Groundstates of the Choquard equations with a sign-changing self-interaction potential

Abstract

We consider a nonlinear Choquard equation $$-\Delta u+u=(V\ast |u{|}^p)|u{|}^{p-2}u\text{in }{\mathbb{R}}^N,$$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p=2$ and $V$ being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $u\in H^1({\mathbb{R}}^N)\setminus \{0\}$ by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.

16 pages

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Cited by in corpus (3)

• The planar Schrödinger-Poisson system with a positive potential
• Existence and symmetry of solutions to 2-D Schrödinger-Newton equations
• Existence and axial symmetry of minimal action odd solutions for 2-D Schrödinger-Newton equation