ABSTRACT
In this paper, we establish a Harnack type inequality for p(x)-harmonic functions when the exponent p(x) makes a jump at a hyperplane and satisfies log-condition otherwise. We demonstrate that in this situation the Harnack inequality in the classical form is not valid. The form of the Harnack inequality we obtain is weaker than the classical one but still powerful enough to imply the Hölder continuity.
Acknowledgements
The authors thank the unanymous referee for his careful reading of this manuscript and valuable remarks.
Notes
No potential conflict of interest was reported by the authors.
Dedicated to the memory of Vasily V. Zhikov, who was a great Mathematician, Friend and Teacher.