Limits as p → a of p-laplacian eigenvalue problems perturbed with a concave or convex term

We investigate the asymptotic behaviour as p -> infinity of sequences of positive weak solutions of the equation {-Delta(p)u = lambda u(p-1) + u(q(p)-1) in Omega, u = 0 on partial derivative Omega, where lambda > 0 and either 1 < q(p) < p or p < q(p), with lim(p ->infinity) q(p)/p = Q not equal 1. Uniform limits are characterized as positive viscosity solutions of the problem {min {vertical bar del u(x)vertical bar - max {Lambda u(x), u(Q)(x)}, -Delta(infinity)u(x)} = 0 in Omega, u = 0 on partial derivative Omega. for appropriate values of Lambda > 0. Due to the decoupling of the nonlinearity under the limit process, the limit problem exhibits an intermediate behavior between an eigenvalue problem and a problem with a power-like right-hand side. Existence and non-existence results for both the original and the limit problems are obtained.