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

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).

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Nonsmooth critical point theory and nonlinear elliptic equations at resonance

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700002202 ◽

2000 ◽

Vol 69 (2) ◽

pp. 245-271 ◽

Cited By ~ 68

Author(s):

Nikolaos C. Kourogenis ◽

Nikolaos S. Papageorgiou

Keyword(s):

Critical Point ◽

Critical Point Theory ◽

Existence Theorems ◽

Nonlinear Elliptic Equations ◽

Point Theory ◽

Energy Functional ◽

Existence Theory ◽

Nonsmooth Critical Point Theory ◽

Nonsmooth Critical Point ◽

The Right

AbstractIn this paper we complete two tasks. First we extend the nonsmooth critical point theory of Chang to the case where the energy functional satisfies only the weaker nonsmooth Cerami condition and we also relax the boundary conditions. Then we study semilinear and quasilinear equations (involving the p-Laplacian). Using a variational approach we establish the existence of one and of multiple solutions. In simple existence theorems, we allow the right hand side to be discontinuous. In that case in order to have an existence theory, we pass to a multivalued approximation of the original problem by, roughly speaking, filling in the gaps at the discontinuity points.

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Existence and subharmonicity of solutions for nonsmooth $ p $-Laplacian systems

AIMS Mathematics ◽

10.3934/math.2021636 ◽

2021 ◽

Vol 6 (10) ◽

pp. 10947-10963

Author(s):

Yan Ning ◽

◽

Daowei Lu ◽

Anmin Mao ◽

Keyword(s):

Variational Methods ◽

Critical Point ◽

Potential Function ◽

Critical Point Theory ◽

Point Theory ◽

Periodic Systems ◽

Nonsmooth Critical Point Theory ◽

Nonsmooth Critical Point ◽

Locally Lipschitz

<abstract><p>In this paper we study nonlinear periodic systems driven by the vectorial $ p $-Laplacian with a nonsmooth locally Lipschitz potential function. Using variational methods based on nonsmooth critical point theory, some existence of periodic and subharmonic results are obtained, which improve and extend related works.</p></abstract>

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Existence and Multiplicity Results for A Fourth-Order Boundary Value Problem

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.1515/aicu-2015-0021 ◽

2015 ◽

Vol 0 (0) ◽

Author(s):

Nemat Nyamoradi ◽

Mohamad Rasoul Hamidi

Keyword(s):

Variational Method ◽

Boundary Value Problem ◽

Fourth Order ◽

Critical Point Theory ◽

Boundary Value ◽

Point Theory ◽

Nonsmooth Critical Point Theory ◽

Multiplicity Results ◽

Existence And Multiplicity ◽

Nonsmooth Critical Point

Abstract In this paper we consider a class of a fourth-order boundary value problem. Using a variational method based on nonsmooth critical point theory, we prove the existence and multiplicity of solutions.

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Existence theorems for elliptic hemivariational inequalities involving thep-Laplacian

Abstract and Applied Analysis ◽

10.1155/s1085337502000908 ◽

2002 ◽

Vol 7 (5) ◽

pp. 259-277 ◽

Cited By ~ 5

Author(s):

Nikolaos C. Kourogenis ◽

Nikolaos S. Papageorgiou

Keyword(s):

Existence Theorems ◽

Principal Eigenvalue ◽

Point Theory ◽

Hemivariational Inequalities ◽

Nonsmooth Critical Point Theory ◽

Energy Functionals ◽

At Resonance ◽

Nonsmooth Critical Point ◽

Locally Lipschitz ◽

Generalized Potential

We study quasilinear hemivariational inequalities involving thep-Laplacian. We prove two existence theorems. In the first, we allow “crossing” of the principal eigenvalue by the generalized potential, while in the second, we incorporate problems at resonance. Our approach is based on the nonsmooth critical point theory for locally Lipschitz energy functionals.

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Solutions for nonlinear variational inequalities with a nonsmooth potential

Abstract and Applied Analysis ◽

10.1155/s1085337504312017 ◽

2004 ◽

Vol 2004 (8) ◽

pp. 635-649 ◽

Cited By ~ 13

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.

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Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

Abstract and Applied Analysis ◽

10.1155/s1085337500000269 ◽

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.

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

Abstract and Applied Analysis ◽

10.1155/2012/971243 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 1

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.

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MULTIPLE SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH A DISCONTINUOUS NONLINEARITY

Analysis and Applications ◽

10.1142/s021953050600067x ◽

2006 ◽

Vol 04 (01) ◽

pp. 1-18 ◽

Cited By ~ 12

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.

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Multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.05.097 ◽

2011 ◽

Vol 218 (2) ◽

pp. 532-541 ◽

Cited By ~ 2

Author(s):

Sami Aouaoui

Keyword(s):

Critical Point ◽

Critical Point Theory ◽

Point Theory ◽

Multiplicity Result ◽

Theory Approach ◽

Nonsmooth Critical Point Theory ◽

Nonsmooth Critical Point ◽

Anisotropic Equation

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Existence of Positive Solutions for Nonlinear Noncoercive Hemivariational Inequalities

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-034-6 ◽

2007 ◽

Vol 50 (3) ◽

pp. 356-364 ◽

Cited By ~ 1

Author(s):

Michael E. Filippakis ◽

Nikolaos S. Papageorgiou

Keyword(s):

Positive Solutions ◽

Point Theory ◽

Hemivariational Inequalities ◽

Nonsmooth Critical Point Theory ◽

Multiplicity Results ◽

Nonlinear Elliptic ◽

Existence Of Positive Solutions ◽

Nonsmooth Critical Point ◽

Linear Problems ◽

Asymptotically Linear Problems

AbstractIn this paper we investigate the existence of positive solutions for nonlinear elliptic problems driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality). Under asymptotic conditions that make the Euler functional indefinite and incorporate in our framework the asymptotically linear problems, using a variational approach based on nonsmooth critical point theory, we obtain positive smooth solutions. Our analysis also leads naturally to multiplicity results.

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