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, 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

[1]

Pietro d’Avenia, Lorenzo Pisani, Gaetano Siciliano. Klein-Gordon-Maxwell systems in a bounded domain. Discrete & Continuous Dynamical Systems, 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, 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, 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]

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

[8]

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

[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, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117

[12]

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

[13]

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

[14]

Hartmut Pecher. Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3303-3321. doi: 10.3934/cpaa.2020146

[15]

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

[16]

Hartmut Pecher. Improved well-posedness results for the Maxwell-Klein-Gordon system in 2D. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021091

[17]

Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292

[18]

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

[19]

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

[20]

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

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (6)

[Back to Top]