Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space

Zhiqian He; Liangying Miao; Zhiqian He; Liangying Miao

doi:10.3934/math.2020249

AIMS Mathematics

2020, Volume 5, Issue 4: 3840-3850. doi: 10.3934/math.2020249

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Research article

Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space

Zhiqian He ^{1
,
,},
Liangying Miao ²

1.
Department of Basic Teaching and Research, Qinghai University, Xining 810016, P. R. China
2.
School of Mathematics and Statistics, Qinghai Nationalities University, Xining 810007, P. R. China

Received: 31 December 2019 Accepted: 12 April 2020 Published: 22 April 2020
MSC : 34B18, 34C23, 35J60

In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation $ \left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) \gt 0, - 1 \lt x \lt 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right. $ where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.
- mean curvature equation,
- positive solutions,
- multiplicity,
- uniqueness,
- cone
Citation: Zhiqian He, Liangying Miao. Uniqueness and multiplicity of positive solutions for one-dimensional prescribed mean curvature equation in Minkowski space[J]. AIMS Mathematics, 2020, 5(4): 3840-3850. doi: 10.3934/math.2020249

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Abstract

In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation $ \left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) \gt 0, - 1 \lt x \lt 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right. $ where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.

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