We establish the monotonicity property for the mass of non-pluripolar products on compact K\"ahler manifolds in the spirit of recent results due to Witt Nystr\"om. Building on this, we initiate the variational study of complex Monge-Amp\`ere equations with prescribed singularity. As applications, we prove existence and uniqueness of K\"ahler--Einstein metrics with prescribed singularity, and we also provide the log concavity property of non-pluripolar products with small unbounded locus. 查看全文>>