Year: 2016
Type: article
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 $p<2$.
DOI: 10.1090/proc/13247
Authors: Marco Ghimenti, Vitaly Moroz, Jean Van Schaftingen
Institutions: University of Pisa, Swansea University, UCLouvain
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Award: G1651659333
OpenAlex ID: W2257949331