Israel Journal of Mathematics

, Volume 210, Issue 1, pp 233–244 | Cite as

Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros

Article

Abstract

In this paper we consider the semilinear elliptic problem
$$\{ _{u = 0{\text{ on }}\partial \Omega }^{ - \vartriangle u = \lambda f(u){\text{ in }}\Omega }$$
where f is a nonnegative, locally Lipschitz continuous function with r positive zeros, Ω is a smooth bounded domain and λ > 0 is a parameter. We show that for large enough λ there exist 2r positive solutions, irrespective of the behavior of f at zero or infinity, provided only that f verifies a suitable non-integrability condition near each of its zeros, thereby generalizing previous known results. The construction of the solutions rely on the sub- and supersolutions method and topological degree arguments, together with the use of a new Liouville theorem which is an extension of recent results to this type of nonlinearities.

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© Hebrew University of Jerusalem 2015

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa LagunaSpain
  2. 2.Universidad de La LagunaLa LagunaSpain
  3. 3.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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