topology of the zero set of V for small \(\epsilon\).