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Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space

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Abstract

Using bifurcation method, we investigate the existence, nonexistence and multiplicity of positive solutions for the following Dirichlet problem involving mean curvature operator in Minkowski space

$$\begin{aligned} \left\{ \begin{array}{lll} -\text {div}\left( \frac{\nabla v}{\sqrt{1-\vert \nabla v\vert ^2}}\right) = \lambda f(\vert x\vert ,v) &{}\quad \text {in}\,\, B_R(0),\\ v=0&{}\quad \text {on}\,\, \partial B_R(0). \end{array} \right. \end{aligned}$$

We managed to determine the intervals of the parameter \(\lambda \) in which the above problem has zero, one or two positive radial solutions corresponding to sublinear, linear, and superlinear nonlinearities f at zero respectively. We also studied the asymptotic behaviors of positive radial solutions as \(\lambda \rightarrow +\infty \).

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Acknowledgments

The author is very grateful to the anonymous referee for his/her very valuable suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, 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 P. Rabinowitz.

Research supported by NSF 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. Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space. Calc. Var. 55, 72 (2016). https://doi.org/10.1007/s00526-016-1012-9

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