On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states

Abstract

Consider the Hénon equation with the homogeneous Neumann boundary condition−Δu+u=|x|αup,u>0inΩ,∂u∂ν=0 on ∂Ω, where Ω⊂B(0,1)⊂RN,N≥2 and ∂Ω∩∂B(0,1)≠∅. We are concerned on the asymptotic behavior of ground state solutions as the parameter α→∞. As α→∞, 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 (N−1)−dimensional measure |∂Ω∩∂B(0,1)|N−1 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 |∂Ω∩∂B(0,1)|N−1. In particular, the critical exponent 2⁎=2(N−1)N−2 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 p∈(1,2⁎−1) and a smooth domain Ω.