Morse指数与带权的椭圆方程的Liouville型定理

Morse Indices and Liouville Type Theorems for Weighted Elliptic Equations

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-1)u,x∈R^N与半空间R_+~N={x∈R^N:x_N>0}上带权的半线性椭圆型问题-△u=|x|~α|u|^(p-1)u,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.

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.