Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations
Morse index and symmetry-breaking for positive solutions of one-dimensional Hénon type equations
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DOI：
10.1016/j.jde.2013.05.029
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发表时间：
2013-10
影响因子：
2.4
通讯作者：
Satoshi Tanaka
中科院分区：
文献类型：
--
作者：
Satoshi Tanaka
本文研究了如下两点边值问题正解的莫尔斯指数和破缺性，其中h∈C[− 1，1]<$C1（[− 1，1]\{0}），h（x）>0，h（−x）=h（x）在[− 1，1]\{0}上，f∈C1[0，∞），f（s）>0（s>0），f（0）=0.一维Hénon方程的问题是一个典型的例子，其中l ≠ 0且p>1。这个问题总是有唯一的正偶数解。众所周知，当l=0时，不存在正的非偶解，当l>0足够大时，存在正的非偶解.本文的结果表明，当l（p−1）≥ 4时，正最小能量解的莫尔斯指数等于1，正偶解的莫尔斯指数等于2，因此正最小能量解是非偶的，并出现破缺现象.当l ≠ 0且p>1充分小时，不存在正的非偶解，且偶正解的莫尔斯指数等于1.
In this paper, the Morse index and the symmetry-breaking for positive solutions of the following two-point boundary value problem are studied, where h∈C[−1,1]∩C1([−1,1]\{0}), h(x)>0, h(−x)=h(x) on [−1,1]\{0}, f∈C1[0,∞), f(s)>0 for s>0, and f(0)=0. The problem for the one-dimensional Hénon equation is a typical example, where l⩾0 and p>1. This problem always has the unique positive even solution. It is well-known that if l=0, then there is no positive non-even solution, and if l>0 is sufficiently large, then there exist positive non-even solutions. The result in this paper shows that if l(p−1)⩾4, then the Morse index of the positive least energy solution equals 1 and the Morse index of the positive even solution equals 2, and hence the positive least energy solution is non-even and symmetry-breaking phenomena occur. It is also shown that if l⩾0 and p>1 are sufficiently small, then there is no positive non-even solution and the Morse index of the even positive solution equals 1.