Nonlinear Elliptic Differential Equations with Multivalued Nonlinearities

Proceedings Mathematical Sciences

Nikolaos S Papageorgiou¹  Raffaella Servadei¹  Antonella Fiacca²  Nikolaos Matzakos³ 

[1] $$;Department of Mathematics, University of Perugia, Via Vanvitelli , Perugia 0, Italy$$;Department of Mathematics, National Technical University, Zografou Campus, Athens 0, Greece$$

关键词: Upper solution;    lower solution;    order interval;    truncation function;    pseudomonotone operator;    coercive operator;    extremal solution;    Yosida approximation;    nonsmooth Palais–Smale condition;    critical point;    eigenvalue problem.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper we study nonlinear elliptic boundary value problems with monotone and nonmonotone multivalued nonlinearities. First we consider the case of monotone nonlinearities. In the first result we assume that the multivalued nonlinearity is defined on all $mathbb{R}$. Assuming the existence of an upper and of a lower solution, we prove the existence of a solution between them. Also for a special version of the problem, we prove the existence of extremal solutions in the order interval formed by the upper and lower solutions. Then we drop the requirement that the monotone nonlinearity is defined on all of $mathbb{R}$. This case is important because it covers variational inequalities. Using the theory of operators of monotone type we show that the problem has a solution. Finally in the last part we consider an eigenvalue problem with a nonmonotone multivalued nonlinearity. Using the critical point theory for nonsmooth locally Lipschitz functionals we prove the existence of at least two nontrivial solutions (multiplicity theorem).

【 授权许可】

Unknown