The variable exponent De Giorgi class and the variable exponent Hölder continuity

Ya-Ru Yuan; Zhi-Yuan Cui; Dun Zhao

doi:10.3934/cpaa.2024073

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2025, Volume 24, Issue 1: 1-14. Doi: 10.3934/cpaa.2024073

This issue Previous Article Next Article Infinitely many solutions for a Schrödinger equation with sign-changing potential and nonlinear term

The variable exponent De Giorgi class and the variable exponent Hölder continuity

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

^*Corresponding author: Dun Zhao
^*Corresponding author: Dun Zhao

Received: November 2023

Revised: July 2024

Early access: August 2024

Published: January 2025

The third author is supported by [NSFC under the grant Nos. 12075102 and 11971212].
In memory of the 80th anniversary of the birth of Professor Xianling Fan..

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Abstract

For a bounded domain $\Omega$ in $\mathbb{R}^{n}$ ( $n \!\ge\! 2$ ), we prove that the variable exponent De Giorgi class $\mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta)$ is a subset of the variable exponent Hölder space $C^{0, \alpha(x)}_{loc}(\Omega)$ . It gives an improvement for the earlier result, which states that $\mathscr{B}_{m(x)} (\Omega, M, \gamma, \gamma_1, \delta)\subset C^{0, \alpha}_{loc}(\Omega)$ , where $\alpha$ is a constant. This result can be applied to get $C^{0, \alpha(x)}_{loc}(\Omega)$ regularity for certain solutions of variational problems and nonlinear elliptic equations with variable exponents.

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Mathematics Subject Classification: 26A16, 35B65.

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