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


Authors: Minghe Pei and Libo Wang
Journal: Proc. Amer. Math. Soc. 145 (2017), 4423-4430
MSC (2010): Primary 35J93, 35J75, 35A20
DOI: https://doi.org/10.1090/proc/13587
Published electronically: April 28, 2017
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Abstract: The existence of positive radial solution is obtained for a mean curvature equation in Minkowski space of the form

$\displaystyle \left \{ \begin {array}{ll} \hbox {div}(\frac {\nabla v}{\sqrt {1... ... \Omega ; \\ v=0\quad \textrm {on}\quad \partial \Omega , \end{array} \right . $

where $ \Omega $ is a unit ball in $ \mathbb{R}^N$, $ f(r,u)$ has singularities at $ u=0$, $ r=0$ and/or $ r=1$. The main tool is the perturbation technique and nonlinear alternative of Leray-Schauder type. The interesting point is that the nonlinear term $ f(r,u)$ at $ u=0$ may be strongly singular.

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Additional Information

Minghe Pei
Affiliation: School of Mathematics and Statistics, Beihua University, JiLin City 132013, People’s Republic of China
Email: peiminghe@163.com

Libo Wang
Affiliation: School of Mathematics and Statistics, Beihua University, JiLin City 132013, People’s Republic of China
Email: wlb$_$math@163.com

DOI: https://doi.org/10.1090/proc/13587
Keywords: Mean curvature equation, strongly singular Dirichlet problem, radial solutions, nonlinear alternative of Leray-Schauder type
Received by editor(s): July 31, 2016
Received by editor(s) in revised form: November 12, 2016
Published electronically: April 28, 2017
Additional Notes: This project was sponsored by the Education Department of JiLin Province of China ([2016]45)
Communicated by: Joachim Krieger
Article copyright: © Copyright 2017 American Mathematical Society