Construction of infinitely many solutions for a critical Choquard equation via local Pohožaev identities

Abstract

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -Δu+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ where $(x^{\prime },x^{\prime \prime })\in {\mathbb{R}}^2\times {\mathbb{R}}^4$, $V(|x^{\prime }|,x^{\prime \prime })$ is a bounded nonnegative function in ${\mathbb{R}}^{+}\times {\mathbb{R}}^4$, and $\ast$ stands for the standard convolution. The equation is critical in the sense of the Hardy-Littlewood-Sobolev inequality. By applying a finite dimensional reduction argument and developing novel local Pohožaev identities, we prove that if the function $r^{2}V(r,x^{\prime \prime })$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.