Today we study a beautiful version of the method of moving planes. This method just invented around 2003 by W.X. Chen, C.M. Li and B. Ou and published in Comm. Pure Appl. Math. around 2006 (for the paper, please go here). In that paper, the author proved among other things that every positive regular solution of the integral equation
is radially symmetric and monotone about some point and therefore assumes the form
with some constant and for some and . These solutions in case and have the following shape.
This integral equation is also closely related to the following family of semilinear PDEs
For a given real number we define
We also denote by the reflection of domain about the plane .
Lemma. For any solution of the PDE, we have
It is also true for
the Kelvin-type transform of for any .
we use the method of moving planes to prove that following
Theorem. Let be a positive solution of the PDE. Then it must be radially symmetric and monotone decreasing about some point.
Outline of proof.
Step 1. Define
In this step, we show that the method of moving planes can be run. To this purpose, we show that there exists a sufficiently negative large value of such that in , i.e. must be empty.
Clearly by definition of , for any we have
It follows from the Hardy-Littlewood-Sobolev inequality that there is a constant such that
for any . By the assumption we can choose sufficiently large such that for we have
Step 2. Now we move the plane to the right as long as the following condition
Suppose that at we have
we show that the plane can be moved further to the right. More precisely, there exists an (depending on , ) such that
It follows from our lemma that
Then obviously has measure zero and
It is worth noticing that
Now the locally integrable of together with the Lebesgue Dominated Convergence theorem imply
for all . Again we have
which implies within the method of moving planes can run.
Step 3. One the method of moving planes stops at some point , we get that
for all . Consequently, has no singularity at the origin. We are then in a position to apply the second fundamental lemma considered here to and to obtain the form of solution. Otherwise, is symmetric w.r.t. the plane . Since the direction can be chosen arbitrary, we deduce that is symmetric and monotone about the origin and thus so is .
Remark. Since we do not assume any asymptotic behavior of near infinity (just assume locally integrable), we are not able to carry out the method of moving planes directly on . To overcome this difficulty, we consider . The advantage here is that may have singularity at the origin and no singularity at infinity.