Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space

Ruyun Ma; Man Xu

doi:10.3934/dcdsb.2018271

Article Contents

2019, Volume 24, Issue 6: 2701-2718. Doi: 10.3934/dcdsb.2018271

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Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space

Ruyun Ma^, and
Man Xu

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Ruyun Ma
* Corresponding author: Ruyun Ma

Received: January 2018

Revised: May 2018

Early access: October 2018

Published: June 2019

The first author is supported by NSF grant NSFC (No.11671322)

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Abstract

In this paper we study global bifurcation phenomena for the Dirichlet problem associated with the prescribed mean curvature equation in Minkowski space

$\left\{ \begin{array}{l} -\text{div}\big(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\big) = λ f(x,u,\nabla u)\ \ \ \ \ \ & \text{in}\ Ω,\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u = 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ & \text{on}\ \partial Ω.\\\end{array} \right.$

Here $Ω$ is a bounded regular domain in $\mathbb{R}^N$ , the function $f$ satisfies the Carathéodory conditions, and $f$ is either superlinear or sublinear in $u$ at $0$ . The proof of our main results are based upon bifurcation techniques.

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Mathematics Subject Classification: Primary: 35J25; Secondary: 47H11, 47J10, 34B18.

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