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.

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.

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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.

We now consider the Pohozaev identity for some integral equations. We start with the following equation

where is the volume of the unit sphere in and and are a smooth function in and a constant, respectively.

Theorem. Suppose is a solution of the above integral equation such that is absolutiely integrable over . And if one setsthen

and the following identity holds

.

This theorem was due to Xu X.W. from the paper published in *J. Funct. Anal.* (2005). When , it was due to Cheng and Lin from this paper published in *Math. Ann.* (1997).

Finiteness for is just the assumption of the integrability of the function . Here we mainly need to show the identity holds true.

We know consider another type of equation, precisely, we consider the positive solution to the following

in . Following is what we need to prove.

Theorem. The following identityholds.

Let us start with the Pohozaev identity for semilinear elliptic equation with polygonal nonlinear terms of the form

over an open, star-shaped domain . We also assume is identical to zero on the boundary .

We multiply the PDE by and integrate over to find

.

The term on the left is just