2014, 34(6): 2535-2560. doi: 10.3934/dcds.2014.34.2535

The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

1. 

Dipartimento di Matematica Applicata, Università di Pisa, Via F. Buonarroti 1/c, 56127 Pisa

2. 

Dipartimento di Matematica, Università di Pisa, via F. Buonarroti 1/c, 56127 Pisa, Italy

3. 

Dipartimento SBAI, Università di Roma "La Sapienza", via Antonio Scarpa 16, 00161 Roma, Italy

Received  February 2013 Revised  June 2013 Published  December 2013

Given a 3-dimensional Riemannian manifold $(M,g)$, we investigate the existence of positive solutions of the Klein-Gordon-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+au=u^{p-1}+\omega^{2}(qv-1)^{2}u & \text{in }M\\ -\Delta_{g}v+(1+q^{2}u^{2})v=qu^{2} & \text{in }M \end{array}\right. $$ and Schrödinger-Maxwell system $$ \left\{ \begin{array}{cc} -\varepsilon^{2}\Delta_{g}u+u+\omega uv=u^{p-1} & \text{in }M\\ -\Delta_{g}v+v=qu^{2} & \text{in }M \end{array}\right. $$ when $p\in(2,6). $ We prove that if $\varepsilon$ is small enough, any stable critical point $\xi_0$ of the scalar curvature of $g$ generates a positive solution $(u_\varepsilon,v_\varepsilon)$ to both the systems such that $u_\varepsilon$ concentrates at $\xi_0$ as $\varepsilon$ goes to zero.
Citation: Marco Ghimenti, Anna Maria Micheletti, Angela Pistoia. The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535
References:
[1]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum,, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1131. doi: 10.1017/S0308210500027268.

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^n$,, Progress in Mathematics, (2006).

[3]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X.

[4]

A. Azzollini, P. D'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779. doi: 10.1016/j.anihpc.2009.11.012.

[5]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057.

[6]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, Topol. Methods Nonlinear Anal., 35 (2010), 33.

[7]

J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations,, Proc. London Math. Soc., 107 (2013), 303.

[8]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283.

[9]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409. doi: 10.1142/S0129055X02001168.

[10]

V. Benci and D. Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 20 (2009), 243. doi: 10.4171/RLM/546.

[11]

D. Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations,, Nonlinear Anal., 58 (2004), 733. doi: 10.1016/j.na.2003.05.001.

[12]

Y. Choquet-Bruhat, Solution globale des équations de Maxwell-Dirac-Klein-Gordon,, Rend. Circ. Mat. Palermo (2), 31 (1982), 267. doi: 10.1007/BF02844359.

[13]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893. doi: 10.1017/S030821050000353X.

[14]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307.

[15]

T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball,, J. Differential Equations, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009.

[16]

T. D'Aprile and J. Wei, Clustered solutions around harmonic centers to a coupled elliptic system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 605. doi: 10.1016/j.anihpc.2006.04.003.

[17]

P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 ().

[18]

P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell system in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. doi: 10.3934/dcds.2010.26.135.

[19]

P. D'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.02.111.

[20]

E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations,, Phys. D., 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0.

[21]

O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831. doi: 10.1142/S0219199710004007.

[22]

M. Ghimenti and A. M. Micheletti, Number and profile of low energy solutions for singularly perturbed Klein Gordon Maxwell systems on a Riemannian manifold,, preprint, ().

[23]

M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems,, in press, (2012).

[24]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$,, in Mathematical Analysis and Applications, (1981), 369.

[25]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Grundlehren der Mathematischen Wissenschaften, (1977).

[26]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573.

[27]

H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal., 67 (2007), 1445. doi: 10.1016/j.na.2006.07.029.

[28]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4.

[29]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[30]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Comm. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0.

[31]

N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger,, Int. Math. Res. Not., 2003 (): 697. doi: 10.1155/S107379280320310X.

[32]

A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds,, Calc. Var. Partial Differential Equations, 34 (2009), 233. doi: 10.1007/s00526-008-0183-4.

[33]

A. M. Micheletti and A. Pistoia, Generic properties of critical points of the scalar curvature for a Riemannian manifold,, Proc. Amer. Math. Soc., 138 (2010), 3277. doi: 10.1090/S0002-9939-10-10382-7.

