The Classical Boundary Blow-Up Solutions for a Class of Gaussian Curvature Equations

Abstract

In this article, we consider the Gaussian curvature problem

$$\begin{aligned} \frac{\hbox {det}(D^{2}u)}{(1+|\nabla u|^{2})^{\frac{N+2}{2}}}=b(x)f(u)g(|\nabla u|)\;\hbox {in}\;\Omega ,\,u=+\infty \;\hbox {on}\;\partial \Omega , \end{aligned}$$

where \(\Omega \) is a bounded smooth uniformly convex domain in \({\mathbb {R}}^{N}\) with \(N\ge 2\), \(b\in \mathrm C^{\infty }(\Omega )\) is positive in \(\Omega \) and may be singular or vanish on \(\partial \Omega \), \(f\in C^{\infty }[0, +\infty )\) (or \(f\in C^{\infty }({\mathbb {R}})\)) is positive and increasing on \([0, +\infty )\) \((\hbox {or } {\mathbb {R}})\), \(g\in C^{\infty }[0, +\infty )\) is positive on \([0, +\infty )\). We first establish the existence and global estimates of \(C^{\infty }\)-strictly convex solutions to this problem by constructing some fine coupling (limit) structures on f and g. Our results (Theorems 2.1–2.3) clarify the influence of properties of b (on the boundary \(\partial \Omega \)) on the existence and global estimates, and reveal the relationship between the solutions of the above problem and some corresponding problems (some details see page 6–7). Then, the nonexistence of convex solutions and strictly convex solutions are also obtained (see Theorems 2.4 and C). Finally, we study the principal expansions of strictly convex solutions near \(\partial \Omega \) by analyzing some coupling structure and using the Karamata regular and rapid variation theories.