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

and let

and

.

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 .

Set

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

and

.

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

which implies

.

**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

.

Let

.

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.

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