加权退化椭圆方程非负解的Liouville型定理

Liouville Type Theorem for Nonnegative Solutions of Weighted Degenerate Elliptic Equations

研究了关于以下退化椭圆方程的非负C^(2)解的Liouville型定理:Δ_(G)u+h(z)u^(p)=0,z∈R^(N)=R^(n)×R^(m),其中h(z)是非负C^(2)函数,Δ_(G)u=Δ_(x)u+|x|^(2α)Δ_(y)u,α≥0.通过构造辅助函数,运用Green公式、散度定理和Young不等式等对辅助函数进行非线性能量估计,证明了当n=1,m>2,0<α≤1/3(或0<α≤1/m),1<p≤Q+2/Q-2(或1<p≤(Q-1)(N+2)/(Q-2)(N-1))且h(z)满足一定条件时,方程只有零解u(z)≡0.

In this paper we study the Liouville type theorem of nonnegative C^(2) solutions for the following degenerate elliptic equation Δ_(G)u+h(z)u^(p)=0,z∈R^(N)=R^(n)×R^(m),where h(z)is a nonnegative C^(2) function,Δ_(G)u=Δ_(x)u+|x|^(2α)Δ_(y)u,α≥0.By constructing auxiliary functions,using Green’s formula,divergence theorem and Young’s inequality to estimate the nonlinear energy of auxiliary function,we obtain the following result:when n=1,m>2,0<α≤1/3(or 0<α≤1/m),1<p≤Q+2/Q-2(or 1<p≤(Q-1)(N+2)/(Q-2)(N-1))and h(z)satisfies some conditions,then the equation has only zero solution u(z)≡0.