\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 144, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/144\hfil Systems of Monge-Amp\'ere equations] {Nontrivial convex solutions for systems of Monge-Amp\'ere equations via global bifurcation} \author[Z. Qi \hfil EJDE-2016/144\hfilneg] {Zexin Qi} \address{Zexin Qi \newline School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, China} \email{qizedong@126.com} \thanks{Submitted December 21, 2015. Published June 14, 2016.} \subjclass[2010]{35J47, 35J96, 35B32} \keywords{Monge-Amp\'ere equations; elliptic system; global bifurcation} \begin{abstract} We obtain existence results for some systems of Monge-Amp\'ere equations, using bifurcation theorems of Krasnosell'ski-Rabinowitz type. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We study the system of coupled Monge-Amp\'ere equations, \begin{equation}\label{1.1} \begin{gathered} \det D^2u=f(u,v), \quad x\in \Omega, \\ \det D^2v=g(u,v), \quad x\in \Omega, \\ u=v=0, \quad x\in\partial \Omega, \end{gathered} \end{equation} where $\Omega$ is a bounded, smooth and strictly convex domain in $\mathbb{R}^n$, and $\det D^2u$ stands for the determinant of Hessian matrix of $u$. We will restrict system \eqref{1.1} to be elliptic and search for nontrivial convex solutions, thus we suppose the functions $f$ and $g$ to be nonnegative. Monge-Amp\'ere equations have received considerable attention in the previous decades, because of their important applications in geometry and other scientific fields. However, systems coupled by Monge-Amp\'ere equations have only been considered in recent years, see for example \cite{m1,w1,z1}. Wang \cite{w1} studied the system \begin{equation}\label{1.2} \begin{gathered} \det D^2u_1={f(-u_2)}, \quad \text{in } B,\\ \det D^2u_2={g(-u_1)}, \quad \text{in } B, \\ u_1=u_2=0, \quad \text{on } \partial B, \end{gathered} \end{equation} with $B:=\{x\in\mathbb{R}^n: |x|<1\}$. Under suitable assumptions on $f$ and $g$, the author found nontrivial radial convex solutions for \eqref{1.2}, using ODE techniques together with fixed point theorems in a cone. More precisely, he obtained \begin{theorem}[{\cite[Theorem 1.1]{w1}}] \label{thmA} Suppose $f,g:[0,\infty)\to [0,\infty)$ are continuous. \begin{itemize} \item[(a)] If $f_0=g_0=0$ and $f_{\infty}=g_{\infty}=\infty$, then \eqref{1.2} has at least one nontrivial radial convex solution. \item[(b)] If $f_0=g_0=\infty$ and $f_{\infty}=g_{\infty}=0$, then \eqref{1.2} has at least one nontrivial radial convex solution. \end{itemize} where \[ f_0:=\lim _{x\to0^{+}}\frac{f(x)}{x^n}, \quad f_{\infty}:=\lim _{x\to\infty}\frac{f(x)}{x^n}. \] \end{theorem} The above theorem implies the existence of a radial convex solution for the system \begin{equation}\label{1.3} \begin{gathered} \det D^2u_1={(-u_2)}^\alpha, \quad \text{in } B, \\ \det D^2u_2={(-u_1)}^\beta, \quad \text{in } B \\ u_1<0,\quad u_2<0,\quad \text{in } B,\\ u_1=u_2=0, \quad \text{on }\partial B \end{gathered} \end{equation} if one of the following two conditions holds: (1) $\alpha > n, \beta > n $, (2) $\alpha < n, \beta < n $. Theorem \ref{thmA} was improved later in \cite{z1} by a decoupling method. For example, for system \eqref{1.3}, the authors in \cite{z1} proved that it has a radial convex solution if and only if $\alpha>0, \beta>0$ and $\alpha\beta\neq n^2$. Moreover, as $\alpha\beta= n^2$, for the eigenvalue problem \begin{equation}\label{1.4} \begin{gathered} \det D^2u_1=\lambda{(-u_2)}^\alpha, \quad \text{in } \Omega, \\ \det D^2u_2=\mu{(-u_1)}^\beta, \quad \text{in } \Omega, \\ u_1<0, u_2<0, \quad \text{in } \Omega,\\ u_1=u_2=0, \quad \text{on } \partial \Omega, \end{gathered} \end{equation} with positive parameters $\lambda$ and $\mu$. They used a nonlinear version of Krein-Rutman theorem developed in \cite{j1} to obtain the following result. \begin{theorem}[{\cite[Theorem 1.4]{z1}}] \label{thmB} Suppose $\Omega\subset\mathbb{R}^n$ is a bounded, smooth and strictly convex domain. If $\alpha>0, \beta>0$ and $\alpha\beta=n^2$, then system \eqref{1.4} admits a convex solution if and only if $\lambda\mu^\frac{\alpha}{n}=C$, where $C$ is a positive constant depending on $n$, $\alpha$ and $\Omega$. \end{theorem} The motivation of this article come from a corollary of Theorem \ref{thmB} (see Lemma \ref{lem2.1} below), as well as the work in \cite{j1}, where Jacobsen investigated global bifurcation problems for a class of fully nonlinear elliptic equations, including the Monge-Amp\'ere equation. As byproducts, Jacobsen obtaied some interesting existence results. We will show that, under suitable assumptions on the functions $f$ and $g$, one can generalize part of the work in \cite{j1} to get new existence results for problem \eqref{1.1}, see Theorem \ref{thm3.8} and \ref{thm4.6} below. This article is organized as follows. In Section 2, we give some preliminaries. In Section 3 and 4, we study two bifurcation problems, where we obtain the main results in this paper. \section{Preliminaries} Unless otherwise stated $\Omega$ is supposed to be a bounded, smooth and strictly convex domain in $\mathbb{R}^n$. Let us recall the Monge-Amp\'ere operator $\mathcal{M}: C^2(\Omega)\to