Two Whyburn type topological theorems and its applications to Monge-Amp\`{e}re equations

TL;DR

This paper corrects a topological lemma, establishes new limit theorems, and applies them to analyze the existence and multiplicity of solutions for a Monge-Ampère equation using bifurcation theory.

Contribution

It introduces corrected topological theorems and applies them to derive global bifurcation results for Monge-Ampère equations, identifying parameter intervals for solutions.

Findings

Intervals of λ with solutions exist

Multiple solutions found for certain λ

Nonexistence regions identified

Abstract

In this paper we correct a gap of Whyburn type topological lemma and establish two superior limit theorems. As the applications of our Whyburn type topological theorems, we study the following Monge-Amp\`{e}re equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=\lambda^N a(x)f(-u)\,\, &\text{in}\,\, \Omega,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial \Omega. \end{array} \right.\nonumber \end{eqnarray} We establish global bifurcation results for the problem. We find intervals of $\lambda$ for the existence, multiplicity and nonexistence of strictly convex solutions for this problem.