Singular Solutions to the Complex Monge-Amp\`ere Equation

We present an explicit pluripotential and viscosity solution to the complex Monge-Amp\`ere equation with constant right-hand side on D× Cⁿ⁻¹\,(n≥ 2), which lies merely in W1,2loc∩ W2,1loc and is not even Dini continuous. Additionally, we exhibit two families of explicit entire toric solutions on Cⁿ with continuous H\"older exponent α∈(0,1) and W2,ploc exponent p∈ (1,2).