Schr\"odinger equation in dimension two with competing logarithmic self-interaction

TL;DR

This paper investigates a two-dimensional Schr"odinger equation with competing logarithmic nonlocal interactions, establishing the existence of infinitely many radially symmetric solutions through variational methods despite the challenges posed by the nonlocal terms and indefinite kernels.

Contribution

It introduces a novel variational framework to handle the competing nonlocal logarithmic interactions in 2D Schr"odinger equations, proving the existence of multiple solutions.

Findings

Infinitely many radially symmetric solutions found.

Successfully manages nonlocal terms with indefinite kernels.

Develops a new functional setting for finite energy solutions.

Abstract

In this paper we study the equation \[ -\Delta u +(\log |\cdot|*|u|^2)u=(\log|\cdot|*|u|^q)|u|^{q-2}u, \qquad \hbox{ in }\mathbb{R}^2, \] 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.