Infinitely many non-radial solutions to a critical Choquard equation

TL;DR

This paper proves the existence of infinitely many non-radial solutions to a critical Choquard equation with symmetric potential in high dimensions, using finite dimensional reduction techniques.

Contribution

It demonstrates the existence of infinitely many non-radial solutions for a class of critical Choquard equations with symmetric potentials, extending previous results to non-radial solutions.

Findings

Infinitely many non-radial solutions exist under certain potential conditions.

Solutions can have arbitrarily large energies.

The method applies finite dimensional reduction to a critical nonlocal PDE.

Abstract

In this paper we study a class of critical Choquard equations with a symmetric potential, i.e. we consider the equation $$-\Delta u +V(|x|) u =\left(|x|^{-\mu}* |u|^{2^\star_\mu}\right)|u|^{2^\star_\mu-2}u,\quad\mbox{in}\quad\mathbb R^N$$ where $V(|x|)$ is a bounded, nonnegative and symmetric potential in $\mathbb R^N$ with $N\geq 5$, $0<\mu\leq 4$, $*$ stands for the standard convolution and $2^\star_\mu:=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy - Littlewood - Sobolev inequality. By applying a finite dimensional reduction method we prove that if $r^2V(r)$ has a local maximum point or local minimum point $r_0>0$ with $V(r_0)>0$ then the problem has infinitely many non-radial solutions with arbitrary large energies.