Semilinear elliptic equations and nonlinearities with zeros

Abstract

In this paper we consider the semilinear elliptic problem {−Δu=λf(u)in Ω,u=0on ∂Ω, where f is a nonnegative, locally Lipschitz continuous function, Ω is a smooth bounded domain and λ>0 is a parameter. Under the assumption that f has an isolated positive zero α such that f(t)(t−α)N+2N−2 is decreasing in (α,α+δ), for some small δ>0, we show that for large enough λ there exist at least two positive solutions uλ<vλ, verifying ‖uλ‖∞<α<‖vλ‖∞ and uλ,vλ→α uniformly on compact subsets of Ω as λ→+∞. The existence of these solutions holds independently of the behavior of f near zero or infinity.