Neumann problems of a class of elliptic equations with doubly critical Sobolev exponents
type: Item
instance of: scholarly article
title: Neumann problems of a class of elliptic equations with doubly critical Sobolev exponents
zbMATH Open document ID: 1130.35066
DOI: 10.1016/S0252-9602(17)30247-3
author: Pigong Han
published in: Acta Mathematica Scientia. Series B. (English Edition)
publication date: 2006-01-24T00:00:00Z
review text: The paper deals with the following Neumann problem \[ -\Delta u+u=| u| ^{2^*-2}u+\mu | u| ^{q-2}u\;\;in\;\;\Omega,\;\;\frac{\partial u}{\partial\gamma}=| u| ^{s^*-2}u\;\;on\;\;\partial\Omega, \] where \(\Omega\) is a bounded smooth domain in \(R^N\), \(N\geq 3\), \(2^*=2N/(N-2)\), \(s^*=2(N-1)/(N-2)\), \(1<q<2\), \(\mu>0\) and \(\gamma\) denotes the outward normal to the boundary \(\partial\Omega\). The non trivial solutions of this problem are non zero critical points of the functional \[ J(u)=\int_\Omega\biggl(\frac{1}{2}| \nabla u| ^2+\frac{1}{2}u^2-\frac{1}{2^*}| u| ^{2^*}-\frac{\mu}{q}| u| ^q\biggr)\,dx-\frac{1}{s^*}\int_{\partial\Omega} | u| ^{s^*}\,ds,\quad u\in H^1(\Omega). \] By using variational methods, the author proves that there exists \(\mu^*>0\) such that, for every \(0<\mu<\mu^*\) this problem has a sequence of solutions \(u_k\in H^1(\Omega)\) such that \(J(u_k)<0\) and \(J(u_k)\to 0\) as \(k\to\infty\).
reviewed by: Giovanni Porru
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 5000928
zbMATH Keywords: Neumann problem
zbMATH Keywords: Semilinear elliptic equations
zbMATH Keywords: Critical Sobolev exponent
MaRDI profile type: Publication
full work available at URL: s0252-9602(17)30247-3
OpenAlex ID: W2783559674
description: scientific article; zbMATH DE number 5000928