Ngô Quốc Anh

May 1, 2011

Method of moving planes: The cylindrically symmetric of solutions of critical Hardy–Sobolev operators

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 5:22

Recently, I have read a paper entitled “Classification of solutions of a critical Hardy-Sobolev operator” published in J. Differential Equations [here].

In that paper, the authors try to classify all positive solutions for the following equation

\displaystyle -\Delta u(x)=\frac{u(x)^\frac{n}{n-2}}{|y|} \quad \text{ in } \mathbb R^n

where x=(y,z)\in \mathbb R^k \times \mathbb R^{n-k}, and u \in \mathcal D^{1,2}(\mathbb R^n).

The main result of the paper can be formulated as follows

Theorem. Let u_0 be the function given by

\displaystyle u_0(x)=u_0(y,z)=c_{n,k}\left( (1+|y|)^2+|z|^2\right)^{-\frac{n-2}{2}}

where

\displaystyle c_{n,k}=\big((n-2)(k-1)\big)^\frac{n-2}{2}.

Then u is a solution to the equation above if and only if

\displaystyle u(y,z)=\lambda^\frac{n-2}{2}u_0(\lambda y, \lambda z+z_0)

for some \lambda>0 and some z_0 \in \mathbb R^{n-k}.

First, they use the method of moving planes to prove the cylindrically symmetric of solutions. Thanks to this symmetry, the equation reduces to an elliptic equation in the positive cone in R^2 which eventually leads to a complete identification of all the solutions of the equation. We skip the detailed discussion here and refer the reader to the original paper.

This result has recently been generalized by Cao and Li.

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