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 the 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.
IntroductionConsider the following Neumann problemwhere Ω ⊂ R N is a bounded smooth domain and N ≥ 2.is the p(x)-Laplacian operator which is the variable exponent version of the p-Laplacian. Also, g ∈ C(Ω) and satisfies ∂Ω g = 0. Note that this latter condition is necessary, since otherwise problem (1.1) has no solution.The variable exponent p satisfies the following hypothesiswhere D is a compactly supported subdomain of Ω, with Lipschitz boundary.Moreover, p ∈ C 1 (Ω \ D) withIn the literature, most of the times the variable exponent p(·) is assumed to be bounded. Recently, the limits p(x) → ∞ have been studied in several problems where the p(x)-Laplacian is involved. See for instance [20] or [23] and the references therein. On the other hand, when p is constant the limits p → ∞ in problems with the p-Laplacian were first studied in [5], in which the physical motivation was given as well. On both cases the notion of infinity Laplacian arises naturally as the limit case.2000 Mathematics Subject Classification. 35J20, 35J60, 35J70.
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