Monotone operator methods for a class of nonlocal multi-phase variable exponent problems

TL;DR

This paper introduces new methods using monotone operators to prove the existence of solutions for complex nonlocal multi-phase variable exponent problems within a novel Musielak-Orlicz Sobolev space framework.

Contribution

It develops two distinct monotone operator approaches to establish solutions for two types of nonlinear nonlocal problems in a new functional setting.

Findings

Existence of solutions for both problem types.

Application of monotone operator methods in Musielak-Orlicz spaces.

Extension to nonlocal multi-phase variable exponent problems.

Abstract

In this paper, we study a class of nonlocal multi-phase variable exponent problems within the framework of a newly introduced Musielak-Orlicz Sobolev space. We consider two problems, each distinguished by the type of nonlinearity it includes. To establish the existence of at least one nontrivial solution for each problem, we employ two different monotone operator methods.