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

Azeb Alghanemi (), Slim Chaabane, Hichem Chtioui and Abdellahi Soumaré
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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

Mathematics, 2023, vol. 11, issue 8, 1-26

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.

Keywords: nonlinear elliptic problems; critical nonlinearities; variational structures; mixed boundary conditions; topological methods (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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