In this paper we obtain new upper bound estimates for the number of solutions of the congruence
$$\begin{equation}
x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U},
\end{equation}$$
for certain ranges of
H and |
${\mathcal U}$
|, where
${\mathcal U}$
is a subset of the field of residue classes modulo
p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence
$$\begin{equation}
x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L
is at most
$p^{\frac{1}{3}-c}$
uniformly over positive integers
n, λ and
L, for some absolute constant
c > 0. This implies, in particular, that if
f(
x) ∈
$\mathbb{Z}$
[
x] is a fixed polynomial without multiple roots in
$\mathbb{C}$
, then the congruence
xf(x) ≡ 1 (mod
p),
x ∈
$\mathbb{N}$
,
x ⩽
p, has at most
$p^{\frac{1}{3}-c}$
solutions as
p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo
p can be represented in the form
xgy (mod
p) with positive integers
x <
p5/8+ϵ and
y <
p3/8. Here
g denotes a primitive root modulo
p. We also prove that almost all the residue classes modulo
p can be represented in the form
xyzgt (mod
p) with positive integers
x, y, z, t < p1/4+ϵ.