Electron. J. Diff. Equ., Vol. 2016 (2016), No. 180, pp. 1-7.

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

Ruyun Ma, Tianlan Chen

Abstract:
In this article, we consider the Dirichlet problem for the prescribed mean curvature equation in the Minkowski space,
$$\displaylines{ -\text{div}\Big(\frac {\nabla u}{\sqrt{1-|\nabla u|^2}}\Big) =\lambda f(u) \quad \text{in } B_R,\cr u=0 \quad \text{on } \partial B_R, }$$
where $B_R:=\{x\in \mathbb{R}^N: |x|< R\}$, $\lambda>0$ 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 $\lambda$.

Submitted June 3, 2016. Published July 7, 2016.
Math Subject Classifications: 35B15, 34K28, 34L30, 35J60, 35J65.
Key Words: Dirichlet problem; Minkowski-curvature; positive solutions; variational methods.

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Ruyun Ma
Department of Mathematics
Northwest Normal University
Lanzhou 730070, China
email: mary@nwnu.edu.cn
Tianlan Chen
Northwest Normal University
Lanzhou 730070, China
email: chentianlan511@126.com

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