Relation between CDF and PDF

Question

I want to prove there is no random variable X that satisfies the following relation: $$f_X(x) = \bar{F}_X(x)\bar{F}_X(-x)F_X(x)$$ where $f_X$, $\bar{F}_X$ and $F_X$ are the PDF, Complementary CDF and CDF of random variable X, respectively. Any idea is really appreciated.

Note: Pointmass at $+\infty$ is indeed a solution, I want to show there is no solution with full support.

Note: The followings has a unique solution which is the logistic distribution. $$f_X(x) = \bar{F}_X(x)\bar{F}_X(-x)$$

Answer 1

Partial answer

If the support is bounded from below, there is $x_0$ such that $F_X(x_0)=0$. But you have also $F_X'(x)=f_X(x)\leq F_X(x)$, which then implies $F_X\equiv 0$, contradiction.