Mathematics

CLC number: O175

∑-Shaped Connected Component of Positive Solutions for One- Dimensional Prescribed Mean Curvature Equation in Minkowski Space

Minkowski 空间中一维给定平均曲率方程正解的∑-型连通分支

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

^† Corresponding author. E-mail: luyq8610@126.com

Received: 5 December 2024

Abstract

In this work, we demonstrate that the existence of an ∑-shaped connected component within the set of positive solutions for the one-dimensional prescribed mean curvature equation in Minkowski space $\{\begin{array}{l}-{\left(\frac{u^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}=\lambda h(t)f(u(t)),t\in (\mathrm{0,1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0},\end{array}$ with boundary conditions having parameter in two cases $f(\mathrm{0})=\mathrm{0}$ and $f(\mathrm{0})>\mathrm{0}$ by using upper and lower solution method, where $\lambda >\mathrm{0}$ is a parameter, $f\in C^{\mathrm{2}}([\mathrm{0},\mathrm{\infty }),\mathbb{R})$ is monotonically increasing and $\underset{u\to {\mathrm{1}}^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u)}{\mathrm{1}-u}=\mathrm{0}$, $h\in C^{\mathrm{1}}([\mathrm{0,1}],(\mathrm{0},\mathrm{\infty }))$ is a nonincreasing function and $h(t)>\mathrm{1}$.

摘要

运用上下解方法证明 Minkowski 空间中一维给定平均曲率方程 $\{\begin{array}{l}-{\left(\frac{u^{\text{'}}(t)}{\sqrt[]{\mathrm{1}-(u^{\text{'}}{(t))}^{\mathrm{2}}}}\right)}^{\mathrm{\text{'}}}=\lambda h(t)f(u(t)),t\in (\mathrm{0}\mathrm{1}),\\ u(\mathrm{0})=\mathrm{0},u^{\text{'}}(\mathrm{1})+\sqrt[]{\lambda }u(\mathrm{1})=\mathrm{0},\end{array}$ $f(\mathrm{0})=\mathrm{0}$在 和 $f(\mathrm{0})>\mathrm{0}$ 两种情形下正解集$\sum$型连通分支的存在性，其中，$\lambda >\mathrm{0}$ 为参数，$f\in C^{\mathrm{2}}([\mathrm{0},\mathrm{\infty }),\mathbb{R})$ 单调递增且满足 $\underset{u\to {\mathrm{1}}^{-}}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{f(u)}{\mathrm{1}-u}=\mathrm{0}$，$h\in C^{\mathrm{1}}([\mathrm{0,1}],(\mathrm{0},\mathrm{\infty }))$ 是单调递减函数且 $h(t)>\mathrm{1}$.