BIFURCATION OF RADIAL POSITIVE SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE PRESCRIBED MEAN CURVATURE EQUATION IN THE FRIEDMANN-LEMAîTRE-ROBERTSON-WALKER SPACETIME

Man Xu; Yanyun Li; Ting Wang

doi:10.11948/20250207

2026 Volume 16 Issue 3

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Man Xu, Yanyun Li, Ting Wang. BIFURCATION OF RADIAL POSITIVE SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE PRESCRIBED MEAN CURVATURE EQUATION IN THE FRIEDMANN-LEMAîTRE-ROBERTSON-WALKER SPACETIME[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1552-1572. doi: 10.11948/20250207

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Man Xu, Yanyun Li, Ting Wang. BIFURCATION OF RADIAL POSITIVE SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE PRESCRIBED MEAN CURVATURE EQUATION IN THE FRIEDMANN-LEMAîTRE-ROBERTSON-WALKER SPACETIME[J]. Journal of Applied Analysis & Computation, 2026, 16(3): 1552-1572. doi: 10.11948/20250207

BIFURCATION OF RADIAL POSITIVE SOLUTIONS OF THE DIRICHLET PROBLEM FOR THE PRESCRIBED MEAN CURVATURE EQUATION IN THE FRIEDMANN-LEMAîTRE-ROBERTSON-WALKER SPACETIME

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Author Bio: Email: liyanyun0203@163.com(Y. Li); Email: 17874158737@163.com(T. Wang)
Corresponding author: Email: xmannwnu@126.com(M. Xu)
Fund Project: The authors were supported by the Lanzhou Youth Science and Technology Talents Innovation Project (2024-QN-61) and the Young Doctor Project of the Department of Education of Gansu Province (2026-QB-019)

Abstract

Abstract

In this paper, we investigate the bifurcation of radial positive solutions of the nonlinear Dirichlet problem associated with the prescribed mean curvature equation in the Friedmann-Lemaître-Robertson-Walker spacetime
$ \left\{ \begin{aligned} &-\mathrm{div}\Big(\frac{\nabla v}{\sqrt{1-|\nabla v |^2}}\Big)=\lambda\Big(\frac{Nf'(\varphi^{-1}(v))}{\sqrt{1-|\nabla v|^2}}-Nf(\varphi^{-1}(v))H(|x|, \varphi^{-1}(v))\Big), \ \text{in}\ \mathcal{B}, \\ &v=0, \ \ \text{on}\ \partial \mathcal{B}, \ \end{aligned} \right. $
where $ \mathcal{B} $ is the unit ball in $ \mathbb{R}^{N} $, $ \lambda $ is a positive parameter, the function $ f $ belongs to $ C^{\infty}(I) $ and satisfies $ f>0 $, $ I $ is an open interval in $ \mathbb{R} $, $ \varphi $ is the function defined by $ \varphi(s)=\int_{0}^s\frac{dt}{f(t)} $, $ \varphi^{-1} $ represents the inverse function of $ \varphi $, $ |\cdot| $ denotes the Euclidean norm in $ \mathbb{R}^{N} $, and the function $ H:[0, 1]\times I\to\mathbb{R} $ is continuous, which is referred to as the mean curvature function. Our findings demonstrate the existence of at least one, two or three radial positive solutions to the aforementioned problem. The proofs are mainly based on the directions of the bifurcation.
- Bifurcation /
- radial positive solutions /
- singular ϕ-Laplacian /
- mean curvature equation /
- Friedmann-Lemaître-Robertson-Walker spacetime
MSC: 34B08, 34B18, 34C23

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