Existence of infinitely many solutions for a critical Hartree type equation with potential: local Pohožaev identities methods

Abstract

This paper deals with the following equation $$-Δu =K(|x'|, x'')\Big(|x|^{-α}\ast (K(|x'|, x'')|u|^{2^{\ast}_α})\Big) |u|^{2^{\ast}_α-2}u\quad\mbox{in}\ \mathbb{R}^N,$$ where $N\ge 5$, $\alpha >5-\frac{6}{N-2}$, $2_{\alpha }^{\ast }=\frac{2N-\alpha }{N-2}$ is the so-called upper critical exponent in the Hardy-Littlewood-Sobolev inequality and $K(|x^{\prime }|,x^{\prime \prime })$, where $(x^{\prime },x^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^{N-2}$, is bounded and nonnegative. Under proper assumptions on the potential function $K$, we obtain the existence of infinitely many solutions for the nonlocal critical equation by using a finite dimensional reduction argument and local Pohožaev identities. It is a remarkable fact that the order of the Riesz potential influences the existence/non-existence of solutions.