The weak upper and lower solution method for quasilinear elliptic equations with generalized subdifferentiable perturbations

Abstract

A variational approach to the method of upper and lower solution is suggested which allows to treat nonlinear elliptic boundary value problems with Baire-measurable lower order nonlinearities. To this end an associated multivalued setting of the problem is considered. First we prove the existence of solutions of a ‘truncated’ auxiliary problem which is related to the minimization of a nonsmooth functional whose critical points are shown to be solutions of this auxiliary problem. Then it is shown that any solution of the auxiliary problem solves the original one. The existence of critical points of the functional under consideration is proved by showing that it satisfies a generalized Palais-Smale condition which is suggested by the Variational Principle of Ekeland.

Keywords:

• elliptic boundary value problems
• discontinuous nonlinearities
• generalized subdifferential
• upper and lower solutions
• critical points
• generalized Palais-Smale condition