C(\Omega)$, $\mathcal{M}[u]=\det D^2u$. Since it is $n$-homogeneous, the eigenvalue problem for a single Monge-Amp\'ere equation with Dirichlet boundary condition is described as \begin{gather*} \det D^2u =| \lambda u|^n, \quad x \in\Omega, \\ u =0, \quad x \in\partial \Omega. \end{gather*} It is known that there exists a unique positive $\lambda=\lambda_0(\Omega)$ such that the above problem admits nonzero convex solutions. In the literature, $\lambda_0(\Omega)$ is called the first eigenvalue, or the principal eigenvalue, of the Monge-Amp\'ere operator corresponding to $\Omega$, see references \cite{j1,l1,t2}. As for systems, we have the following lemma. \begin{lemma}[{\cite[Corollary 1.5]{z1}}] \label{lem2.1} The eigenvalue problem \begin{equation}\label{eigenvalue problem} \begin{gathered} \det D^2u={|\lambda v|}^n, \quad x\in \Omega, \\ \det D^2v={|\lambda u|}^n, \quad x\in\Omega, \\ u = v = 0, \quad x\in\partial \Omega \end{gathered} \end{equation} admits nonzero convex solutions if and only if $|\lambda|=\lambda_0(\Omega)$. \end{lemma} Thanks to Lemma \ref{lem2.1}, we are able to study global bifurcation problems for some systems of Monge-Amp\'ere equations. Before we do this in the next sections, let us make some preparations first. We begin with some notations and terminologies that will be used later. As in reference \cite{j1}, we will use the following terminologies. Let $Z$ be a real Banach space with a cone $P\subset Z$. The cone $P$ induces a partial order via $u\preceq v \Leftrightarrow v-u \in P$. Let $A_0: Z \to Z$. \begin{itemize} \item $A_0$ is called homogeneous if it is positively homogeneous with degree 1. \item $A_0$ is monotone if it satisfies $x\preceq y\Rightarrow A_0(x)\preceq A_0(y)$. \end{itemize} Now we recall a result due to Trudinger. As a special case of Trudinger \cite[Theorem 1.1]{t1}, in the second paragraph on p. 1253, we have \begin{lemma} \label{lem2.2} Let $\Omega$ be a strictly convex bounded domain in $\mathbb{R}^n$, $\psi \in C(\overline\Omega)$ with $\psi\geq 0$, $\phi \in C(\overline{\Omega})$. Then there exists a unique admissible weak solution $u \in C^{1}(\Omega)\cap C(\overline\Omega)$ of the equation \begin{gather*} \det D^2u = \psi, \quad x \in\Omega, \\ u = \phi, \quad x \in \partial \Omega. \end{gather*} \end{lemma} \begin{remark} \label{rmk2.3} \rm The definition of admissible weak solution coincides with the Aleksandrov sense weak solution (please see \cite[page 1252-1253]{t1}), thus the admissible weak solutions occurred in the rest of the paper are also Aleksandrov solutions. For the notion of Aleksandrov solution, see \cite[Definition 1.1.1, Theorem 1.1.13 and Definition 1.2.1]{g1}. \end{remark} By Lemma \ref{lem2.2}, we can define a solution operator as follows. Denote $C(\overline{\Omega})$ to be the usual Banach space of continuous functions with sup-norm. Define a cone $$ K_0:=\{u\in C(\overline{\Omega}):u(x)\leq 0,\forall x \in \Omega\}, $$ and an operator $$ T:C(\overline{\Omega})\to C(\overline{\Omega}), \quad T(f)=u, $$ where $u$ is the unique admissible weak solution of \begin{equation}\label{definition of solution operator} \begin{gathered} \det D^2u =|f(x)|, \quad x \in\Omega, \\ u = 0, \quad x \in\partial \Omega. \end{gathered} \end{equation} Note the Monge-Amp\'ere operator is $n$-hessian (see \cite{c1}), so the solution operator defined by \eqref{definition of solution operator} coincides with $T_{n}$, where $T_k(k=1,2,\cdots,n)$ are solution operators for $k$-hessian equations defined in Section 3.1 in \cite{j1}. By \cite[Proposition 3.2]{j1}, $T: C(\overline{\Omega}) \to C(\overline{\Omega})$ is completely continuous. Let us define another operator. Take the Banach space $E:=C(\overline{\Omega})\times C(\overline{\Omega})$, with norm $\|(u,v)\|:=\|u\|_{\infty}+\|v\|_{\infty}$. Define a cone $$ K=\{(u,v)\in E:u(x)\leq 0,\;v(x)\leq 0,\;\forall x \in \Omega\}. $$ We remark that we don't distinguish the writing of norms in $E$ and $C(\overline{\Omega})$, and both are denoted by $\|\cdot\|$ in the paper. Define $$ A:E\to E,\; A(u,v)=(w,z), $$ where $(w,z)$ is the unique admissible weak solution pair of \begin{equation}\label{solution operator A} \begin{gathered} \det D^2w={|v|}^n, \quad x\in \Omega, \\ \det D^2z={|u|}^n, \quad x\in\Omega, \\ w = z = 0, \quad x\in\partial \Omega. \end{gathered} \end{equation} By the completely continuity of $T$ it is easy to see $A$ is also completely continuous from $E$ to $E$. Now $E$ is a real Banach space, and the cone $K$ induced a partial order on $E$ via $(u_1,v_1)\preceq (u_2,v_2)\Leftrightarrow(u_2-u_1,v_2-v_1)\in K$. It is readily checked that $A$ is homogeneous; by \cite[Lemma 1.4.6]{g1}, the comparison principle, we see $T$ is monotone, so is $A$, i.e., $(u_1,v_1)\preceq (u_2,v_2)\Rightarrow A(u_1,v_1)\preceq A(u_2,v_2)$. Properties of the operators $T$ and $A$ can be summarized as follows. \begin{lemma} \label{lem2.4} $T: C(\overline{\Omega})\to C(\overline{\Omega})$ is a completely continuous, monotone operator. $A: E\to E$ is a completely continuous, monotone operator; moreover, it is homogeneous. \end{lemma} In this article, we take $b_0$ a fixed number such that $b_0>\lambda_0$. We have the