Multiple positive solutions for Dirichlet problem of prescribed mean curvature equations in Minkowski spaces

Abstract

In this article, we consider the Dirichlet problem for the prescribed mean curvature equation in the Minkowski space, $$\displaylines{ -\hbox{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) =\lambda f(u) \quad \text{in } B_R,\cr u=0 \quad \text{on } \partial B_R, }$$ where $B_R:=\{x\in \mathbb{R}^N: |x|0$ is a parameter and $f:[0, \infty)\to\mathbb{R}$ is continuous. We apply some standard variational techniques to show how changes in the sign of f lead to multiple positive solutions of the above problem for sufficiently large $\lambda$.

Keywords

• Dirichlet problem
• Minkowski-curvature
• positive solutions
• variational methods