On the existence and asymptotic behaviour of bounded positive entire solutions for quasilinear elliptic problems

Abstract

We establish new results concerning the existence and asymptotic behaviour of solutions for the nonlinear elliptic problem where Δ p u = div(|∇u| p−2∇u), with 1 < p < N, denotes the p-Laplacian operator and f : ℝ N × (0, ∞) → ℝ is a suitable continuous function. The principal aim of this article is to study the case 0 < l < ∞, because the extreme cases l = 0 and l = ∞ have been intensely studied in recent years. The main tools we use to prove the principal results are the method of lower and upper solutions, an argument of penalization and a technique of monotonization–regularization of the nonlinearity f.