Example 7 Let $n(n \geqslant 2)$ be a fixed integer.
(1) Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$

holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$;
(2) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
(40th IMO)
Explain: First, find the range of values for $c$, then provide a proof, and finally point out the necessary and sufficient conditions for the equality to hold when $c$ takes its minimum value.
$$
\begin{array}{l}
\text { When } x_{1}=x_{2}=1, x_{3}=x_{4}=\cdots=x_{n}=0 \text {, } \\
c \geqslant \frac{1}{16} \times 1 \times 1 \times\left(1^{2}+1^{2}\right)=\frac{1}{8} .
\end{array}
$$

Below is the proof: the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant \frac{1}{8}\left(\sum_{i=1}^{n} x_{i}\right)^{4}
$$

holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$.
Notice that
$$
\begin{array}{l}
\left(\sum_{i=1}^{n} x_{i}\right)^{4}=\left(\sum_{k=1}^{n} x_{k}^{2}+2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\right)^{2} \\
\geqslant 4 \sum_{k=1}^{n} x_{k}^{2} \cdot 2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j} \\
=8 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j} \cdot \sum_{k=1}^{n} x_{k}^{2} \\
\geqslant 8 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) .
\end{array}
$$

Therefore, $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant \frac{1}{8}\left(\sum_{i=1}^{n} x_{i}\right)^{4}$.
Thus, the minimum value of $c$ is $\frac{1}{8}$, and the necessary and sufficient condition for the equality to hold is that two of the $x_{i}$ are equal, and the remaining $n-2$ $x_{i}$ are all 0.

Note: This problem is a typical example of the method "first derive the conclusion from simple cases, then provide a proof." There are other methods to solve this problem, such as the method of "gradual adjustment and continuous approximation."