We now consider another kind of problem involving biharmonic operator. Let us assume a solution of the equation

in . We shall prove the following result

Theorem. The following identityholds.

*Proof*. Multiplying the PDE by on the both sides

and integrating the resulting equation over the ball , one has

.

Clearly, by the divergence theorem, one gets

Regarding to the first term we still use the divergence theorem, however, we are dealing with the biharmonic operator which is defined to be . We have

Note that

Therefore

.

We actually can estimate further as follows

which follows

.

Hence

Multiply the PDE by on both sides and integrating it over the ball together with the fact that

we obtain

Combining all gives

Cf. Exact solutions of nonlinear conformally invariant integral equations in , *Adv. Math.* **194** (2005), no. 2, pp. 485-503.

See also:

- The Pohozaev identity: Integral equation with exponential nonlinearity
- The Pohozaev identity: Semilinear elliptic problem with polygonal nonlinear
- The Pohozaev identity: Semilinear elliptic problem with exponential nonlinearity

Hello!

I am interested in the Pohozaev identity but not good at it. Please say more about its applications! It seems that it is basically used to prove non-existence?

Thanks.

Comment by unknown — May 3, 2011 @ 21:54

You are right, this is a great tool for proving non-existence result. The simplest example, originally due to Pohozaev, can be found here http://wp.me/p4uAr-Nu.

Comment by Ngô Quốc Anh — May 3, 2011 @ 22:28