Infinitely many non-radial solutions for a Choquard equation

Fashun Gao,Minbo Yang

doi:10.1515/anona-2022-0224

Abstract

Abstract In this article, we consider the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -\Delta u+V\left(| x| )u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{| u(y){| }^{2}}{| x-y| }{\rm{d}}y\right)u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where V ( r ) V\left(r) is a positive bounded function. Under some proper assumptions on V ( r ) V\left(r) , we are able to establish the existence of infinitely many non-radial solutions.

Highlights

Introduction and main resultsIn the past two decades, many authors have devoted to the study of existence, multiplicity, and properties of the solutions of the non-linear Choquard equation (1.1),−Δu + V (x)u = (∣x∣−μ ∗ ∣u∣2)u, in N. (1.1)In a early paper [1], Lieb proved that the ground state U of the equation−Δu + u = (∣x∣−1 ∗ ∣u∣2)u in 3, (1.2)is radial and unique up to translations
While Lions [2] showed the existence of a sequence of radially symmetric solutions via variational methods
We introduce the energy functional associated with equation (1.4) by

Summary

Introduction and main results

In the past two decades, many authors have devoted to the study of existence, multiplicity, and properties of the solutions of the non-linear Choquard equation (1.1),. In [3,4], the authors proved, if u is a ground state of equation (1.2), u is either positive or negative and there exist x0 ∈ 3 and a monotone function v ∈ C∞(0, ∞) such that for every x ∈ 3, u(x) = v(∣x − x0∣). Chen [9] proved that the ground state solution U is non-degenerate, i.e., the kernel of the linearized equation. The main result of this article is to establish the existence of infinitely many non-radial solution for (1.4) under assumption (V ). To prove the main results, we will adopt the idea introduced by Wei and Yan in [15] to use the unique ground state U of equation (1.2) to build up the approximate solutions for (1.4) with large number of bumps near the infinity.

Preliminaries

The reduction argument