A Topological Approach to Singular Double-Phase Equations with Variable Exponents

TL;DR

This paper introduces a topological method to establish the existence of solutions for singular double-phase equations with variable exponents in a novel Musielak-Orlicz Sobolev space, considering gradient-dependent nonlinearities.

Contribution

It develops a new topological framework using Leray-Schauder degree for singular double-phase problems with variable exponents and gradient dependence.

Findings

Proves existence of at least one nontrivial solution

Extends the theory to equations with gradient-dependent nonlinearities

Introduces a new Musielak-Orlicz Sobolev space setting

Abstract

In the present paper, we study a singular double phase variable exponent Dirichlet problem in the setting of a new Musielak-Orlicz Sobolev space with the nonlinearity (the external source) having gradient dependence (so-called convection terms). We apply a topological existence result incorporating the Leray-Schauder degree and homotopy mapping together to prove the existence of at least one nontrivial solution.