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Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions
1. | Dipartimento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy |
2. | Laboratoire Amiénois de Mathématique Fondamentale et Appliquée, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039 AMIENS, France |
We analyze existence, multiplicity and oscillatory behavior of positive radial solutions to a class of quasilinear equations governed by the Lorentz-Minkowski mean curvature operator. The equation is set in a ball or an annulus of $ \mathbb R^N $, is subject to homogeneous Neumann boundary conditions, and involves a nonlinear term on which we do not impose any growth condition at infinity. The main tool that we use is the shooting method for ODEs.
References:
[1] |
A. 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. |
[2] |
A. Azzollini,
On a prescribed mean curvature equation in Lorentz-Minkowski space, J. Math. Pures Appl., 106 (2016), 1122-1140.
doi: 10.1016/j.matpur.2016.04.003. |
[3] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.
doi: 10.1007/BF01211061. |
[4] |
C. Bereanu, P. Jebelean and 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.
doi: 10.1090/S0002-9939-08-09612-3. |
[5] |
C. Bereanu, P. 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. |
[6] |
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.
doi: 10.1016/j.jfa.2012.10.010. |
[7] |
C. Bereanu and J. Mawhin,
Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
[8] |
D. Bonheure, J.-B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, Calc. Var. Partial Differential Equations, 56 (2017), Art. 74, 35 pp.
doi: 10.1007/s00526-017-1163-3. |
[9] |
D. Bonheure, F. Colasuonno and J. Földes,
On the Born-Infeld equation for electrostatic fields with a superposition of point charges, Ann. Mat. Pura Appl., 198 (2019), 749-772.
doi: 10.1007/s10231-018-0796-y. |
[10] |
D. Bonheure, P. d'Avenia and A. Pomponio,
On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877-906.
doi: 10.1007/s00220-016-2586-y. |
[11] |
D. Bonheure, M. Grossi, B. Noris and S. Terracini,
Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations, 261 (2016), 455-504.
doi: 10.1016/j.jde.2016.03.016. |
[12] |
D. Bonheure, C. Grumiau and C. Troestler,
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions, Nonlinear Anal., 147 (2016), 236-273.
doi: 10.1016/j.na.2016.09.010. |
[13] |
D. Bonheure and A. Iacopetti,
On the regularity of the minimizer of the electrostatic Born-Infeld energy, Arch. Ration. Mech. Anal., 232 (2019), 697-725.
doi: 10.1007/s00205-018-1331-4. |
[14] |
D. Bonheure, B. Noris and T. Weth,
Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588.
doi: 10.1016/j.anihpc.2012.02.002. |
[15] |
A. Boscaggin, F. Colasuonno and 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), http://dx.doi.org/10.1017/prm.2018.143. |
[16] |
A. Boscaggin, F. Colasuonno and 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, http://dx.doi.org/10.1051/cocv/2016064.
doi: 10.1051/cocv/2017074. |
[17] |
A. Boscaggin and G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, preprint, arXiv: 1805.06659. |
[18] |
A. Boscaggin and M. Garrione, Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball, Commun. Contemp. Math., 21 (2019), 1850006, 18 pp.
doi: 10.1142/S0219199718500062. |
[19] |
I. Coelho, C. Corsato, F. Obersnel and P. Omari,
Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.
doi: 10.1515/ans-2012-0310. |
[20] |
I. Coelho, C. Corsato and 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.
doi: 10.12775/TMNA.2014.034. |
[21] |
F. Colasuonno and B. Noris,
A $p$-Laplacian supercritical Neumann problem, Discrete Contin. Dyn. Syst., 37 (2017), 3025-3057.
doi: 10.3934/dcds.2017130. |
[22] |
F. Colasuonno and B. Noris,
Radial positive solutions for $p$-Laplacian supercritical Neumann problems, Bruno Pini Mathematical Analysis Seminar 2017, Bruno Pini Math. Anal. Semin., Univ. Bologna, Alma Mater Stud., Bologna, 8 (2017), 55-72.
