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
where , and .
The main result of the paper can be formulated as follows
Theorem. Let be the function given by
Then is a solution to the equation above if and only if
for some and some .
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 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.