Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\mathbb{R}^ N\) with nonlinearities in the critical growth range
type: Item
instance of: scholarly article
title: Semilinear Dirichlet problems for the \(N\)-Laplacian in \(\mathbb{R}^ N\) with nonlinearities in the critical growth range
zbMATH Open document ID: 0858.35043
author: João Marcos do Ó
published in: Differential and Integral Equations
publication date: 1997-03-31T00:00:00Z
review text: This paper studies the homogeneous Dirichlet problem for equations of the form \(\text{div}(|Du|^{n-2}Du)= f(x,u)\) in a smooth bounded domain \(\Omega\) in \(\mathbb{R}^n\) with \(n\geq 2\). The critical growth range here is on \(f\), which is a nonnegative, continuous function. It is assumed that there is a constant \(\alpha_0\) such that \(f(x,u)= o(\exp(\alpha|u|^{n/(n-2)}))\) as \(u\to\infty\) for \(\alpha>\alpha_0\) and \(\exp(\alpha|u|^{n/(n-1)})= o(f(x,u))\) for \(\alpha<\alpha_0\). In addition, certain technical conditions are made on \(f\) which imply that the problem has a positive solution, but the important element here is that \(f\) grows faster than any polynomial as \(u\to\infty\). Thus the ``standard'' methods for analyzing critical growth problems must be modified. The analysis here is quite clever, and the results improve previous attempts by various authors to study this problem.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 896469
zbMATH Keywords: critical nonlinearity
zbMATH Keywords: positive solution
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 896469