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

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

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

Dengfeng Lü ^1,

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

Received October 2015 Revised January 2016 Published July 2016

Abstract
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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 $\mu\rightarrow +\infty$.

Keywords: ground state solution., variational method, magnetic field, potential well, Nonlinear Choquard equation.

Mathematics Subject Classification: Primary: 35J60, 35J10; Secondary: 35Q5.

Citation: Dengfeng Lü. Existence and concentration behavior of ground state solutions for magnetic nonlinear Choquard equations. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1781-1795. doi: 10.3934/cpaa.2016014

References:

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[2]	A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. 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, Ann. I. H. Poincaré-AN, 27 (2010), 1205-1226. doi: 10.1016/j.anihpc.2010.05.003.
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[5]	T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. 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, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027.
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[14]	Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Ann. Mat. Pura Appl., 190 (2011), 427-451. doi: 10.1007/s10231-010-0157-y.
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[16]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.
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[19]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
[21]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. 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, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.
[23]	M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.
[24]	R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600. doi: 10.1007/BF02105068.
[25]	S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.
[26]	J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905. 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, J. Math. Anal. Appl., 403 (2013), 680-694. doi: 10.1016/j.jmaa.2013.02.062.

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References:

[1]	N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.
[2]	A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271. 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, Ann. I. H. Poincaré-AN, 27 (2010), 1205-1226. doi: 10.1016/j.anihpc.2010.05.003.
[4]	S. Barile, A multiplicity result for singular NLS equations with magnetic potentials, Nonlinear Anal., 68 (2008), 3525-3540. 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 $R^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741. 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, J. Differential Equations, 222 (2006), 381-424. doi: 10.1016/j.jde.2005.06.027.
[7]	S. Cingolani, Semiclassical stationary states of nonlinear Schrödinger equation with external magnetic field, J. Differential Equations, 188 (2003), 52-79. doi: 10.1016/S0022-0396(02)00058-X.
[8]	S. Cingolani, M. Clapp and S. Secchi, Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248. doi: 10.1007/s00033-011-0166-8.
[9]	S. Cingolani, M. Clapp and S. Secchi, Intertwining semiclassical solutions to a Schrödinger-Newton system, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 891-908.
[10]	S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations, 160 (2000), 118-138. doi: 10.1006/jdeq.1999.3662.
[11]	S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrödinger equation with electromagnetic fields, J. Math. Anal. Appl., 275 (2002), 108-130. 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, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 973-1009. doi: 10.1017/S0308210509000584.
[13]	M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15. doi: 10.1016/j.jmaa.2013.04.081.
[14]	Y. Ding and Z.-Q. Wang, Bound states of nonlinear Schrödinger equations with magnetic fields, Ann. Mat. Pura Appl., 190 (2011), 427-451. 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, J. Differential Equations, 251 (2011), 3500-3521. doi: 10.1016/j.jde.2011.08.038.
[16]	E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1977), 93-105.
[17]	E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Mathematics 14, AMS, 2001. doi: 10.1090/gsm/014.
[18]	E. H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
[19]	P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.
[21]	L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467. 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, J. Funct. Anal., 265 (2013), 153-184. doi: 10.1016/j.jfa.2013.04.007.
[23]	M. Nolasco, Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential, Commun. Pure Appl. Anal., 9 (2010), 1411-1419. doi: 10.3934/cpaa.2010.9.1411.
[24]	R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 28 (1996), 581-600. doi: 10.1007/BF02105068.
[25]	S. Secchi, A note on Schrödinger-Newton systems with decaying electric potential, Nonlinear Anal., 72 (2010), 3842-3856. doi: 10.1016/j.na.2010.01.021.
[26]	J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905. 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, J. Math. Anal. Appl., 403 (2013), 680-694. doi: 10.1016/j.jmaa.2013.02.062.

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