Abstract
In this paper, Liouville-type theorems of nonnegative solutions for some elliptic integral systems are considered. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality instead of Maximum Principle.
Highlights
U y q1 v y q2 x y N v y p1 u y p2 x y N (2)
The integral systems are closely related to the following systems of differential equations in N
We study Problem (1) and Problem (2) by virtue of the moving plane method and obtain the following theorems of non
Summary
Every positive smooth solution of PDE (3)(or (4)) multiplied by a constant satisfies (1) (or (2)) respectively This equivalence between integral and PDE systems can be verified as in the proof of Theorem 1 in [1]. Theorem 1.1: Let (u, v) be a nonnegative solution of Problem (1) and The remaining cases can be handled in the same way We leave this to the interested reader. Let us emphasize that considerable attention has been drawn to Liouville-type results and existence of positive solutions for general nonlinear elliptic equations and systems, and that numerous related works are devoted to some of its variants, such as more general quasilinear operators, and domains.
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