Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers

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 congruences form _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$$ $$(x_1x_2x_3,p)=1$$ . (In fact, we consider smoothed version this problem.) If $$n\rightarrow \infty $$ establish an formula (and thereby existence such solutions) under condition $$N\gg q^{1/2+\varepsilon }$$ these coefficients allowed vary n, show holds if q^{11/18+\varepsilon The latter should compared result by Heath-Brown who established non-zero $$N \gg q^{5/8+\varepsilon for odd square-free moduli q.