General comparison principle for quasilinear elliptic inclusions

The main goal of this paper is to prove existence and comparison results for elliptic differential inclusions governed by a quasilinear elliptic operator and a multivalued function given by Clarke’s generalized gradient of some locally Lipschitz function. These kinds of problems have been treated in the past by various authors including the authors of this paper. However, in all the works we are aware of, additional assumptions on the structure of the elliptic operator and/or the generalized Clarke’s gradient are needed to get comparison results in terms of sub-supersolutions. Comparison principles were obtained recently, e.g., in the case where the elliptic operator is of potential type, or Clarke’s gradient is required to satisfy some one-sided growth condition, or the sub-supersolutions are supposed to satisfy additional properties. The novelty of this paper is that we are able to obtain a comparison principle without assuming any of the above restrictions. To the best of our knowledge this is the first mathematical treatment of the considered elliptic inclusion in its full generality. The obtained results of this paper complement the development of the sub-supersolution method for nonsmooth problems presented in a recent monograph by S. Carl, Vy K. Le and D. Motreanu.