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We study the interplay of superconductivity and disorder by solvingnumerically the Bogoliubov-de-Gennes equations in a two dimensional lattice ofsize $80\times80$ which makes possible to investigate the weak-coupling limit.In contrast with results in the strong coupling region, we observe enhancementof superconductivity and intriguing multifractal-like features such as a broadlog-normal spatial distribution of the order parameter, a parabolic singularityspectrum, and level statistics consistent with those of a disordered metal atthe Anderson transition. The calculation of the superfluid density, includingsmall phase fluctuations, reveals that, despite this intricate spatialstructure, phase coherence still holds for sufficiently weak disorder. It onlybreaks down for stronger disorder but before the insulating transition takesplace.

Enhanced, phase coherent, multifractal-like, two-dimensional superconductivity