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

Mustafa Avci

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.

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

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.
Paper Structure (3 sections, 10 theorems, 41 equations)

This paper contains 3 sections, 10 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Mathematical Background and Auxiliary Results
  3. Main Results

Key Result

Proposition 2.1

If $u,u_{n}\in L^{h(x)}(\Omega)$, we have

Theorems & Definitions (12)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Definition 3.1
  • Example 3.2
  • ...and 2 more