Suppose $p_1,p_2,\dots,p_n \in [0,1]$ and they satisfies $$ \sum_{j=1}^n p_j = 1 $$ and $$ \sum_{j=1}^n p_j^2 = C $$ with a given constant $C \in [1/n,1]$. The problem is to find the minimum of $$ -\sum_{j=1}^n p_j \log p_j .$$ Obviously, this problem could be solved by method of Lagrange multipliers. Since I believe that this problem is pretty typical, I have a question:

Is there any literature (paper or book) which solved this problem?