Why should the input value on a DPA attack be non-constant?
The Pearson product-moment correlation coefficient is what you evaluate, in general, when you perform DPA (CPA would be a better name for this, but DPA is used also in this context).

It is defined as:
$\rho(X,Y) = \frac{cov(X,Y)}{\sigma_X\sigma_Y}$

Where $X,Y$ are your vectors, $cov$ is the covariance and $\sigma_X$ is the standard deviation of X.

In your case, one of the vectors should be a vector of power values at the same time instant for all traces and the other vector should be the hamming weight of your hypothesis.

In your case the second vector is made by the same value. Therefore its standard deviation is 0 and then the correlation is not defined (denominator is zero).

Note that DPA is much better than SPA because by using multiple traces it somehow average out other factors (i.e. noise), but for this you also need different input values (I would say the more the better).