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 $$-\Delta u+u=(I_{\alpha }\ast |u{|}^2)u\text{in }{\mathbb{R}}^N,$$ 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 $$-\Delta u+u=(I_{\alpha }\ast |u{|}^p)|u{|}^{p-2}u\text{in }{\mathbb{R}}^N$$ 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

Cited by in corpus (4)

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