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A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense ofHarvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian isnonnegative. We show that the associated Monge-Ampere equation can be solved onany compact Kahler manifold. As a consequence we prove the existence ofsolutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems forbalanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.

The Monge-Amp\`ere equation for (n-1)-plurisubharmonic functions on a compact K\"ahler manifold