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

Abstract

Consider the Henon equation with the homogeneous Neumann boundary condition -Delta u + u = |x|(alpha)u(p), u . 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega subset of B(0,1) subset of R-N, N >= 2 and partial derivative Omega boolean AND partial derivative B(0,1) # (sic). We are concerned on the asymptotic behavior of ground state solutions as the parameter alpha -> infinity. As alpha -> infinity, the non autonomous term Ixr is getting singular near vertical bar x vertical bar(alpha) = 1. The singular behavior of vertical bar x vertical bar(alpha) for large alpha > 0 forces the solution to blow up. Depending subtly on the (N - 1)-dimensional measure vertical bar partial derivative Omega boolean AND partial derivative B(0,1)vertical bar(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 vertical bar partial derivative Omega boolean AND partial derivative B(0, 1)vertical bar(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 is an element of (1,2* -1) and a smooth domain Omega. (C) 2018 Elsevier Inc. All rights reserved.