Separation structure of positive radial solutions of a semilinear elliptic equation in Rn
Separation structure of positive radial solutions of a semilinear elliptic equation in Rn
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DOI：
10.1016/s0022-0396(03)00172-4
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发表时间：
2003-11
影响因子：
2.4
通讯作者：
Soohyun Bae
中科院分区：
文献类型：
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作者：
Soohyun Bae
本文研究了以下几个相互关联的问题：（1）研究了椭圆型方程Δu+K（|X|）up=0（在Rn中）。当r− K（r）以适当的衰减率随着r→∞（其中k>−2）而减小到一个正常数时，当n和p足够大时，详细描述了径向解在∞附近的渐近行为。当方程具有分离结构（任意两个径向正解互不相交）时，所得的渐近性态决定了作为相应半线性热方程定常状态的径向正解的稳定性的自然拓扑. (ii)利用正则解的局部分离性，证明了当初始值为0的解全局存在时奇异解的存在性。(iii)通过Kelvin变换，正则解的分离结构反映了奇异解的分离结构。我们提出了一个新的指数，这是研究奇异解的分离和相交的关键。
This paper deals with the following inter-connected subjects: (i) Separation of positive radial solutions is studied for the elliptic equation Δu+K(|x|)up=0 in Rn. In case r−ℓK(r) is decreasing, with suitable decay rate, to a positive constant as r→∞ for some ℓ>−2, the asymptotic behavior near ∞ of radial solutions is described in detail when n and p are large enough. When the equation has separation structure (any two positive radial solutions do not intersect), the obtained asymptotic behaviors decide natural topology for stability of positive radial solutions regarded as steady states of the corresponding semilinear heat equation. (ii) By using local separation of regular solutions, we establish the existence of a singular solution when every solution with positive initial data at 0 exists globally. (iii) Through Kelvin's transform, separation structure of regular solutions reflects that of singular solutions. We present a new exponent, which is critical in the study of separation and intersection of singular solutions.