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

作　　者：韦冬瑜 WEI Dong-yu(School of Mathematics and Statistics,Guangxi Normal University,Guilin 541006,China)

机构地区：[1]广西师范大学数学与统计学院,广西桂林541006

出　　处：《南宁师范大学学报（自然科学版）》2022年第1期12-21,共10页Journal of Nanning Normal University：Natural Science Edition

基　　金：国家自然科学基金(11861016)。

摘　　要：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.

关 键 词：Grushin算子 LIOUVILLE型定理 Green公式 散度定理 YOUNG不等式 

分 类 号：O175.25[理学—数学]