Existence of entire radial large solutions for a class of Monge-Ampère type equations and systems. (English) Zbl 1454.35145

Zhang, Zhijun; Liu, Hanxue

Rocky Mt. J. Math. 50, No. 5, 1893-1899 (2020).

Summary: This paper is mainly concerned with existence of entire positive radial large solutions for a class of Monge-Ampère type equations:
\[ \det D^2 u (x)-\alpha \Delta u = a (|x|)f(u), \quad x \in \mathbb{R}^N ,\] and systems:
\[ \begin{aligned} \det D^2 u (x)-\alpha \Delta u & = a (|x|) f(v), \quad & x \in \mathbb{R}^N, \\ \det D^2 v (x)-\beta \Delta v & = b (|x|) g(u), \quad & x \in \mathbb{R}^N, \end{aligned}\] where \(\det D^2 u\) is the so-called Monge-Ampère operator, \(\Delta\) is the classical Laplace operator, \(N \geq 2\), \(\alpha, \beta\) are positive constants, \(f, g : [0, \infty) \to [0, \infty)\) are continuous and nondecreasing, and \(a, b : \mathbb{R}^N \to [0, \infty)\) are continuous.

MSC:

35J60	Nonlinear elliptic equations
35J96	Monge-Ampère equations
35B08	Entire solutions to PDEs

Keywords:

singular Monge-Ampère equation; entire radial large solutions

References:

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