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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.

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