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

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In this paper we study a non-homogeneous Neumann problem, where the p(x)-Laplacian is involved and p=∞ in a subdomain. By considering a suitable sequence pₖ of bounded 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 show that the limit of the uₖ 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=∞. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.