Solutions for fractional operator problem via local Pohozaev identities

Abstract

We consider the following fractional Schr\"{o}dinger equation involving critical exponent: \begin{equation*} \left\{\begin{array}{ll} (-\Delta)^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 $s\in(\frac{1}{2}, 1)$, $(y',y'')\in \mathbb{R}^2\times \mathbb{R}^{N-2}$, $V(|y'|,y'')$ 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.