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Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems

Author

Listed:
  • Azeb Alghanemi

    (Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia)

  • Slim Chaabane

    (Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax 3018, Tunisia)

  • Hichem Chtioui

    (Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax 3018, Tunisia)

  • Abdellahi Soumaré

    (Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax 3018, Tunisia)

Abstract

We consider a nonlinear variational elliptic problem with critical nonlinearity on a bounded domain of R n , n ≥ 3 and mixed Dirichlet–Neumann boundary conditions. We study the effect of the domain’s topology on the existence of solutions as Bahri–Coron did in their famous work on the homogeneous Dirichlet problem. However, due to the influence of the part of the boundary where the Neumann condition is prescribed, the blow-up picture in the present setting is more complicated and makes the mixed boundary problems different with respect to the homogeneous ones. Such complexity imposes modification of the argument of Bahri–Coron and demands new constructions and extra ideas.

Suggested Citation

  • Azeb Alghanemi & Slim Chaabane & Hichem Chtioui & Abdellahi Soumaré, 2023. "Towards a Proof of Bahri–Coron’s Type Theorem for Mixed Boundary Value Problems," Mathematics, MDPI, vol. 11(8), pages 1-26, April.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:8:p:1955-:d:1128819
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