Multiplicity of solutions for the Minkowski-curvature equation via shooting method

Alberto Boscaggin; Francesca Colasuonno; Benedetta Noris

doi:10.6092/issn.2240-2829/10577

Authors

Alberto Boscaggin Dipartimento di Matematica "Giuseppe Peano" Università di Torino via Carlo Alberto 10, 10123 Torino
Francesca Colasuonno Dipartimento di Matematica "Giuseppe Peano" Università di Torino via Carlo Alberto 10, 10123 Torino http://orcid.org/0000-0003-2671-029X
In 2012, She obtained the PhD in Mathematics at the University of Bari. In the period 2013-2018, She was a Postdoctoral Fellow first at the Polytechnic University of Turin, then at the Université Libre de Bruxelles, and finally at the University of Bologna. Since March 2018 She is an Untenured Assistant Professor at the Department of Mathematics of the University of Turin.
Benedetta Noris Laboratoire Amiénois de Mathématique Fondamentale et Appliquée Université de Picardie Jules Verne 33 rue Saint-Leu, 80039 AMIENS

DOI:

https://doi.org/10.6092/issn.2240-2829/10577

Keywords:

Lorentz-Minkowski mean curvature operator, Shooting method, Existence and multiplicity, Oscillatory solutions, Neumann boundary conditions

Abstract

In this paper we prove the existence and the multiplicity of radial positive oscillatory solutions for a nonlinear problem governed by the mean curvature operator in the Lorentz-Minkowski space. The problem is set in an N-dimensional ball and is subject to Neumann boundary conditions. The main tool used is the shooting method for ODEs.

References

A. Azzollini. Ground state solution for a problem with mean curvature operator in Minkowski space. J. Funct. Anal., 266 (2014) 2086-2095.

A. Azzollini. On a prescribed mean curvature equation in Lorentz-Minkowski space. J. Math. Pures Appl., 106 (2016) 1122-1140.

R. Bartnik, L. Simon. Spacelike hypersurfaces with prescribed boundary values and mean curvature. Comm. Math. Phys., 87 (1982/83) 131-152.

C. Bereanu, P. Jebelean, J. Mawhin. Radial solutions for some nonlinear problems involving mean curvature operators in Euclidean and Minkowski spaces. Proc. Amer. Math. Soc., 137 (2009) 161- 169.

C. Bereanu, P. Jebelean, 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.

C. Bereanu, P. Jebelean, P. J. Torres. Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space. J. Funct. Anal., 264 (2013) 270-287.

C. Bereanu, J. Mawhin. Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian. J. Differential Equations, 243 (2007) 536-557.

D. Bonheure, F. Colasuonno, J. F ̈oldes. On the Born-Infeld equation for electrostatic fields with a superposition of point charges. Ann. Mat. Pura Appl., 198 (2019) 749-772.

D. Bonheure, P. d’Avenia, A. Pomponio. On the electrostatic Born-Infeld equation with extended charges. Comm. Math. Phys., 346 (2016) 877-906.

D. Bonheure, A. Iacopetti. On the regularity of the minimizer of the electrostatic Born-Infeld energy. Arch. Ration. Mech. Anal., 232 (2019) 697-725.

M. Born, L. Infeld. Foundations of the new field theory. Proc. R. Soc. Lond. Ser. A, 144 (1934) 425-451.

A. Boscaggin, F. Colasuonno, B. Noris. Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions. ESAIM Control Optim. Calc. Var., 24 (2018) 1625-1644.

A. Boscaggin, F. Colasuonno, B. Noris. A priori bounds and multiplicity of positive solutions for a p-Laplacian Neumann problem with sub-critical growth. Proc. Roy. Soc. Edinburgh Sect. A, (2019)

doi: 10.1017/prm.2018.143.

A. Boscaggin, F. Colasuonno, B. Noris. Positive radial solutions for the Minkowski-curvature equation with neumann boundary conditions. Discrete Contin. Dyn. Syst. Ser. S, (2020) doi:10.3934/dcdss.2020150.

A. Boscaggin, G. Feltrin. Positive periodic solutions to an indefinite Minkowski-curvature equation. arXiv preprint arXiv:1805.06659, (2018).

A. Boscaggin, M. Garrione. Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball. Commun. Contemp. Math., 21 1850006, 18, (2019).

A. Boscaggin, F. Zanolin. Pairs of nodal solutions for a class of nonlinear problems with one-sided growth conditions. Adv. Nonlinear Stud., 13 (2013) 13-53.

I. Coelho, C. Corsato, F. Obersnel, P. Omari. Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. Adv. Nonlinear Stud., 12 (2012) 621-638.

I. Coelho, C. Corsato, S. Rivetti. Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball. Topol. Methods Nonlinear Anal., 44 (2014) 23-39.

C. Corsato, F. Obersnel, P. Omari, S. Rivetti. Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space. J. Math. Anal. Appl., 405 (2013) 227-239.

G. Dai, J. Wang. Nodal solutions to problem with mean curvature operator in Minkowski space.

Differential Integral Equations, 30 (2017) 463-480.

K. Ecker. Area maximizing hypersurfaces in Minkowski space having an isolated singularity.

Manuscripta Math., 56 (1986) 375-397.

C. Gerhardt. H-surfaces in Lorentzian manifolds. Comm. Math. Phys., 89 (1983) 523-553.

J. Mawhin. Resonance problems for some non-autonomous ordinary differential equations. In Stability and bifurcation theory for non-autonomous differential equations vol. 2065 of Lecture Notes in Math., 103-184, Springer, Heidelberg (2013).

W. Reichel, W. Walter. Sturm-Liouville type problems for the p-Laplacian under asymptotic non- resonance conditions. J. Differential Equations, 156 (1999) 50-70.

Multiplicity of solutions for the Minkowski-curvature equation via shooting method

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