On quasilinear Brezis-Nirenberg type problems with weights
type: Item
instance of: scholarly article
title: On quasilinear Brezis-Nirenberg type problems with weights
zbMATH Open document ID: 1209.35055
author: Cecilia Yarur
author: Marta Garcia-Huidobro
published in: Advances in Differential Equations
publication date: 2010-07-09T00:00:00Z
review text: The authors consider radially symmetric positive solutions of the \(p\)-Laplacian problem \[ \begin{cases} -\Delta_p u =\lambda C(| x|)| u|^{p-2}u +B(| x|)| u|^{q-2}u, &\text{in } B_R(0)\setminus\{0\},\\ u=0&\text{on } \partial B_R(0),\\ \lim_{| x|\to0}| x|^{N-1}|\nabla u(x)|^{p-1}=0, \end{cases}\tag{1} \] where \(B_R(0)\) denotes, for \(R>0\), the open ball in \(\mathbb{R}^N\) of radius \(R\), centered at \(0\). It is assumed that \(q\geq p>1\), \(N\geq p\), and that the weights \(B,C\) are positive and such that \(r^{N-1}B(r)\) and \(r^{N-1}C(r)\) are in \(L^1(0,R)\). First a critical exponent \(p^*\) is defined that depends on \(B\) and \(p\) and controls the compactness of embeddings of certain weighted Sobolev spaces. The existence of a smallest eigenvalue \(\lambda_1\), which is positive, and a corresponding eigenfunction \(\varphi_1\) for the nonlinear eigenvalue problem \[ \begin{cases} -\Delta_p u =\lambda C(| x|)| u|^{p-2}u &\text{in } B_R(0)\setminus\{0\},\\ u=0&\text{on } \partial B_R(0),\\ \lim_{| x|\to0}| x|^{N-1}|\nabla u(x)|^{p-1}=0, \end{cases}\tag{2} \] is proved under appropriate conditions on \(p\) and \(C\). With these notions, and imposing additional assumptions on the weights \(B\) and \(C\) and the exponents \(p\) and \(q\), the authors prove existence and nonexistence results roughly of the following form: The subcritical problem (1), where \(q<p^*\), has a nontrivial, radially symmetric, and nonnegative solution if and only if \(\lambda<\lambda_1\). In the critical case \(q=p^*\), if \(p\) and \(q\) satisfy a condition that restricts the dimension \(N\), there exist \(0<\lambda^*< \lambda^{**}<\lambda_1\) such that (1) has a nontrivial, radially symmetric, and nonnegative solution if \(\lambda\in(\lambda^{**},\lambda_1)\), and it has no such solution if \(\lambda\in(0,\lambda^*)\). Finally, (1) has no such solution if \(q\) is supercritical, i.e. \(q>p^*\), and \(R\) and \(\lambda>0\) are small enough. The dimension restriction in the criti…
reviewed by: Nils Ackermann
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35B07
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 5734996
zbMATH Keywords: Brezis-Nirenberg type results
zbMATH Keywords: critical exponent
zbMATH Keywords: nonlinear eigenvalue
zbMATH Keywords: positive radial solutions
zbMATH Keywords: quasilinear elliptic equation
zbMATH Keywords: weight function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5734996