following crucial result essentially given by Jacobsen\cite{j1}, and give a proof for completeness. \begin{lemma}\label{lem2.5} For Leray-Schauder degree, we have \begin{equation}\label{critical degree} \deg(\operatorname{id}-b_0A(\cdot,\cdot),B_{r}(0,0),0)=0, \quad \forall r>0. \end{equation} \end{lemma} \begin{proof} First we note, by Lemma \ref{lem2.1}, that the degree in \eqref{critical degree} is well defined for any $r>0$, and it is independent with the value of $r$. We argue by contradiction. Let $(u_0,v_0)$ be a nonzero solution pair of \eqref{eigenvalue problem} corresponding to $\lambda_0$, then we have \begin{equation}\label{15} \lambda_0A(u_0,v_0)=(u_0,v_0). \end{equation} Fix $\overline{r}>0$. By the continuity of Leray-Schauder degree, we can choose $\epsilon>0$ small, such that $$ \deg(\operatorname{id}-b_0A(\cdot,\cdot),B_{\overline{r}}(0,0),0) =\deg(\operatorname{id}-b_0A((\cdot,\cdot)+\epsilon(u_0,v_0)),B_{\overline{r}}(0,0),0). $$ When \eqref{critical degree} not true, we obtain \[ \deg(\operatorname{id}-b_0A((\cdot,\cdot)+\epsilon(u_0,v_0)), B_{\overline{r}}(0,0),0)\neq0, \] which implies the existence of $(\overline{u},\overline{v})\in B_{\overline{r}}(0,0)$ such that \begin{equation}\label{17} (\overline{u},\overline{v})=b_0A((\overline{u},\overline{v}) +\epsilon(u_0,v_0)). \end{equation} Recall the partial order induced by $K$ in $E$, we have $(\overline{u},\overline{v})\preceq(\overline{u},\overline{v})+\epsilon(u_0,v_0)$. Since $A$ is monotone, we obtain \begin{equation}\label{18} A(\overline{u},\overline{v})\preceq A((\overline{u},\overline{v}) +\epsilon(u_0,v_0)). \end{equation} Equations \eqref{17} and \eqref{18} give us \begin{equation}\label{19} A(\overline{u},\overline{v})\preceq\frac{(\overline{u},\overline{v})}{b_0}. \end{equation} On the other hand, from $\epsilon(u_0,v_0)\preceq(\overline{u},\overline{v})+\epsilon(u_0,v_0)$, we have $$ A(\epsilon(u_0,v_0))\preceq A((\overline{u},\overline{v})+\epsilon(u_0,v_0)); $$ using \eqref{17} again, we reach \begin{equation}\label{21} b_0 A(\epsilon(u_0,v_0))\preceq (\overline{u},\overline{v}). \end{equation} By \eqref{15}, \eqref{21} and the homogeneity of $A$, \begin{equation}\label{first operation} \frac{b_0 \epsilon (u_0,v_0)}{\lambda_0}\preceq (\overline{u},\overline{v}). \end{equation} Now operate $A$ on both sides of \eqref{first operation}, we have \begin{equation}\label{22} \frac{b_0 \epsilon A(u_0,v_0)}{\lambda_0}\preceq A(\overline{u},\overline{v}). \end{equation} Combining \eqref{22} with \eqref{15} and \eqref{19}, we deduce \begin{equation}\label{second operation} \frac{b_0^2 \epsilon (u_0,v_0)}{\lambda_0^2}\preceq (\overline{u},\overline{v}). \end{equation} Noticing \eqref{first operation} and \eqref{second operation}, one can prove by induction that \[ \frac{b_0^n \epsilon (u_0,v_0)}{\lambda_0^n}\preceq(\overline{u},\overline{v}), \quad\forall n\in \mathbb{N}. \] So \[ (u_0,v_0)\preceq \Big(\frac{\lambda_0}{b_0}\Big)^n\cdot\frac{(\overline{u}, \overline{v})}{\epsilon},\quad \forall n\in \mathbb{N}. \] Letting $n\to\infty$, from $b_0>\lambda_0>0$ we obtain $(u_0,v_0)\preceq (0,0)$. Thus $-(u_0,v_0)\in K$, giving $(u_0,v_0)\in K\cap(-K)= \{(0,0)\}$, a contradiction with $(u_0,v_0)\neq (0,0)$. This finishes the proof of the lemma. \end{proof} \section{Global bifurcation} Our basic assumption on $f$ and $g$ is \begin{itemize} \item[(A1)] $f,g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}_{+}:=[0,+\infty)$ are continuous. \end{itemize} We seek nontrivial solutions to \eqref{1.1}. The approach used is motivated by \cite{j1}. More precisely, we embed \eqref{1.1} into the one-parameter family of problems \begin{equation}\label{bifurcation problem} \begin{gathered} \det D^2u={|\lambda v|}^n+f(u,v), \quad x\in \Omega, \\ \det D^2v={|\lambda u|}^n+g(u,v), \quad x\in\Omega, \\ u = v = 0, \quad x\in\partial \Omega, \end{gathered} \end{equation} and consider the behavior of global bifurcation or global asymptotic bifurcation continuum. By continuum we shall mean a closed connected set. We associated to \eqref{bifurcation problem} the solution operator $H:\mathbb{R}\times E\to E, H(\lambda,(u,v))=(w,z)$, where $(w,z)$ is the unique solution pair of \begin{equation}\label{solution operator H} \begin{gathered} \det D^2w={|\lambda v|}^n+f(u,v), \quad x\in \Omega, \\ \det D^2z={|\lambda u|}^n+g(u,v), \quad x\in\Omega, \\ w = z = 0, \quad x\in\partial \Omega. \end{gathered} \end{equation} Using the operator $T$ defined in Section 2, we can write $H=(H_1,H_2)$, where $H_1(\lambda,(u,v))=T({|\lambda v|}^n+f(u,v))$, $H_2(\lambda,(u,v))=T({|\lambda u|}^n+g(u,v))$. By assumption (A1) and the completely continuity of $T$ (see Lemma \ref{lem2.4}), it is easy to check $H_1$ and $H_2$ are both completely continuous. So $H$ is completely continuous. Define $h:\mathbb{R}\times E\to E, h(\lambda ,(u,v))=(u,v)-H(\lambda ,(u,v))$ and consider the equation \begin{equation}\label{bifurcation equation} h(\lambda ,(u,v))=0. \end{equation} We see that $\lambda$, $u$ and $v$ satisfy \eqref{bifurcation problem} if and only if $(\lambda,(u,v))$ is a solution of \eqref{bifurcation equation}. Note that \eqref{bifurcation problem} can be seen as a