Complex Variables and Elliptic Equations ( IF 0.6 ) Pub Date : 2020-09-22 , DOI: 10.1080/17476933.2020.1816981 Phuong Le 1, 2, 3 , Anh Tuan Duong 4 , Nhu Thang Nguyen 4
We study the Liouville-type theorems for the sub-elliptic inequality and for the corresponding system where , and is a strongly degenerate operator of the form Here the functions satisfy certain conditions such that is homogeneous of degree 2 with respect to a group of dilations in .
We first prove that the scalar inequality has no positive solution provided where Q is the homogeneous dimension of associated to the operator .
Then, we establish the nonexistence of positive solutions of the corresponding system in one of following cases:
or ,
p, q>0 and ,
p, q>0, pq>1 and .
Our proofs are based on a maximum principle argument combined with a refinement of Souto's reduction.
中文翻译:
涉及 Δλ-Laplacian 的亚椭圆系统的 Liouville 型定理
我们研究了亚椭圆不等式的 Liouville 型定理 以及对应的系统 在哪里 , 和 是形式的强退化运算符 这里的功能 满足某些条件使得 相对于一组膨胀是 2 次齐次的 .
我们首先证明标量不等式没有正解提供 其中Q是齐次维数 与运营商相关联 .
然后,我们在以下情况之一中建立相应系统的正解不存在:
或者 ,
p , q > 0 和,
p , q >0, pq >1 和.
我们的证明基于最大原理论证并结合了 Souto 约简的改进。