# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, \emph{Stud. Appl. Math.}, 57 (1977), 93.   Google Scholar [17] E. H. Lieb and M. Loss, Analysis,, 2nd Edition, (2001).  doi: 10.1090/gsm/014.  Google Scholar [18] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, \emph{Comm. Math. Phys.}, 53 (1977), 185.   Google Scholar [19] P. L. Lions, The Choquard equation and related questions,, \emph{Nonlinear Anal.}, 4 (1980), 1063.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] R. Penrose, On gravity's role in quantum state reduction,, \emph{Gen. Relativity Gravitation}, 28 (1996), 581.  doi: 10.1007/BF02105068.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, \emph{Stud. Appl. Math.}, 57 (1977), 93.   Google Scholar [17] E. H. Lieb and M. Loss, Analysis,, 2nd Edition, (2001).  doi: 10.1090/gsm/014.  Google Scholar [18] E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems,, \emph{Comm. Math. Phys.}, 53 (1977), 185.   Google Scholar [19] P. L. Lions, The Choquard equation and related questions,, \emph{Nonlinear Anal.}, 4 (1980), 1063.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [24] R. Penrose, On gravity's role in quantum state reduction,, \emph{Gen. Relativity Gravitation}, 28 (1996), 581.  doi: 10.1007/BF02105068.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276 [2] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [3] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [4] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [5] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [6] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [7] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [8] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448 [9] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [10] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [11] José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376 [12] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [13] Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264 [14] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [15] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [16] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [17] Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248 [18] Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382 [19] Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013 [20] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

2019 Impact Factor: 1.105