A Harnack inequality for a transmission problem withp(x)-Laplacian

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.