Liouville-type theorems for coupled-drift Monge-Amp\`ere equations

We study entire solutions and periodic correctors for the coupled-drift Monge-Amp\`ere equation $$\det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0.$$ For $V\equiv0$, we obtain a sharp classification of the whole-space solvability regimes: all entire smooth strictly convex solutions are quadratic when $a=b=0$; no such solution exists when $a\neq0$ and $a\cdot b\le0$; and non-quadratic entire solutions exist when $a=0$ and $b\neq0$, or when $a\cdot b>0$. The main new ingredient is a scalar maximum-principle argument valid in every dimension $n\ge2$, which proves that the null case $a\neq0$, $a\cdot b=0$ admits no entire smooth strictly convex solution. For periodic $V$, we establish the existence and uniqueness for the drifted cell problem $$\det(A+D^2\psi) = \exp\{-a\cdot D\psi+V-c_A\}, \quad A+D^2\psi>0 \quad\text{on }\mathbb T^n.$$ We also prove that any asymptotically quadratic entire solution must satisfy $b=Aa$. If its remainder is bounded, then the solution is the corresponding quadratic-periodic corrector up to an additive constant.