Boundedness of solutions of the non-uniformly convex, non-standard growth Laplacian

Abstract

We study the variable exponent p(·)-Laplace equation

when p attains the value 1. We prove a removability result and local boundedness for its solutions and obtain a weak continuity result. Our results also apply to (𝒜, ℬ)-solutions of p(·)-growth.

Keywords:

• non-standard growth
• variable exponent
• Laplace equation
• Dirichlet energy
• local boundedness
• solution
• Hausdorff measure
• capacity

AMS Subject Classifications::

• Primary 35J60
• 35B20
• Secondary 35J25

Acknowledgements

We would like to thank the referees for their comments, especially for noticing the short-coming in the original proof of Theorem 3.2. This research was supported by the Academy of Finland, INTAS and the Emil Aaltonen Foundation.