|
[23] |
C. Corsato, F. Obersnel, P. Omari and 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.
doi: 10.1016/j.jmaa.2013.04.003. |
[24] |
G. W. Dai and J. Wang,
Nodal solutions to problem with mean curvature operator in Minkowski space, Differential Integral Equations, 30 (2017), 463-480.
|
[25] |
E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, Concordia University, (2012), http://cmvl.cs.concordia.ca/auto/. |
[26] |
K. Ecker,
Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math., 56 (1986), 375-397.
doi: 10.1007/BF01168501. |
[27] |
C. Gerhardt,
$H$-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553.
doi: 10.1007/BF01214742. |
[28] |
J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. |
[29] |
Y. Q. Lu, T. L. Chen and R. Y. Ma,
On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2649-2662.
doi: 10.3934/dcdsb.2016066. |
[30] |
J. Mawhin,
Resonance problems for some non-autonomous ordinary differential equations, Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2065 (2013), 103-184.
doi: 10.1007/978-3-642-32906-7_3. |
[31] |
E. Montefusco and P. Pucci,
Existence of radial ground states for quasilinear elliptic equations, Adv. Differential Equations, 6 (2001), 959-986.
|
[32] |
P. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic equations in the exponential case, Indiana Univ. Math. J., 47 (1998), 529-539.
doi: 10.1512/iumj.1998.47.2045. |
[33] |
P. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.
doi: 10.1512/iumj.1998.47.1517. |
[34] |
W. Reichel and W. Walter,
Sturm-Liouville type problems for the $p$-Laplacian under asymptotic non-resonance conditions, J. Differential Equations, 156 (1999), 50-70.
doi: 10.1006/jdeq.1998.3611. |
show all references
References:
[1] |
A. 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. |
[2] |
A. Azzollini,
On a prescribed mean curvature equation in Lorentz-Minkowski space, J. Math. Pures Appl., 106 (2016), 1122-1140.
doi: 10.1016/j.matpur.2016.04.003. |
[3] |
R. Bartnik and L. Simon,
Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982/83), 131-152.
doi: 10.1007/BF01211061. |
[4] |
C. Bereanu, P. Jebelean and 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.
doi: 10.1090/S0002-9939-08-09612-3. |
[5] |
C. Bereanu, P. 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. |
[6] |
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.
doi: 10.1016/j.jfa.2012.10.010. |
[7] |
C. Bereanu and J. Mawhin,
Existence and multiplicity results for some nonlinear problems with singular $\phi$-Laplacian, J. Differential Equations, 243 (2007), 536-557.
doi: 10.1016/j.jde.2007.05.014. |
[8] |
D. Bonheure, J.-B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, Calc. Var. Partial Differential Equations, 56 (2017), Art. 74, 35 pp.
doi: 10.1007/s00526-017-1163-3. |
[9] |
D. Bonheure, F. Colasuonno and J. Földes,
On the Born-Infeld equation for electrostatic fields with a superposition of point charges, Ann. Mat. Pura Appl., 198 (2019), 749-772.
doi: 10.1007/s10231-018-0796-y. |
[10] |
D. Bonheure, P. d'Avenia and A. Pomponio,
On the electrostatic Born-Infeld equation with extended charges, Comm. Math. Phys., 346 (2016), 877-906.
doi: 10.1007/s00220-016-2586-y. |
[11] |
D. Bonheure, M. Grossi, B. Noris and S. Terracini,
Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations, 261 (2016), 455-504.
doi: 10.1016/j.jde.2016.03.016. |
[12] |
D. Bonheure, C. Grumiau and C. Troestler,
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions, Nonlinear Anal., 147 (2016), 236-273.
