Separation structure of positive radial solutions of a semilinear elliptic equation in Rn

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 Kelvinʹs 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.