Existence of entire radial large solutions for a class of Monge–Ampère type equations and systems

Abstract

This paper is mainly concerned with existence of entire positive radial large solutions for a class of Monge–Ampère type equations:

$$det{D}^{2}u\left(x\right)-\alpha \mathrm{\Delta }u=a\left(\left|x\right|\right)f\left(u\right),x\in {ℝ}^N,$$

and systems:

$$\begin{array}{ccccc}\hfill det{D}^{2}u\left(x\right)-\alpha \mathrm{\Delta }u& =a\left(\left|x\right|\right)f\left(v\right),\hfill & \hfill & x\in {ℝ}^N,\hfill & \hfill \\ \hfill det{D}^{2}v\left(x\right)-\beta \mathrm{\Delta }v& =b\left(\left|x\right|\right)g\left(u\right),\hfill & \hfill & x\in {ℝ}^N,\hfill \end{array}$$

where $det{D}^{2}u$ is the so-called Monge–Ampère operator, $\mathrm{\Delta }$ is the classical Laplace operator, $N\ge 2$, $\alpha ,\beta$ are positive constants, $f,g:\left[0,\infty \right)\to \left[0,\infty \right)$ are continuous and nondecreasing, and $a,b:{ℝ}^N\to \left[0,\infty \right)$ are continuous.