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

Let p > 5 be a fixed prime and assume that a(1), a(2), a(3) are coprime to p. We study the asymptotic behavior of small solutions of congruences of the form a(1)x(1)(2) + alpha(2)x(2)(2) + alpha(3)x(3)(2) 0 mod q with q = p(n), where max{vertical bar x(1)vertical bar, vertical bar x(2)vertical bar, vertical bar x(3)vertical bar} <= N and (x(1)x(2)x(3), p) = 1. (In fact, we consider a smoothed version of this problem.) If a(1), a(2), a(3) ( )are fixed and n -> infinity, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition N >> q(1/2+epsilon). If these coefficients are allowed to vary with n, we show that this formula holds if N >> q(11/18+epsilon). The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition N >> q(5/8+epsilon) for odd square-free moduli q.