Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros

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.