V. V. Motreanu

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.

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.

On p -Laplace equations with concave terms and asymmetric perturbations

We consider a nonlinear Dirichlet problem driven by the p -Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is ( p − 1)-superlinear on ℝ + and ( p − 1)-(sub)linear on ℝ − . Using variational methods based on the critical point theory together with truncation techniques, Ekelands variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

A version of Zhong's coercivity result for a general class of nonsmooth functionals

A version of Zhongs coercivity result (1997) is established for nonsmooth functionals expressed as a sum Φ

Periodic solutions for nonautonomous systems with nonsmooth quadratic or superquadratic potential

We study a semilinear nonautonomous second order periodic system with a nonsmooth potential function which exhibits a quadratic or superquadratic growth. We establish the existence of a solution, using minimax methods of the nonsmooth critical point theory.

On a non-smooth eigenvalue problem in Orlicz–Sobolev spaces

This article studies a non-smooth eigenvalue problem for a Dirichlet boundary value inclusion on a bounded domain Ω which involves a φ-Laplacian and the generalized gradient in the sense of Clarke of a locally Lipschitz function depending also on the points in Ω. Specifically, the existence of a sequence of eigensolutions satisfying in addition certain asymptotic and locational properties is established. The approach relies on an approximation process in a suitable Orlicz–Sobolev space by eigenvalue problems in finite-dimensional spaces for which one can apply a finite-dimensional, non-smooth version of the Ljusternik–Schnirelman theorem. As a byproduct of our analysis, a version of Aubin–Clarkes theorem in Orlicz spaces is obtained.

Coerciveness Property for a Class of Non-Smooth Functionals

The paper establishes a general coerciveness property for a class of non-smooth functionals satisfying an appropriate Palais-Smale condition. This result is obtained by applying an abstract principle supplying qualitative information concerning the asymptotic behaviour of a non-smooth functional. Comparison with other results in this field is provided.

Embeddings of weighted Sobolev spaces and degenerate Dirichlet problems involving the weighted p-Laplacian

We deal with two Dirichlet boundary value problems involving the weighted p-Laplacian. We introduce double weighted Sobolev spaces as solution spaces of the considered problems. We show the existence of weak solutions of these problems under an assumption of compact embedding involving the double weighted Sobolev spaces. Also, we establish sufficient conditions for fulfilling this embedding assumption.

Nonsmooth Variational Problems in the Limit Case and Duality

The paper contains a duality result and two existence theorems for nonsmooth boundary value problems, with unbounded constraints, in the limit case. Examples illustrate the abstract results.

Generic existence of nondegenerate homoclinic solutions

The paper focuses on the homoclinic solutions of a general second order Hamiltonian system. By applying an abstract parametric transversality result, it is shown that generically the problem admits finitely many homoclinic solutions. These solutions are nondegenerate in the sense that they correspond to nondegenerate critical points of the associated functional and depend smoothly on the vector field.