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

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

* 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.

Citation: Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. BereanuP. 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.  Google Scholar

[5]

C. BereanuP. 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.  Google Scholar

[6]

C. BereanuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. BonheureF. 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.  Google Scholar

[10]

D. BonheureP. 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.  Google Scholar

[11]

D. BonheureM. GrossiB. 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.  Google Scholar

[12]

D. BonheureC. 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.  Google Scholar

[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.  Google Scholar

[14]

D. BonheureB. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

A. Boscaggin and G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, preprint, arXiv: 1805.06659. Google Scholar

[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.  Google Scholar

[19]

I. CoelhoC. CorsatoF. 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.  Google Scholar

[20]

I. CoelhoC. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

C. CorsatoF. ObersnelP. 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.  Google Scholar

[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.   Google Scholar

[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/. Google Scholar

[26]

K. Ecker, Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math., 56 (1986), 375-397.  doi: 10.1007/BF01168501.  Google Scholar

[27]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553.  doi: 10.1007/BF01214742.  Google Scholar

[28]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969.  Google Scholar

[29]

Y. Q. LuT. 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.  Google Scholar

[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.  Google Scholar

[31]

E. Montefusco and P. Pucci, Existence of radial ground states for quasilinear elliptic equations, Adv. Differential Equations, 6 (2001), 959-986.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

C. BereanuP. 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.  Google Scholar

[5]

C. BereanuP. 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.  Google Scholar

[6]

C. BereanuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

D. BonheureF. 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.  Google Scholar

[10]

D. BonheureP. 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.  Google Scholar

[11]

D. BonheureM. GrossiB. 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.  Google Scholar

[12]

D. BonheureC. 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.  Google Scholar

[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.  Google Scholar

[14]

D. BonheureB. 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

A. Boscaggin and G. Feltrin, Positive periodic solutions to an indefinite Minkowski-curvature equation, preprint, arXiv: 1805.06659. Google Scholar

[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.  Google Scholar

[19]

I. CoelhoC. CorsatoF. 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.  Google Scholar

[20]

I. CoelhoC. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[23]

C. CorsatoF. ObersnelP. 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.  Google Scholar

[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.   Google Scholar

[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/. Google Scholar

[26]

K. Ecker, Area maximizing hypersurfaces in Minkowski space having an isolated singularity, Manuscripta Math., 56 (1986), 375-397.  doi: 10.1007/BF01168501.  Google Scholar

[27]

C. Gerhardt, $H$-surfaces in Lorentzian manifolds, Comm. Math. Phys., 89 (1983), 523-553.  doi: 10.1007/BF01214742.  Google Scholar

[28]

J. K. Hale, Ordinary Differential Equations, Pure and Applied Mathematics, Vol. XXI. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969.  Google Scholar

[29]

Y. Q. LuT. 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.  Google Scholar

[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.  Google Scholar

[31]

E. Montefusco and P. Pucci, Existence of radial ground states for quasilinear elliptic equations, Adv. Differential Equations, 6 (2001), 959-986.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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)
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 $
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 $
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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