Mei-qiang FENG
In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations $$ \left \{ \begin{array}{l} \text{det}\ D^2u_1=\lambda h_1(|x|)f_1(-u_2), \qquad \text{in} \ \ \Omega,\\ \text{det}\ D^2u_2=\lambda h_2(|x|)f_2(-u_1), \qquad \text{in} \ \ \Omega,\\ u_1=u_2=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on} \ \ \partial \Omega \end{array} \right. $$ for a certain range of $\lambda >0$, $h_i$ are weight functions, $f_i$ are continuous functions with possible singularity at $0$ and satisfy a combined $N$-superlinear growth at $\infty$, where $i\in \{1,2\}$, $\Omega$ is the unit ball in $\mathbb{R}^N$. We establish the existence of a nontrivial radial convex solution for small $\lambda$, multiplicity results of nontrivial radial convex solutions for certain ranges of $\lambda$, and nonexistence results of nontrivial radial solutions for the case $\lambda\gg 1$. The asymptotic behavior of nontrivial radial convex solutions is also considered.
