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 \ \ array{ll - div [ (A₀ (x) + A (x) |u|^ps) | u|^p-2 u + s\ A (x) |u|^ps-2 u\ | u|ᵖ &\\ =\ |u|^p (s + 1) -2 u + g (x, u) & in, \\ u = 0 & on, array. \] where is a bounded domain in RN, N 2, 1 1/p, both the coefficients A₀ (x) and A (x) are in L^ () and far away from 0, R, and the ``perturbation'' term g (x, t) is a Carath\'eodory function on R which grows as |t|^r-1 with 1 r < p (s + 1) and is such that g (x, t) |t|^p-2 t as t 0. By introducing suitable thresholds for the parameters and, which are related to the coefficients A₀ (x), respectively A (x), under suitable hypotheses on g (x, t), the existence of a nontrivial weak solution is proved if either is large enough with small enough or is small enough with 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.