Fibered nonlinearities for $p(x)$-Laplace equations

In $\R^m\times\R^{n-m}$, endowed with coordinates $X=(x,y)$, we consider the PDE $$-{\rm div} \big(\alpha(\x) |\nabla u(\X)|^{p(x)-2}\nabla u(\X)\big)=f(x,u(\X)).$$ We prove a geometric inequality and a symmetry result.