Problemas elípticos semilineares envolvendo não linearidade com zero positivo.

Abstract

In this work, we consider the following class of semilinear elliptie problems {−Δu=λf(u) em Ω, u=0 sobre ∂Ω, where Ω C ℝN (N ≥ 3) is a smooth domam, λ > 0 is a parameter, and f is a locally Lipschitz and non-negative function with an isolated zero α > 0, Regarding the nonlinearity, we assume only growth hypotheses in a neighborhood of its zero By dispensing with the need for growth eonditions at infinity or at the origin, we show that, for X sufficiently large, there exist at least two positive Solutions uλ<vλ, satisfying ∥uλ∥∞<α<∥vλ∥∞ and uλ,vλ→ α uniformly on compact subsets of Ω as λ →+∞. For the existence of the ürst solution, we employ the method of weak sub and super Solutions eombined with the theory of viscosity Solutions, The seeond solution is obtained using variational methods and truneation teehniques.