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

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.