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

Following is our main result

**Lemma** 1. The following identities

hold.

*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.

**Lemma** 2. The following identities

hold.

*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 identity

holds.

*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.

See also:

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