scholarly journals Multiple Solutions for a Class of Differential Inclusion System Involving the(p(x),q(x))-Laplacian

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Bin Ge ◽  
Ji-Hong Shen
Keyword(s):  
Boundary Condition ◽  
Differential Inclusion ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Point Theory ◽  
Nontrivial Solutions ◽  
Locally Lipschitz Functions ◽  
Nonsmooth Critical Point Theory ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz

We consider a differential inclusion system involving the(p(x),q(x))-Laplacian with Dirichlet boundary condition on a bounded domain and obtain two nontrivial solutions under appropriate hypotheses. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions.

scholarly journals Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

2000 ◽  
Vol 5 (2) ◽  
pp. 119-135
Author(s):  
Nikolaos C. Kourogenis ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Principle ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Point Theory ◽  
Existence Theory ◽  
Nontrivial Solutions ◽  
Nonsmooth Critical Point Theory ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz ◽  
Ps Condition

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and a generalized version of the Ekeland variational principle. At the end of the paper we show that the nonsmooth Palais-Smale (PS)-condition implies the coercivity of the functional, extending this way a well-known result of the “smooth” case.

MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH A DISCONTINUOUS NONLINEARITY

2006 ◽  
Vol 04 (01) ◽  
pp. 1-18 ◽  
Author(s):  
MICHAEL E. FILIPPAKIS ◽  
NIKOLAOS S. PAPAGEORGIOU
Keyword(s):  
Nonlinear Elliptic Equation ◽  
Point Theory ◽  
Discontinuous Nonlinearity ◽  
Locally Lipschitz Functions ◽  
Nonlinear Elliptic Problems ◽  
Nonsmooth Critical Point Theory ◽  
Nonlinear Elliptic ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz ◽  
Truncation Techniques

We consider a nonlinear elliptic equation driven by the p-Laplacian with a discontinuous nonlinearity. Such problems have a "multivalued" and a "single-valued" interpretation. We are interested in the latter and we prove the existence of at least two distinct solutions, both smooth and one strictly positive. Our approach is variational based on the nonsmooth critical point theory for locally Lipschitz functions, coupled with penalization and truncation techniques.

Multiple Solutions for Resonant Hemivariational Inequalities via Minimax Methods

2009 ◽  
Vol 9 (3) ◽  
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’ Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Multiple Solutions ◽  
Principal Eigenvalue ◽  
Point Theory ◽  
Hemivariational Inequalities ◽  
Dirichlet Problems ◽  
Nonsmooth Critical Point Theory ◽  
Minimax Methods ◽  
Nonsmooth Critical Point ◽  
Multiplicity Theorem ◽  
The Right

AbstractIn this paper we consider nonlinear Dirichlet problems driven by the p-Laplacian differential operator with a nonsmooth potential (hemivariational inequalities). We assume that the problem is resonant at infinity with respect to λ1 > 0 (the principal eigenvalue of the Dirichlet p-Lapalcian) from the right. Using minimax methods based on the nonsmooth critical point theory we prove an existence and a multiplicity theorem.

Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance

2001 ◽  
Vol 131 (5) ◽  
pp. 1091-1111 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Principle ◽  
Critical Point Theory ◽  
Multiple Solutions ◽  
Existence Theorems ◽  
Point Theory ◽  
Hemivariational Inequalities ◽  
Locally Lipschitz Functions ◽  
Strong Resonance ◽  
At Resonance ◽  
Locally Lipschitz

In this paper we consider quasilinear hemivariational inequalities at resonance. We obtain existence theorems using Landesman-Lazer-type conditions and multiplicity theorems for problems with strong resonance at infinity. Our method of proof is based on the non-smooth critical point theory for locally Lipschitz functions and on a generalized version of the Ekeland variational principle.

scholarly journals Nonlinear Periodic Systems with thep-Laplacian: Existence and Multiplicity Results

2007 ◽  
Vol 2007 ◽  
pp. 1-23
Author(s):  
Francesca Papalini
Keyword(s):  
Existence Theorems ◽  
Point Theory ◽  
Energy Functional ◽  
Periodic Systems ◽  
Multiplicity Result ◽  
Nonsmooth Critical Point Theory ◽  
Multiplicity Results ◽  
Existence And Multiplicity ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz

We study second-order nonlinear periodic systems driven by the vectorp-Laplacian with a nonsmooth, locally Lipschitz potential function. Under minimal and natural hypotheses on the potential and using variational methods based on the nonsmooth critical point theory, we prove existence theorems and a multiplicity result. We conclude the paper with an existence theorem for the scalar problem, in which the energy functional is indefinite (unbounded from both above and below).

scholarly journals Solutions for nonlinear variational inequalities with a nonsmooth potential

2004 ◽  
Vol 2004 (8) ◽  
pp. 635-649 ◽  
Author(s):  
Michael E. Filippakis ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Inequality ◽  
Variational Inequalities ◽  
Elliptic Equations ◽  
Lower Semicontinuous ◽  
Point Theory ◽  
Nonsmooth Critical Point Theory ◽  
Nonsmooth Critical Point ◽  
Locally Lipschitz ◽  
Proper Convex ◽  
Nonsmooth Potential

First we examine a resonant variational inequality driven by thep-Laplacian and with a nonsmooth potential. We prove the existence of a nontrivial solution. Then we use this existence theorem to obtain nontrivial positive solutions for a class of resonant elliptic equations involving thep-Laplacian and a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory for functionals of the formφ=φ1+φ2withφ1locally Lipschitz andφ2proper, convex, lower semicontinuous.

scholarly journals Multiple Solutions for Nonhomogeneous Neumann Differential Inclusion Problems by thep(x)-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qing-Mei Zhou
Keyword(s):  
Variational Method ◽  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Nontrivial Solutions ◽  
Weierstrass Theorem ◽  
Locally Lipschitz Functions ◽  
Inclusion Problems ◽  
Neumann Problems ◽  
Locally Lipschitz ◽  
Nonlinear Neumann Problems

A class of nonlinear Neumann problems driven byp(x)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality) was considered. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass theorem and Mountain Pass theorem are used to prove the existence of at least two nontrivial solutions.

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