Existence theorems for elliptic Monge-Ampère equations in the plane
Two dimensional Monge-Ampère equations of the form \[ (1)\quad u_{xx}u_{yy}-u^ 2_{xy}-\alpha (u_{xx}+u_{yy})=f(| (x,y)|,u,| \nabla u|) \] (where \(\alpha\geq 0\) is a constant) are studied in the entire plane \({\mathbb{R}}^ 2\). Sufficient conditions are given under which (1) has infinitely many positive radially symmetric entire solutions which are strictly convex in \({\mathbb{R}}^ 2\) and asymptotic to constant multiples of \((| x| +| y|)\) if \(\alpha =0\) or \(x^ 2+y^ 2\) if \(\alpha >0\) as \(| x| +| y| \to +\infty\).
infinitely many positive radially symmetric entire solutions
scientific article; zbMATH DE number 4192237