[34]

D. Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: Looking for solitary waves,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1519. doi: 10.1098/rspa.2003.1267.

[35]

L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain,, Appl. Math. Lett., 21 (2008), 521. doi: 10.1016/j.aml.2007.06.005.

[36]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. doi: 10.1142/S0218202505003939.

[37]

G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system,, J. Math. Anal. Appl., 365 (2010), 288. doi: 10.1016/j.jmaa.2009.10.061.

[38]

Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbbR^3$,, Discrete Contin. Dyn. Syst., 18 (2007), 809. doi: 10.3934/dcds.2007.18.809.

show all references

References:
[1]

A. Ambrosetti and M. Badiale, Variational perturbative methods and bifurcation of bound states from the essential spectrum,, Proc. Roy. Soc. Edinburgh Sect. A, 128 (1998), 1131. doi: 10.1017/S0308210500027268.

[2]

A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^n$,, Progress in Mathematics, (2006).

[3]

A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X.

[4]

A. Azzollini, P. D'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 779. doi: 10.1016/j.anihpc.2009.11.012.

[5]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations,, J. Math. Anal. Appl., 345 (2008), 90. doi: 10.1016/j.jmaa.2008.03.057.

[6]

A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, Topol. Methods Nonlinear Anal., 35 (2010), 33.

[7]

J. Bellazzini, L. Jeanjean and T. Luo, Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations,, Proc. London Math. Soc., 107 (2013), 303.

[8]

V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283.

[9]

V. Benci and D. Fortunato, Solitary waves of the nonlinear Klein-Gordon field equation coupled with the Maxwell equations,, Rev. Math. Phys., 14 (2002), 409. doi: 10.1142/S0129055X02001168.

[10]

V. Benci and D. Fortunato, Existence of hylomorphic solitary waves in Klein-Gordon and in Klein-Gordon-Maxwell equations,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 20 (2009), 243. doi: 10.4171/RLM/546.

[11]

D. Cassani, Existence and non-existence of solitary waves for the critical Klein-Gordon equation coupled with Maxwell's equations,, Nonlinear Anal., 58 (2004), 733. doi: 10.1016/j.na.2003.05.001.

[12]

Y. Choquet-Bruhat, Solution globale des équations de Maxwell-Dirac-Klein-Gordon,, Rend. Circ. Mat. Palermo (2), 31 (1982), 267. doi: 10.1007/BF02844359.

[13]

T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 893. doi: 10.1017/S030821050000353X.

[14]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307.

[15]

T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball,, J. Differential Equations, 226 (2006), 269. doi: 10.1016/j.jde.2005.12.009.

[16]

T. D'Aprile and J. Wei, Clustered solutions around harmonic centers to a coupled elliptic system,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 605. doi: 10.1016/j.anihpc.2006.04.003.

[17]

P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 ().

[18]

P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell system in a bounded domain,, Discrete Contin. Dyn. Syst., 26 (2010), 135. doi: 10.3934/dcds.2010.26.135.

[19]

P. D'Avenia, L. Pisani and G. Siciliano, Dirichlet and Neumann problems for Klein-Gordon-Maxwell systems,, Nonlinear Anal., 71 (2009). doi: 10.1016/j.na.2009.02.111.

[20]

E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations,, Phys. D., 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0.

[21]

O. Druet and E. Hebey, Existence and a priori bounds for electrostatic Klein-Gordon-Maxwell systems in fully inhomogeneous spaces,, Commun. Contemp. Math., 12 (2010), 831. doi: 10.1142/S0219199710004007.

[22]

M. Ghimenti and A. M. Micheletti, Number and profile of low energy solutions for singularly perturbed Klein Gordon Maxwell systems on a Riemannian manifold,, preprint, ().

[23]

M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems,, in press, (2012).

[24]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbbR^n$,, in Mathematical Analysis and Applications, (1981), 369.

[25]

D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Grundlehren der Mathematischen Wissenschaften, (1977).

[26]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials,, Adv. Nonlinear Stud., 8 (2008), 573.

