Double phase anisotropic variational problems and combined effects of reaction and absorption terms
type: Item
instance of: scholarly article
title: Double phase anisotropic variational problems and combined effects of reaction and absorption terms
zbMATH Open document ID: 1404.35191
author: Qihu Zhang
author: Vicenţiu D. Rădulescu
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2018-09-18T00:00:00Z
review text: The authors are interested in finding multiple solutions to quasilinear elliptic equations given by \[ -\text{div\,}A(x,\nabla u)+V(x)|u|^{\alpha(x)-2}u=f(x,u) \quad \text{in }\mathbb{R}^N, \] where \(A:\mathbb{R}^N\times \mathbb{R}^N\to \mathbb{R}^N\) and \(f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}\) satisfy sutiable structure conditions. The problem corresponds to double phase anisotropic phenomena in the sense that the differential operator has behaviors like \(|\xi|^{q(x)-2}\xi\) for small \(|\xi|\) and behaviors like \(|\xi|^{p(x)-2}\xi\) for large \(|\xi|\), where \(1<\alpha(\cdot)\leq p(\cdot)< q(\cdot)<N\). The aim of the authors is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Their results extend previous works from cases where the exponents \(p\) and \(q\) are constant to the case of functions \(p(\cdot)\) and \(q(\cdot)\). In addition, the authors are also able to weaken some of the hypotheses in these papers and overcome the lack of compactness by using the Cerami compactness condition.
reviewed by: Patrick Winkert
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 6938394
zbMATH Keywords: variable exponent elliptic operators
zbMATH Keywords: integral functionals
zbMATH Keywords: variable exponent Orlicz-Sobolev spaces
zbMATH Keywords: critical points
MaRDI profile type: Publication
full work available at URL: j.matpur.2018.06.015
OpenAlex ID: W2809336018
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Wikidata QID: Q129679343
DOI: 10.1016/J.MATPUR.2018.06.015
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