Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

Alberto Boscaggin; Francesca Colasuonno; Benedetta Noris

doi:10.3934/dcdss.2020150

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July 2020, 13(7): 1921-1933. doi: 10.3934/dcdss.2020150

Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions

Alberto Boscaggin ^1, , Francesca Colasuonno ^1,, and Benedetta Noris ^2,

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

^* Corresponding author: Francesca Colasuonno

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday, with great esteem

Received June 2018 Revised February 2019 Published November 2019

Fund Project: This work was partially supported by the INdAM - GNAMPA Project 2019 "Il modello di Born-Infeld per l'elettromagnetismo nonlineare: esistenza, regolarità e molteplicità di soluzioni". A. Boscaggin and B. Noris acknowledge also the support of the project ERC Advanced Grant 2013 n. 339958: "Complex Patterns for Strongly Interacting Dynamical Systems – COMPAT"

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.

Keywords: Lorentz-Minkowski mean curvature operator, shooting method, existence and multiplicity, oscillating solutions, Neumann boundary conditions.

Mathematics Subject Classification: Primary: 35J62, 35B05, 35A24, 34B18; Secondary: 35B09.

Citation: Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150

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.

Figure 1. Number of half-turns performed by the solutions of the Cauchy problem $ (u(R_1), v(R_1)) = (d, 0) $ associated with (5) varying with the initial condition $ u(R_1) = d $. The existence of a bound from above $ d^* $ for the initial data $ d $ in correspondence to which the solutions of the Cauchy problem perform at least one half-turn in the phase plane is a consequence of the fact that $ |u'|\le 1 $, see (5)

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Figure 2. (a) Partial bifurcation diagram in dimension $ N = 1 $, with $ R_1 = 0 $, $ R_2 = 1 $, and $ s_0 = 1 $. (b) Graphs of eight solutions belonging to the four branches represented in (a). The colour of each solution is the same as the branch it belongs to. For each branch we have selected two solutions, one with $ u(0)>1 $ and the other with $ u(0)<1 $. The solutions displayed correspond to different values of $ q $

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Figure 3. (a) Partial bifurcation diagram in a unit disk (i.e., $ R_1 = 0 $, $ R_2 = 1 $, and $ N = 2 $), with $ s_0 = 1 $. (b) Solutions corresponding to $ q = 70 $

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Figure 4. A solution $ (u_d, v_d) $, with $ 0<d<s_0 $, in the phase plane $ (u, v) $. The solution is also given in polar coordinates $ (\theta_d, \rho_d) $ with $ \alpha = 1 $. It can be noted from the picture that $ u_d(\bar r) = s_0 $ if and only if $ \cos\theta_d(\bar r) = 0 $ and that $ v_d(r) = 0 $ for some $ r\in [R_1, R_2] $ if and only if $ \sin\theta_d(r) = 0 $

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