中文摘要：

该文研究全空间R^N上带权的半线性椭圆型方程 -△u=｜x｜^α｜u｜^p-1u,x∈R^N 与半空间R＋^N={x∈R^N：xN〉0}上带权的半线性椭圆型问题 -△u=｜x｜^α｜u｜^p-1u,x∈R＋^N,u｜ R＋^N=0 的Liouville型定理,其中N≥3,α〉-2.证明了,当1〈p〈N＋2α＋2/N-2时,上述问题的Morse指数有限的有界解只能是零解．

英文摘要：

This paper is concerned with Liouville type theorems for weighted semilinear elliptic equations -△u=｜x｜^α｜u｜^p-1u,x∈R^N and -△u=｜x｜^α｜u｜^p-1u,x∈R＋^N,u｜ R＋^N=0 where N ≥ 3 and α 〉 -2. We prove that the bounded solutions of the above problems with finite Morse indices are zero when 1〈p〈N＋2α＋2/N-2.