Addendum to: Stable and Finite Morse Index Solutions for Dirichlet Problems with Small Diffusion in a Degenerate Case and Problems with Infinite Boundary Values

摘要

1 Introduction. This is a short addendum to [1], where we studied positive solutions for small positive ε of -ε~2△u = f(u) in Ω, u = 0 in some cases where f has an isolated positive zero which is not simple, and where we emphasised the difference with the case where the zeros are simple. Here Ω is a smooth bounded domain in R~N. With hindsight, (and this has been confirmed by several comments to the author) we were rather too brief about a moving plane argument in [1]. Here we explain this argument (and generalise the result a little). Note that these limit problems which are half space problems with infinite boundary value arise in many other cases where the nonlinearity has a degenerate zero.