Critical Points of Kirchhoff-Routh-Type Functions

For $2\le N\in \N$ and $\G_i\in \R\setminus{0}$ we proof that functions of the form $$ H_\G (p_1,\dots, p_N) = \sum_{i\ne j} \G_i\G_j G(p_i,p_j) + \sum_{i=1}^N \G_i^2 R(p_i) , $$ admit critical points under various circumstances. The $p_i$ will either belong to an open, bounded subset $\gO\subset \R^d$ with smooth boundary for $d \ge 3$ or to a compact, two dimensional, riemanian manifold $(\gS ,g )$. Furthermore, $G$ is a (Dirichlet) Green's function of the negative Laplacian $-\gD$ associated to $\gO$ or $(\gS, g)$ and $R$ is its Robin's function.\ For the case of an open set, we also consider the function $\gr$ that is the least eigenvalue of the matrix $$ (M(x_1,\dots,x_N))_{i,j=1}^N := \begin{cases} -G(x_i , x_j),& i\ne j\ R(x_i),& i=j . \end{cases} $$ To achieve the critical points, we also calculate appropriate approximations of the Green's function and Robin's function when close to their singularities.