Least action nodal solutions for the quadratic Choquard equation

TL;DR

This paper establishes the existence of a minimal action nodal solution for the quadratic Choquard equation by analyzing the limit of solutions for nonlinear variants as the nonlinearity approaches quadratic form.

Contribution

It introduces a novel approach to construct minimal action nodal solutions for the quadratic Choquard equation via limit processes from nonlinear cases.

Findings

Existence of minimal action nodal solutions for p > 2

Non-existence of such solutions for p < 2

Solution constructed as limit of nonlinear solutions as p approaches 2

Abstract

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $$ -\Delta u + u = \big(I_\alpha \ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,$$ where $I_\alpha$ is the Riesz potential of order $\alpha\in(0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations $$ -\Delta u + u = \big(I_\alpha \ast |u|^p\big)|u|^{p-2}u \quad\text{in }\; \mathbb R^N$$ when $p\searrow 2$. The existence of minimal action nodal solutions for $p>2$ can be proved using a variational minimax procedure over Nehari nodal set. No minimal action nodal solutions exist when $p<2$.