August 2018, 38(8): 3899-3911. doi: 10.3934/dcds.2018169

Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases

Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy

Received  September 2017 Revised  March 2018 Published  May 2018

This paper deals with the prescribed mean curvature equations
$ - {\text{div}}\left( {\frac{{\nabla u}}{{\sqrt {1 \pm |\nabla u{|^2}} }}} \right) = g(u){\text{ }}\;\;\;\;{\text{in }}{\mathbb{R}^N},$
both in the Euclidean case, with the sign "+", and in the Lorentz-Minkowski case, with the sign "-", for N ≥ 1 under the assumption g'(0)>0. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if N = 1, while they are radial symmetric and decay to zero at infinity with their derivatives, if N≥ 2.
Citation: Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169
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, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1122-1140. doi: 10.1016/j.matpur.2016.04.003.

[3]

A. AzzolliniP. d'Avenia and A. Pomponio, Quasilinear elliptic equations in $\mathbb{R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213. doi: 10.1007/s00526-012-0578-0.

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[5]

H. BerestyckiP. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.

[6]

D. BonheureA. Derlet and C. De Coster, Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 259-284.

[7]

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.

[8]

M. Born and L. Infeld, Foundations of the new field theory, Nature, 132 (1933), 1004.

[9]

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

[10]

M. Conti and F. Gazzola, Existence of ground states and free-boundary problems for the prescribed mean-curvature equation, Adv. Differential Equations, 7 (2002), 667-694.

[11]

C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, Università degli Studi di Trieste, Trieste, 2015.

[12]

M. del Pino and I. Guerra, Ground states of a prescribed mean curvature equation, J. Differential Equations, 241 (2007), 112-129. doi: 10.1016/j.jde.2007.06.010.

[13]

G. Evequoz, A dual approach in Orlicz spaces for the nonlinear Helmholtz equation, Z. Angew. Math. Phys., 66 (2015), 2995-3015. doi: 10.1007/s00033-015-0572-4.

[14]

G. Evequoz and T. Weth, Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation, Journal of Fixed Point Theory and Applications, 19 (2017), 475-502. doi: 10.1007/s11784-016-0362-4.

[15]

G. Evequoz and T. Weth, Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Ration. Mech. Anal., 211 (2014), 359-388. doi: 10.1007/s00205-013-0664-2.

[16]

G. Evequoz and T. Weth, Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728. doi: 10.1016/j.aim.2015.04.017.

[17]

D. FortunatoL. Orsina and L. Pisani, Born-Infeld type equations for electrostatic fields, J. Math. Phys., 43 (2002), 5698-5706. doi: 10.1063/1.1508433.

[18]

B. FranchiE. Lanconelli and J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equations in $\mathbb{R}^n$, Adv. Math., 118 (1996), 177-243. doi: 10.1006/aima.1996.0021.

[19]

N. FukagaiM. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}N$, Funkcial. Ekvac., 49 (2006), 235-267. doi: 10.1619/fesi.49.235.

[20]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-295. doi: 10.4310/MAA.2008.v15.n3.a3.

[21]

T. Kusano and C. A. Swanson, Radial entire solutions of a class of quasilinear elliptic equations, J. Differential Equations, 83 (1990), 379-399. doi: 10.1016/0022-0396(90)90064-V.

[22]

R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz Equations Z. Angew. Math. Phys., 68 (2017), Art. 121, 19 pp. doi: 10.1007/s00033-017-0859-8.

[23]

W.-M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, 5 (1985), 171-185.

[24]

W.-M. Ni and J. Serrin, Existence and Non-existence theorems for quasi-linear partial differential equations, The anomalous case, Accad. Naz. Lincei, Convegni Dei Lincei, 77 (1986), 231-257.

[25]

L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700. doi: 10.1090/S0002-9939-1987-0894440-8.

[26]

A. Pomponio and T. Watanabe, Some quasilinear elliptic equations involving multiple $p$-Laplacians, to appear on Indiana Univ. Math. J.

[27]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

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, Journal de Mathématiques Pures et Appliquées, 106 (2016), 1122-1140. doi: 10.1016/j.matpur.2016.04.003.

