A Liouville theorem for solutions of degenerate Monge-Amp\`ere equations

In this paper, we give a new proof of a celebrated theorem of J"orgens which states that every classical convex solution of $$\det {\mathrm{\nabla }}^{2}u(x)=1in{\mathbb{R}}^2$$ has to be a second order polynomial. Our arguments do not use complex analysis, and can be applied to establish such Liouville type theorems for solutions of a class of degenerate Monge-Amp`ere equations. We prove that every convex generalized (or Alexandrov) solution of $$\det {\mathrm{\nabla }}^{2}u(x_1,x_2)=\mathrm{\mid }{x}_1{\mathrm{\mid }}^{\alpha }in{\mathbb{R}}^2,$$ where $\alpha >-1$, has to be $$u(x_1,x_2)=\frac{a}{(\alpha +2)(\alpha +1)}\mathrm{\mid }{x}_1{\mathrm{\mid }}^{2+\alpha }+\frac{a{b}^2}{2}{x}_1^2+b{x}_1{x}_2+\frac{1}{2a}{x}_2^2+\mathrm{\ell }(x_1,x_2)$$ for some constants $a>0$, $b$ and a linear function $\mathrm{\ell }(x_1,x_2)$. This work is motivated by the Weyl problem with nonnegative Gauss curvature.