Existence of solutions for a class of elliptic equations in \(\mathbb R^N\) with vanishing potentials
type: Item
instance of: scholarly article
title: Existence of solutions for a class of elliptic equations in \(\mathbb R^N\) with vanishing potentials
zbMATH Open document ID: 1250.35103
author: Marco A. Souto
author: Claudianor O. Alves
published in: Journal of Differential Equations
publication date: 2012-05-11T00:00:00Z
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 6033465
zbMATH Keywords: semilinear elliptic problem
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: positive solution
reviewed by: Huansong Zhou
MaRDI profile type: Publication
full work available at URL: j.jde.2012.01.025
OpenAlex ID: W2198655720
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DOI: 10.1016/J.JDE.2012.01.025
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review text: In this paper, the authors study the following semilinear elliptic problem: NEWLINE\[NEWLINE-\Delta u + V(x)u =f(s) \text{\;in \;} \mathbb{R}^N, \tag{P}NEWLINE\]NEWLINE where \(V(x) \geq 0\) for all \(x\in \mathbb{R}^N\), \(V(x)\) is bounded above by a positive constant on \(B_1(0)\) and there are \(\Lambda >0\), \(R>1\) such that NEWLINE\[NEWLINE\frac{1}{R^4} \lim\inf_{|x|\geq R} |x|^4 V(x) \geq \Lambda.NEWLINE\]NEWLINE Under certain conditons on \(f(s)\), such as, \(f(s)\) is superlinear and with subcritial growth on \(s\), and by applying a version of the mountain pass theorem, the authors prove that there exists \(\Lambda^*>0\) such that problem (P) has a positive solution for all \(\Lambda \geq \Lambda^*\).
description: scientific article; zbMATH DE number 6033465