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

Let \(\Omega \) $\mathrm{\Omega }$ be a smooth, bounded, strictly convex domain in \( \mathbb {R}^N \, (N\ge 2)\) ${\mathbb{R}}^N(N\ge 2)$ . Assume \(K,\ f\) $K,f$ and g are smooth positive functions and K(x) may be singular near \(\partial \Omega \) $\partial \mathrm{\Omega }$ . 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 \(\begin{aligned} M[u]=K(x)[f(u)+g(u)|\nabla u|^q] \text{ for } x \in \Omega ,\; u(x)\rightarrow +\infty \text{ as } \textrm{dist}(x,\partial \Omega )\rightarrow 0, \end{aligned}\) $$\begin{array}{c}\hfill M[u]=K(x)[f(u)+g(u)|\mathrm{\nabla }u{|}^q]\text{for}x\in \mathrm{\Omega },u(x)\to +\infty \text{as dist}(x,\partial \mathrm{\Omega })\to 0,\end{array}$$ where \(M[u]=\det \, (u_{x_{i}x_{j}})\) $M[u]=det(u_{x_i{x}_j})$ is the Monge-Ampère operator and \(0\le q<N+1\) $0\le q<N+1$ . 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 \(K,\ f\) $K,f$ and g. We present some examples to illustrate the applicability of our main results.