Existence results for a borderline case of a class of p-Laplacian problems

The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically "linear" problem {- div [A(0)(x) + A(x)|u|(ps )|del u|(p-2 )del u] + s A(x)|u|(ps-2)u |del u|(p) ( )= mu|u|(p(s+1)-2)u + g(x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a bounded domain in R-N, N >= 2, 1 < p < N, s > 1/p, both the coefficients A(0)(x) and A(x) are in L-infinity(Omega) and far away from 0, mu is an element of R, and the "perturbation" term g(x, t) is a Caratheodory function on Omega x R which grows as |t|(r-1) with 1 <= r < p(s + 1) and is such that g(x, t) approximate to nu|t|(p-2)t as t -> 0. By introducing suitable thresholds for the parameters nu and mu, which are related to the coefficients A(0)(x), respectively A(x), under suitable hypotheses on g(x, t), the existence of a nontrivial weak solution is proved if either nu is large enough with mu small enough or nu is small enough with mu large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.