Positive radial solutions of a mean curvature equation in Lorentz–Minkowski space with strong singularity

ABSTRACT

The existence and uniqueness of positive radial solutions are obtained for a mean curvature equation in Lorentz–Minkowski space of the form $\begin{array}{rl}& \text{div}\left(\frac{\mathrm{\nabla }v}{\sqrt{1-|\mathrm{\nabla }v{|}^2}}\right)+f(|x|,v)=0\mathrm{in}\mathrm{\Omega },\\ & v=0\mathrm{on}\mathrm{\partial \Omega },\end{array}$ where Ω is a unit ball in ${\mathbb{R}}^N$, $f(r,u)$ may be singular at r=0 and/or r=1, and strongly singular at u=0. The main tool is the perturbation technique and Schauder fixed point theorem.

Keywords:

• Mean curvature equation
• strong singularity
• radial solution
• Schauder fixed point theorem
• existence and uniqueness

AMS Subject Classifications:

• 35J93
• 35J75
• 35A20

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project is sponsored by the Education Department of JiLin Province of China ([2016]45).