Spectrum,bifurcation and hypersurfaces of prescribed k -th mean curvature in Minkowski space

Abstract

By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k-th mean curvature problem in Minkowski spacetime $\begin{matrix} \{\begin{array}{ll} {(r^{N - k} {(\frac{v^{'}}{\sqrt{1 - v^{' 2}}})}^{k})}^{'} = λ \frac{N}{C_{N}^{k}} r^{N - 1} H_{k} (r, v) & in (0, R), \\ | v^{'} | < 1 & in (0, R), \\ v^{'} (0) = v (R) = 0 . \end{array} \end{matrix}$ As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ.

Keywords

Spectrum Bifurcation Nodal solutions A priori bounds Asymptotic behaviors One-sign solution

Get full access to this article

View all access options for this article.

References

R.A.

Admas, Sobolev Spaces, Academic Press, New-York, 1975.

Azzollini, Ground state solution for a problem with mean curvature operator in Minkowski space, J. Funct. Anal. 266 (2014), 2086–2095. doi:10.1016/j.jfa.2013.10.002.

Bartnik, Existence of maximal surfaces in asymptotically flat spacetimes, Comm. Math. Phys. 94 (1984), 155–175. doi:10.1007/BF01209300.

Bartnik, Regularity of variational maximal surfaces, Acta Math. 161 (1988), 145–181. doi:10.1007/BF02392297.

Bartnik and

Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982), 131–152. doi:10.1007/BF01211061.

Bayard, Problème de Dirichlet pour la courbure scalaire dans

R^{3, 1}

, C.R. Acad. Sci. Paris Sér. I Math. 332 (2001), 219–222. doi:10.1016/S0764-4442(00)01807-3.

Bayard, Dirichlet problem for spacelike hypersurfaces with prescribed scalar curvature in

R^{n, 1}

, Calc. Var. Partial Differential Equations 18 (2003), 1–30. doi:10.1007/s00526-002-0178-5.

Bayard and

Delanoë, Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), 903–915. doi:10.1016/j.anihpc.2008.03.008.

Bereanu,

Jebelean and

Mawhin, Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces, Proc. Amer. Math. Soc. 137 (2009), 161–169. doi:10.1090/S0002-9939-08-09612-3.

10.

Bereanu,

Jebelean and

Mawhin, The Dirichlet problem with mean curvature operator in Minkowski space-a variational approach, Adv. Nonlinear Stud. 14 (2014), 315–326. doi:10.1515/ans-2014-0204.

11.

Bereanu,

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. doi:10.1016/j.jfa.2012.10.010.

12.

Bereanu,

Jebelean and

P.J.

Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal. 265 (2013), 644–659. doi:10.1016/j.jfa.2013.04.006.

13.

Calabi, Examples of Berstein problems for some nonlinear equations, in: Proc. Sym. Global Analysis, Berkeley, Univ. of Calif., 1968.

14.

S.-Y.

Cheng and

S.-T.

Yau, Maximal spacelike hypersurfaces in the Lorentz–Minkowski spaces, Ann. of Math. 104 (1976), 407–419. doi:10.2307/1970963.

15.

Corsato,

Obersnel,

Omari and

Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl. 405 (2013), 227–239. doi:10.1016/j.jmaa.2013.04.003.

16.

Corsato,

Obersnel,

Omari and

Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space, Discrete Contin. Dyn. Syst. Special Issue (2013), 159–168.

17.

Dai, Bifurcation and positive solutions for problem with mean curvature operator in Minkowski space, Calc. Var. Partial Differential Equations 55 (2016), 72. doi:10.1007/s00526-016-1012-9.

18.

Dai, Two Whyburn type topological theorems and its applications to Monge–Ampère equations, Calc. Var. Partial Differential Equations 55 (2016), 97. doi:10.1007/s00526-016-1029-0.

19.

Dai, Global bifurcation for problem with mean curvature operator on general domain, NoDEA Nonlinear Differential Equations Appl. 24 (2017), 30. doi:10.1007/s00030-017-0454-x.

20.

Dai, Bifurcation and admissible solutions for the Hessian equation, J. Funct. Anal. 273 (2017), 3200–3240. doi:10.1016/j.jfa.2017.08.001.

21.

Dai, Bifurcation and nonnegative solutions for problem with mean curvature operator on general domain, Indiana Univ. Math. J. 67 (2018), 2103–2121. doi:10.1512/iumj.2018.67.7546.

22.

Dai, Generalized limit theorem and bifurcation for problems with Pucci’s operator, Topol. Methods Nonlinear Anal. 56 (2020), 229–261.

