Multiple entire solutions of fractional Laplacian Schrödinger equations

Abstract

$ \begin{align*} \begin{cases} (-\Delta)^s u+V(x)u = f(x,u), \; \; \; x\in \mathbb{R}^N ,\\ u\in H^{s}(\mathbb {R}^N) , \\ \end{cases} \end{align*} $ where both $ V(x) $ and $ f(x, u) $ are periodic in $ x $, $ 0 $ belongs to a spectral gap of the operator $ (-\Delta)^s+V $ and $ f(x, u) $ is subcritical in $ u $. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $ f $, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions.

Keywords

• fractional schrödinger equation
• cerami sequence
• infinitely many geometrically distinct solutions