Let us now consider the following equation
where is a real number satisfying .
Lieb [here] proved among other things that there exist maximizing functions for the Hardy-Littlewoord-Sobolev inequality on
When and , the Euler-Langrange equation for a maximizing is nothing but our integral equation.
Having discussion of fractional Laplacian [here] one can easily see that this integral equation is also closely related to the following family of semilinear PDEs
The classification of solution to the integral equation was done by Chen, Li and Ou [here] published in Comm. Pure App. Math. around 2006 via the integral form of the method of moving planes. Our goal is to derive a new approach based on the integral form of the method of moving spheres. This result was due to Zhang and Hao [here] published in J. Math. Anal. Appl. around 2008.
Theorem. Let be a positive solution to the integral equation. Then is radially symmetric and has the form
for some constants and .
Outline of the proof. Let be a positive function on , for and we define
Lemma 1. For any solution of the integral equation, we have
This lemma has the same form of the lemma considered in this entry. In fact, the proof is straightforward.
Lemma 2. For , there exists such that
for any and .
This lemma tells us that we can run the method of moving spheres. The idea is as follows: it follows from the form of our solution that our solution is indeed monotone decreasing. Thus starting from a point it must be possible to find such a so that outside a sphere centered at with suitable radius, the attitude of function is lower that that at . The proof is similar to the proof of step 1 in this entry.
Next for each , we define
Lemma 3. If for some then
The proof of this lemma is similar to the proof of step 2 in this entry. We do it by contradiction argument. Having all discussion above, we are able to complete the proof of theorem. In fact, w shall prove that is finite for all .
Proof of theorem. If there exists some such that , then by Lemma 3,
By the definition of ,
Multiply the above by and let we get
It now follows from the Lemma 3 that
for any . Thus gives
by the second fundamental lemma [here]. If for any then
It now follows from the first fundamental lemma [here] that is constant which contradiction with the integral equation.