Asymptotics for a variational problem with critical growth and slightly positive Dirichlet data

✍ Scribed by Shigeru Moriyama

• Book ID: 103848654
• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 845 KB
• Volume: 70
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2008.02.001

✦ Synopsis

Let Ω be a bounded smooth domain in R N , N ≥ 3. We consider the variational problem inf Ω |∇u| 2 for the admissible class

Caffarelli and Spruck [L.A. Caffarelli, J. Spruck, Variational problems with critical Sobolev growth and positive Dirichlet data, Indiana Univ. Math. J. 39 (1990) 1-18] proved the existence of the solution u γ ,ε satisfying

for λ γ ,ε > 0. We prove that the solution concentrates at exactly one interior point as ε goes to zero. Furthermore we study the exact rate and location of the blowing up.