AIMS Mathematics, 2020, 5(4): 3840-3850. doi: 10.3934/math.2020249.

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

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

In this paper, we study the uniqueness and multiplicity of positive solutions of one-dimensional prescribed mean curvature equation \begin{equation*}\left\{ \begin{array}{l} - \left({\frac{{u'}}{{\sqrt {1 - u{'^2}} }}} \right)' = \lambda f\left(u \right), \\ u\left(x \right) > 0, - 1 < x < 1, \\ u\left({ - 1} \right) = u\left(1 \right) = 0, \end{array} \right.\end{equation*} where $\lambda$ is a positive parameter. The main tool is the fixed point index in cones.
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Keywords 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. AIMS Mathematics, 2020, 5(4): 3840-3850. doi: 10.3934/math.2020249

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