The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

TL;DR

This paper develops a method using sub- and supersolutions to establish the existence of multiple solutions for complex variable exponent double phase elliptic systems with nonlinear boundary conditions, broadening the scope of solvable problems.

Contribution

It introduces a novel enclosure and existence framework for coupled systems with nonlinear boundaries using trapping regions, applicable under general data assumptions.

Findings

Established existence of infinitely many solutions.

Extended results to homogeneous Dirichlet boundary conditions.

Provided a new approach for variable exponent double phase systems.

Abstract

In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sup- and supersolutions. Under very general assumptions on the data we then apply our result to get infinitely many solutions. Moreover, we also discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.