perturbation of the eigenvalue problem \eqref{eigenvalue problem}. For our purpose in this section, the perturbation terms also need to satisfy: \begin{itemize} \item[(A2)] $f(s,t)=o({(|s|+|t|)}^n), g(s,t)=o({(|s|+|t|)}^n)$, as $|s|+|t|\to0$; \item[(A3)] either $\frac{f(s,t)}{{(|s|+|t|)}^n}\to\infty$ or $\frac{g(s,t)}{{(|s|+|t|)}^n}\to\infty$, as $|s|+|t|\to\infty$. \end{itemize} Under assumptions of (A1) and (A2), one has $f(0,0)=g(0,0)=0$, thus \eqref{bifurcation equation} admits trivial solution branch $\mathbb{R}\times (0,0)$. In order to obtain a nontrivial branch of solutions to \eqref{bifurcation equation}, we need the following bifurcation theorem of Krasnosell'ski-Rabinowitz type. \begin{theorem}[global bifurcation, \cite{s1}] \label{thm3.1} Let $Y$ be a Banach space, let $F:\mathbb{R}\times Y\to Y$ be completely continuous, such that $F(\lambda,0)=0$, for all $\lambda\in\mathbb{R}$. Suppose there exist constants $a,b\in\mathbb{R}$, with $a0$, sufficiently small, such that \begin{itemize} \item[(1)] $\deg(\operatorname{id}-H(0,(\cdot,\cdot)),B_{r}(0,0),0)=1$, \item[(2)] $\deg(\operatorname{id}-H(b_0,(\cdot,\cdot)),B_{r}(0,0),0)=0$. \end{itemize} \end{lemma} \begin{proof} First of all, by Lemma \ref{lem3.2}, $(0,(0,0))$ and $(b_0,(0,0))$ are not bifurcation points for \eqref{bifurcation equation}, so one can take $r>0$ sufficiently small, such that the degrees in assertions (1) and (2) are well defined. Let $\tilde{b}\in \{0,b_0\}$. Define a homotopic mapping $F_{\tilde{b}}:[0,1]\times E\to E, F_{\tilde{b}}(t,(u,v))=(w,z)$, where $(w,z)$ is the unique solution pair of \begin{gather*} \det D^2w={|\tilde{b}v|}^n+tf(u,v), \quad x\in\Omega,\\ \det D^2z={|\tilde{b}u|}^n+tg(u,v), \quad x\in\Omega, \\ w = z = 0, \quad x\in\partial \Omega. \end{gather*} By the complete continuity of $T$, $F_{\tilde{b}}:[0,1]\times E\to E$ is completely continuous. We claim when $r>0$ is sufficiently small, $\deg(\operatorname{id}-F_{\tilde{b}}(t,(\cdot,\cdot)),B_{r}(0,0),0)$ is well defined for all $t\in[0,1]$. If this were not true, then there exist $\{t_m\}\subset[0,1]$ with $t_m\to t_0\in[0,1]$, and $\{(u_m,v_m)\}\subset E$ with $\|(u_m,v_m)\|=r_m>0$, $r_m\to0$, such that $(u_m,v_m)=F_{\tilde{b}}(t_m,(u_m,v_m))$, i.e., \begin{gather*} \det D^2u_m={|\tilde{b}v_m|}^n+t_mf(u_m,v_m), \quad x\in\Omega, \\ \det D^2v_m={|\tilde{b}u_m|}^n+t_mg(u_m,v_m), \quad x\in\Omega,\\ u_m = v_m = 0, \quad x\in\partial \Omega. \end{gather*} By mimicking the rest proof after \eqref{8} in Lemma \ref{lem3.2}, one reaches again $|\tilde{b}|=\lambda_0$, a contradiction with $\tilde{b}\in \{0,b_0\}$. So for $r>0$ sufficiently small we have \[ (u,v)\neq F_{\tilde{b}}(t,(u,v)),\quad \forall(u,v)\in \partial B_{r}(0,0),\;\forall t\in [0,1]. \] This implies $F_{\tilde{b}}$ is a degree-preserving homotopic mapping. We distinguish the following two cases. \smallskip \noindent\textbf{Case $\tilde{b}=0$.} For $r>0$ small, we have \begin{align*} \deg(\operatorname{id}-F_0(1,(\cdot,\cdot)),B_{r}(0,0),0) &=\deg(\operatorname{id}-F_0(0,(\cdot,\cdot)),B_{r}(0,0),0)\\ &=\deg(\operatorname{id},B_{r}(0,0),0)=1. \end{align*} Since $F_0(1,(\cdot,\cdot))=H(0,(\cdot,\cdot))$, assertion (1) is valid. \smallskip \noindent\textbf{Case $\tilde{b}=b_0$.} For $r>0$ small, we have \begin{equation} \begin{aligned} \deg(\operatorname{id}-H(b_0,(\cdot,\cdot)),B_{r}(0,0),0) &=\deg(\operatorname{id}-F_{b_0}(1,(\cdot,\cdot)),B_{r}(0,0),0)\\ &=\deg(\operatorname{id}-F_{b_0}(0,(\cdot,\cdot)),B_{r}(0,0),0)\\ &=\deg(\operatorname{id}-A(b_0(\cdot,\cdot)),B_{r}(0,0),0)\\ &=\deg(\operatorname{id}-b_0A(\cdot,\cdot),B_{r}(0,0),0). \end{aligned} \label{13} \end{equation} By Lemma \ref{lem2.5} and \eqref{13}, we see assertion (2) is also valid. \end{proof} Now let us recall a known blow-up result. Since the Monge-Amp\'ere operator is $n$-hessian, we have a special case of Jacobsen \cite[Lemma 5.1]{j1}. \begin{lemma} \label{lem3.4} Let $\{v_m\}\subset C(\overline{\Omega})$ be a collection of admissible weak solutions to the Dirichlet problem \begin{gather*} \det D^2v_m =g_m, \quad x \in\Omega, \\ v_m = 0,\quad x \in\partial \Omega, \end{gather*} where $g_m:\Omega\to\mathbb{R}$ form a collection of nonnegative continuous functions. If $g_m(x)\to\infty$, uniformly on some compact sub-domain of $\Omega$, then $\|v_m\|\to\infty$. \end{lemma} Using this lemma, we can establish some priori bounds concerning solutions of \eqref{bifurcation equation}. \begin{lemma} \label{lem3.5} Under assumption {\rm (A1)}, there exists $M_1>0$, such that any solution $(\lambda,(u,v))$ of \eqref{bifurcation equation} with $(u,v)\neq (0,0)$ must satisfy $|\lambda|\leq M_1$. \end{lemma} \begin{proof} We argue by contradiction. If the conclusion of Lemma \ref{lem3.5} is false, then there exists $\{(\lambda_k,(u_k,v_k))\}$, solving \eqref{bifurcation equation} for each $k$, such that $\|u_k\|+\|v_k\|>0$, and $|\lambda_k|\to\infty$ as $k\to\infty$. Let \[ \widetilde{u}_k:=\frac{u_k}{\|u_k\|+\|v_k\|}, \quad \widetilde{v}_k:= \frac{v_k}{\|u_k\|+\|v_k\|}, \] and let $\Omega'$ be a compact sub-domain of $\Omega$. Since $\|\widetilde{u}_k\|+\|\widetilde{v}_k\|=1$, we may assume, without loss of generality, that there exists