N. S. Papageorgiou

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).

On the convergence of solutions of multivalued parabolic equations and applications

Abstract In this paper we examine parametric nonlinear parabolic problems with multivalued terms. Using a general notion of G-convergence for such operators we prove a convergence theorem for the solution sets of the corresponding Cauchy–Dirichlet problem. We also study a related minimax control problem.