Eigenvalue, bifurcation, existence and nonexistence of solutions for Monge-Amp\`{e}re equations

TL;DR

This paper investigates eigenvalue problems for Monge-Ampère equations, establishing bifurcation results, solution existence intervals, and conditions for convex solutions in various domains.

Contribution

It introduces new bifurcation results for Monge-Ampère equations with specific nonlinearities, expanding understanding of solution structures and existence conditions.

Findings

Bifurcation at the first eigenvalue $\\lambda_1$

Existence of two distinct unbounded solution continua

Conditions for convex solutions in general domains

Abstract

In this paper we study the following eigenvalue boundary value problem for Monge-Amp\`{e}re equations: {equation} \{{array}{l} \det(D^2u)=\lambda^N f(-u)\,\, \text{in}\,\, \Omega, u=0,\,\text{on}\,\, \partial \Omega. {array}. {equation} We establish the unilateral 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 $g:\mathbb{R}\rightarrow\mathbb{R}$, we show that $(\lambda_1,0)$ is a bifurcation point of the problem and there are two distinct unbounded continua of one-sign 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 solutions for this problem in unit ball. Moreover, we also get some results on the existence and nonexistence of convex solutions for this problem in general domain by domain comparison method.