Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. III.
type: Item
instance of: scholarly article
title: Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. III.
zbMATH Open document ID: 1046.35043
DOI: 10.1016/S0022-0396(03)00088-3
author: Alberto Ferrero
author: Filippo Gazzola
published in: Journal of Differential Equations
publication date: 2004-03-29T00:00:00Z
review text: This is a third part of a series of very interesting papers [Part I, \textit{F. Gazzola} and \textit{J. Serrin}, Ann. Inst. H. Poincaré, Anal. Non Lineaire 19, 477--504 (2002; Zbl 1013.35031); Part II: \textit{F. Gazzola}, \textit{L. A. Peletier}, \textit{P. Pucci}, \textit{J. Serrin}, Ann. Inst. H. Poincaré, Anal. Non Linéaire 20, 947--974 (2003; Zbl 1086.35041)] devoted to the study of asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing parameters. Precisely, set \(\Delta_mu=\text{ div\,}(| \nabla u| ^{m-2}\nabla u)\) for the \(m\)-Laplacian, and let \(m^*=nm/(n-m)\) be the critical Sobolev exponent for the embedding \(D^{1,m}({\mathbb R}^n)\subset L^{m^*}({\mathbb R}^n)\). Consider the quasilinear degenerate elliptic equation \[ -\Delta_mu=-\delta u^{q-1}+u^{p-1}\quad \text{ in}\;{\mathbb R}^n \eqno{(1)} \] with \(\delta>0,\) \(n>m>1\) and \(1<q<p<m^*\). A ground state of \((1)\) is defined as a \(C^1({\mathbb R}^n)\) nonnegative nontrivial radial distribution solution which vanishes at infinity. The equation \((1)\) admits a unique ground state for all \(\delta,p,q\) in the given range while, if either \(p=m^*\) and \(\delta>0\) or \(\delta=0\) and \(p\in(q,m^*)\) then there are no ground states of \((1)\). Finally, if both \(\delta=0\) and \(p=m^*\) then \((1)\) admits an one-parameter family of ground states given by the formula \[ U_d(x)=d\left(1+D\left(d^{m/(n-m)}| x| \right)^{m/(m-1)}\right)^{(m-n)/m},\quad d>0 \eqno{(2)} \] where \(D=D_{n,m}=(m-1)/(n-m)n^{1/(m-1)}\) and \(U_d(0)=d.\) The main goal of the paper under review is the study of behavior of ground states in the borderline cases when \(p\uparrow m^*\) and/or \(\delta\downarrow 0.\) Also \(q=m\) is a limit case because, as proven in [\textit{B. Franchi}, \textit{E. Lanconelli} and \textit{J. Serrin}, Adv. Math. 118, 177--243 (1996; Zbl 0853.35035)], if \(q<m\) then the ground state of \((1)\) has compact support while if \(q\geq m\) it remains positive in \({\ma…
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 2062715
zbMATH Keywords: degenerate quasilinear elliptic operators
zbMATH Keywords: \(m\)-Laplacian
zbMATH Keywords: critical growth
zbMATH Keywords: ground states
zbMATH Keywords: asymptotic behavior
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2062715