Harnack's inequality for the $p(x)$-Laplacian with a two-phase exponent $p(x)$

One considers solutions of the $p(x)$-Laplacian equation in a neighborhood of a point $x_0$ on a hyperplane $\Sigma$. It is assumed that the exponent $p(x)$ possesses a logarithmic continuity modulus as $x_0$ is approached from one of the half-spaces separated by $\Sigma$. A version of the Harnack inequality is proved for these solutions.