کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4610106 | 1338544 | 2015 | 29 صفحه PDF | دانلود رایگان |
• We find a curvature K(r)K(r) with a unique minimum, but the scalar curvature equation has many ground states with fast decay.
• We give an example where the derivative of a zero of the Melnikov function decides between a unique or multiple homoclinics.
• We derive a Melnikov approach which requires just Lipschitz regularity and allows very degenerate zeroes.
This paper is devoted to the study of positive radial solutions of the scalar curvature equation, i.e.Δu(x)+K(|x|)uσ−1(x)=0Δu(x)+K(|x|)uσ−1(x)=0 where σ=2nn−2 and we assume that K(|x|)=k(|x|ε)K(|x|)=k(|x|ε) and k(r)∈C1k(r)∈C1 is bounded and ε>0ε>0 is small. It is known that we have at least a ground state with fast decay for each positive critical point of k for ε small enough. In fact if the critical point k(r0)k(r0) is unique and it is a maximum we also have uniqueness; surprisingly we show that if k(r0)k(r0) is a minimum we have an arbitrarily large number of ground states with fast decay. The results are obtained using Fowler transformation and developing a dynamical approach inspired by Melnikov theory. We emphasize that the presence of subharmonic solutions arising from zeroes of Melnikov functions has not appeared previously, as far as we are aware.
Journal: Journal of Differential Equations - Volume 259, Issue 8, 15 October 2015, Pages 4327–4355