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

Marco Ghimenti; Anna Maria Micheletti; Angela Pistoia

doi:10.3934/dcds.2014.34.2535

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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

Marco Ghimenti ^1, , Anna Maria Micheletti ^2, and Angela Pistoia ^3,

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

Abstract
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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.

Keywords: scalar curvature, Lyapunov-Schmidt reduction., Klein-Gordon-Maxwell systems, Scrhödinger-Maxwell systems, Riemannian manifolds.

Mathematics Subject Classification: 35J60, 35J20, 35B40, 58E30, 81V1.

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, 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), 1131-1161. doi: 10.1017/S0308210500027268. Google Scholar
[2]	A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^n$, Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. Google Scholar
[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. 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), 779-791. 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), 90-108. 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), 33-42. 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), 303-339. Google Scholar
[8]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. 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), 409-420. 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), 243-279. 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), 733-747. 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), 267-288. 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), 893-906. 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), 307-322. Google Scholar
[15]	T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. 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), 605-628. 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), 135-149. 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), e1985-e1995. 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), 371-373. 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), 831-869. 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, Proceedings of the Workshop on Nonlinear Differential Equations, 2012, arXiv:1303.6627. 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, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. Google Scholar
[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. 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), 573-595. Google Scholar
[27]	H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. 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), 19-44. 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), 243-266. 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), 123-136. 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), 233-265. 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), 3277-3284. 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), 1519-1527. 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), 521-528. 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), 141-164. doi: 10.1142/S0218202505003939. Google Scholar
[37]	G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. 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), 809-816. doi: 10.3934/dcds.2007.18.809. Google Scholar

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), 1131-1161. doi: 10.1017/S0308210500027268. Google Scholar
[2]	A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbbR^n$, Progress in Mathematics, 240, Birkhäuser Verlag, Basel, 2006. Google Scholar
[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. 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), 779-791. 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), 90-108. 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), 33-42. 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), 303-339. Google Scholar
[8]	V. Benci and D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293. 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), 409-420. 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), 243-279. 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), 733-747. 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), 267-288. 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), 893-906. 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), 307-322. Google Scholar
[15]	T. D'Aprile and J. Wei, Layered solutions for a semilinear elliptic system in a ball, J. Differential Equations, 226 (2006), 269-294. 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), 605-628. 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), 135-149. 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), e1985-e1995. 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), 371-373. 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), 831-869. 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, Proceedings of the Workshop on Nonlinear Differential Equations, 2012, arXiv:1303.6627. 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, Part A, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981, 369-402. Google Scholar
[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. 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), 573-595. Google Scholar
[27]	H. Kikuchi, On the existence of solutions for a elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 67 (2007), 1445-1456. 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), 19-44. 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), 243-266. 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), 123-136. 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), 233-265. 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), 3277-3284. 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), 1519-1527. 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), 521-528. 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), 141-164. doi: 10.1142/S0218202505003939. Google Scholar
[37]	G. Siciliano, Multiple positive solutions for a Schrödinger-Poisson-Slater system, J. Math. Anal. Appl., 365 (2010), 288-299. 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), 809-816. doi: 10.3934/dcds.2007.18.809. Google Scholar

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

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