Leszek Gasiński

Nonsmooth critical point theory and nonlinear boundary value problems

MATHEMATICAL BACKGROUND Sobolev Spaces Set-Valued Analysis Nonsmooth Analysis Nonlinear Operators Elliptic Differential Equations Remarks CRITICAL POINT THEORY Locally Lipschitz Functionals Constrained Locally Lipschitz Functionals Perturbations of Locally Lipschitz Functionals Local Linking and Extensions Continuous Functionals Multivalued Functionals Remarks ORDINARY DIFFERENTIAL EQUATIONS Dirichlet Problems Periodic Problems Nonlinear Boundary Conditions Variational Methods Method of Upper and Lower Solutions Positive Solutions and Other Methods Hamiltonian Inclusions Remarks ELLIPTIC EQUATIONS Problems at Resonance Neumann Problems Problems with an Area-Type Term Strongly Nonlinear Problems Method of Upper and Lower Solutions Multiplicity Results Positive Solutions Problems with Discontinuous Nonlinearities Remarks APPENDIX Set Theory and Topology Measure Theory Functional Analysis Nonlinear Analysis List of Symbols References

Existence and Multiplicity of Solutions for Neumann p-Laplacian-Type Equations

Abstract We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.

An Existence Theorem for Wave-Type Hyperbolic Hemivariational Inequalities

In this paper we prove the existence of solutions for a hyperbolic hemivariationalinequality of the form u″ + Bu + ∂j (u) ∋ f where B is a linear elliptic operator and ∂j is the Clarke subdifferential of a locally Lipschitz function j. Our result is based on the parabolic regularization method.

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.

An existence theorem for nonlinear hemivariational inequalities at resonance

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.

Existence of solutions for hyperbolic hemivariational inequalities

In this paper we study a hyperbolic hemivariational inequality with a nonlinear, pseudomonotone operator depending on the derivative of an unknown function and a linear, monotone operator depending on an unknown function. Using the surjectivity result for L-pseudomonotone operators, an existence result for such inequalities is proved.

Nonlinear hemivariational inequalities at resonance

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.

Optimal Shape Design Problems for a Class of Systems Described by Parabolic Hemivariational Inequality

Optimal shape design problems for systems governed by a parabolic hemivariational inequality are considered. A general existence result for this problem is established by the mapping method.

Nontrivial Solutions for Resonant Hemivariational Inequalities

We study a resonant semilinear elliptic hemivariational inequality. Under some assumptions of strong resonance on the Clarke subdifferential of the superpotential, and using nonsmooth critical point theory, the existence of a nontrivial solution of the problem is shown.

Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping

Abstract In this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form u″+Au′+Bu+∂j(u)∋f, where B is a linear elliptic operator and A is linear and nonnegative (not necessarily coercive).