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

Abstract

Abstract In this paper we study a Neumann problem with non-homogeneous boundary condition, where the p ( x )-Laplacian is involved and p = ∞ 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 = ∞. Finally, we formulate the problem of which this limit function is a solution in the viscosity sense.