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.

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.

Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Variational Methods ◽  
Neumann Problem ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Solutions ◽  
Asymmetric Reaction ◽  
Neumann Problems ◽  
Nonlinear Neumann Problem ◽  
Asymmetric Behaviour ◽  
Nonlinear Neumann Problems

AbstractWe consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

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.

Infinitely Many Solutions for a Non-homogeneous Differential Inclusion with Lack of Compactness

2019 ◽  
Vol 19 (3) ◽  
pp. 625-637 ◽  
Author(s):  
Bin Ge ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Variational Principle ◽  
Differential Inclusion ◽  
Energy Functional ◽  
Mountain Pass Lemma ◽  
Infinitely Many Solutions ◽  
Locally Lipschitz Functions ◽  
Fountain Theorem ◽  
Inclusion Problems ◽  
Locally Lipschitz ◽  
Symmetric Mountain Pass Lemma

Abstract In this paper, we consider the following class of differential inclusion problems in {\mathbb{R}^{N}} involving the {p(x)} -Laplacian: -\Delta_{p(x)}u+V(x)\lvert u\rvert^{p(x)-2}u\in a(x)\partial F(x,u)\quad\text{% in}\ \mathbb{R}^{N}. We are concerned with a multiplicity property, and our arguments combine the variational principle for locally Lipschitz functions with the properties of the generalized Lebesgue–Sobolev space. Applying the nonsmooth symmetric mountain pass lemma and the fountain theorem, we establish conditions such that the associated energy functional possesses infinitely many critical points, and then we obtain infinitely many solutions.

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.

scholarly journals Nontrivial solutions for a multivalued problem with strong resonance

1996 ◽  
Vol 38 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Vicenţiu D. Rădulescu
Keyword(s):  
Critical Point ◽  
Banach Spaces ◽  
Mountain Pass Theorem ◽  
Point Theory ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Reflexive Banach Spaces ◽  
Strong Resonance ◽  
Locally Lipschitz ◽  
Critical Point Theorems

The Mountain-Pass Theorem of Ambrosetti and Rabinowitz (see [1]) and the Saddle Point Theorem of Rabinowitz (see [21]) are very important tools in the critical point theory of C1-functional. That is why it is natural to ask us what happens if the functional fails to be differentiable. The first who considered such a case were Aubin and Clarke (see [6]) and Chang (see [12]),who gave suitable variants of the Mountain-Pass Theorem for locally Lipschitz functionals which are denned on reflexive Banach spaces. For this aim they replaced the usual gradient with a generalized one, which was firstly defined by Clarke (see [13], [14]).As observed by Brezis (see [12, p. 114]), these abstract critical point theorems remain valid in non-reflexive Banach spaces.

scholarly journals Multiple Solutions for Second-Order Sturm–Liouville Boundary Value Problems with Subquadratic Potentials at Zero

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Dan Liu ◽  
Xuejun Zhang ◽  
Mingliang Song
Keyword(s):  
Multiple Solutions ◽  
Mountain Pass Theorem ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Linking Theorem ◽  
Infinitely Many Solutions ◽  
Nontrivial Solutions ◽  
Symmetric Mountain Pass Theorem ◽  
Sturm Liouville

We deal with the following Sturm–Liouville boundary value problem: − P t x ′ t ′ + B t x t = λ ∇ x V t , x ,     a.e.   t ∈ 0,1 x 0 cos    α − P 0 x ′ 0 sin    α = 0 x 1 cos    β − P 1 x ′ 1 sin    β = 0 Under the subquadratic condition at zero, we obtain the existence of two nontrivial solutions and infinitely many solutions by means of the linking theorem of Schechter and the symmetric mountain pass theorem of Kajikiya. Applying the results to Sturm–Liouville equations satisfying the mixed boundary value conditions or the Neumann boundary value conditions, we obtain some new theorems and give some examples to illustrate the validity of our results.

Multiplicity Results for Nonlinear Neumann Problems

2006 ◽  
Vol 58 (1) ◽  
pp. 64-92 ◽  
Author(s):  
Michael Filippakis ◽  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Neumann Boundary ◽  
Point Theory ◽  
Nontrivial Solutions ◽  
Zero Eigenvalue ◽  
Multiplicity Results ◽  
Neumann Problems ◽  
Nonlinear Elliptic ◽  
Nonsmooth Problems ◽  
Neumann Type ◽  
Nonlinear Neumann Problems

AbstractIn this paper we study nonlinear elliptic problems of Neumann type driven by the p-Laplacian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a C1-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-p-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative p-Laplacian with Neumann boundary condition.

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