Eigenvalue, bifurcation and convex solutions for Monge-Ampère equations

Abstract

In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations \det(D^2u)=\lambda^N f(-u) \text{in } \Omega, u=0 \text{on } \partial \Omega. We establish global bifurcation results for the problem with $f(u)=u^N+g(u)$ and $\Omega$ being the unit ball of $\mathbb{R}^N$. More precisely, under some natural hypotheses on the perturbation function $g\colon[0,+\infty)\rightarrow[0,+\infty)$, we show that $(\lambda_1,0)$ is a bifurcation point of the problem and there exists an unbounded continuum of convex solutions, where $\lambda_1$ is the first eigenvalue of the problem with $f(u)=u^N$. As the applications of the above results, we consider with determining interval of $\lambda$, in which there exist convex solutions for this problem in unit ball. Moreover, we also get some results about the existence and nonexistence of convex solutions for this problem on general domain by domain comparison method.