# American Institute of Mathematical Sciences

June  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 and Continuous Dynamical Systems, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535
##### References:
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Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations, Topol. Methods Nonlinear Anal., 35 (2010), 33-42. [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), 303-339. [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. [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), 409-420. 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), 243-279. 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), 733-747. 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), 267-288. 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), 893-906. 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), 307-322. [15] T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. 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), 605-628. 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, 13 pp. [18] P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell system in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149. 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), e1985-e1995. doi: 10.1016/j.na.2009.02.111. [20] E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D., 18 (1986), 371-373. 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), 831-869. 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, arXiv:1303.6498. [23] M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems, in press, Proceedings of the Workshop on Nonlinear Differential Equations, 2012, arXiv:1303.6627. [24] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. [25] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 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), 573-595. [27] H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. 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), 19-44. doi: 10.1215/S0012-7094-94-07402-4. [29] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. 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), 123-136. 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, 697-734. 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), 233-265. 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), 3277-3284. 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), 1519-1527. 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), 521-528. 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), 141-164. doi: 10.1142/S0218202505003939. [37] G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. 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 $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816. doi: 10.3934/dcds.2007.18.809.

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##### 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), 1131-1161. doi: 10.1017/S0308210500027268. [2] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^{N}$, Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404. 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), 779-791. 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), 90-108. 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), 33-42. [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), 303-339. [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. [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), 409-420. 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), 243-279. 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), 733-747. 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), 267-288. 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), 893-906. 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), 307-322. [15] T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. 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), 605-628. 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, 13 pp. [18] P. D'Avenia, L. Pisani and G. Siciliano, Klein-Gordon-Maxwell system in a bounded domain, Discrete Contin. Dyn. Syst., 26 (2010), 135-149. 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), e1985-e1995. doi: 10.1016/j.na.2009.02.111. [20] E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations, Phys. D., 18 (1986), 371-373. 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), 831-869. 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, arXiv:1303.6498. [23] M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems, in press, Proceedings of the Workshop on Nonlinear Differential Equations, 2012, arXiv:1303.6627. [24] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{N}$, in Mathematical Analysis and Applications, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. [25] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 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), 573-595. [27] H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. 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), 19-44. doi: 10.1215/S0012-7094-94-07402-4. [29] M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbb{R}^{N}$, Arch. Rational Mech. Anal., 105 (1989), 243-266. 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), 123-136. 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, 697-734. 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), 233-265. 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), 3277-3284. 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), 1519-1527. 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), 521-528. 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), 141-164. doi: 10.1142/S0218202505003939. [37] G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. 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 $\mathbb{R}^3$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816. doi: 10.3934/dcds.2007.18.809.
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