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

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.