一个超临界椭圆问题的径向奇异正解(英文)

Positive Radial Singular Solution of a Supercritical Elliptic Problem

对非线性椭圆问题正解的研究具有实际的物理意义，其研究方法主要有拓扑度理论和变分方法．当非线性项是次临界超线性增长时，极小极大定理是最为有力的工具．即使该超线性项是临界增长的，仍可在某能量面以下重建紧性以保证极小极大定理是适用的．但是，当该超线性项是超临界增长时，由于没有嵌入定理，Ｓｏｂｏｌｅｖ 空间上的变分方法失效．目前，对超临界问题的研究仅限于讨论其径向对称解，这时该问题转化为常微分方程．注意到球域上该问题的正解均是径向对称的，使用基于压缩映像的不动点定理，通过一系列详细的估计，证明了该问题的具某种奇异性的径向正解的存在惟一性．

The problem of elliptic equation involving supercritical growth is one of the most popular problems in study of elliptic problem. One of the reasons is the demand of practical applications, and the other is that either the method of studying or the properties of the solutions is intrinsically different from those of critical and subcritical cases. In this paper the existence and uniqueness of positive radial singular solution of a supercritical elliptic problem is considered on the unit ball. It is well known that all positive classical solutions are radially symmetric, so some ODE techniques can be used to prove the existence and uniqueness of positive radial singular solution of the problem by contraction principle.