A Neumann problem involving the p(x)-Laplacian with p = infinity in a subdomain

摘要

In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p(x)-Laplacian is involved and p = infinity in a subdomain. By considering a suitable sequence p(k) of bounded variable exponents such that p(k) -> p and replacing p with p(k) in the original problem, we prove the existence of a solution u(k) for each of those intermediate ones. We show that the limit of (u(k)) exists and after giving a variational characterization of it in the part of the domain where p is bounded, we show that it is a viscosity solution in the part where p = infinity. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.