Dijkstra running time complexity is =>
$O\left ( E . T_{dk} + V . T_{em} \right )$
where $T_{dk}$ and $T_{em}$ are the decrease key and extract minimum operations of vertex set Q respectively.
For AVL tree, all the operations are $O\left ( Log V \right )$
Hence, Time complexity to run dijkstra becomes =>
=> $O\left ( E.|Log V| + V . | Log V| \right )$