Dumitru Motreanu

Minimax theorems and qualitative properties of the solutions of hemivariational inequalities

Preface. 1. Elements of Nonsmooth Analysis. Hemivariational Inequalities. 2. Nonsmooth Critical Point Theory. 3. Minimax Methods for Variational-Hemivariational Inequalities. 4. Eigenvalue Problems for Hemivariational Inequalities. 5. Multiple Solutions of Eigenvalue Problems for Hemivariational Inequalities. 6. Eigenvalue Problems for Hemivariational Inequalities on the Sphere. 7. Resonant Eigenvalue Problems for Hemivariational Inequalities. 8. Double Eigenvalue Problems for Hemivariational Inequalities. 9. Periodic and Dynamic Problems.

Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems

Preface. Introduction. 1. Elements of Nonsmooth Analysis. 1. Generalized Gradients of Locally Lipschitz Functionals. 2. Palais Smale Condition and Coerciveness for a Class of Nonsmooth Functionals. 3. Nonsmooth Analysis in the Sense of Degiovanni. 2. Variational Methods. 1. Critical Point Theory for Locally Lipschitz Functionals. 2. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals. 3. A Critical Point Theory in Metric Spaces. 3. Variational Methods. 1. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals in the Limit Case. 2. Examples. 4. Multivalued Elliptic Problems in Variational Form. 1. Multiplicity for Locally Lipschitz Periodic Functionals. 2. The Multivalued Forced-pendulum Problem. 3. Hemivariational Inequalities Associated to Multivalued Problems with Strong Resonance. 4. A Parallel Nonsmooth Critical Point Theory. Approach to Stationary Schrodinger Type Equations in Constraints. 8. Non-Symmetric Perturbations of Symmetric Eigenvalue Problems. 1. Non-Symmetric Perturbations of Eigenvalue Problems for Periodic Hemivariational Inequalities with Constraints. 2. Perturbations of Double of Eigenvalue Problems for General Hemivariational Inequalities with Constraints. 3. location of Solutions by Minimax Methods of Variational Hemivariational Inequalities. 10. Nonsmooth Evolution Problems. 1. First Order Evolution Variational Inequalities. 2. Second Order Evolution Variational Inequalities. 3. Stability Problems for Evolution Variational Inequalities. 11. Inequality Problems in BV and Geometric Applications. 1. The General Framework. 2. Area Type Functionals. 3. A Result of Clark Type. 4. An Inequality Problem with Superlinear Potential.

Extremal solutions of quasilinear parabolic inclusions with generalized Clarke's gradient

Abstract In this paper we consider an initial boundary value problem for a parabolic inclusion whose multivalued nonlinearity is characterized by Clarkes generalized gradient of some locally Lipschitz function, and whose elliptic operator may be a general quasilinear operator of Leray–Lions type. Recently, extremality results have been obtained in case that the governing multivalued term is of special structure such as, multifunctions given by the usual subdifferential of convex functions or subgradients of so-called dc-functions. The main goal of this paper is to prove the existence of extremal solutions within a sector of appropriately defined upper and lower solutions for quasilinear parabolic inclusions with general Clarkes gradient. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison, truncation, and special test function techniques as well as tools from nonsmooth analysis.

Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator

We consider a nonlinear Neumann problem driven by a nonhomogeneous quasilinear degenerate elliptic differential operator div a(x,∇u), a special case of which is the p-Laplacian. The reaction term is a Carathéodory function f(x, s) which exhibits subcritical growth in s. Using variational methods based on the mountain pass and second deformation theorems, together with truncation and minimization techniques, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). A crucial tool in our analysis is a result of independent interest which we prove here and which relates W 1,p and C1 local minimizers of a C1-functional constructed with the general differential operator div a(x,∇u).

A class of variational–hemivariational inequalities of elliptic type

This paper is devoted to the existence of solutions for variational–hemivariational inequalities of elliptic type, with a higher order quasilinear principal part, at resonance as well as at nonresonance. The approach relies on the use of pseudomonotone operators. By means of the notion of Clarkes generalized gradient and the properties of the first eigenfunction of the quasilinear principal part, we also build a Landesman–Lazer theory in the nonsmooth framework of variational–hemivariational inequalities of elliptic type.

Double Eigenvalue Problems for Hemivariational Inequalities

The aim of the present paper is to study a new type of eigenvalue problem, called a double eigenvalue problem, which arises in hemivariational inequalities related to nonconvex nonsmooth energy functionals. The paper provides existence results as well as some qualitative properties for the solutions to double eigenvalue problems for hemivariational inequalities under the presence of given nonlinear compact operators which are not necessarily of a variational structure. It presents three different approaches to such problems: minimization, minimax methods and (sub) critical point theory on a sphere. Applications illustrate the theory.

Multiple nontrivial solutions for nonlinear eigenvalue problems

In this paper we study a nonlinear eigenvalue problem driven by the p-Laplacian. Assuming for the right-hand side nonlinearity only unilateral and sign conditions near zero, we prove the existence of three nontrivial solutions, two of which have constant sign (one is strictly positive and the other is strictly negative), while the third one belongs to the order interval formed by the two opposite constant sign solutions. The approach relies on a combination of variational and minimization methods coupled with the construction of upper-lower solutions. The framework of the paper incorporates problems with concave-convex nonlinearities.

Existence Results For Inequality Problems

We establish several existence results of Hartman-Stampacchia type for hemivariational inequalities on bounded and convex sets in a real reflexive Banach space. We also study the cases of coercive and noncoercive variational-hemivariational inequalities. Two applications on nonmonotone laws in networks and nonconvex semipermeability illustrate the abstract results.

MULTIPLE SOLUTIONS FOR A DIRICHLET PROBLEM WITH p -LAPLACIAN AND SET-VALUED NONLINEARITY

The existence of a negative solution, of a positive solution, and of a sign-changing solution to a Dirichlet eigenvalue problem with p-Laplacian and multi-valued nonlinearity is investigated via suband supersolution methods as well as variational techniques for nonsmooth functions.

POSITIVE SOLUTIONS AND MULTIPLE SOLUTIONS AT NON-RESONANCE, RESONANCE AND NEAR RESONANCE FOR HEMIVARIATIONAL INEQUALITIES WITH p-LAPLACIAN

In this paper we study eigenvalue problems for hemivariational inequalities driven by the p-Laplacian differential operator. We prove the existence of positive smooth solutions for both non-resonant and resonant problems at the principal eigenvalue of the negative p-Laplacian with homogeneous Dirichlet boundary condition. We also examine problems which are near resonance both from the left and from the right of the principal eigenvalue. For nearly resonant from the right problems we also prove a multiplicity result.