Limits as p → ∞ of p-Laplacian concave-convex problems

We study the behavior as p -> infinity of the sequel of positive weak solutions of the concave-convex problem {-div(|del u|(p-2)del u) = lambda u(q(p)) + u(r(p)) in Omega u > 0 in Omega u = 0 on partial derivative Omega, where Omega subset of R-n is a bounded domain, lambda > 0 and the exponents q, r satisfy lim(p -> infinity) q(p)/p - 1 = Q, lim(p -> infinity) r(p)/p - 1 = R, with 0 < Q < 1 < R. We characterize any positive uniform limit of a sequence of weak solutions of (P) as a viscosity solution of min{|del u(Lambda)| - max{Lambda u(Lambda)(Q), u(Lambda)(R)},- Delta(infinity)u(Lambda) = 0 in Omega. Notice that the limit process decouples the nonlinearity. We obtain existence, nonexistence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter Lambda. (C) 2011 Elsevier Ltd. All rights reserved.