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Positive weak solutions of a double-phase variable exponent problem with a fractional-Hardy-type singular potential and superlinear nonlinearity

Mustafa Avci

TL;DR

The paper addresses the existence of positive weak solutions for a singular double-phase PDE with variable exponents and a fractional-Hardy-type potential. It develops a variational framework in Musielak-Orlicz Sobolev spaces, introducing a two-phase energy functional $\mathcal{I}$ that incorporates the double-phase operator and the singular term, and proves differentiability and Mountain-Pass geometry under Ambrosetti-Rabinowitz-type conditions. By establishing the Palais-Smale condition and applying the Mountain-Pass theorem, the authors obtain a nontrivial positive weak solution, further strengthened by a truncation approach and the strong minimum principle to ensure positivity. The work extends double-phase and Hardy-type potential analysis to a variable-exponent, non-autonomous Musielak-Orlicz setting, providing explicit inequalities and embedding results to control the singular term and guarantee existence.

Abstract

In the present paper, we study a double-phase variable exponent problem which is set up within a variational framework including a singular potential of fractional-Hardy-type. We employ the Mountain-Pass theorem and the strong minimum principle to obtain the existence of at least one nontrivial positive weak solution.

Positive weak solutions of a double-phase variable exponent problem with a fractional-Hardy-type singular potential and superlinear nonlinearity

TL;DR

The paper addresses the existence of positive weak solutions for a singular double-phase PDE with variable exponents and a fractional-Hardy-type potential. It develops a variational framework in Musielak-Orlicz Sobolev spaces, introducing a two-phase energy functional that incorporates the double-phase operator and the singular term, and proves differentiability and Mountain-Pass geometry under Ambrosetti-Rabinowitz-type conditions. By establishing the Palais-Smale condition and applying the Mountain-Pass theorem, the authors obtain a nontrivial positive weak solution, further strengthened by a truncation approach and the strong minimum principle to ensure positivity. The work extends double-phase and Hardy-type potential analysis to a variable-exponent, non-autonomous Musielak-Orlicz setting, providing explicit inequalities and embedding results to control the singular term and guarantee existence.

Abstract

In the present paper, we study a double-phase variable exponent problem which is set up within a variational framework including a singular potential of fractional-Hardy-type. We employ the Mountain-Pass theorem and the strong minimum principle to obtain the existence of at least one nontrivial positive weak solution.
Paper Structure (3 sections, 13 theorems, 33 equations)

This paper contains 3 sections, 13 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Mathematical Background and Preliminaries
  3. Variational Framework and Main Results

Key Result

Proposition 2.1

For any $u\in L^{h( x) }(\Omega)$ and $v\in L^{h^{\prime}(x)}(\Omega)$, we have where $L^{h^{\prime }(x) }(\Omega)$ is conjugate space of $L^{h( x) }(\Omega)$ such that $\frac{1}{h( x) }+\frac{1}{h^{\prime}( x)}=1$.

Theorems & Definitions (15)

  • Proposition 2.1: Hölder Inequality
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Lemma 2.9
  • Definition 2.10
  • ...and 5 more