Counting fundamental solutions to the Pell equation with prescribed size

The cardinality of the set of $D⩽x$ for which the fundamental solution of the Pell equation $t^2-D{u}^2=1$ is less than $D^{\frac{1}{2}+\alpha }$ with $\alpha \in [\frac{1}{2},1]$ is studied and certain lower bounds are obtained, improving previous results of Fouvry by introducing the $q$-analogue of van der Corput method to algebraic exponential sums with smooth moduli.