Strictly Convex Solutions to the Singular Boundary Blow-Up Monge-Ampère Problems: Existence and Asymptotic Behavior

Abstract

Let \(\Omega \) be a smooth, bounded, strictly convex domain in \( \mathbb {R}^N \, (N\ge 2)\). Assume \(K,\ f\) and g are smooth positive functions and K(x) may be singular near \(\partial \Omega \). When K satisfies suitable conditions, we provide sufficient and necessary conditions on f and g for the existence of strictly convex solutions to the singular boundary blow-up Monge-Ampère problem

$$\begin{aligned} M[u]=K(x)[f(u)+g(u)|\nabla u|^q] \text{ for } x \in \Omega ,\; u(x)\rightarrow +\infty \text{ as } \textrm{dist}(x,\partial \Omega )\rightarrow 0, \end{aligned}$$

where \(M[u]=\det \, (u_{x_{i}x_{j}})\) is the Monge-Ampère operator and \(0\le q<N+1\). Two nonexistence results of strictly convex solution are also considered when K has strong singularity. In addition, we analyze the boundary asymptotic behavior of such solution by finding new structure conditions on \(K,\ f\) and g. We present some examples to illustrate the applicability of our main results.