Schrödinger equation in dimension two with competing logarithmic self-interaction

Abstract

In this paper we study the equation \[ -Δu +(\log |\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2, \] where $8/3<q<4$. By means of variational arguments, we find infinitely many radially symmetric classical solutions. The main difficulties rely on the competition between the two nonlocal terms and on the presence of logarithmic kernels, which have not a prescribed sign. In addition, in order to find finite energy solutions, a suitable functional setting analysis is required.

arXiv comment20 pages

References in corpus (5)

• Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics
• A guide to the Choquard equation
• Existence of groundstates for a class of nonlinear Choquard equations
• The planar Schrödinger-Poisson system with a positive potential
• Nonlinear scalar field equation with competing nonlocal terms