Let $N>1$ and \begin{equation*} f_N\left(l_1,\dots,l_{N-1}\right)=\sum\limits_{i=1}^{N-1}\left[l_i-\log\left(l_i\right)\right]-\sum_{0<j<i<N}\log\left(l_i-l_j\right) \end{equation*} Let $\left(l_1^{\left(0\right)},\dots,l_{N-1}^{\left(0\right)}\right)$ such that $0<l_1^{\left(0\right)}<\cdots<l_{N-1}^{\left(0\right)}$ minimize $f_N$. Then what is $l_{N-1}^{\left(0\right)}$ as a function of $N$. Even an asymptotic answer for large $N$ would be great.
This question came up in finding the equilibrium positions of a physical system.
I have already written a code to minimise this using gradient descent.