Mountain pass on a closed convex set
type: Item
instance of: scholarly article
title: Mountain pass on a closed convex set
zbMATH Open document ID: 0895.58055
author: Ma Li
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-09-14T00:00:00Z
review text: The author proves a version of the mountain pass theorem for a functional defined on a closed convex set of a Banach space. Such a functional does not necessarily satisfy the Palais-Smale condition. This abstract result is applied to the study of the semilinear Dirichlet problem \[ \begin{cases} -\Delta u+u^2 =f(x) \quad & \text{in } \Omega,\\ u=0 \quad & \text{on } \partial \Omega,\end{cases} \tag{*} \] where \(\Omega\) is a piecewise smooth and bounded domain in \(\mathbb{R}^n\), \(n<6\). The problem (*) was studied by K. C. Chang and J. Nirenberg who proved the existence of a negative and a nontrivial solution for (*). In this paper the author shows the existence of a positive solution for (*).
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1005673
zbMATH Keywords: convex sets
zbMATH Keywords: Dirichlet boundary problem
zbMATH Keywords: mountain pass theorem
reviewed by: Antonio Masiello
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5227
OpenAlex ID: W1989249605
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DOI: 10.1006/JMAA.1997.5227
description: scientific article; zbMATH DE number 1005673