For symmetric functions, people ask *Do symmetric problems have symmetric solutions?*, e.g., [3] and [4].
The answer is no in general. However, solutions of symmetric problems often exhibit certain symmetry.
In [1], for a class of symmetric polynomials, the global minimum under some conditions
is attained at some points with $|\{x_1, x_2, \cdots, x_n\} |\le 2$, i.e., at most two distinct components.
In [2], for a linear combination of elementary symmetric polynomials, under some conditions, each of the local extrema ($n$-dimensional vector)
has at most $k$ distinct components. 

However, I am curious about whether there are examples (for **differentiable** symmetric functions under some conditions) in which all components of the global minimizer are **distinct**, if not impossible.

First, let us see an example. Under the conditions $x, y, z > 0$ and $xyz = 1$,
the global minimum of $$g(x,y,z) = \frac{\sin \frac{\pi x}{2} }{x} + \frac{\sin \frac{\pi y}{2}}{y} + \frac{\sin \frac{\pi z}{2}}{z}$$
is attained at some points $(x_0, y_0, z_0)$ with exactly two of $x_0, y_0, z_0$ being equal,
for example $(x_0, x_0, \frac{1}{x_0^2})$ where $x_0 \approx 2.852$ and the minimum $g_{\min} \approx 0.878$.
Also $g(1,1,1) = 3 > g_{\min}$ and $\lim_{\min(x,y,z)\to 0^{+}} g(x,y,z) > 0.884 > g_{\min}$.

In the problem above, the components of the global minimizer is not distinct.
Now, suppose $f: (0, \infty) \rightarrow \mathbb{R}$ is a differentiable function.
Let $F(x, y, z) = f(x) + f(y) + f(z)$.
I want to find some examples of $f$ such that under the conditions $x, y, z>0$ and $xyz=1$,
the global minimum of $F(x,y,z)$ is attained at some point $(x_0, y_0, z_0)$
with none of $x_0, y_0, z_0$ being equal, if not impossible.

By the way, for cyclic symmetric functions, I found examples in which the global minimum
is attained at some point with distinct components. For example,
let
$$F_1(a, b, c) = \frac{a^2b + 2a^2c + 2ab^2 + b^3 + 31abc}{(a+b+50c)(a+b+c)^2}$$
and let $G(a,b,c) = F_1(a, b, c) + F_1(b, c, a) + F_1(c, a, b)$.
Then the minimum of $G(a, b, c)$ under the conditions $a, b, c \ge 0$ and $a+b+c=3$ is not achieved at $(1, 1, 1)$ or $abc=0$.
Indeed, we have $G(1,1,1) = 37/156 \approx 0.2372$ and
$$G(a, 3-a, 0) = \frac{49a^4-8094a^3+45900a^2-66177a-4050}{9(49a+3)(49a-150)} > 0.21, \quad \forall 0\le a\le 3.$$
However, $G(1/2, 1/8, 19/8) = 1018835/4907936 \approx 0.2076$; 
Actually, the global minimum is attained at some point with distinct components  (also none of them is zero).

Reference:

[1] Vasile Cirtoaje, “The Equal Variable Method”, J. Inequal. Pure and Appl. Math., 8(1), 2007.
Online: https://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf

[2] Alexander Kovacec, et. al., “A note on extrema of linear combinations of elementary symmetric functions”,
Linear and Multilinear Algebra, Volume 60, 2012 - Issue 2.

[3] R. F. Rinehart, "On Extrema of Functions which Satisfy Certain Symmetry Conditions",
The American Mathematical Monthly, Vol. 47, No. 3 (Mar., 1940), pp. 145-152.

[4] William C. Waterhouse, “Do Symmetric Problems Have Symmetric Solutions?”, The American Mathematical Monthly, vol. 90, 1983, pp. 378-387.