Existence of entire solutions to the Monge-Ampère equation (Q2879442)

In the paper the existence of entire convex solutions of the Monge-Ampère equation NEWLINE\[NEWLINE \det D^2 u =f NEWLINE\]NEWLINE in \(\mathbb R ^n \) is proven under the following assumption. The nonnegative function \(f\) satisfies NEWLINE\[NEWLINE \int _E f\, dx \leq b\int _{E\;2} f\, dx NEWLINE\]NEWLINE for some constant \(b\) and any ellipsoid centered at the origin. Given any ellipsoid \(E\) there exists an entire solution \(u\) of the equation such that \(u(0)=0, \;Du(0)=0\), and the ellipsoid of minimal volume containing the sublevel set of \(u\) of level 1 is similar to \(E\). If the assumption on \(f\) above holds for ellipsoids centered at any point then the growth estimate NEWLINE\[NEWLINE C_1 |x|^{1+a}\leq u(x) \leq C_2 |x|^{1+b} NEWLINE\]NEWLINE holds for some positive costants \(a,b, C_1 , C_2\) and \(|x|>1\). The theorem generalizes a result of [\textit{K.-S. Chou} and \textit{X.-J. Wang}, Commun. Pure Appl. Math. 49, No. 5, 529--539 (1996; Zbl 0851.35035)], where \(f\) was bounded and bounded away from zero.NEWLINENEWLINEUsing the Legendre transform one can apply the result to the equation related to Gauss curvature flows NEWLINE\[NEWLINE \det D^2 u =(1+|Du|^2 )^{ (n/2 -1 -a)}. NEWLINE\]NEWLINE It follows that for \(a\geq 2\) there are infinitely many smooth, convex but not rotationally symmetric solutions of this equation. For \(a> n+1\) the solutions are entire.