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

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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

$$\begin{aligned} \left\{ \begin{array}{lll} \det \left( D^2u\right) =\lambda ^N a(x)f(-u)\,\, &{}\quad \text {in}\,\, \Omega ,\\ u=0~~~~~~~\,\,&{}\quad \text {on}\,\, \partial \Omega . \end{array} \right. \end{aligned}$$

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.

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Acknowledgments

The author is very grateful to an anonymous referee for his or her very valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, 116024, Dalian, People’s Republic of China

    Guowei Dai

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  1. Guowei Dai
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Correspondence to Guowei Dai.

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Communicated by L. Caffarelli.

Research supported by NNSF of China (Nos. 11261052, 11401477), the Fundamental Research Funds for the Central Universities (No. DUT15RC(3)018) and Research project of science and technology of Liaoning Provincial Education Department (No. ZX20150135).

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Dai, G. Two Whyburn type topological theorems and its applications to Monge–Ampère equations. Calc. Var. 55, 97 (2016). https://doi.org/10.1007/s00526-016-1029-0

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