On an elliptic equation with exponential growth
type: Item
instance of: scholarly article
title: On an elliptic equation with exponential growth
zbMATH Open document ID: 0887.35055
author: J. A. Aguilar Crespo
author: Ireneo Peral
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-05-05T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108406
review text: The authors study existence and multiplicity of solutions of the homogeneous Dirichlet problem for the equation \(-\Delta_p u=\lambda V(x)e^u\) in a bounded domain \(\Omega\subset\mathbb R^N\), where \(-\Delta_p\) is the \(p\)-Laplacian, \(p>1\), \(\lambda>0\), \(V\in L^q(\Omega)\), \(q\geq1\) if \(p>N\), \(q>N/p\) if \(p\leq N\). The existence for \(\lambda\) small is shown using the Banach fixed point theorem. If, in addition, \(V\) changes sign and \(p\geq N\) then the Mountain Pass Theorem is used in order to obtain a second solution. The authors find also sufficient conditions for existence and nonexistence of positive solutions. Moreover, they study the existence of a minimal solution for \(V\geq0\) and \(p<N\) and they analyze radial solutions in the case \(V(x)=|x|^{-\alpha}\), \(\alpha<p\), \(\Omega\) being the unit ball.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 1021768
zbMATH Keywords: multiplicity
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: exponential growth
zbMATH Keywords: minimal solution
zbMATH Keywords: nonexistence of positive solutions
reviewed by: Pavol Quittner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1021768