Solutions for fractional operator problem via local Pohozaev identities

Abstract

We consider the following fractional Schrödinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-Δ)^s u+V(|y'|,y'')u=u^{2^*_s-1} \ \hbox{ in } \ \mathbb{R}^N, \\ u>0, \ y \in \mathbb{R}^N, \end{array}\right. \end{equation*} where $s\in (\frac{1}{2},1)$, $(y^{\prime },y^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$, $V(|y^{\prime }|,y^{\prime \prime })$ is a bounded nonnegative function with a weaker symmetry condition. We prove the existence of infinitely many solutions for the above problem by a finite dimensional reduction method combining various Pohazaev identies.