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

Abstract

We consider a nonlinear Choquard equationwhen 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 ∈ H 1 (R N ) \ {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.

Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System

“…In [11], Bonheure, Van Schaftingen and the first author obtained sharp decay estimates of this unique positive solution to the logarithmic Choquard equation (1.3) and they showed the nondegeneracy of the unique positive ground state. We also mention the recent paper [6] for the existence of the ground state of (1.3), with a = 0, γ = 1, via relaxed problems.…”

Trudinger–Moser‐type inequality with logarithmic convolution potentials

“…In particular, when $$\Omega$$ is the entire space, the relevance of the logarithmic convolution kernel in () is due to the fact that it represents, up to a constant, the fundamental solution of $$-\Delta$$. Therefore, in the case where $$\Omega =\mathbb {R}^2$$, this kernel arises in a reduction of planar Schrödinger–Poisson systems to a single integro‐differential equation, see, for example, [3, 7, 8 13, 14, 16 20, 21, 24, 25] and () below.…”

Ground state solutions to logarithmic Choquard equations in R3

“…The logarithmic nonlinearity originates from some physical models like in dynamic of thin films of viscous fluids (see). In, Battaglia and Van Schaftingen proved the existence of a nontrival ground state solution by solving a relaxed problem based on a constrained minimization. When , Guo and Wu obtained a mountain‐pass solution and a ground state solution of .…”