23.

Dai,

Han and

Ma, Unilateral global bifurcation and nodal solutions for the p-Laplacian with sign-changing weight, Complex Var. Elliptic Equ. 59 (2014), 847–862. doi:10.1080/17476933.2013.791686.

24.

Dai,

Romero and

P.J.

Torres, Global bifurcation of solutions of the mean curvature spacelike equation in certain Friedmann–Lemaître–Robertson–Walker spacetimes, J. Differential Equations 264 (2018), 7242–7269. doi:10.1016/j.jde.2018.02.014.

25.

Dai and

Wang, Nodal solutions to problem with mean curvature operator in Minkowski space, Differential Integral Equations 30 (2017), 463–480. doi:10.57262/die/1489802422.

26.

Deimling, Nonlinear Functional Analysis, Springer-Verlag, New-York, 1985.

27.

Delanoë, The Dirichlet problem for an equation of given Lorentz–Gauss curvature, Ukrainian Math. J. 42 (1990), 1538–1545. doi:10.1007/BF01060827.

28.

Fuente,

Romero and

P.J.

Torres, Existence and extendibility of rotationally symmetric graphs with a prescribed higher mean curvature function in Euclidean and Minkowski spaces, J. Math. Anal. Appl. 446 (2017), 1046–1059. doi:10.1016/j.jmaa.2016.09.022.

29.

Gerhardt, H-Surfaces in Lorentzian manifolds, Comm. Math. Phys. 89 (1983), 523–553. doi:10.1007/BF01214742.

30.

Gerhardt, Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. Reine Angew. Math. 554 (2003), 157–199.

31.

Guan, The Dirichlet problem for Monge–Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature, Trans. Amer. Math. Soc. 350 (1998), 4955–4971. doi:10.1090/S0002-9947-98-02079-0.

32.

Guan,

H.Y.

Jian and

R.M.

Schoen, Entire spacelike hypersurfaces of prescribed Gauss curvature in Minkowski space, J. Reine Angew. Math. 595 (2006), 167–188.

33.

Huang,

H.Y.

Jian and

Su, Spacelike hypersurfaces of prescribed Gauss–Kronecker curvature in exterior domains, Acta Math. Sin. (Engl. Ser.) 25 (2009), 491–502. doi:10.1007/s10114-008-6010-1.

34.

Kichenassamy and

Veron, Singular solutions of the p-Laplace equation, Math. Ann. 275 (1985), 599–615. doi:10.1007/BF01459140.

35.

A.M.

Li, Spacelike hypersurfaces with constant Gauss–Kronecker curvature in the Minkowski space, Arch. Math. 64 (1995), 534–551. doi:10.1007/BF01195136.

36.

Lindqvist, Univ. Jyväskylä, Report 161, in: Notes on the p-Laplace Equation, 2nd edn, 2017.

37.

I.P.

Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, 1950.

38.

Quaas, Existence of positive solutions to a “semilinear” equation involving the Pucci’s operator in a convex domain, Differential Integral Equations 17 (2004), 481–494.

39.

Rabinowitz, Théorie du degré topologique et applications à des problèmes aux limites non linéares, Lectures Notes Lab. Analyse Numérique Universite PARIS VI (1975).

40.

P.H.

Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513. doi:10.1016/0022-1236(71)90030-9.

41.

P.H.

Rabinowitz, On bifurcation from infinity, J. Funct. Anal. 14 (1973), 462–475. doi:10.1016/0022-1236(73)90051-7.

42.

Szulkin, Ljusternik–Schnirelmann theory on

C^{1}

-manifolds, Ann. I. H. Poincaré, Anal. non linéaire 5 (1988), 119–139. doi:10.1016/s0294-1449(16)30348-1.

43.

A.E.

Treibergs, Entire spacelike hypersurfaces of constant mean curvature in Minkowski space, Invent. Math. 66 (1982), 39–56. doi:10.1007/BF01404755.

44.

Urbas, The Dirichlet problem for the equation of prescribed scalar curvature in Minkowski space, Calc. Var. Partial Differential Equations 18 (2003), 307–316. doi:10.1007/s00526-003-0206-0.

45.

Urbas, Interior curvature bounds for spacelike hypersurfaces of prescribed k-th mean curvature, Comm. Anal. Geom. 11 (2003), 235–261. doi:10.4310/CAG.2003.v11.n2.a4.

46.

G.T.

Whyburn, Topological Analysis, Princeton University Press, Princeton, 1958.