Local bifurcation structure and stability of the mean curvature equation in the static spacetime

We consider the curvature equation in the static spacetime, $$ \text{div} \Big(\frac{f(x)\nabla u}{\sqrt{1-f^2(x)| \nabla u|^2}}\Big) +\frac{\nabla u \nabla f(x)}{\sqrt{1-f^2(x)| \nabla u|^2}}=\lambda NH \quad\text{in }\Omega, $$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N \geq 1\); the function \(H\) gives the mean curvature. We investigate the local bifurcation structure and stability of the solutions to this equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/48/abstr.html