A Neumann problem for the p(x)-Laplacian with p = 1 in a subdomain

In this paper we study a Neumann problem with non-homogeneous boundary conditions for the p(x)-Laplacian. In particular we assume that p(⋅) is a step function defined in a domain Ω and equals to 1 in a subdomain Ω1 and 2 in its complementary Ω2. By considering a suitable sequence pk of variable exponents such that pk→p and replacing p with pk in the original problem, we prove the existence of a solution uk for each of those intermediate ones. We also show, that under a hypothesis concerning the boundary data g, the limit of the sequence (uk) is a function u, which belongs to the space of functions of bounded variation and is a solution to the original p(⋅)-problem.