Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives
type: Item
instance of: scholarly article
title: Exact multiplicity results for variational inequalities involving pointwise constraints on the derivatives
zbMATH Open document ID: 1046.35022
DOI: 10.1016/S0362-546X(02)00178-5
author: Riccardo Molle
author: Donato Passaseo
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb R^ n\), let \(\psi \in H^1_0(\Omega)\) and let \(K_\psi\) denote the set of all \(u\in H^1_0(\Omega)\) such that \(\Delta u \leq \Delta\psi\) in the weak sense. The authors examine the problem of finding an element \(u\in K_\psi\) such that \[ \int_{\Omega} [Du\cdot D(u-v)-g(x,u)(v-u)+h(u-v)]\,dx \geq 0 \] for all \(v \in K_\psi\) for given functions \(g\) and \(h\). If \(K_\psi\) is the set of \(u \in H^1_0\) such that \(u \geq \psi\) in \(\Omega\), then this problem is the familiar obstacle problem, but the constraint on the Laplacian of the solution leads to rather different behavior. In particular, the current problem behaves like the corresponding elliptic boundary value problem \[ \Delta u+g(u)=h \text{ in } \Omega, \quad u=0 \text{ on } \partial\Omega. \] If \(\lambda_j\) denotes the \(j\)th eigenvalue of the Dirichlet Laplacian on \(\Omega\), then, for the boundary value problem, the number of solutions depends on the behavior of \(g(x,t)/t\) as \(t\to\pm\infty\). Here, the behavior as \(t\to-\infty\) is replaced by the assumption that the solution lies in \(K_\psi\). Specifically, if \(\lim_{t\to\infty} g(x,t)/t >\lambda_1\) and if \(e_1\) is the normalized eigenfunction of the Dirichlet Laplacian with eigenvalue \(\lambda_1\), then, for any \(\tilde h \in L^2\), there is a number \(\tau_0\) such that the original problem with \(h=\tilde h +\tau e_1\) has no solution if \(\tau<\tau_0\), at least one solution if \(\tau=\tau_0\), and at least two solutions if \(\tau>\tau_0\).
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J85
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49J20
zbMATH DE Number: 1880106
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: unilateral constraints
zbMATH Keywords: variational methods
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: supersolutions
zbMATH Keywords: jumping nonlinearities
zbMATH Keywords: exact number of solutions
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880106