Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential.
type: Item
instance of: scholarly article
title: Multiple solutions for nonlinear elliptic equations at resonance with a nonsmooth potential.
zbMATH Open document ID: 1086.35050
author: Nikolaos S. Papageorgiou
author: Dumitru Motreanu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-03-29T00:00:00Z
review text: The authors study the following nonlinear elliptic problem \[ \begin{gathered} -\text{div}(\| Dx(z)\|^{p- 2}Dx(z))- \lambda,\;| x(z)|^{p- 2}x(z)\in \partial_j(z, x(z))\text{ a.e. on }Z,\\ x|_\Gamma= 0,\end{gathered}\tag{1} \] where \(Z\in\mathbb{R}^N\) is a bounded domain with a \(C^{1,\alpha}\)-boundary \(\Gamma\) \((\alpha\in (0,1))\) and \(j(z,\cdot)\) is a locally Lipschitz not necessarily smooth potential, with \(\partial j(z,\cdot)\) being the subdifferential in the sense of Clarke. The goal of the authors is to establish the existence of at least two distinct solutions for (1) assuming a different asymptotic behaviour of \(\partial j(z,\cdot)\) at \(\pm\infty\). Moreover, the authors show that one of the solutions is smooth and strictly positive.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35P30
Mathematics Subject Classification ID: 35R70
zbMATH DE Number: 2062878
zbMATH Keywords: Clarke subdifferential
zbMATH Keywords: nonsmooth Palais-Smale condition
zbMATH Keywords: Ekeland variational principle
zbMATH Keywords: nonlinear regularity
zbMATH Keywords: nonsmooth mountain pass theorem
zbMATH Keywords: resonant problem
zbMATH Keywords: \(p\)-Laplacian
zbMATH Keywords: principal eigenvalue
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: j.na.2003.11.011
OpenAlex ID: W2042890004
cites work: Q3140494
cites work: Variational methods for non-differentiable functionals and their applications to partial differential equations
cites work: A New Approach to Lagrange Multipliers
cites work: Q4000274
cites work: Well-posed optimization problems
cites work: Q3029673
cites work: An existence theorem for nonlinear hemivariational inequalities at resonance
cites work: Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance
cites work: Eigenvalue problems for variational-hemivariational inequalities at resonance
cites work: Multiple nontrivial solutions of semilinear strongly resonant elliptic equations
cites work: Three solutions for a strongly resonant elliptic problem
cites work: Multiple Nontrivial Solutions of Semilinear Elliptic Equations
cites work: Existence of multiple solutions for quasilinear and semilinear elliptic equations
cites work: Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities
cites work: Q4367585
cites work: Q4942053
cites work: Q4761442
cites work: Q4139197
cites work: Hemivariational inequalities with the potential crossing the first eigenvalue.
cites work: Linear and quasilinear elliptic equations
cites work: Multiple solutions of semilinear elliptic problems at resonance
cites work: Q4078579
cites work: Boundary regularity for solutions of degenerate elliptic equations
cites work: On the Equation div( | ∇u | p-2 ∇u) + λ | u | p-2 u = 0
cites work: On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems
cites work: Asymptotically linear elliptic equations without nonresonance conditions
cites work: A minimax approach to the eigenvalue problem of hemivariational inequalities and applications
cites work: Minimax theorems and qualitative properties of the solutions of hemivariational inequalities
cites work: Q4355191
cites work: Q4323869
cites work: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems
cites work: A strong maximum principle for some quasilinear elliptic equations
DOI: 10.1016/J.NA.2003.11.011
description: scientific article; zbMATH DE number 2062878