Positive solutions of the Dirichlet problem for the prescribed mean curvature equation
type: Item
instance of: scholarly article
title: Positive solutions of the Dirichlet problem for the prescribed mean curvature equation
zbMATH Open document ID: 1207.35145
author: Franco Obersnel
author: Pierpaolo Omari
published in: Journal of Differential Equations
publication date: 2010-09-29T00:00:00Z
review text: The authors consider the Dirichlet problem for the prescribed mean curvature equation \[ - \text{div}\big(\nabla u/ \sqrt{1+|\nabla u|^2}\big)= \lambda f(x,u) \quad\text{in }\Omega, \qquad u=0 \quad\text{on }\partial\Omega \tag \(*\) \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) (\(N\geq 2\)) with Lipschitz boundary, \(f:\Omega\times\mathbb R\rightarrow\mathbb R\) is a Carathéodory function, and \(\lambda\) is a real parameter. Solutions are taken in the bounded variation sense, namely, \(u\in \text{BV}(\Omega)\) is a solution of \((*)\) if \(f(\cdot,u)\in L^p(\Omega)\) for some \(p>N\) and \[ {\mathcal I}(v)-{\mathcal I}(u) \geq \lambda \int_\Omega f(x,u)(v-u)\,dx \] for every \(v\in \text{BV}(\Omega)\), where \({\mathcal I}:\text{BV}(\Omega)\rightarrow\mathbb R\) is defined by \[ {\mathcal I}(w) = \int_\Omega \sqrt{1+|\nabla w|^2} + \int_{\partial\Omega} | w_{|\partial\Omega}| \,d{\mathcal H}^{n-1}. \] In some cases stronger notions of solution are also considered. The authors are interested in conditions on \(F(x,s)=\int_0^s f(x,t)\,dt\) and on \(\lambda\) guaranteeing the existence and multiplicity of positive solutions of \((*)\). The proofs use variational methods and combine critical point theory, lower and upper solution methods, and elliptic regularization. The authors present an exhaustive list of results, including, as far as possible, all the possible configurations of limits of \(F(x,s)/s^2\) at zero and \(F(x,s)/s\) at infinity, which yield the existence of one, two, three or infinitely many positive solutions of \((*)\). Considering these limits at zero and infinity is suggested by the fact that the mean curvature operator behaves like the Laplacian near zero and like the 1-Laplacian \(\Delta_1u=\text{div}(\nabla u/|\nabla u|)\) near infinity.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J93
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 5793318
zbMATH Keywords: prescribed mean curvature equation
zbMATH Keywords: bounded variation solution
zbMATH Keywords: weak solution
zbMATH Keywords: strong solution
zbMATH Keywords: positive solution
zbMATH Keywords: existence
zbMATH Keywords: multiplicity
zbMATH Keywords: variational methods
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: regularization
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: j.jde.2010.07.001
OpenAlex ID: W2036529126
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DOI: 10.1016/J.JDE.2010.07.001
description: scientific article; zbMATH DE number 5793318