Radial solutions for a Neumann problem with mean curvature on exterior domain

Abstract

We are concerned with the Neumann problem with Minkowski-curvature operator (P) $\{\begin{array}{ll}{\displaystyle -\mathrm{div}\left(\frac{\mathrm{\nabla }u(x)}{\sqrt{1-|\mathrm{\nabla }u(x){|}^2}}\right)=g(u(x))}& \mathrm{in}\mathrm{\Omega },\\ {\displaystyle u(x)>0}& \mathrm{in}\mathrm{\Omega },\\ {\displaystyle u\to 0}& \mathrm{as}|x|\to \mathrm{\infty },\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=-\sigma }& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}$(P) where Ω is an exterior domain in ${\mathbb{R}}^N$ and $\frac{\mathrm{\partial }}{\mathrm{\partial }\nu }$ denotes the normal interior derivative on $\mathrm{\partial }\mathrm{\Omega }$. We have shown the existence of radial positive solutions of (P) when g satisfies some suitable assumptions. The proof of our main result is based upon the shooting method.

Keywords:

• Mean curvature operator
• exterior domain
• positive radial solutions
• Shooting method

2020 Mathematics Subject Classifications:

• 35J57
• 35J66
• 34B18

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 12061064].