Existence results for some nonlinear elliptic system with lack of compactness
type: Item
instance of: scholarly article
title: Existence results for some nonlinear elliptic system with lack of compactness
zbMATH Open document ID: 1087.35043
DOI: 10.1016/S0362-546X(02)00151-7
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The author investigates the existence of a weak solution of the system \[ -\Delta_p u=u\left| u\right|^{\alpha-1}\left| v\right|^{\beta+1}+f, \; -\Delta_p v=v\left| u\right|^{\alpha+1}\left| v\right|^{\beta-1}+g \] subject to homogeneous Dirichlet boundary conditions on a smooth bounded domain \(\Omega\subset {\mathbb R}^n\). It is supposed that \(\alpha,\;\beta>-1\), \({{(\alpha+1)(n-p)}\over{np}}+{{(\beta+1)(n- q)}\over{nq}}=1\), and \(\max\{p,q\}<\alpha+\beta+2\). The main result guarantees the existence of a solution, if \(f\in W^{-1,p'}(\Omega)\) and \(g\in W^{-1,q'}(\Omega)\) are both nonzero, and \(\left\| f\right\|_{-1,p'}+\left\| g\right\|_{-1,q'}\) is sufficiently small. The proof utilizes local minimization of an adapted problem. Convergence is obtained by concentration-compactness arguments. It should be noted that the nonlinearities involve critical Sobolev exponents.
reviewed by: Georg Hetzer
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35J55
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 1879994
zbMATH Keywords: critical exponent
zbMATH Keywords: elliptic system
zbMATH Keywords: pseudo-Laplacian
zbMATH Keywords: non-compactness
zbMATH Keywords: local minimization
author: J. Velin
MaRDI profile type: Publication
cites work: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations
cites work: Q4748596
cites work: Q4338532
cites work: The concentration-compactness principle in the calculus of variations. The limit case. I
cites work: Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations
cites work: Q5516356
cites work: On nonhomogeneous elliptic equations involving critical Sobolev exponent
cites work: Résultats d'existence et de non-existence pour la solution positive et bornée d'une e.d.p. elliptique non linéaire
cites work: Q3986711
cites work: Q4296478
full work available at URL: s0362-546x(02)00151-7
OpenAlex ID: W1965284225
description: scientific article; zbMATH DE number 1879994