a $\gamma>0$ such that \begin{equation}\label{positive lower bound of vk} \|\widetilde{v}_k\|\geq\gamma. \end{equation} Moreover, by \cite[Lemma 5.10]{j1}, there exists $\eta>0$ such that \begin{equation}\label{another lower bound} |\widetilde{v}_k(x)|\geq\eta\|\widetilde{v}_k\|, \quad \forall x \in\Omega'. \end{equation} As $(\lambda_k,(u_k,v_k))$ solves \eqref{bifurcation equation}, we have \begin{gather*} \det D^2u_k={|\lambda_k v_k|}^n+f(u_k,v_k),\quad x\in\Omega,\\ u_k= 0,\quad x\in\partial \Omega. \end{gather*} Dividing the above equation by $(\|u_k\|+\|v_k\|)^n$, we obtain \begin{equation}\label{divided equation} \begin{gathered} \det D^2\widetilde{u}_k=\psi_k, \quad x\in \Omega, \\ \widetilde{u}_k = 0, \quad x\in\partial \Omega, \end{gathered} \end{equation} where \begin{equation}\label{definition of psi} \psi_k:={|\lambda_k \widetilde{v}_k|}^n +\frac{f(u_k,v_k)}{(\|u_k\|+\|v_k\|)^n}. \end{equation} Equations \eqref{definition of psi}, \eqref{positive lower bound of vk} and \eqref{another lower bound} yield that, for $x\in\Omega'$, \begin{equation}\label{blow up of psi} \psi_k(x)\geq{|\lambda_k \widetilde{v}_k|}^n \geq{|\lambda_k\eta\gamma|}^n\to\infty, k\to\infty. \end{equation} By Lemma \ref{lem3.4}, we deduce from \eqref{divided equation} and \eqref{blow up of psi} that $\|\widetilde{u}_k\|\to\infty$ as $k\to\infty$, a contradiction with $\|\widetilde{u}_k\|\leq 1$. \end{proof} \begin{lemma}\label{lem3.6} Under assumptions {\rm (A1)} and {\rm (A3)}, there exists $M_2>0$, such that a solution $(\lambda,(u,v))$ of \eqref{bifurcation equation} must satisfy $\|u\|+\|v\|\leq M_2$. \end{lemma} \begin{proof} Without loss of generality, we assume the first alternative of (A3) holds, i.e., \begin{equation}\label{first alternative of A3} \frac{f(s,t)}{{(|s|+|t|)}^n}\to\infty, \quad\text{as } |s|+|t|\to\infty. \end{equation} We argue by contradiction. If the conclusion of Lemma \ref{lem3.6} is false, then there exists $\{(\lambda_k,(u_k,v_k))\}$, solving \eqref{bifurcation equation} for each $k$, such that $\|u_k\|+\|v_k\|>0$, $\|u_k\|+\|v_k\|\to\infty$ as $k\to\infty$. Now we have \begin{gather*} \det D^2u_k={|\lambda_k v_k|}^n+f(u_k,v_k),\quad x\in\Omega,\\ u_k= 0,\quad x\in\partial \Omega. \end{gather*} Divide the above equation by $(\|u_k\|+\|v_k\|)^n$, and denote \[ \varphi_k(x):=\frac{{|\lambda_k v_k(x)|}^n+f(u_k(x),v_k(x))}{(\|u_k\|+\|v_k\|)^n}, \] we reach \begin{equation}\label{5} \begin{gathered} \det D^2\Big(\frac{u_k}{\|u_k\|+\|v_k\|}\Big)=\varphi_k, \quad x \in \Omega, \\ \frac{u_k}{\|u_k\|+\|v_k\|} = 0, \quad x \in\partial \Omega. \end{gathered} \end{equation} Note that \begin{equation}\label{6} \varphi_k(x)\geq \frac{f(u_k,v_k)}{(|u_k|+|v_k|)^n} \Big(\frac{|u_k|+|v_k|}{\|u_k\|+\|v_k\|}\Big)^n, \forall x \in \Omega. \end{equation} Let $\Omega'$ be a compact sub-domain of $\Omega$. By \cite[Lemma 5.10]{j1}, there exists $\delta>0$, such that $|u_k(x)|\geq\delta\|u_k\|$ and $|v_k(x)|\geq\delta\|v_k\|$ for any $x \in\Omega'$. Thus $|u_k(x)|+|v_k(x)|\geq\delta(\|u_k\|+\|v_k\|)$ for any $x \in\Omega'$. Since $\|u_k\|+\|v_k\|\to\infty$, we see it holds uniformly for $x \in\Omega'$ that $|u_k(x)|+|v_k(x)|\to\infty$. Using these facts and \eqref{first alternative of A3}, we deduce from \eqref{6} that for $x \in\Omega'$, it holds uniformly \begin{equation}\label{7} \varphi_k(x)\to\infty , \quad\text{as } k\to\infty. \end{equation} By \eqref{5} and \eqref{7}, we infer from Lemma \ref{lem3.4} that \[ \frac{\|u_k\|}{\|u_k\|+\|v_k\|}\to\infty ,\quad \text{as } k\to\infty, \] which contradicts \[ \frac{\|u_k\|}{\|u_k\|+\|v_k\|}\leq1, \quad \forall k\in\mathbb{N}. \] \end{proof} We are in a position to give the main results in this section. Recall that, by continuum we mean a closed connected set. \begin{theorem} \label{thm3.7} Under assumptions {\rm (A1)--(A3)}, there exists a bounded continuum of solutions to \eqref{bifurcation equation} bifurcating from $(\lambda_0,(0,0))$ in $\mathbb{R}\times E$. This continuum connects $(\lambda_0,(0,0))$ to $(-\lambda_0,(0,0))$. It is nontrivial in the sense that it intersects the trivial solution branch of \eqref{bifurcation equation} only at $(\pm\lambda_0,(0,0))$. \end{theorem} \begin{proof} Let us apply Theorem \ref{thm3.1} to the Banach space $E$ and the operator $H$. By Lemmas \ref{lem3.2} and \ref{lem3.3}, we infer from Theorem \ref{thm3.1} that there exists a nontrivial branch of solutions to \eqref{bifurcation equation}, say $\overline{\mathcal{C}}$, bifurcating from $(\lambda_0,(0,0))$, and it holds $([0,b_0]\times(0,0))\cap\overline{\mathcal{C}}=(\lambda_0,(0,0))$. Furthermore, either \begin{enumerate} \item $\overline{\mathcal{C}}$ is unbounded in $\mathbb{R}\times E$, or \item $\overline{\mathcal{C}}\cap [(\mathbb{R}\backslash[0,b_0])\times(0,0)]\neq\emptyset$. \end{enumerate} By Lemmas \ref{lem3.5} and \ref{lem3.6}, $\overline{\mathcal{C}}$ must be bounded in $\mathbb{R}\times E$, so $\overline{\mathcal{C}}$ must connect to another bifurcation point. By Lemma \ref{lem3.2}, $\overline{\mathcal{C}}$ connects $(\lambda_0,(0,0))$ to $(-\lambda_0,(0,0))$, and it cannot intersect the trivial solution branch of \eqref{bifurcation equation} at points other