Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.     

Periodic solutions for second order differential inclusions with the scalar p-Laplacian

We study periodic problems driven by the scalar p-Laplacian with a multivalued right-hand side nonlinearity. We prove two existence theorems. In the first, we assume nonuniform nonresonance conditions between two successive eigenvalues of the negative p-Laplacian with periodic boundary conditions. In the second, we employ certain Landesman-Lazer type conditions. Our approach is based on degree theory.     

Multiple nontrivial solutions for nonlinear periodic problems with the p-Laplacian

We consider a nonlinear periodic problem driven by the scalar p-Laplacian with a nonsmooth potential (hemivariational inequality). Using the degree theory for multivalued perturbations of (S)₊-operators and the spectrum of a class of weighted eigenvalue problems for the scalar p-Laplacian, we prove the existence of at least three distinct nontrivial solutions, two of which have constant sign.     

Three nontrivial solutions for the -Laplacian with a nonsmooth potential

We consider a nonlinear elliptic problem driven by the partial p-Laplacian and with a nonsmooth potential (hemivariational inequality). Using variational techniques based on nonsmooth analysis and degree theoretic arguments for operators of the monotone type, we establish the existence of at least three distinct nontrivial smooth solutions.     

Solvability of nonlinear variational-hemivariational inequalities

In this paper we study nonlinear elliptic differential equations driven by the p-Laplacian with unilateral constraints produced by the combined effects of a monotone term and of a nonmonotone term (variational-hemivariational inequality). Our approach is variational and uses the subdifferential theory of nonsmooth functions and the theory of accretive and monotone operators. Also using these ideas and a special choice of the monotone term, we prove the existence of a strictly positive smooth solution for a class of nonlinear equations with nonsmooth potential (hemivariational inequality).     

Multiple solutions for a system of equations with p-Laplacian

In the paper, by using of the Limit Index, we prove a theorem applying to get multiple critical values of some strongly indefinite nonsmooth functionals, and then we apply it to a system of equations involving the p-Laplacian.     

Pairs of positive solutions for the periodic scalar p-Laplacian

We study a nonlinear periodic problem driven by the scalar p-Laplacian and having a nonsmooth potential (hemivariational inequality). Using a combination of variational techniques and degree-theoretic methods based on a degree map for certain multivalued perturbations of (S)₊operators, we establish the existence of two positive solutions.     

Positive solutions for nonlinear periodic problems

We consider a nonlinear periodic problem driven by the scalar p-Laplacian and with a nonsmooth potential. Using the degree map for multivalued perturbations of (S)₊-operators and the spectrum of a weighted eigenvalue problem for the scalar periodic p-Laplacian, we prove the existence of a strictly positive solution. Michael E. Filippakis: Researcher supported by a grant of the National Scholarship Foundation of Greece (I.K.Y.)     

Existence of solutions for some nonlinear problems with p-Laplacian like operators

The purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p-Laplacian like operators.     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

Three positive solutions for the one-dimensional p-Laplacian

In this paper, nonlinear two point boundary value problems with p-Laplacian operators subject to Dirichlet boundary condition and nonlinear boundary conditions are studied. We show the existence of three positive solutions by the five functionals fixed point theorem.     

Multiple positive solutions of the one-dimensional p-Laplacian

In this work we investigate the existence of positive solutions of the p-Laplacian, using the quadrature method. We prove the existence of multiple solutions of the one-dimensional p-Laplacian for α?0, and determine their exact number for α=0.     

Existence of positive solutions for the p-Laplacian with p-gradient term

In this paper, we study the existence of positive solutions for the p-Laplacian involving a p-gradient term. Due to the non-variational structure and the fact that the nonlinearity may be critical or supercritical, the variational method is no longer valid. Taking advantage of global C1,α estimates and the Liouville type theorems, we employ the blow-up argument to obtain the a priori estimates on solutions, and finally obtain the existence result based on the Krasnoselskii fixed point theorem.     

Existence and multiplicity results for Dirichlet problems with p-Laplacian

In this paper we apply Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems.     

Existence and multiplicity of positive solutions for singular quasilinear problems

In this work we combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular p-Laplacian problems. In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.     

Eigenvalue problems for nonlinear elliptic equations with unilateral constraints

In this paper we study eigenvalue problems for hemivariational and variational inequalities driven by the p$p$-Laplacian differential operator. Using topological methods (based on multivalued versions of the Leray–Schauder alternative principle) and variational methods (based on the nonsmooth critical point theory), we prove existence and multiplicity results for the eigenvalue problems that we examine.     

Landesman-Lazer type conditions for a system of p-Laplacian like operators

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems.     

A multiplicity theorem for hemivariational inequalities with a p -Laplacian-like differential operator

We consider a parametric nonlinear elliptic inclusion with a multivalued p$p$-Laplacian-like differential operator and a nonsmooth potential (hemivariational inequality). Using a variational approach based on the nonsmooth critical point theory, we show that for all the values of the parameter in an open half-line, the problem admits at least two nontrivial solutions. Our result extends a recent one by Kristály, Lisei, and Varga [A. Kristály, H. Lisei, C. Varga, Multiple solutions for p$p$-Laplacian type operator, Nonlinear Anal. 68 (5) (2008) 1375–1381].     

Nonlinear elliptic problems of Neumann-type

In this paper we study a nonlinear elliptic differential equation driven by thep-Laplacian with a multivalued boundary condition of the Neumann type. Using techniques from the theory of maximal monotone operators and a theorem of the range of the sum of monotone operators, we prove the existence of a (strong) solution.     

Existence of infinitely many periodic solutions for ordinary p-Laplacian systems

In this paper, we study the existence and multiplicity of non-trivial periodic solutions of ordinary p-Laplacian systems by using the minimax technique in critical point theory. We also give an example to illustrate that the obtained results are new even in the case p=2.