Abstract
In this paper, we study a conjecture of J.Serrin and give a partial generalized result of the work of de Figueiredo and Felmer about Liouville type Theorem for non-negative solutions for an elliptic system. We use a new type of moving plane method introduced by Chen-Li-Ou. Our new ingredient is the use of Stein-Weiss inequality.
Highlights
In this note, we study the nonnegative solutions of the system of integral equations in Rn u(x) = v(x) = Rn Rn |x |x − −y|μ−n v(y)q dy, y|μ−nu(y)pdy. (1)with n ≥ 2, 0 < μ < n, and 1 < p, q ≤ (n + μ)/(n − μ) but not both equal to (n + μ)/(n − μ)
With n ≥ 2, 0 < μ < n, and 1 < p, q ≤ (n + μ)/(n − μ) but not both equal to (n + μ)/(n − μ). This integral system is closely related to the following system of differential equations in Rn
We believe that the critical hyperbola in the conjecture is closely related to the famous Hardy-Littlewood-Sobolev inequality [10] and its generalization
Summary
This integral system is closely related to the following system of differential equations in Rn We believe that the critical hyperbola in the conjecture is closely related to the famous Hardy-Littlewood-Sobolev inequality [10] and its generalization. Liouville theorem, moving plane method, Stein-Weiss inequality. Our result below can be considered as a generalization of the work of de Figueiredo and Felmer [8] about Liouville type Theorem.
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