September  2016, 15(5): 1781-1795. doi: 10.3934/cpaa.2016014

Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations

1. 

School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, China

Received  October 2015 Revised  January 2016 Published  July 2016

In the present paper, we consider the following magnetic nonlinear Choquard equation \begin{eqnarray} \big(-i\nabla+A(x)\big)^{2}u+\big( g_{0}(x)+\mu g(x)\big)u=\big(|x|^{-\alpha}*|u|^{p}\big)|u|^{p-2}u,\\ u\in H^{1}(\mathbb{R}^{N},\mathbb{C}), \end{eqnarray} where $N\geq 3$, $\alpha\in (0,N)$, $p\in(\frac{2N-\alpha}{N}, \frac{2N-\alpha}{N-2})$, $A(x): {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}^{N}$ is a magnetic vector potential, $\mu>0$ is a parameter, $g_{0}(x)$ and $g(x)$ are real valued electric potential functions on ${\mathbb{R}}^{N}$. Under some suitable conditions, we show that there exists $\mu^{*}>0$ such that the above equation has at least one ground state solution for $\mu\geq\mu^{*}$. Moreover, the concentration behavior of solutions is also studied as $\mu\rightarrow +\infty$.
Citation: Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014
References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, \emph{Math. Z.}, 248 (2004), 423. doi: 10.1007/s00209-004-0663-y.

[2]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152.

[3]

N. Ba, Y. Deng and S. Peng, Multi-peak bound states for Schröinger equations with compactly supported or unbounded potentials,, \emph{Ann. I. H. Poincar\'e-AN}, 27 (2010), 1205. doi: 10.1016/j.anihpc.2010.05.003.

[4]

S. Barile, A multiplicity result for singular NLS equations with magnetic potentials,, \emph{Nonlinear Anal.}, 68 (2008), 3525. doi: 10.1016/j.na.2007.03.044.

[5]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$,, \emph{Comm. Partial Differential Equations}, 20 (1995), 1725. doi: 10.1080/03605309508821149.

[6]

D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields,, \emph{J. Differential Equations}, 222 (2006), 381. doi: 10.1016/j.jde.2005.06.027.

[7]

S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field,, \emph{J. Differential Equations}, 188 (2003), 52. doi: 10.1016/S0022-0396(02)00058-X.

[8]

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation,, \emph{Z. Angew. Math. Phys.}, 63 (2012), 233. doi: 10.1007/s00033-011-0166-8.

[9]

S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 6 (2013), 891.

[10]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, \emph{J. Differential Equations}, 160 (2000), 118. doi: 10.1006/jdeq.1999.3662.

[11]

S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields,, \emph{J. Math. Anal. Appl.}, 275 (2002), 108. doi: 10.1016/S0022-247X(02)00278-0.

[12]

S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 140 (2010), 973. doi: 10.1017/S0308210509000584.

[13]

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation,, \emph{J. Math. Anal. Appl.}, 407 (2013), 1. doi: 10.1016/j.jmaa.2013.04.081.

[14]

Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields,, \emph{Ann. Mat. Pura Appl.}, 190 (2011), 427. doi: 10.1007/s10231-010-0157-y.

[15]

G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields,, \emph{J. Differential Equations}, 251 (2011), 3500. doi: 10.1016/j.jde.2011.08.038.

[16]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, \emph{Stud. Appl. Math.}, 57 (1977), 93.

[17]

E. H. Lieb and M. Loss, Analysis,, 2nd Edition, (2001). doi: 10.1090/gsm/014.

[18]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, \emph{Comm. Math. Phys.}, 53 (1977), 185.

[19]

P. L. Lions, The Choquard equation and related questions,, \emph{Nonlinear Anal.}, 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4.

[20]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 109.

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 455. doi: 10.1007/s00205-008-0208-3.

[22]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics,, \emph{J. Funct. Anal.}, 265 (2013), 153. doi: 10.1016/j.jfa.2013.04.007.

[23]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 1411. doi: 10.3934/cpaa.2010.9.1411.

[24]

R. Penrose, On gravity's role in quantum state reduction,, \emph{Gen. Relativity Gravitation}, 28 (1996), 581. doi: 10.1007/BF02105068.

[25]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential,, \emph{Nonlinear Anal.}, 72 (2010), 3842. doi: 10.1016/j.na.2010.01.021.

