Existence of weak solutions to borderline double-phase problems with logarithmic convection term

TL;DR

This paper proves the existence of weak solutions for a class of complex double-phase elliptic equations with logarithmic convection, using advanced functional analysis techniques in generalized Orlicz spaces.

Contribution

It introduces a novel approach employing pseudo-monotone operators and modular function spaces to establish weak solutions for borderline double-phase problems.

Findings

Existence of weak solutions under certain conditions

Extension to broader classes of double-phase problems

Application of generalized Orlicz space techniques

Abstract

In this study, we devote our attention to the question of clarifying the existence of a weak solution to a class of quasilinear double-phase elliptic equations with logarithmic convection terms under some appropriate assumptions on data. The proof is based on the surjectivity theorem for the pseudo-monotone operators and modular function spaces and embedding theorems in generalized Orlicz spaces. Our approach in this paper can be extended naturally to a larger class of unbalanced double-phase problems with logarithmic perturbation and gradient dependence on the right-hand sides.