Advertisement

Advances in the Study of Singular Semilinear Elliptic Problems

  • Daniela Giachetti
  • Pedro J. Martínez-Aparicio
  • François Murat
Chapter
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 8)

Abstract

In this paper we deal with some results concerning semilinear elliptic singular problems with Dirichlet boundary conditions. The problem becomes singular where the solution u vanishes. The model of this kind of problems is
$$\displaystyle\begin{array}{rcl} \left \{\begin{array}{@{}l@{\quad }l@{}} u \geq 0 \quad &\mbox{ in }\varOmega, \\ -div\,A(x)Du = F(x,u)\quad &\mbox{ in}\;\varOmega, \\ u = 0 \quad &\mbox{ on}\;\partial \varOmega,\\ \quad \end{array} \right.& & {}\\ \end{array}$$
where Ω is a bounded open set of \(\mathbb{R}^{N}\), N ≥ 1, A is a coercive matrix with coefficients in \(L^{\infty }(\varOmega )\) and \(F: (x,s) \in \varOmega \times [0,+\infty [\rightarrow F(x,s) \in [0,+\infty ]\) is a Carathéodory function which is singular at s = 0. Our aim is to study the meaning of the assumptions made on the singular function F(x, s) in the papers [Giachetti et al., J. Math. Pures Appl. (2016, in press); Giachetti et al., Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0 (Preprint, 2016); Giachetti et al., Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes (Preprint, 2016)], to extend some uniqueness results of the solution given in the same papers, and to prove the \(L^{\infty }\)-regularity of the solutions under some regularity assumption on the data.

Notes

Acknowledgements

The authors would like to thank their own institutions (Dipartimento di Scienze di Base e Applicate per l’Ingegneria della Facoltà di Ingegneria Civile e Industriale di Sapienza Università di Roma, Departamento de Matemática Aplicada y Estadística de la Universidad Politécnica de Cartagena, Laboratoire Jacques-Louis Lions de l’Université Pierre et Marie Curie Paris VI et du CNRS) for providing the support of reciprocal visits which allowed them to perform the present work. The work of Pedro J. Martínez-Aparicio has been partially supported by the grant MTM2015-68210-P of the Spanish Ministerio de Economía y Competitivided (MINECO), the FQM-116 grant of the Junta de Andalucía and the grant Programa de Apoyo a la Investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 19461/PI/14.

References

  1. 1.
    Boccardo, L., Casado-Díaz, J.: Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence. Asymptot. Anal. 86, 1–15 (2014)MathSciNetMATHGoogle Scholar
  2. 2.
    Boccardo, L., Orsina, L.: Semilinear elliptic equations with singular nonlinearities. Calc. Var. Partial Differential Equations 37, 363–380 (2010)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: A semilinear elliptic equation with a mild singularity at u = 0: existence and homogenization. J. Math. Pures Appl. (2016). doi:10.1016/j.matpur.2016.04.007Google Scholar
  4. 4.
    Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: Definition, existence, stability and uniqueness of the solution to a semilinear elliptic problem with a strong singularity at u = 0. Preprint (2016)Google Scholar
  5. 5.
    Giachetti, D., Martínez-Aparicio, P.J., Murat, F.: Homogenization of a Dirichlet semilinear elliptic problem with a strong singularity at u = 0 in a domain with many small holes. Preprint (2016)Google Scholar
  6. 6.
    Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier Grenoble 15, 189–258 (1965)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Daniela Giachetti
    • 1
  • Pedro J. Martínez-Aparicio
    • 2
  • François Murat
    • 3
  1. 1.Facoltà di Ingegneria Civile e Industriale, Dipartimento di Scienze di Base e Applicate per l’IngegneriaSapienza Università di RomaRomaItaly
  2. 2.Departamento de Matemática Aplicada y EstadísticaUniversidad Politécnica de CartagenaCartagenaSpain
  3. 3.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie et CNRSParis Cedex 05France

Personalised recommendations