## Rocky Mountain Journal of Mathematics

### Global structure of positive solutions for problem with mean curvature operator on an annular domain

#### Abstract

We study the global structure of positive solutions of the following mean curvature equation in the Minkowski space $-\div \bigg (\frac {\nabla u}{\sqrt {1-\vert \nabla u\vert ^2}}\bigg )= \lambda f(x,u),$ on an annular domain with the Robin boundary condition. According to the behavior of $f$ near $0$, we obtain the existence and multiplicity of positive solutions for this problem.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 6 (2018), 1799-1814.

Dates
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/euclid.rmjm/1543028438

Digital Object Identifier
doi:10.1216/RMJ-2018-48-6-1799

Mathematical Reviews number (MathSciNet)
MR3879302

Zentralblatt MATH identifier
06987225

#### Citation

Cao, Xiaofei; Dai, Guowei; Zhang, Ning. Global structure of positive solutions for problem with mean curvature operator on an annular domain. Rocky Mountain J. Math. 48 (2018), no. 6, 1799--1814. doi:10.1216/RMJ-2018-48-6-1799. https://projecteuclid.org/euclid.rmjm/1543028438

#### References

• R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982), 131–152.
• C. Bereanu, P. Jebelean and P.J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal. 264 (2013), 270–287.
• ––––, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), 644–659.
• E. Calabi, Examples of Berstein problems for some nonlinear equations, Proc. Sym. Global Analysis, University of California, Berkeley, 1968.
• S.-Y. Cheng and S.-T. Yau, Maximal spacelike hypersurfaces in the Lorentz-Minkowski spaces, Ann. Math. 104 (1976), 407–419.
• G. Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. 55 (2016), 72.
• ––––, Bifurcation and nonnegative solutions for problem with mean curvature operator on general domains, Indiana Univ. Math. J., http://www.iumj.indiana.edu /IUMJ/Preprints/7546.pdf.
• K. Deimling, Nonlinear functional analysis, Springer-Verlag, New-York, 1987.
• E.L. Ince, Ordinary differential equations, Dover Publications, Inc., New York, 1927.
• R. Ma, H. Gao and Y. Lu, Global structure of radial positive solutions for a prescribed mean curvature problem in a ball, J. Funct. Anal. 270 (2016), 2430–2455.
• K. Schmitt and R. Thompson, Nonlinear analysis and differential equations: An introduction, Univ. Utah Lect. Notes, University of Utah Press, Salt Lake City, 2004.
• A.E. Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), 39–56.
• W. Walter, Ordinary differential equations, Springer, New York, 1998.