Abstract
In this paper we consider the semilinear elliptic problem
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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García-Melián, J., Iturriaga, L. Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros. Isr. J. Math. 210, 233–244 (2015). https://doi.org/10.1007/s11856-015-1251-z
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DOI: https://doi.org/10.1007/s11856-015-1251-z