Differential and Integral Equations

Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions

Franco Obersnel and Pierpaolo Omari

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Abstract

We discuss existence and multiplicity of bounded variation solutions of the mixed problem for the prescribed mean curvature equation \begin{equation*} -{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = f(x,u) \hbox{\, in $\Omega$}, \quad u=0 \hbox{\, on $\Gamma_{D}$}, \quad \partial u / \partial \nu =0 \hbox{\, on $ \Gamma_{N}$}, \end{equation*} where $\Gamma_{D} $ is an open subset of $\partial \Omega$ and $\Gamma_{N}=\partial \Omega\setminus \Gamma_{D}$. Our approach is based on variational techniques and a lower and upper solutions method specially developed for this problem.

Article information

Source
Differential Integral Equations Volume 22, Number 9/10 (2009), 853-880.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019512

Mathematical Reviews number (MathSciNet)
MR2553060

Zentralblatt MATH identifier
1240.35131

Subjects
Primary: 35J93: Quasilinear elliptic equations with mean curvature operator
Secondary: 35J20: Variational methods for second-order elliptic equations

Citation

Obersnel, Franco; Omari, Pierpaolo. Existence and multiplicity results for the prescribed mean curvature equation via lower and upper solutions. Differential Integral Equations 22 (2009), no. 9/10, 853--880. https://projecteuclid.org/euclid.die/1356019512.


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