# 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 & Continuous Dynamical Systems - A, 2014, 34 (6) : 2535-2560. doi: 10.3934/dcds.2014.34.2535
##### References:
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Appl., 20 (2009), 243. doi: 10.4171/RLM/546. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar [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. Google Scholar [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. Google Scholar [17] P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 (). Google Scholar [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. Google Scholar [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. Google Scholar [20] E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations,, Phys. D., 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0. Google Scholar [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. Google Scholar [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, (). Google Scholar [23] M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems,, in press, (2012). Google Scholar [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. Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Grundlehren der Mathematischen Wissenschaften, (1977). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. doi: 10.1017/S0308210500027268. Google Scholar [2] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^n$,, Progress in Mathematics, (2006). Google Scholar [3] A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem,, Commun. Contemp. Math., 10 (2008), 391. doi: 10.1142/S021919970800282X. Google Scholar [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. Google Scholar [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. Google Scholar [6] A. Azzollini and A. Pomponio, Ground state solutions for the nonlinear Klein-Gordon-Maxwell equations,, Topol. Methods Nonlinear Anal., 35 (2010), 33. Google Scholar [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. Google Scholar [8] V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations,, Topol. Methods Nonlinear Anal., 11 (1998), 283. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein-Gordon-Maxwell equations,, Adv. Nonlinear Stud., 4 (2004), 307. Google Scholar [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. Google Scholar [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. Google Scholar [17] P. D'Avenia and L. Pisani, Nonlinear Klein-Gordon equations coupled with Born-Infeld type equations,, Electron. J. Differential Equations, 2002 (). Google Scholar [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. Google Scholar [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. Google Scholar [20] E. Deumens, The Klein-Gordon-Maxwell nonlinear systems of equations,, Phys. D., 18 (1986), 371. doi: 10.1016/0167-2789(86)90201-0. Google Scholar [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. Google Scholar [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, (). Google Scholar [23] M. Ghimenti and A. M. Micheletti, Low energy solutions for the semiclassical limit of Schrödinger Maxwell systems,, in press, (2012). Google Scholar [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. Google Scholar [25] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order,, Grundlehren der Mathematischen Wissenschaften, (1977). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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