I want to find a maximum for $\sum_{i = 1}^n x_i$ subject to $\sum_{i = 1}^n x_i^{\alpha} \leqslant c$ where $c > 0$, $x_i > 0$ and $\alpha$ is a positive number greater than zero and not equal to 1.
This is an extension to my previous question found here which involved finding a solution when when $\alpha = 2$ by using the Cauchy-Schwarz inequality.
If $\alpha > 1$, the feasible region is convex. By symmetry the maximum will occur at a point where all $x_i$ are equal, namely $x_i = (c/n)^{1/\alpha}$.
If $\alpha < 1$, the function $\sum_{i=1}^n x_i^\alpha$ is concave on $\{x: x_i \ge 0, \sum_{i=1}^n x_i = s\}$. It follows that the maximum (if you allow $x_i \ge 0$ rather than $> 0$) will occur at an extreme point, namely one $x_i = c^{1/\alpha}$, the others $0$. If you insist on $x_i > 0$, there is no maximum but you can get arbitrarily close to $c^{1/\alpha}$.
