On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary.
type: Item
instance of: scholarly article
title: On the nonlinear Neumann problem involving the critical Sobolev exponent on the boundary.
zbMATH Open document ID: 1074.35038
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: This paper concerns the nonlinear Neumann problem \[ -\Delta u=Q(x)| u| ^{2^*-2}u\quad \text{in }\Omega, \] \[ \frac{\partial}{\partial\nu}u(x)=P(x)| u| ^{q-2}u\quad\text{on }\partial\Omega,\qquad u>0\quad\text{on }\Omega, \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega\), \(\nu\) is the outward normal to \(\partial\Omega,\) \(2^*=2N/(N-2)\), \(N\geq 3\), is the usual critical Sobolev exponent, \(q=2(N-1)/(N-2)\) is the critical Sobolev exponent for the trace embedding of \(H^1\) into \(L^q(\partial\Omega)\), \(Q(x)\) and \(P(x)\) are smooth functions. The author observes that if both \(Q\) and \(P\) are positive then the problem does not have solution. First, the case \(Q\equiv 0\) is discussed. \(P\) is assumed to change sign on \(\partial\Omega\) and moreover \[ \int_{\partial\Omega}P(x)\,dS<0. \] Furthermore, \(P\) is assumed to attain its maximum value at a point \(y\in\partial\Omega\) with \(H(y)>0\), and moreover \(| P(x)-P(y)| =o(| x-y| )\) for \(x\) near \(y\). Here \(H(y)\) denotes the mean curvature of \(\partial\Omega\) at the point \(y\), and \(o\) has the usual meaning. Under all the above assumptions, by using a constrained variational problem, it is proved that the Neumann problem has a solution. Next, the case \(Q(x)\geq 0\), \(Q(x)\not\equiv0\) is discussed. It is assumed that \(Q(x)=0\) for \(x\in\{\partial\Omega:P(x)>0\}\). Under some suitable additional conditions on \(P\) and \(Q\), by using the mountain-pass principle, it is proved the existence of a solution of the Neumann problem.
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 47J30
zbMATH DE Number: 2056389
zbMATH Keywords: Neumann problem
zbMATH Keywords: critical Sobolev exponent
zbMATH Keywords: least energy solutions
author: Jan Chabrowski
reviewed by: Giovanni Porru
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.036
OpenAlex ID: W1966652784
cites work: Q4863664
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DOI: 10.1016/J.JMAA.2003.10.036
description: scientific article; zbMATH DE number 2056389