Israel Journal of Mathematics

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

Multiplicity of solutions for some semilinear problems involving nonlinearities with zeros



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Alarcón, J. García-Melián and A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equations involving the Laplacian, Ann. Scuola Norm. Sup. Pisa, to appear. DOI: 10.2422/2036-2145.201402_007.Google Scholar
  2. [2]
    S. Alarcón, L. Iturriaga and A. Quaas, Existence and multiplicity results for Pucci’s operators involving nonlinearities with zeros, Calculus of Variations and Partial Differential Equations 45 (2012), 443–454.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Communications in Partial Differential Equations 36 (2011), 2011–2047.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Communications on Pure and Applied Mathematics 34 (1981), 525–598.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Communications in Partial Differential Equations 6 (1981), 883–901.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    L. Iturriaga, S. Lorca and E. Massa, Positive solutions for the p-Laplacian involving critical and supercritical nonlinearities with zeros, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire 27 (2010), 763–771.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    L. Iturriaga, E. Massa, J. Sánchez and P. Ubilla, Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros, Journal of Differential Equations 248 (2010), 309–327.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review 24 (1982), 441–467.MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci’s Operator, Communications in Partial Differential Equations 31 (2006), 987–1003.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© 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

Personalised recommendations