数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (4): 1213-1244.doi: 10.1016/S0252-9602(11)60311-1

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ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

 Zakaria Boucheche1, Ridha Yacoub1, Hichem Chtioui2   

  1. 1.Institut Préparatoire aux Etudes d'Ingénieur de Monastir, Avenue Ibn Al Jazzar, 5019 Monastir, Tunisia;2.Département de Mathématiques, Faculté|des Sciences de Sfax, Route Soukra, Sfax, Tunisia
  • 收稿日期:2009-08-27 修回日期:2010-04-08 出版日期:2011-07-20 发布日期:2011-07-20

ON A BI-HARMONIC EQUATION INVOLVING CRITICAL EXPONENT: EXISTENCE AND MULTIPLICITY RESULTS

 Zakaria Boucheche1, Ridha Yacoub1, Hichem Chtioui2   

  1. 1.Institut Préparatoire aux Etudes d'Ingénieur de Monastir, Avenue Ibn Al Jazzar, 5019 Monastir, Tunisia;2.Département de Mathématiques, Faculté|des Sciences de Sfax, Route Soukra, Sfax, Tunisia
  • Received:2009-08-27 Revised:2010-04-08 Online:2011-07-20 Published:2011-07-20

摘要:

In this paper,  we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier
boundary conditions: Δ2 u=K(x)up, u>0 in Ω, Δuu=0 on ∂Ω, where Ω is a smooth domain in Rn, n≥5, and p+1 = 2n/n-4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers  lower bounds for the number of solutions. As further applications of this  Morse theoretical approach, we prove more existence results.

关键词: critical points at infinity, Palais-Smale condition, Morse lemma at infinity, Morse inequalities

Abstract:

In this paper,  we consider the problem of existence as well as multiplicity results for a bi-harmonic equation under the Navier
boundary conditions: Δ2 u=K(x)up, u>0 in Ω, Δuu=0 on ∂Ω, where Ω is a smooth domain in Rn, n≥5, and p+1 = 2n/n-4 is the critical Sobolev exponent. We obtain highlightly a new criterion of existence, which provides existence results for a dense subset of positive functions, and generalizes Bahri-Coron type criterion in dimension six. Our argument gives also estimates on the Morse index of the obtained solutions and extends some known results. Moreover, it provides, for generic K, Morse inequalities at infinity, which delivers  lower bounds for the number of solutions. As further applications of this  Morse theoretical approach, we prove more existence results.

Key words: critical points at infinity, Palais-Smale condition, Morse lemma at infinity, Morse inequalities

中图分类号: 

  • 53C15