Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896655 | Journal of Functional Analysis | 2018 | 44 Pages |
Abstract
Consider the Hénon equation with the homogeneous Neumann boundary conditionâÎu+u=|x|αup,u>0inΩ,âuâν=0 on âΩ, where ΩâB(0,1)âRN,Nâ¥2 and âΩâ©âB(0,1)â â
. We are concerned on the asymptotic behavior of ground state solutions as the parameter αââ. As αââ, the non-autonomous term |x|α is getting singular near |x|=1. The singular behavior of |x|α for large α>0 forces the solution to blow up. Depending subtly on the (Nâ1)âdimensional measure |âΩâ©âB(0,1)|Nâ1 and the nonlinear growth rate p, there are many different types of limiting profiles. To catch the asymptotic profiles, we take different types of renormalization depending on p and |âΩâ©âB(0,1)|Nâ1. In particular, the critical exponent 2â=2(Nâ1)Nâ2 for the Sobolev trace embedding plays a crucial role in the renormalization process. This is quite contrasted with the case of Dirichlet problems, where there is only one type of limiting profile for any pâ(1,2ââ1) and a smooth domain Ω.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Jaeyoung Byeon, Zhi-Qiang Wang,