Two Whyburn type topological theorems and its applications to Monge-Ampère equations

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-Ampère equation \begin{eqnarray} \left\{ \begin{array}{lll} \det\left(D^2u\right)=λ^N a(x)f(-u)\,\, &\text{in}\,\, Ω,\\ u=0~~~~~~~~~~~~~~~~~~~~~~\,\,&\text{on}\,\, \partial Ω. \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.

arXiv admin note: substantial text overlap with arXiv:1207.6669