[3]

A. AzzolliniP. d'Avenia and A. Pomponio, Quasilinear elliptic equations in $\mathbb{R}^N$ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations, 49 (2014), 197-213. doi: 10.1007/s00526-012-0578-0.

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.

[5]

H. BerestyckiP. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $\mathbb{R}^N$, Indiana Univ. Math. J., 30 (1981), 141-157. doi: 10.1512/iumj.1981.30.30012.

[6]

D. BonheureA. Derlet and C. De Coster, Infinitely many radial solutions of a mean curvature equation in Lorentz-Minkowski space, Rend. Istit. Mat. Univ. Trieste, 44 (2012), 259-284.

[7]

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.

[8]

M. Born and L. Infeld, Foundations of the new field theory, Nature, 132 (1933), 1004.

[9]

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

[10]

M. Conti and F. Gazzola, Existence of ground states and free-boundary problems for the prescribed mean-curvature equation, Adv. Differential Equations, 7 (2002), 667-694.

[11]

C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, Università degli Studi di Trieste, Trieste, 2015.

[12]

M. del Pino and I. Guerra, Ground states of a prescribed mean curvature equation, J. Differential Equations, 241 (2007), 112-129. doi: 10.1016/j.jde.2007.06.010.

[13]

G. Evequoz, A dual approach in Orlicz spaces for the nonlinear Helmholtz equation, Z. Angew. Math. Phys., 66 (2015), 2995-3015. doi: 10.1007/s00033-015-0572-4.

[14]

G. Evequoz and T. Weth, Branch continuation inside the essential spectrum for the nonlinear Schrödinger equation, Journal of Fixed Point Theory and Applications, 19 (2017), 475-502. doi: 10.1007/s11784-016-0362-4.

[15]

G. Evequoz and T. Weth, Real solutions to the nonlinear Helmholtz equation with local nonlinearity, Arch. Ration. Mech. Anal., 211 (2014), 359-388. doi: 10.1007/s00205-013-0664-2.

[16]

G. Evequoz and T. Weth, Dual variational methods and nonvanishing for the nonlinear Helmholtz equation, Adv. Math., 280 (2015), 690-728. doi: 10.1016/j.aim.2015.04.017.

[17]

D. FortunatoL. Orsina and L. Pisani, Born-Infeld type equations for electrostatic fields, J. Math. Phys., 43 (2002), 5698-5706. doi: 10.1063/1.1508433.

[18]

B. FranchiE. Lanconelli and J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equations in $\mathbb{R}^n$, Adv. Math., 118 (1996), 177-243. doi: 10.1006/aima.1996.0021.

[19]

N. FukagaiM. Ito and K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}N$, Funkcial. Ekvac., 49 (2006), 235-267. doi: 10.1619/fesi.49.235.

[20]

C. Gui and F. Zhou, Asymptotic behavior of oscillating radial solutions to certain nonlinear equations, Methods Appl. Anal., 15 (2008), 285-295. doi: 10.4310/MAA.2008.v15.n3.a3.

[21]

T. Kusano and C. A. Swanson, Radial entire solutions of a class of quasilinear elliptic equations, J. Differential Equations, 83 (1990), 379-399. doi: 10.1016/0022-0396(90)90064-V.

[22]

R. Mandel, E. Montefusco and B. Pellacci, Oscillating solutions for nonlinear Helmholtz Equations Z. Angew. Math. Phys., 68 (2017), Art. 121, 19 pp. doi: 10.1007/s00033-017-0859-8.

[23]

W.-M. Ni and J. Serrin, Non-existence theorems for quasilinear partial differential equations, Rend. Circ. Mat. Palermo, 5 (1985), 171-185.

[24]

W.-M. Ni and J. Serrin, Existence and Non-existence theorems for quasi-linear partial differential equations, The anomalous case, Accad. Naz. Lincei, Convegni Dei Lincei, 77 (1986), 231-257.

[25]

L. A. Peletier and J. Serrin, Ground states for the prescribed mean curvature equation, Proc. Amer. Math. Soc., 100 (1987), 694-700. doi: 10.1090/S0002-9939-1987-0894440-8.

[26]

A. Pomponio and T. Watanabe, Some quasilinear elliptic equations involving multiple $p$-Laplacians, to appear on Indiana Univ. Math. J.

[27]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

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