On a singular and nonhomogeneous \(N\)-Laplacian equation involving critical growth
type: Item
instance of: scholarly article
title: On a singular and nonhomogeneous \(N\)-Laplacian equation involving critical growth
zbMATH Open document ID: 1241.35088
author: Manassés De Souza
author: João Marcos do Ó
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-05-10T00:00:00Z
review text: The following quasilinear elliptic equation \[ -\triangle_N u:=-\text{div}(|\nabla u|^{N-2}\nabla u)=|x|^{-a}f(x,u) +h(x), x \in \Omega, \;u\in W^{1,N}_0(\Omega), \;a\in [0,N), \] is considered, where \(\Omega \subset \mathbb{R}^N \;(N\geq 3)\) is a smooth bounded domain and \(h(x)\not\equiv 0\) is in the dual space of \(W^{1,N}_0(\Omega)\). By using the Ekeland's variational principle and the mountain pass theorem, the authors prove that, if the norm of \(h(x)\) is small enough, the above problem possesses at least two solutions which have negative energy and positive energy, respectively. Here the nonlinear term \(f(x,u)\) may have critical growth in the sense of \(-\triangle_N u\) in \(\mathbb{R}^N\). Moreover, the authors also prove that the two solutions for the problem are nonnegative (nonpositive) if \(h(x)\geq 0\) (\(h(x)\leq 0\), respectively).
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5886411
zbMATH Keywords: variational method
zbMATH Keywords: mountain pass theorem
zbMATH Keywords: \(N\)-Laplacian equation
zbMATH Keywords: Trudinger-Moser inequality
zbMATH Keywords: critical points
zbMATH Keywords: critical growth
reviewed by: Huansong Zhou
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.03.028
OpenAlex ID: W2076246183
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DOI: 10.1016/J.JMAA.2011.03.028
description: scientific article; zbMATH DE number 5886411