[26]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations,, \emph{J. Math. Phys.}, 50 (2009). doi: 10.1063/1.3060169.

[27]

M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities,, \emph{J. Math. Anal. Appl.}, 403 (2013), 680. doi: 10.1016/j.jmaa.2013.02.062.

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part,, \emph{Math. Z.}, 248 (2004), 423. doi: 10.1007/s00209-004-0663-y.

[2]

A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 253. doi: 10.1007/s002050100152.

[3]

N. Ba, Y. Deng and S. Peng, Multi-peak bound states for Schröinger equations with compactly supported or unbounded potentials,, \emph{Ann. I. H. Poincar\'e-AN}, 27 (2010), 1205. doi: 10.1016/j.anihpc.2010.05.003.

[4]

S. Barile, A multiplicity result for singular NLS equations with magnetic potentials,, \emph{Nonlinear Anal.}, 68 (2008), 3525. doi: 10.1016/j.na.2007.03.044.

[5]

T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$,, \emph{Comm. Partial Differential Equations}, 20 (1995), 1725. doi: 10.1080/03605309508821149.

[6]

D. Cao and Z. Tang, Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields,, \emph{J. Differential Equations}, 222 (2006), 381. doi: 10.1016/j.jde.2005.06.027.

[7]

S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field,, \emph{J. Differential Equations}, 188 (2003), 52. doi: 10.1016/S0022-0396(02)00058-X.

[8]

S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation,, \emph{Z. Angew. Math. Phys.}, 63 (2012), 233. doi: 10.1007/s00033-011-0166-8.

[9]

S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system,, \emph{Discrete Contin. Dyn. Syst. Ser. S}, 6 (2013), 891.

[10]

S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,, \emph{J. Differential Equations}, 160 (2000), 118. doi: 10.1006/jdeq.1999.3662.

[11]

S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields,, \emph{J. Math. Anal. Appl.}, 275 (2002), 108. doi: 10.1016/S0022-247X(02)00278-0.

[12]

S. Cingolani, S. Secchi and M. Squassina, Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 140 (2010), 973. doi: 10.1017/S0308210509000584.

[13]

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation,, \emph{J. Math. Anal. Appl.}, 407 (2013), 1. doi: 10.1016/j.jmaa.2013.04.081.

[14]

Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields,, \emph{Ann. Mat. Pura Appl.}, 190 (2011), 427. doi: 10.1007/s10231-010-0157-y.

[15]

G. Li, S. Peng and C. Wang, Infinitely many solutions for nonlinear Schrödinger equations with electromagnetic fields,, \emph{J. Differential Equations}, 251 (2011), 3500. doi: 10.1016/j.jde.2011.08.038.

[16]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, \emph{Stud. Appl. Math.}, 57 (1977), 93.

[17]

E. H. Lieb and M. Loss, Analysis,, 2nd Edition, (2001). doi: 10.1090/gsm/014.

[18]

E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, \emph{Comm. Math. Phys.}, 53 (1977), 185.

[19]

P. L. Lions, The Choquard equation and related questions,, \emph{Nonlinear Anal.}, 4 (1980), 1063. doi: 10.1016/0362-546X(80)90016-4.

[20]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. Part I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 109.

[21]

L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation,, \emph{Arch. Ration. Mech. Anal.}, 195 (2010), 455. doi: 10.1007/s00205-008-0208-3.

[22]

V. Moroz and J. Van Schaftingen, Ground states of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics,, \emph{J. Funct. Anal.}, 265 (2013), 153. doi: 10.1016/j.jfa.2013.04.007.

[23]

M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential,, \emph{Commun. Pure Appl. Anal.}, 9 (2010), 1411. doi: 10.3934/cpaa.2010.9.1411.

[24]

R. Penrose, On gravity's role in quantum state reduction,, \emph{Gen. Relativity Gravitation}, 28 (1996), 581. doi: 10.1007/BF02105068.

[25]

S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential,, \emph{Nonlinear Anal.}, 72 (2010), 3842. doi: 10.1016/j.na.2010.01.021.

[26]

J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations,, \emph{J. Math. Phys.}, 50 (2009). doi: 10.1063/1.3060169.

[27]

M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities,, \emph{J. Math. Anal. Appl.}, 403 (2013), 680. doi: 10.1016/j.jmaa.2013.02.062.

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