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

In this article, we consider the Dirichlet problem for the prescribed mean curvature equation in the Minkowski space,
 -\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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