  Let $p>5$ be a fixed prime and assume that $\alpha_1,\alpha_2,\alpha_3$ are
coprime to $p$. We study the asymptotic behavior of small solutions of
congruences of the form $\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv
0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and
$(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.)
If $\alpha_1,\alpha_2,\alpha_3$ are fixed and $n\rightarrow \infty$, we
establish an asymptotic formula (and thereby the existence of such solutions)
under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are
allowed to vary with $n$, we show that this formula holds if $N\gg
q^{11/18+\varepsilon}$. The latter should be compared with a result by
Heath-Brown who established the existence of non-zero solutions under the
condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$.
