A free boundary problem for quasilinear elliptic equations in exterior domains

Abstract

We prove the existence of weak solutions that belong to some local Sobolev space of a quasilinear elliptic problem in an exterior domain. The free-boundary problem considered here arises from the discontinuous behavior of the nonlinearity involved. The method of upper and lower solutions, extremality results for quasilinear elliptic problems in bounded domains, gradient estimates and abstract fixed point principles in partially ordered sets are the main tools used in the proof of our main result. An application to a superlinear discontinuous elliptic problem with the p-Laplacian is given.