# American Institute of Mathematical Sciences

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. Azzollini, P. 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. Berestycki, P. 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. Bonheure, A. 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. 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. [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. Fortunato, L. 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. Franchi, E. 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. Fukagai, M. 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. Azzollini, P. 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. Berestycki, P. 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. Bonheure, A. 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. 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. [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. Fortunato, L. 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. Franchi, E. 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. Fukagai, M. 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.
 [1] Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155 [2] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [3] Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367 [4] Chiara Corsato, Colette De Coster, Franco Obersnel, Pierpaolo Omari, Alessandro Soranzo. A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 213-256. doi: 10.3934/dcdss.2018013 [5] Franco Obersnel, Pierpaolo Omari. On a result of C.V. Coffman and W.K. Ziemer about the prescribed mean curvature equation. Conference Publications, 2011, 2011 (Special) : 1138-1147. doi: 10.3934/proc.2011.2011.1138 [6] Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441 [7] Chiara Corsato, Colette De Coster, Pierpaolo Omari. Radially symmetric solutions of an anisotropic mean curvature equation modeling the corneal shape. Conference Publications, 2015, 2015 (special) : 297-303. doi: 10.3934/proc.2015.0297 [8] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [9] Kin Ming Hui. Existence of self-similar solutions of the inverse mean curvature flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 863-880. doi: 10.3934/dcds.2019036 [10] Yuxia Guo, Jianjun Nie. Infinitely many non-radial solutions for the prescribed curvature problem of fractional operator. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6873-6898. doi: 10.3934/dcds.2016099 [11] Jinju Xu. A new proof of gradient estimates for mean curvature equations with oblique boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1719-1742. doi: 10.3934/cpaa.2016010 [12] Matteo Novaga, Enrico Valdinoci. Closed curves of prescribed curvature and a pinning effect. Networks & Heterogeneous Media, 2011, 6 (1) : 77-88. doi: 10.3934/nhm.2011.6.77 [13] Yong Huang, Lu Xu. Two problems related to prescribed curvature measures. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1975-1986. doi: 10.3934/dcds.2013.33.1975 [14] Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615 [15] Christophe Cheverry, Mekki Houbad. A class of large amplitude oscillating solutions for three dimensional Euler equations. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1661-1697. doi: 10.3934/cpaa.2012.11.1661 [16] Türker Özsarı. Blow-up of solutions of nonlinear Schrödinger equations with oscillating nonlinearities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 539-558. doi: 10.3934/cpaa.2019027 [17] Diego Castellaneta, Alberto Farina, Enrico Valdinoci. A pointwise gradient estimate for solutions of singular and degenerate pde's in possibly unbounded domains with nonnegative mean curvature. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1983-2003. doi: 10.3934/cpaa.2012.11.1983 [18] Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-18. doi: 10.3934/dcdsb.2018271 [19] Brittany Froese Hamfeldt. Convergent approximation of non-continuous surfaces of prescribed Gaussian curvature. Communications on Pure & Applied Analysis, 2018, 17 (2) : 671-707. doi: 10.3934/cpaa.2018036 [20] Neil S. Trudinger. On the local theory of prescribed Jacobian equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1663-1681. doi: 10.3934/dcds.2014.34.1663

2017 Impact Factor: 1.179