Least action nodal solutions for the quadratic Choquard equation

  • Ghimenti M
  • Moroz V
  • Van Schaftingen J
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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 $p<2$.

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APA

Ghimenti, M., Moroz, V., & Van Schaftingen, J. (2016). Least action nodal solutions for the quadratic Choquard equation. Proceedings of the American Mathematical Society, 145(2), 737–747. https://doi.org/10.1090/proc/13247

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