In this topic we consider the analysis of solutions of the following system entitled Toda system

Following is our main result

Lemma1. The following identitieshold.

*Proof*. Multiply the first equation by and integrate over the ball

.

With the fact that

the RHS can be estimated as follows

The left hand side also can be estimated, we refer the reader to this topic for details

.

Thus we firstly obtain

Similarly, we also obtain

The proof follows.

Lemma2. The following identitieshold.

*Proof*. Multiply the first equation by and integrate over the ball

.

Regarding to the RHS, we get

Involving the LHS,

Thus

Similarly,

The proof follows.

We are now in a position to derive the Pohozaev identity for Toda system in .

Theorem. The following identityholds.

*Proof*. The proof is just an application of Lemmas 1 and 2, we omit it here.

As an application to this theorem, we are going to derive a priori estimate for solutions of Toda systems in . Firstly, we introduce the following notations. Denote

.

We then have the following result.

Corollary. We have.

*Proof*. By the theorem above and by letting the proof follows.

**Remark**. In this entry, we introduce a new techique to deal with

so actually lots of Pohozaev identities mentioned before can be simplified a little bit.

- The Pohozaev identity: Elliptic problem with biharmonic operator
- 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

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