than $(\pm\lambda_0,(0,0))$. \end{proof} \begin{theorem} \label{thm3.8} Assume the functions $f$ and $g$ satisfy {\rm (A1)--(A3)}, then \eqref{bifurcation problem} admits at least a nontrivial convex solution for all $\lambda\in (-\lambda_0,\lambda_0)$. In particular, \eqref{1.1} admits at least a nontrivial convex solution. \end{theorem} \begin{proof} By Theorem \ref{thm3.7}, there exists a bounded continuum of solutions to \eqref{bifurcation equation} that is nontrivial, and it connects $(-\lambda_0,(0,0))$ to $(\lambda_0,(0,0))$ in $\mathbb{R}\times E$. Since it is connected, for arbitrarily fixed $\widehat{\lambda}\in (-\lambda_0,\lambda_0)$, the continuum must cross $\lambda=\widehat{\lambda}$ at a point, say $(\widehat{\lambda},(u,v))$. By Lemma \ref{lem3.2}, $\widehat{\lambda}$ is not a bifurcation value, thus $(u,v)\neq(0,0)$, and it is a nontrivial convex solution for \eqref{bifurcation problem} with $\lambda=\widehat{\lambda}$. \end{proof} \section{Global asymptotic bifurcation} Besides (A1), we also need the following assumptions on $f$ and $g$: \begin{itemize} \item[(A4)] either $\frac{f(s,t)}{{(|s|+|t|)}^n}\to\infty$, or $\frac{g(s,t)}{{(|s|+|t|)}^n}\to\infty$, as $|s|+|t|\to0$; \item[(A5)] $\frac{f(s,t)}{{(|s|+|t|)}^n}\to0$, and $\frac{g(s,t)}{{(|s|+|t|)}^n}\to0$, as $|s|+|t|\to\infty$. \end{itemize} In this section, we study global asymptotic bifurcation problems for \eqref{bifurcation equation}. Our analysis is based on the theorem below. \begin{theorem}[Global asymptotic bifurcation, \cite{s1}] \label{thm4.1} Let $Y$ be a Banach space, let $F:\mathbb{R}\times Y\to Y$ be completely continuous. Suppose there exist constants $a,b\in \mathbb{R}$, with $a0$ such that \[ F(a,y)\neq y \neq F(b,y), \] for all $y\in Y$ with $\|y\|\geq M$. Furthermore, assume that \[ \deg(\operatorname{id}-F(a,\cdot),B_{R}(0),0) \neq \deg(\operatorname{id}-F(b,\cdot),B_{R}(0),0), \] for $R>M$. Then there exists at least one continuum $\mathcal{C}$ of solutions to \emph{\eqref{23}} that is unbounded in $[a,b]\times Y$ and either \begin{enumerate} \item $\mathcal{C}$ in unbounded in the $\lambda$ direction, or else, \item there exists an interval $[c,d]$ such that $(a,b)\cap (c,d)=\emptyset$, and $\mathcal{C}$ bifurcates from infinity in $[c,d]\times Y$. \end{enumerate} \end{theorem} In Theorem \ref{thm4.1}, to say $\mathcal{C}$ bifurcates from infinity in $[c,d]\times Y$, we mean there exist $\nu\in[c,d]$ and a sequence $\{(\lambda_k,y_k)\}\subseteq\mathcal{C}$, such that $\lambda_k\to\nu$ and $\|y_k\|_{Y}\to\infty$ as $ k\to\infty$. We shall apply Theorem \ref{thm4.1} to the Banach space $E$ and the operator $H$ after we collect some lemmas. \begin{lemma} \label{lem4.2} Under assumptions {\rm (A1)} and {\rm (A5)}, a necessary condition for $\mu$ to be an asymptotic bifurcation value of \eqref{bifurcation equation} is $|\mu|=\lambda_0$. \end{lemma} \begin{proof} Suppose $\mu$ is an asymptotic bifurcation value for \eqref{bifurcation equation}, i.e., there exists a sequence $\{(\lambda_k,(u_k,v_k))\}$ such that $\|u_k\|+\|v_k\|\to\infty, \lambda_k\to\mu$ as $ k\to\infty$, and it satisfies $h(\lambda_k,(u_k,v_k))=0, \forall k\in\mathbb{N}$. Thus we have \begin{equation}\label{24} \begin{gathered} \det D^2u_k={|\lambda_k v_k|}^n+f(u_k,v_k), \quad x\in \Omega, \\ \det D^2v_k={|\lambda_k u_k|}^n+g(u_k,v_k), \quad x\in\Omega, \\ u_k = v_k = 0, \quad x\in\partial \Omega. \end{gathered} \end{equation} Divide \eqref{24} by $(\|u_k\|+\|v_k\|)^n$, and denote \begin{gather*} \overline{u}_k=\frac{u_k}{\|u_k\|+\|v_k\|}, \quad \overline{v}_k= \frac{v_k}{\|u_k\|+\|v_k\|}, \\ \overline{f}_k=\frac{f(u_k,v_k)}{(\|u_k\|+\|v_k\|)^n}, \quad \overline{g}_k=\frac{g(u_k,v_k)}{(\|u_k\|+\|v_k\|)^n}, \end{gather*} we obtain \begin{equation}\label{25} \begin{gathered} \det D^2\overline{u}_k=|\lambda_k\overline{v}_k|^n+\overline{f}_k, \quad x\in \Omega, \\ \det D^2\overline{v}_k=|\lambda_k\overline{u}_k|^n+\overline{g}_k, \quad x\in \Omega, \\ \overline{u}_k = \overline{v}_k = 0, \quad x\in\partial \Omega. \end{gathered} \end{equation} This system can be rewritten as \begin{equation}\label{26} \begin{gathered} \overline{u}_k=T(|\lambda_k\overline{v}_k|^n+\overline{f}_k), \\ \overline{v}_k=T(|\lambda_k\overline{u}_k|^n+\overline{g}_k). \end{gathered} \end{equation} We claim that $\overline{f}_k(x)\to0$, uniformly for $x\in\Omega$, as $k\to\infty$. Indeed, by (A5), for each $\epsilon>0$, there exists $M_0>0$, such that if $|(s,t)|:=|s|+|t|>M_0$, then \begin{equation}\label{1002} \frac{f(s,t)}{(|s|+|t|)^n}<\epsilon. \end{equation} For this $M_0>0$, denote $f_0:=\max_{|(s,t)|\leq M_0}f(s,t)$, then for large $k$, \begin{equation}\label{1003} \frac{f_0}{(\|u_k\|+\|v_k\|)^n}<\epsilon. \end{equation} By \eqref{1002}, \eqref{1003} and (A1), we deduce that for $k$ sufficiently large, $$ 0\leq\overline{f}_k(x)<\epsilon, \quad \forall x\in\Omega. $$ So our claim holds. Similarly, $\overline{g}_k(x)\to0$, uniformly for $x\in\Omega$, as $k\to\infty$. By mimicking the counterpart in the proof of Lemma \ref{lem3.2}, one is ready to reach $|\mu|=\lambda_0$. We omit the details. \end{proof} Recall $b_0$ is a fixed number such that $b_0>\lambda_0$. \begin{lemma} \label{lem4.3} Assume {\rm (A1)} and {\rm (A5)}. Then there exists $M>0$, such that for $R>M$, \begin{itemize} \item[(1)] $\deg(\operatorname{id}-H(0,(\cdot,\cdot)),B_{R}(0,0),0)=1$; \item[(2)] $\deg(\operatorname{id}-H(b_0,(\cdot,\cdot)),B_{R}(0,0),0)=0$. \end{itemize} \end{lemma} \begin{proof} By Lemma \ref{lem4.2}, there exists $M>0$ such that for all $(u,v)\in E$ with $\|(u,v)\|\geq M$, $$ H(0,(u,v))\neq (u,v) \neq H(b_0,(u,v)). $$ So when $R>M$, the degrees in the assertions are well defined and independent of $R$. Let $\tilde{b}\in \{0,b_0\}$. Define a homotopic mapping $F_{\tilde{b}}:[0,1]\times E\to E$, be defined by $F_{\tilde{b}}(t,(u,v))=(w,z)$, where $(w,z)$ is the unique solution pair of \begin{gather*} \det D^2w={|\tilde{b}v|}^n+tf(u,v), \quad x\in\Omega, \\ \det D^2z={|\tilde{b}u|}^n+tg(u,v), \quad x\in\Omega, \\ w = z = 0, \quad x\in\partial \Omega. \end{gather*} By the complete continuity of $T$, one verifies $F_{\tilde{b}}:[0,1]\times E\to E$ is completely continuous. We point out that when $R>M$ is sufficient large, the function $\deg(\operatorname{id}-F_{\tilde{b}}(t,(\cdot,\cdot)),B_{R}(0,0),0)$ is well defined for all $t\in[0,1]$. If this were not true, then there exist $\{t_m\}\subset[0,1]$ with $t_m\to t_0\in[0,1]$, and $\{(u_m,v_m)\}\subset E$ with $\|(u_m,v_m)\|=R_m\to+\infty$, such that $(u_m,v_m)=F_{\tilde{b}}(t_m,(u_m,v_m))$, i.e., \begin{gather*} \det D^2u_m={|\tilde{b}v_m|}^n+t_mf(u_m,v_m), \quad x\in\Omega, \\ \det D^2v_m={|\tilde{b}u_m|}^n+t_mg(u_m,v_m), \quad x\in\Omega, \\ u_m = v_m = 0, \quad x\in\partial \Omega. \end{gather*} Divide the above system by $(\|u_m\|+\|v_m\|)^n$, and then follow the arguments used in the proof of Lemma \ref{lem4.2}, one reaches again $|\tilde{b}|=\lambda_0$, a contradiction with $\tilde{b}\in \{0,b_0\}$. So when $R>M$ is sufficiently large, \[ (u,v)\neq F_{\tilde{b}}(t,(u,v)),\quad \forall(u,v)\in \partial B_{R}(0,0),\quad \forall t\in [0,1]. \] This implies $F_{\tilde{b}}$ is a degree-preserving homotopic mapping. We distinguish the following two cases. \smallskip \noindent\textbf{Case $\tilde{b}=0$.} For $R>M$ large, we have \begin{align*} \deg(\operatorname{id}-F_0(1,(\cdot,\cdot)),B_{R}(0,0),0) &=\deg(\operatorname{id}-F_0(0,(\cdot,\cdot)),B_{R}(0,0),0)\\ &=\deg(\operatorname{id},B_{R}(0,0),0) =1. \end{align*} Since $F_0(1,(\cdot,\cdot))=H(0,(\cdot,\cdot))$, assertion (1) is valid. \smallskip \noindent\textbf{Case $\tilde{b}=b_0$.} For $R>M$ large, we have \begin{align*} \deg(\operatorname{id}-H(b_0,(\cdot,\cdot)),B_{R}(0,0),0) &=\deg(\operatorname{id}-F_{b_0}(1,(\cdot,\cdot)),B_{R}(0,0),0)\\ &=\deg(\operatorname{id}-F_{b_0}(0,(\cdot,\cdot)),B_{R}(0,0),0)\\ &=\deg(\operatorname{id}-A(b_0(\cdot,\cdot)),B_{R}(0,0),0)\\ &=\deg(\operatorname{id}-b_0A(\cdot,\cdot),B_{R}(0,0),0). \end{align*} By this equality and Lemma \ref{lem2.5}, we see assertion (2) is also valid. \end{proof} \begin{lemma} \label{lem4.4} Assume {\rm (A1)} and {\rm (A4)}. Then there exists $\varepsilon>0$, such that any solution $(\lambda,(u,v))$ of \eqref{bifurcation equation} with $(u,v)\neq(0,0)$ must satisfy $\|(u,v)\|\geq\varepsilon$. \end{lemma} \begin{proof} Without loss of generality, we assume the first alternative of (A4) holds, i.e., \begin{equation}\label{first alternative of A4} \frac{f(s,t)}{{(|s|+|t|)}^n}\to\infty,\quad\text{as } |s|+|t|\to 0. \end{equation} We argue by contradiction. If the conclusion of Lemma \ref{lem4.4} is false, then there exists $\{(\lambda_k,(u_k,v_k))\}$, solving \eqref{bifurcation equation} for each $k$, such that $(u_k,v_k)\neq(0,0)$, $(u_k,v_k)\to(0,0)$ as $k \to \infty$. Now we have \begin{gather*} \det D^2u_k={|\lambda_k v_k|}^n+f(u_k,v_k),\quad x\in\Omega,\\ u_k= 0,\quad x\in\partial \Omega. \end{gather*} Divide the above equation by $(\|u_k\|+\|v_k\|)^n$, and denote \begin{gather*} \zeta_k(x):=\frac{{|\lambda_k v_k(x)|}^n+f(u_k(x),v_k(x))}{(\|u_k\|+\|v_k\|)^n}, \end{gather*} we reach \begin{equation}\label{31} \begin{gathered} \det D^2\Big(\frac{u_k}{\|u_k\|+\|v_k\|}\Big)=\zeta_k,\quad x \in \Omega, \\ \frac{u_k}{\|u_k\|+\|v_k\|} = 0,\quad x \in\partial \Omega. \end{gathered} \end{equation} Note that \begin{equation}\label{32} \zeta_k(x)\geq \frac{f(u_k,v_k)}{(|u_k|+|v_k|)^n} \Big(\frac{|u_k|+|v_k|}{\|u_k\|+\|v_k\|}\Big)^n, \forall x \in \Omega. \end{equation} Let $\Omega'$ be a compact sub-domain of $\Omega$. By \cite[Lemma 5.10]{j1}, there exists $\widetilde{\delta}>0$, such that $|u_k(x)|\geq\widetilde{\delta}\|u_k\|$ and $|v_k(x)|\geq\widetilde{\delta}\|v_k\|$, for any $x \in\Omega'$. Thus \begin{equation}\label{1005} |u_k(x)|+|v_k(x)|\geq\widetilde{\delta}(\|u_k\|+\|v_k\|),\quad \forall x \in\Omega'. \end{equation} Note $(u_k,v_k)\to(0,0)$ in $E$, so for $x \in\Omega'$, it holds uniformly \begin{equation}\label{1006} |u_k(x)|+|v_k(x)|\to 0,\quad k\to\infty. \end{equation} By \eqref{first alternative of A4}, \eqref{1005} and \eqref{1006}, it is easy to deduce from \eqref{32} that for $x \in\Omega'$, it holds uniformly \begin{equation}\label{33} \zeta_k(x)\to\infty ,\quad k\to\infty. \end{equation} By \eqref{31} and \eqref{33}, we infer from Lemma \ref{lem3.4} that \[ \frac{\|u_k\|}{\|u_k\|+\|v_k\|}\to\infty ,\quad