\min_{1 \leq j \leq l} c_j \right) \, R^{\gamma}.$$

Suppose that for a given $r > 0$, $(x_1, \ldots, x_k) \in U$ satisfies $$\min_{1 \leq i < j \leq k} \|x_i - x_j\| \leq r.$$ Fix $1 \leq i < j \leq k$ such that $\|x_i - x_j\|$ is minimal, then $$\label{eq:energyblowupmin}
f(x_1, \ldots, x_k) \geq \beta \, \frac{w_i \, w_j}{\|x_i - x_j\|^{\delta}} \geq \beta \, w_i \, w_j \, \frac{1}{r^{\delta}} \geq \beta \, \left( \min_{1 \leq i \leq k} w_i^2 \right) \, \frac{1}{r^{\delta}}.$$

Note that $f$ is invariant under translations, so that without loss of generality we can restrict ourselves to solutions with $x_1 = 0$. Let $$E := f((0, 0, \ldots, 0), (1, 0, \ldots, 0), (2, 0, \ldots, 0), \ldots, (k - 1, 0, \ldots, 0))$$ be an upper bound for the minimum value of $f$.

By we know that there exists an $R > 0$ such that $f(0, x_2, \ldots, x_k) > E$ whenever $\max_{i \neq j} \|x_i - x_j\| > R$. On the other hand, by there exists an $r > 0$ such that $f(0, x_2, \ldots, x_k) > E$ whenever $\min_{i \neq j} \|x_i - x_j\| < r$. So if $(0, x_2, \ldots, x_k) 