Least action nodal solutions for the quadratic Choquard equation

Abstract

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation $$ -Δu + u = \big(I_α\ast |u|^2\big)u \quad\text{in }\; \mathbb R^N,$$ where $I_α$ is the Riesz potential of order $α\in(0,N)$. The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations $$ -Δu + u = \big(I_α\ast |u|^p\big)|u|^{p-2}u \quad\text{in }\; \mathbb R^N$$ 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 $p<2$.

11 pages