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

Milena Chermisi, Enrico Valdinoci

In ${\mathbb{R}}^m\times {\mathbb{R}}^{n-m}$, endowed with coordinates $X=(x,y)$, we consider the PDE $$ -{\rm div} \big(α(\x) |\nabla u(\X)|^{p(x)-2}\nabla u(\X)\big)=f(x,u(\X)).$$ We prove a geometri…