  In the present paper we obtain several new results related to the problem of
upper bound estimates for the number of solutions of the congruence $$
x^{x}\equiv \lambda\pmod p;\quad x\in \mathbb{N},\quad x\le p-1, $$ where $p$
is a large prime number, $\lambda$ is an integer corpime to $p$. Our arguments
are based on recent estimates of trigonometric sums over subgroups due to
Shkredov and Shteinikov.