[27]

H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations,, Nonlinear Anal., 67 (2007), 1445. doi: 10.1016/j.na.2006.07.029.

[28]

S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy,, Duke Math. J., 74 (1994), 19. doi: 10.1215/S0012-7094-94-07402-4.

[29]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi: 10.1007/BF00251502.

[30]

N. Masmoudi and K. Nakanishi, Uniqueness of finite energy solutions for Maxwell-Dirac and Maxwell-Klein-Gordon equations,, Comm. Math. Phys., 243 (2003), 123. doi: 10.1007/s00220-003-0951-0.

[31]

N. Masmoudi and K. Nakanishi, Nonrelativistic limit from Maxwell-Klein-Gordon and Maxwell-Dirac to Poisson-Schrödinger,, Int. Math. Res. Not., 2003 (): 697. doi: 10.1155/S107379280320310X.

[32]

A. M. Micheletti and A. Pistoia, The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds,, Calc. Var. Partial Differential Equations, 34 (2009), 233. doi: 10.1007/s00526-008-0183-4.

[33]

A. M. Micheletti and A. Pistoia, Generic properties of critical points of the scalar curvature for a Riemannian manifold,, Proc. Amer. Math. Soc., 138 (2010), 3277. doi: 10.1090/S0002-9939-10-10382-7.

[34]

D. Mugnai, Coupled Klein-Gordon and Born-Infeld-type equations: Looking for solitary waves,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460 (2004), 1519. doi: 10.1098/rspa.2003.1267.

[35]

L. Pisani and G. Siciliano, Note on a Schrödinger-Poisson system in a bounded domain,, Appl. Math. Lett., 21 (2008), 521. doi: 10.1016/j.aml.2007.06.005.

[36]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere,, Math. Models Methods Appl. Sci., 15 (2005), 141. doi: 10.1142/S0218202505003939.

[37]

G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system,, J. Math. Anal. Appl., 365 (2010), 288. doi: 10.1016/j.jmaa.2009.10.061.

[38]

Z. Wang and H.-S. Zhou, Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbbR^3$,, Discrete Contin. Dyn. Syst., 18 (2007), 809. doi: 10.3934/dcds.2007.18.809.

[1]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 135-149. doi: 10.3934/dcds.2010.26.135

[2]

Sitong Chen, Xianhua Tang. Improved results for Klein-Gordon-Maxwell systems with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2333-2348. doi: 10.3934/dcds.2018096

[3]

Pierre-Damien Thizy. Klein-Gordon-Maxwell equations in high dimensions. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1097-1125. doi: 10.3934/cpaa.2015.14.1097

[4]

Percy D. Makita. Nonradial solutions for the Klein-Gordon-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2271-2283. doi: 10.3934/dcds.2012.32.2271

[5]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201

[6]

Paulo Cesar Carrião, Patrícia L. Cunha, Olímpio Hiroshi Miyagaki. Existence results for the Klein-Gordon-Maxwell equations in higher dimensions with critical exponents. Communications on Pure & Applied Analysis, 2011, 10 (2) : 709-718. doi: 10.3934/cpaa.2011.10.709

[7]

Pingzheng Zhang, Jianhua Sun. Clustered layers for the Schrödinger-Maxwell system on a ball. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 657-688. doi: 10.3934/dcds.2006.16.657

[8]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[9]

Ahmed Y. Abdallah. Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (1) : 55-69. doi: 10.3934/cpaa.2006.5.55

[10]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739

[11]

Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117

[12]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034

[13]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669

[14]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573

[15]

Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

[16]

Victor Wasiolek. Uniform global existence and convergence of Euler-Maxwell systems with small parameters. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2007-2021. doi: 10.3934/cpaa.2016025

[17]

Jianjun Yuan. Global solutions of two coupled Maxwell systems in the temporal gauge. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1709-1719. doi: 10.3934/dcds.2016.36.1709

[18]

Anthony M. Bloch, Rohit Gupta, Ilya V. Kolmanovsky. Neighboring extremal optimal control for mechanical systems on Riemannian manifolds. Journal of Geometric Mechanics, 2016, 8 (3) : 257-272. doi: 10.3934/jgm.2016007

[19]

Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002

[20]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

[Back to Top]