k\to\infty, \] which contradicts $\frac{\|u_k\|}{\|u_k\|+\|v_k\|}\leq1$ for all $k\in \mathbb{N}$. \end{proof} Now we can give the main results of this section. \begin{theorem} \label{thm4.5} Assume {\rm (A1), (A4)} and {\rm (A5)}. Then there exists an unbounded continuum of nontrivial solutions to \eqref{bifurcation equation} in $\mathbb{R}\times E$. The continuum bifurcates from infinity at $\mu=\pm\lambda_0$, and it is bounded in the $\lambda$ direction. \end{theorem} \begin{proof} Let us apply Theorem \ref{thm4.1} to the Banach space $E$ and the operator $H$. By Lemma \ref{lem4.2}, there exists $M>0$ such that for $(u,v)\in E$ with $\|(u,v)\|\geq M$, it holds $H(0,(u,v))\neq (u,v) \neq H(b_0,(u,v))$. By Lemma \ref{lem4.3}, we can choose $M>0$ large, so that \[ \deg(\operatorname{id}-H(0,\cdot),B_{R}(0),0)\neq \deg(\operatorname{id}-H(b_0,\cdot),B_{R}(0),0) \] for $R>M$. We infer from Theorem \ref{thm4.1} that there exists a continuum of solutions to \eqref{bifurcation equation}, say $\widetilde{\mathcal{C}}$, that is unbounded in $[0,b_0]\times E$, which forces $\lambda_0$ to be an asymptotic bifurcation value by Lemma \ref{lem4.2}. Furthermore, either \begin{enumerate} \item $\widetilde{\mathcal{C}}$ is unbounded in the $\lambda$ direction, or \item there exist an interval $[c,d]$ such that $(0,b_0)\cap (c,d)=\emptyset$, and $\widetilde{\mathcal{C}}$ bifurcates from infinity in $[c,d]\times E$. \end{enumerate} By Lemma \ref{lem3.5}, $\widetilde{\mathcal{C}}$ is bounded in the $\lambda$ direction, so it must bifurcate from infinity at $-\lambda_0$ by Lemma \ref{lem4.2}. Since $\widetilde{\mathcal{C}}$ is connected and unbounded, we infer from Lemma \ref{lem4.4} that \begin{equation}\label{last} (u,v)\neq(0,0), \quad \forall (\lambda,(u,v))\in\widetilde{\mathcal{C}}. \end{equation} \end{proof} \begin{theorem} \label{thm4.6} Assume {\rm (A1), (A4)} and {\rm (A5)}. Then \eqref{bifurcation problem} has a nontrivial convex solution for all $\lambda\in (-\lambda_0,\lambda_0)$. In particular, \eqref{1.1} admits a nontrivial convex solution. \end{theorem} \begin{proof} Note that the continuum $\widetilde{\mathcal{C}}$ obtained in the proof of Theorem \ref{thm4.5} bifurcates from infinity at $\mu=\pm\lambda_0$. So by connectedness and \eqref{last}, we see \eqref{bifurcation problem} has a nontrivial convex solution for all $\lambda\in (-\lambda_0,\lambda_0)$. \end{proof} \begin{remark}\label{rmk4.8} \rm We say that a solution $(u,v)$ of \eqref{bifurcation problem} is a vector solution if $u\neq0$ and $v\neq0$. When $0<|\lambda|<\lambda_0$, the solutions for \eqref{bifurcation problem} obtained in Theorem \ref{thm3.8} and \ref{thm4.6} are vector solutions, which can be inferred from system \eqref{bifurcation problem} itself and the assumption (A1). Similarly, if $f$ and $g$ are such that $$ f(s,t)>0, \quad g(s,t)>0, \quad \forall (s,t)\neq(0,0), $$ then solutions for \eqref{1.1} obtained in Theorem \ref{thm3.8} and Theorem \ref{thm4.6} are also vector solutions. \end{remark} To illustrate our results for problem \eqref{1.1}, We present the following example: Let $\Omega$ be a bounded, smooth, and strictly convex domain in $\mathbb{R}^n$. If $0n$, then the system \begin{gather*} \det D^2u_1=\lambda (-u_1-u_2)^{p_1}, \quad x\in \Omega, \\ \det D^2u_2=\lambda (-u_1-u_2)^{p_2}, \quad x\in \Omega, \\ u_1=u_2=0, \quad x\in\partial \Omega \end{gather*} admits at least a nontrivial convex solution for any $\lambda>0$. \subsection*{Acknowledgements} This work was supported by Doctoral Starting up Foundation of Henan Normal university (Grant No. 5101019170149). \begin{thebibliography}{00} \bibitem{c1} L. Caffarelli, L. Nirenberg, J. Spruck; \emph{The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian}, Acta Math. \textbf{155} (1985), 261-301. \bibitem{g1} C. Gutierrez; \emph{The Monge-Amp\'ere Equation}, Birkhauser, Basel, 2000. \bibitem{j1} J. Jacobsen; \emph{Global bifurcation problems associated with K-Hessian operators}, Topol. Methods Nonlinear Anal. \textbf{14} (1999), 81-130. \bibitem{l1} P. L. Lions; \emph{Two remarks on Monge-Amp\'ere equations}, Ann. Mat. Pura Appl. \textbf{142}(4) (1985), 263-275. \bibitem{m1} L. Ma, B. Liu; \emph{Symmetry results for classical solutions of Monge-Amp\'ere systems on bounded planar domains}, J. Math. Anal. Appl. \textbf{369}(2) (2010), 678-685. \bibitem{s1} K. Schmitt, R. Thompson; \emph{Nonlinear analysis and differential equations: An introduction}. Univ. of Utah Lecture Notes, Univ. of Utah Press, Salt Lake City, 2004. \bibitem{t1} N. S. Trudinger; \emph{Weak solutions of hessian equations}, Communication in Partial Differential Equations. \textbf{22}(7\&8) (1997), 1251-1261. \bibitem{t2} K. Tso; \emph{On a real Monge-Amp\'ere functional}, Invent Math. \textbf{101} (1990), 425-448. \bibitem{w1} H. Wang; \emph{Convex solutions of systems arising from Monge-Amp\'ere equations}, Electric Journal of Qualitative Theory of Differential Equations, Special Edition I. \textbf{ 26} (2009), 1-8. \bibitem{z1} Z. Zhang, Z. Qi; \emph{On a power-type coupled system of Monge-Ampere equations}, Topol. Methods Nonlinear Anal. \textbf{46}(2) (2015), 717-729. \end{thebibliography} \end{document}