Schrödinger equation in dimension two with competing logarithmic self-interaction

TL;DR

This work investigates a two-dimensional Schrödinger-type equation with competing logarithmic nonlocal interactions driven by $- abla^2 u + ($log$| abla|*|u|^2)u = ($log$| abla|*|u|^q)|u|^{q-2}u$ in ${\bf R}^2$ for $q\in(\tfrac{8}{3},4)$. It introduces a novel weighted energy framework in the space $\mathcal{X}$, leveraging the sign-indefinite log kernel and a Strauss-type radial estimate to obtain finite-energy solutions. Using an augmented-dimension variational scheme and a Cerami-type compactness framework on the radial subspace, it proves the existence of infinitely many radially symmetric classical solutions with diverging energy levels. The approach is further extended to a generalized problem with exponents $1<p<q$ and $2p^2/(p+1)<q<2p$, establishing the existence of infinitely many radial finite-energy solutions in this broader setting.

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.

Theorems & Definitions (51)

• Theorem 1.1
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Remark 2.3
• Remark 2.4
• proof
• Proposition 2.6
• Proposition 2.7