Semilinear elliptic equations and nonlinearities with zeros
Section snippets
Introduction and results
The purpose of this paper is the study of the semilinear elliptic problem where is a nonnegative, locally Lipschitz function defined in is a smooth bounded domain of and will be regarded as a parameter. Our main objective is to analyze the existence and multiplicity of positive classical solutions of (1.1) when is large.
When 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 of (1.1) which lies below . 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 is actually the maximal solution in the order interval . In what follows and throughout the paper, will stand for the distance of a point to the boundary of .
The principal result of this section is the following.
Theorem 3 Assume
The second solution
This section will be devoted to prove the existence of a second solution of (1.1) for large , verifying in and . Since we are not assuming any growth restriction on the function , our first step is to truncate the nonlinearity. Taking given by hypothesis (H) we define for some . By hypothesis is a locally Lipschitz function in . Throughout most of the section, we will analyze the truncated problem
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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