Abstract. We consider the natural Neumann boundary condition for the ∞-Laplacian. We study the limit as p → ∞ of solutions of −∆pup = 0 in a domain Ω with |Dup| p−2 ∂up/∂ν = g on ∂Ω. We obtain a natural minimization problem that is verified by a limit point of {up} and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the Monge-Kantorovich mass transfer problems when the measures are supported on ∂Ω.

In this paper we study the existence of nontrivial solutions of the problem −∆u + u = |u| p−2 u in Ω, ∂u ∂ν = λ|u| q−2 u on ∂Ω, with 1 < q < 2(N −1)/(N −2) and 1 < p ≤ 2N/(N −2). In the concave-convex case, i.e., 1 < q < 2 < p, if λ is small there exist two positive solutions while for λ large there is no positive solution. When p is critical, and q subcritical we obtain existence results using the concentration compactness method. Finally we apply the implicit function theorem to obtain solutions for λ small near u 0 = 1.

We study the Steklov eigenvalue problem for the ∞-Laplacian. To this end we consider the limit as p → ∞ of solutions of −∆ p u p = 0 in a domain Ω with |∇u p | p−2 ∂u p /∂ν = λ|u| p−2 u on ∂Ω. We obtain a limit problem that is satisfied in the viscosity sense and a geometric characterization of the second eigenvalue.

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We prove the uniqueness of positive Wwhere N ≥ 3, 1 < p <