We consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiffusive type. Using variational methods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

In this paper, we study a quasi-static frictional contact problem for a viscoelastic body with damage effect inside the body as well as normal compliance condition and multi-valued friction law on the contact boundary. The considered friction law generalizes Coulomb friction condition into multi-valued setting. The variational–hemi-variational formulation of the problem is derived and arguments of fixed point theory and surjectivity results for pseudo-monotone operators are applied, in order to prove the existence and uniqueness of solution.

In 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 ${{C}^{1}}$ -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.

In this paper we examine periodic problems driven by the scalar $p$-Laplacian. Using non-smooth critical-point theoryand a recent multiplicity result based on local linking (the original smooth version is due to Brezis and Nirenberg),we prove three multiplicity results, the third for semilinear problems with resonance at zero. We also study aquasilinear periodic eigenvalue problem with the parameter near resonance. We prove the existence of three distinctsolutions, extending in this way a semilinear and smooth result of Mawhin and Schmitt.

AMS 2000 Mathematics subject classification: Primary 34C25

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.

We consider a nonlinear hemivariational inequality with the p-Laplacian at resonance. Using an extension of the nonsmooth mountain pass theorem of Chang, which makes use of the Cerami compactness condition, we prove the existence of a nontrivial solution. Our existence results here extends a recent theorem on resonant hemivariational inequalities, by the authors in 1999.

In this paper we consider nonlinear hemivariational inequalities involving the p-Laplacian at resonance. We prove the existence of a nontrivial solution. Our approach is variational based on the critical point theory for nonsmooth, locally Lipschitz functionals due to Chang.