# Threshold in a maximisation problem (with KKT conditions)

I'm looking to maximise with respect to $x_i$ $$L = \sum_{i= 1}^n y_i \frac {x_i^{1 - \epsilon}}{1 - \epsilon}$$ subject to $\sum_{i= 1}^n x_i = B$ and $x_i \ge 0$ for all $i$, where $y_i$, $B$ and $\epsilon$ are fixed parameters and $\epsilon$ is between 0 and 1. The standard KKT conditions approach that I've taken hasn't given me much useful information about the optimal. But I know that my objective function is of a specific (and strictly concave) form, so what I'm trying to do is define a threshhold of the given parameters below which a given $x_i$ will be 0, and above which it will be non-zero. Is this possible to do, and if so how would I go about it? If not, is there another approach I should consider that will give me more useful information about the optimal?