doi: 10.1016/j.na.2016.09.010. |
[13] |
D. Bonheure and A. Iacopetti,
On the regularity of the minimizer of the electrostatic Born-Infeld energy, Arch. Ration. Mech. Anal., 232 (2019), 697-725.
doi: 10.1007/s00205-018-1331-4. |
[14] |
D. Bonheure, B. Noris and T. Weth,
Increasing radial solutions for Neumann problems without growth restrictions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 573-588.
doi: 10.1016/j.anihpc.2012.02.002. |
[15] |
A. Boscaggin, F. Colasuonno and 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), http://dx.doi.org/10.1017/prm.2018.143. |
[16] |
A. Boscaggin, F. Colasuonno and 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, http://dx.doi.org/10.1051/cocv/2016064.
doi: 10.1051/cocv/2017074. |
[17] |
A. Boscaggin and G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, preprint, arXiv: 1805.06659. |
[18] |
A. Boscaggin and M. Garrione, Pairs of nodal solutions for a Minkowski-curvature boundary value problem in a ball, Commun. Contemp. Math., 21 (2019), 1850006, 18 pp.
doi: 10.1142/S0219199718500062. |
[19] |
I. Coelho, C. Corsato, F. Obersnel and P. Omari,
Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.
doi: 10.1515/ans-2012-0310. |
[20] |
I. Coelho, C. Corsato and 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.
doi: 10.12775/TMNA.2014.034. |
[21] |
F. Colasuonno and B. Noris,
A $p$-Laplacian supercritical Neumann problem, Discrete Contin. Dyn. Syst., 37 (2017), 3025-3057.
doi: 10.3934/dcds.2017130. |
[22] |
F. Colasuonno and B. Noris,
Radial positive solutions for $p$-Laplacian supercritical Neumann problems, Bruno Pini Mathematical Analysis Seminar 2017, Bruno Pini Math. Anal. Semin., Univ. Bologna, Alma Mater Stud., Bologna, 8 (2017), 55-72.
|
[23] |
C. Corsato, F. Obersnel, P. Omari and 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.
doi: 10.1016/j.jmaa.2013.04.003. |
[24] |
G. W. Dai and J. Wang,
Nodal solutions to problem with mean curvature operator in Minkowski space, Differential Integral Equations, 30 (2017), 463-480.
|
[25] |
E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and bifurcation software for ordinary differential equations, Concordia University, (2012), http://cmvl.cs.concordia.ca/auto/. |
[26] |
K. Ecker,
Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math., 56 (1986), 375-397.
doi: 10.1007/BF01168501. |
[27] |
C. Gerhardt,
$H$-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553.
doi: 10.1007/BF01214742. |
[28] |
J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. |
[29] |
Y. Q. Lu, T. L. Chen and R. Y. Ma,
On the Bonheure-Noris-Weth conjecture in the case of linearly bounded nonlinearities, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2649-2662.
doi: 10.3934/dcdsb.2016066. |
[30] |
J. Mawhin,
Resonance problems for some non-autonomous ordinary differential equations, Stability and Bifurcation Theory for Non-Autonomous Differential Equations, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Springer, Heidelberg, 2065 (2013), 103-184.
doi: 10.1007/978-3-642-32906-7_3. |
[31] |
E. Montefusco and P. Pucci,
Existence of radial ground states for quasilinear elliptic equations, Adv. Differential Equations, 6 (2001), 959-986.
|
[32] |
P. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic equations in the exponential case, Indiana Univ. Math. J., 47 (1998), 529-539.
doi: 10.1512/iumj.1998.47.2045. |
[33] |
P. Pucci and J. Serrin,
Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J., 47 (1998), 501-528.
doi: 10.1512/iumj.1998.47.1517. |
[34] |
W. Reichel and W. Walter,
Sturm-Liouville type problems for the $p$-Laplacian under asymptotic non-resonance conditions, J. Differential Equations, 156 (1999), 50-70.
doi: 10.1006/jdeq.1998.3611. |




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