# Radial Convex Solutions of a Singular Dirichlet Problem with the Mean Curvature Operator in Minkowski Space

Article

First Online:

- 1 Downloads

## Abstract

In this paper, we study the existence of nontrivial radial convex solutions of a singular Dirichlet problem involving the mean curvature operator in Minkowski space. The proof is based on a well-known fixed point theorem in cones. We deal with more general nonlinear term than those in the literature.

## Key words

radial convex solutions singular Dirichlet problem mean curvature operator fixed point theorem in cones## 2010 MR Subject Classification

34B15 35A20 35J93## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

We would like to show our thanks to Professor Jifeng Chu (Shanghai Normal University) for useful discussions.

## References

- [1]Bartnik R, Simon L. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm Math Phys, 1982–1983,
**87**: 131–152MathSciNetCrossRefzbMATHGoogle Scholar - [2]Bereanu C, Jebelean P, Mawhin J. Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc Amer Math Soc, 2009,
**137**: 161–169MathSciNetCrossRefzbMATHGoogle Scholar - [3]Bereanu C, Jebelean P, Mawhin J. Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math Nachr, 2010,
**283**: 379–391MathSciNetCrossRefzbMATHGoogle Scholar - [4]Bereanu C, Jebelean P, Torres P J. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J Funct Anal 2013,
**264**: 270–287MathSciNetCrossRefzbMATHGoogle Scholar - [5]Bereanu C, Jebelean P, Torres P J. Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. J Funct Anal 2013,
**265**: 644–659MathSciNetCrossRefzbMATHGoogle Scholar - [6]Bereanu C, Jebelean P, Mawhin J. The Dirichlet problem with mean curvature operator in Minkowski space-a variational approach. Adv Nonlinear Stud, 2014,
**14**: 315–326MathSciNetCrossRefzbMATHGoogle Scholar - [7]Chu J, Chen H, O’Regan D. Positive periodic solutions and eigenvalue intervals for systems of second order differential equations. Math Nachr, 2008,
**281**: 1549–1556MathSciNetCrossRefzbMATHGoogle Scholar - [8]Coelho I, Corsato C, Rivetti S. Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol Methods Nonlinear Anal, 2014,
**44**: 23–39MathSciNetCrossRefzbMATHGoogle Scholar - [9]Corsato C, Obersnel F, Omari P, Rivetti S. Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J Math Anal Appl, 2013,
**405**: 227–239MathSciNetCrossRefzbMATHGoogle Scholar - [10]Dai G. Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space. Calc Var Partial Differ Equ, 2016,
**55**(72): 1–17MathSciNetGoogle Scholar - [11]Hu S, Wang H. Convex Solutions of boundary value problems arising from Monge-Ampère equations. Discrete Contin Dyn Syst, 2006,
**16**: 705–720MathSciNetCrossRefzbMATHGoogle Scholar - [12]Infante G, Webb J R L. Nonzero solutions of Hammerstein integral equations with discontinuous kernels. J Math Anal Appl, 2002,
**272**: 30–42MathSciNetCrossRefzbMATHGoogle Scholar - [13]Krasnoselskii M A. Positive Solutions of Operator Equations. Noordhoff: Groningen, 1964Google Scholar
- [14]Lan K Q. Multiple positive solutions of semilinear differential equations with singularities. J London Math Soc, 2001,
**63**: 690–704MathSciNetCrossRefzbMATHGoogle Scholar - [15]Ma R, Gao H, Lu Y. Global structure of radial positive solutions for a prescribed mean curvature problem in a ball. J Funct Anal, 2016,
**270**: 2430–2455MathSciNetCrossRefzbMATHGoogle Scholar - [16]Ma R. Positive solutions for Dirichlet problems involving the mean curvature operator in Minkowski space. Monatsh Math, 2018,
**187**(2): 315–325MathSciNetCrossRefGoogle Scholar - [17]Ma R, Gao H. Multiple positive solutions for a class of semipositone Neumann problems with singular φ-Laplacian. Acta Mathematica Scientia, 2017,
**37B**(5): 1472–1482MathSciNetCrossRefzbMATHGoogle Scholar - [18]Mawhin J. Radial solution of Neumann problem for periodic perturbations of the mean extrinsic curvature operator. Milan J Math, 2011,
**79**: 95–112MathSciNetCrossRefzbMATHGoogle Scholar - [19]Pei M, Wang L. Multiplicity of positive radial solutions of a singular mean curvature equations in Minkowski space. Appl Math Lett 2016,
**60**: 50–55MathSciNetCrossRefzbMATHGoogle Scholar - [20]Pei M, Wang L. Positive radial solutions of a mean curvature equation in Minkowski space with strong singularity. Proc Amer Math Soc, 2017,
**145**: 4423–4430MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019