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

Minh-Phuong Tran, Thanh-Nhan Nguyen

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.

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

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.
Paper Structure (4 sections, 8 theorems, 110 equations)

This paper contains 4 sections, 8 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminary materials
  3. Statement of existence theorem
  4. Proof of main result

Key Result

Lemma 2.3

Let $G$ be a Young function.

Theorems & Definitions (22)

  • Definition 2.1: Young function
  • Definition 2.2: $\Delta_2$ and $\nabla_2$-conditions
  • Lemma 2.3
  • Definition 2.4: Orlicz spaces
  • Definition 2.5: Orlicz-Sobolev spaces
  • Lemma 2.6
  • Definition 2.7: Orlicz-Zygmund spaces $L^p\log^{\alpha} L(\Omega)$
  • Remark 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 12 more