Construction of infinitely many solutions for a critical Choquard equation via local Poho\v{z}aev identities

TL;DR

This paper proves the existence of infinitely many solutions for a critical Choquard equation with axisymmetric potentials in D, using local Pohoeav identities and finite dimensional reduction under certain potential conditions.

Contribution

It introduces a novel approach combining local Pohoeav identities with finite dimensional reduction to establish multiple solutions for a critical nonlinear PDE.

Findings

Existence of infinitely many solutions under topologically nontrivial potential conditions.

Development of new local Pohoeav identities for critical equations.

Application of finite dimensional reduction technique to nonlinear PDEs.

Abstract

In this paper, we study a class of the critical Choquard equations with axisymmetric potentials, $$ -\Delta u+ V(|x'|,x'')u =\Big(|x|^{-4}\ast |u|^{2}\Big)u\hspace{4.14mm}\mbox{in}\hspace{1.14mm} \mathbb{R}^6, $$ where $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{4}$, $V(|x'|, x'')$ is a bounded nonnegative function in $\mathbb{R}^{+}\times\mathbb{R}^{4}$, and $*$ 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 Poho\v{z}aev identities, we prove that if the function $r^2V(r,x'')$ has a topologically nontrivial critical point then the problem admits infinitely many solutions with arbitrary large energies.