Quasilinear elliptic equations with critical growth involving jumping nonlinearities

Abstract

In this paper we deal with the following class of quasilinear elliptic problemswhere 1 ≤ < , Ω ⊂ is bounded with smooth boundary, is a nonlinearity with subcritical growth condition and ∈ ∞ (Ω). This class of equations is inspired by the famous Ambrosetti-Prodi problems and here we prove existence of two solutions when the nonlinearity crosses the first eigenvalue, that means < 1 , and has one sided critical growth. By using some new and delicate estimates for the Talenti functions associated to the critical problem, we invoke variational methods to obtain our results. K E Y W O R D SAmbrosetti-Prodi type result, critical Sobolev exponent, -Laplace operator M S C ( 2 0 1 0 ) 35B33, 35J621094