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 \begin{equation*} -\Delta u + u = \bigl (I_\alpha \ast \|u\|^2\bigr )u \quad \text {in \(\mathbb {R}^N\)}, \end{equation*} 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 \begin{equation*} -\Delta u + u = \bigl (I_\alpha \ast \|u\|^p\bigr )|u|^{p-2}u \quad \text {in \(\mathbb {R}^N\)} \end{equation*} when $p\searrow 2$. The existence of minimal action nodal solutions for $p>2$ can be proved using a variational minimax procedure over a Nehari nodal set. No minimal action nodal solutions exist when $p<2$.