Positive solutions to nonlinear inclusion problems in Orlicz–Sobolev spaces

Abstract

In this paper, we study the existence of positive solutions to a nonlinear elliptic inclusion problem driven by an elliptic differential operator with Dirichlet boundary and a multivalued term depending on the solution and its gradient in reflexive Orlicz–Sobolev spaces by using a linear functional analysis method and a subsolution–supersolution method. We provide some sufficient conditions to ensure that a subsolution and a supersolution exist. An existence theorem for positive solutions of the problem is established.

COMMUNICATED BY:

• Jen-Chih Yao

Keywords:

• Orlicz–Sobolev spaces
• nonlinear elliptic inclusion
• multivalued operator
• subsolution–supersolution
• positive solutions

AMS Subject Classifications:

• 35J25
• 35J55

Acknowledgements

The authors are grateful to the referees for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author is supported by National Natural Science Foundation of China (Grant no. 11871375).