Positive solutions of an obstacle problem
type: Item
instance of: scholarly article
title: Positive solutions of an obstacle problem
zbMATH Open document ID: 0866.49017
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1997-07-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73354
review text: In this paper the author studies the existence of positive solutions of the obstacle problem, with a nonlinear term in the variational inequality. The variational problem is formulated in an exterior domain with a smooth boundary. A positive parameter \(\lambda\) is included in the variational inequality and a discussion is done according to the values of this parameter. It is proved the existence of a value \(0<\lambda^*<+\infty\) such that the problem has at least one positive solution for every \(\lambda\) in \((0,\lambda^*)\) and there is no positive solution for \(\lambda>\lambda^*\). Furthermore, there exists a second value \(\lambda^{**}\), with \(0<\lambda^{**}\leq \lambda^*\), such that the problem has at least two different positive solutions for every \(\lambda\in (0,\lambda^{**})\). Finally, assuming that the obstacle belongs to \(H^{2,q}(\Omega)\), the author proves that any positive solution belongs to the space \(H^{2,q}(\Omega)\cap C^{1,\alpha}(\Omega)\).
reviewed by: Eduardo Casas
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 824220
zbMATH Keywords: positive solutions
zbMATH Keywords: obstacle problem
zbMATH Keywords: variational inequality
zbMATH Keywords: exterior domain
author: Jianfu Yang
MaRDI profile type: Publication
OpenAlex ID: W2321187117
cites work: Q4204518
cites work: Dual variational methods in critical point theory and applications
cites work: Positive solutions of some nonlinear elliptic problems in exterior domains
cites work: Nonlinear scalar field equations. I: Existence of a ground state
cites work: A Relation Between Pointwise Convergence of Functions and Convergence of Functionals
cites work: Q3905599
cites work: Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(R^ n\)
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. II
cites work: A free boundary problem involving limiting Sobolev exponents
cites work: Holes and obstacles
cites work: Q3745373
cites work: Existence of solitary waves in higher dimensions
cites work: Bifurcation in Lp (RN ) for a Semilinear Elliptic Equation
cites work: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems
cites work: Positive solutions of semilinear elliptic problems in exterior domains
cites work: Existence of multiple positive solutions of inhomogeneous semilinear elliptic problems in unbounded domains
DOI: 10.5802/AFST.799
description: scientific article; zbMATH DE number 824220