Existence and concentration of positive solutions for a Schrödinger logarithmic equation
type: Item
instance of: scholarly article
title: Existence and concentration of positive solutions for a Schrödinger logarithmic equation
zbMATH Open document ID: 1411.35090
author: D. C. de Morais Filho
author: Claudianor O. Alves
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 2018-10-30T00:00:00Z
review text: The paper is concerned with the Schrödinger logarithmic equation \[ -\varepsilon ^2 \Delta u +V(x)u= u \log u^2 \quad \text{in }\mathbb{R}^N, \quad u\in H^1(\mathbb{R}^N),\tag{1} \] where \(\varepsilon >0\), \(N\geq 3\) and the continuous function \(V:\mathbb{R}^N \to \mathbb{R}\) satisfies \[ \lim_{|x| \to \infty} V(x)> \inf_{x\in \mathbb{R}^N} V(x)=:V_0. \] The authors prove that (1) has a positive solution \(v_{\varepsilon}\) for all sufficiently small \(\varepsilon>0\), and for each sequence \(\varepsilon _n \to 0\), up to a subsequence, the family \(\{v_{\varepsilon_{n}}\}\) concentrates around a point \(y_0\in \mathbb{R}^N\) such that \(V(y_0)=V_0\). The approach relies on a mountain pass theorem without Palais-Smale condition in the frame of critical point theory for \(C^1\) perturbations of convex lower semicontinuous functionals. The case when \(V\) is periodic or asymptotically periodic is also addressed.
reviewed by: P. Jebelean
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 6968554
zbMATH Keywords: Schrödinger logarithmic equation
zbMATH Keywords: positive solutions
zbMATH Keywords: variational methods
MaRDI profile type: Publication
full work available at URL: s00033-018-1038-2
OpenAlex ID: W2898414676
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DOI: 10.1007/S00033-018-1038-2
description: scientific article; zbMATH DE number 6968554