On Hölder continuity to a function class with nonstandard growth condition

Abstract

With the help of De Giorgi's method, the embedding of the ${\mathfrak{B}}_{m\left(x\right)}(\mathrm{\Omega },M,\gamma ,{\gamma }_1,\delta ,\frac{1}{q})$ class into $C^{0,\alpha }\left(\mathrm{\Omega }\right)$ is proved. This result can be applied to the $C^{0,\alpha }$ continuity to some elliptic equations with variable exponent.

Keywords:

• De Giorgi's method
• Hölder continuity
• variable exponent

AMS Subject Classifications:

• 35J25
• 35J70
• 35J92

Acknowledgement

I am grateful to the anonymous referees for the useful and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Supported by Youth science and technology talent growth project in Guizhou Province for ordinary institutions of higher learning [grant number KY[2021]142].