How hard is Self-DLOG?
The problem about fixed  points of discrete logarithms is  known the Brizolis problem. In particular for the average number of solutions
$$N(p)=\frac{1}{\varphi(p-1)}\sum_g\left|\{0\le x\le p-1:g^x=x\pmod p\}\right|$$ is known that (Grechnikov, 2012) $N(p)=1+S(p)$, where 
$$-C(\varepsilon)p^{-1/4+\varepsilon}\le S(p)\le \exp(C'\mathrm{Li}((\log p)^{c\frac{\log\log\log\log p}{\log\log\log p}})).$$

It is also necessary to say that inversed problem is easier: for a given $x$ one can take $R=x^{x^{-1}\bmod (p-1)}.$