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In this paper we aim at employing a compactness criterion of Da Prato,Malliavin, Nualart for square integrable Brownian functionals to constructunique strong solutions of SDE's under an integrability condition on the driftcoefficient. The obtained solutions turn out to be Malliavin differentiable andare used to derive a Bismut-Elworthy-Li formula for solutions of the Kolmogorovequation.

Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada-Watanabe principle