# American Institute of Mathematical Sciences

June  2019, 39(6): 3365-3398. doi: 10.3934/dcds.2019139

## Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory

 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Zhongwei Tang

Received  August 2018 Published  February 2019

Fund Project: This work is supported by NSFC (11571040, 11671331).

This paper concerns the following nonlinear Choquard equation:
 $$$-\varepsilon^{2}\Delta w+V(x)w = \varepsilon^{-\theta}W(x)(I_\theta*(W|w|^p))|w|^{p-2}w,\quad x\in\mathbb{R}^N, ~~~~~~~~~~~~(*)$$$
where
 $\varepsilon>0,\ N>2,\ I_\theta$
is the Riesz potential with order
 $\theta\in(0,N),\ p\in\big[2,\frac{N+\theta}{N-2}\big),\ \min V>0$
and
 $\inf W>0$
. Under proper assumptions, we explore the existence, concentration, convergence and decay estimate of semiclassical solutions for
 $(\ast)$
. The multiplicity of solutions is established via pseudo-index theory. The existence of sign-changing solutions is achieved by minimizing the energy on Nehari nodal set.
Citation: Min Liu, Zhongwei Tang. Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3365-3398. doi: 10.3934/dcds.2019139
##### References:
 [1] N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.  doi: 10.1016/j.jfa.2005.11.010. [2] C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $\mathbb{R}^2$, J. Differ. Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021. [3] C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009. [4] M. Bahrami, A. Großardt, S. Donadi and A. Bassi, The Schrödinger-Newton equation and its foundations, New J. Phys., 16 (2014), 115007, 17pp.  doi: 10.1088/1367-2630/16/11/115007. [5] V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.1090/S0002-9947-1982-0675067-X. [6] D. Bonheure, S. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.  doi: 10.1016/j.jfa.2017.02.026. [7] 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. [8] 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. [9] Y. Ding and J. Wei, Multiplicity of semiclassical solutions to nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 987-1010.  doi: 10.1007/s11784-017-0410-8. [10] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019. [11] M. Ghimenti, V. Moroz and J. Van Schaftingen, Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747.  doi: 10.1090/proc/13247. [12] D. Giulini and A. Großardt, The Schrödinger equation as a nonrelativistic limit of self-gravitating Klein-Gordon and Dirac fields, Class. Quantum Gravity, 25 (2012), 215010. [13] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University press, 1952. [14] N. S. Landkof, Foundations of Modern Potential Theory-Translated from the Russian by A.P. Doohovskoy, Springer-Verlag Berlin Heidelberg New York, 1972. [15] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. [16] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [17] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., Theory Methods Appl., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [18] 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. [19] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7. [20] 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. [21] V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x. [22] V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2. [23] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [24] S. Pekar, Untersuchung über die Elektronnentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [25] D. Ruiz and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differ. Equations, 264 (2018), 1231-1262.  doi: 10.1016/j.jde.2017.09.034. [26] T. Wang, Existence and nonexistence of nontrivial solutions for Choquard type equations, Electron. J. Differ. Equ., 2016 (2016), Paper No. 3, 17 pp. [27] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp.  doi: 10.1063/1.3060169. [28] M. Willem, Minimax Theorems, Birckhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [29] M. Yang, Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^2$ with critical exponential growth, ESAIM, Control Optim. Calc. Var., 24 (2018), 177-209.  doi: 10.1051/cocv/2017007.

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##### References:
 [1] N. Ackermann, A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.  doi: 10.1016/j.jfa.2005.11.010. [2] C. O. Alves, D. Cassani, C. Tarsi and M. Yang, Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $\mathbb{R}^2$, J. Differ. Equations, 261 (2016), 1933-1972.  doi: 10.1016/j.jde.2016.04.021. [3] C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943-3988.  doi: 10.1016/j.jde.2017.05.009. [4] M. Bahrami, A. Großardt, S. Donadi and A. Bassi, The Schrödinger-Newton equation and its foundations, New J. Phys., 16 (2014), 115007, 17pp.  doi: 10.1088/1367-2630/16/11/115007. [5] V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.  doi: 10.1090/S0002-9947-1982-0675067-X. [6] D. Bonheure, S. Cingolani and J. Van Schaftingen, The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.  doi: 10.1016/j.jfa.2017.02.026. [7] 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. [8] 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. [9] Y. Ding and J. Wei, Multiplicity of semiclassical solutions to nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 987-1010.  doi: 10.1007/s11784-017-0410-8. [10] M. Ghimenti and J. Van Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.  doi: 10.1016/j.jfa.2016.04.019. [11] M. Ghimenti, V. Moroz and J. Van Schaftingen, Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747.  doi: 10.1090/proc/13247. [12] D. Giulini and A. Großardt, The Schrödinger equation as a nonrelativistic limit of self-gravitating Klein-Gordon and Dirac fields, Class. Quantum Gravity, 25 (2012), 215010. [13] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University press, 1952. [14] N. S. Landkof, Foundations of Modern Potential Theory-Translated from the Russian by A.P. Doohovskoy, Springer-Verlag Berlin Heidelberg New York, 1972. [15] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105. [16] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032. [17] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., Theory Methods Appl., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4. [18] 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. [19] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7. [20] 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. [21] V. Moroz and J. Van Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235.  doi: 10.1007/s00526-014-0709-x. [22] V. Moroz and J. Van Schaftingen, Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.  doi: 10.1090/S0002-9947-2014-06289-2. [23] V. Moroz and J. Van Schaftingen, A guide to the Choquard equation, J. Fixed Point Theory Appl., 19 (2017), 773-813.  doi: 10.1007/s11784-016-0373-1. [24] S. Pekar, Untersuchung über die Elektronnentheorie der Kristalle, Akademie Verlag, Berlin, 1954. [25] D. Ruiz and J. Van Schaftingen, Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differ. Equations, 264 (2018), 1231-1262.  doi: 10.1016/j.jde.2017.09.034. [26] T. Wang, Existence and nonexistence of nontrivial solutions for Choquard type equations, Electron. J. Differ. Equ., 2016 (2016), Paper No. 3, 17 pp. [27] J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp.  doi: 10.1063/1.3060169. [28] M. Willem, Minimax Theorems, Birckhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [29] M. Yang, Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^2$ with critical exponential growth, ESAIM, Control Optim. Calc. Var., 24 (2018), 177-209.  doi: 10.1051/cocv/2017007.
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