Elsevier

Nonlinear Analysis

Volume 134, March 2016, Pages 117-126
Nonlinear Analysis

Semilinear elliptic equations and nonlinearities with zeros

https://doi.org/10.1016/j.na.2015.12.025Get rights and content

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)(tα)N+2N2  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,vλα uniformly on compact subsets of Ω as λ+. The existence of these solutions holds independently of the behavior of f near zero or infinity.

Section snippets

Introduction and results

The purpose of this paper is the study of the semilinear elliptic problem {Δu=λf(u)in  Ω,u=0on  Ω, where f is a nonnegative, locally Lipschitz function defined in [0,+),Ω is a smooth bounded domain of RN(N3) and λ>0 will be regarded as a parameter. Our main objective is to analyze the existence and multiplicity of positive classical solutions of (1.1) when λ is large.

When f is positive, it turns out that the behaviors at zero and infinity are important in order to ensure the existence of

The first solution

In this section we will deal with the construction of the positive solution uλ of (1.1) which lies below α. It will be obtained with the well-known method of sub and supersolutions. For its application in the proofs of Section  3, it is to be noted that uλ is actually the maximal solution in the order interval [0,α]. In what follows and throughout the paper, d(x) will stand for the distance of a point x to the boundary of Ω.

The principal result of this section is the following.

Theorem 3

Assume f:[0,+)R

The second solution

This section will be devoted to prove the existence of a second solution vλ of (1.1) for large λ, verifying vλ>uλ in Ω and vλ>α. Since we are not assuming any growth restriction on the function f, our first step is to truncate the nonlinearity. Taking δ given by hypothesis (H) we define f̃(t){f(0)t0,f(t)0tα+δ,f(α+δ)δp(tα)pt>α+δ, for some p(1,N+2N2). By hypothesis f̃ is a locally Lipschitz function in [0,+). Throughout most of the section, we will analyze the truncated problem {Δu=λf̃

Acknowledgments

J. G-M was partially supported by Ministerio de Ciencia e Innovación under grant MTM2011-27998 (Spain) and Conicyt MEC number 80130002. B. B. was partially supported by a postdoctoral fellowship given by Fundación Ramón Areces (Spain) and MTM2013-40846-P, MINECO. L. I. was partially supported by Programa Basal PFB 03, CMM, U. de Chile and Fondecyt grant 1120842.

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