Abstract
We present the quantum phase diagram of a superconducting quantum dots array. Superconducting quantum dots are connected through Josephson junctions. We study the effect of finite-range Josephson coupling and also finite-range interactions between the dots. Our field-theoretical study predicts different kinds of phase near the charge degeneracy point. We predict two different kinds of repulsive Luttinger liquid phase and also the possibilities of different density wave instabilities from the inter-play of long-range interaction between the dots and Cooper pair density. We observe different multicritical points with interesting phases.