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July  2018, 38(7): 3567-3593. doi: 10.3934/dcds.2018151

## On critical Choquard equation with potential well

 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

* Minbo Yang is the corresponding author, he was partially supported by NSFC (11571317) and ZJNSF(LY15A010010)

Received  August 2017 Revised  February 2018 Published  April 2018

Fund Project: Zifei Shen and Fashun Gao were supported by NSFC (11671364).

In this paper we are interested in the following nonlinear Choquard equation
 $-Δ u+(λ V(x)-β)u = \big(|x|^{-μ}* |u|^{2_{μ}^{*}}\big)|u|^{2_{μ}^{*}-2}u\;\;\;\;\;\;\;\;\;\;\mbox{in}\;\; \mathbb{R}^N,$
where
 $λ, β∈\mathbb{R}^+$
,
 $0<μ is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function $V∈ \mathcal{C}(\mathbb{R}^N, \mathbb{R})$such that $Ω : = \mbox{int} V^{-1}(0)$is a nonempty bounded set with smooth boundary. If $β>0$is a constant such that the operator $-Δ +λ V(x)-β$is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int $V^{-1}(0)$for $λ$large enough and also characterize the asymptotic behavior of the solutions as the parameter $λ$goes to infinity. Furthermore, for any $0<β<β_{1}$, we are able to prove the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where $β_{1}$is the first eigenvalue of $-Δ$on $Ω$with Dirichlet boundary condition. Citation: Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151 ##### 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. Google Scholar [2] 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. Google Scholar [3] 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. Differential Equations, 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021. Google Scholar [4] C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988. doi: 10.1016/j.jde.2017.05.009. Google Scholar [5] C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9. Google Scholar [6] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. Google Scholar [7] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. Google Scholar [8] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/PL00001511. Google Scholar [9] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93. doi: 10.1007/BF00375686. Google Scholar [10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [11] B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119 (1993), 179-186. doi: 10.1090/S0002-9939-1993-1145940-X. Google Scholar [12] 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. Google Scholar [13] M. Clapp and Y. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys., 55 (2004), 592-605. doi: 10.1007/s00033-004-1084-9. Google Scholar [14] M. Clapp and Y. Ding, Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Differential Integral Equations, 16 (2003), 981-992. Google Scholar [15] 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. Google Scholar [16] Y. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034. doi: 10.1016/j.jde.2010.03.022. Google Scholar [17] Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manus. Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. Google Scholar [18] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math. Google Scholar [19] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041. doi: 10.1016/j.jmaa.2016.11.015. Google Scholar [20] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., https://doi.org/10.1142/S0219199717500377. doi: 10.1142/S0219199717500377. Google Scholar [21] M. Ghimenti, V. Moroz and J. Van Schaftingen, Least Action nodal solutions for ghe quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747. doi: 10.1090/proc/13247. Google Scholar [22] 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. Google Scholar [23] Y. Guo and Z. Tang, Multi-bump solutions for Schrödinger equation involving critical growth and potential wells, Discrete Contin. Dyn. Syst., 35 (2015), 3393-3415. doi: 10.3934/dcds.2015.35.3393. Google Scholar [24] Y. Guo and Z. Tang, Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells, J. Differential Equations, 259 (2015), 6038-6071. doi: 10.1016/j.jde.2015.07.015. Google Scholar [25] Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. Google Scholar [26] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. Google Scholar [27] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. Google Scholar [28] E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014. Google Scholar [29] P.L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [30] 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. Google Scholar [31] V. Moroz and J. Van Schaftingen, Groundstates 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. Google Scholar [32] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equation: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054. Google Scholar [33] S. Pekar, Untersuchungüber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar [34] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitat., 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar [35] 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 [36] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. Google Scholar [37] J. C. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169. Google Scholar [38] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar show all references ##### 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. Google Scholar [2] 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. Google Scholar [3] 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. Differential Equations, 261 (2016), 1933-1972. doi: 10.1016/j.jde.2016.04.021. Google Scholar [4] C. O. Alves, F. Gao, M. Squassina and M. Yang, Singularly perturbed critical Choquard equations, J. Differential Equations, 263 (2017), 3943-3988. doi: 10.1016/j.jde.2017.05.009. Google Scholar [5] C. O. Alves, A. B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. Partial Differential Equations, 55 (2016), Art. 48, 28 pp. doi: 10.1007/s00526-016-0984-9. Google Scholar [6] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078. Google Scholar [7] T. Bartsch, A. Pankov and Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494. Google Scholar [8] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 51 (2000), 366-384. doi: 10.1007/PL00001511. Google Scholar [9] V. Benci and G. Cerami, The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rational Mech. Anal., 114 (1991), 79-93. doi: 10.1007/BF00375686. Google Scholar [10] H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. doi: 10.1002/cpa.3160360405. Google Scholar [11] B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc., 119 (1993), 179-186. doi: 10.1090/S0002-9939-1993-1145940-X. Google Scholar [12] 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. Google Scholar [13] M. Clapp and Y. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys., 55 (2004), 592-605. doi: 10.1007/s00033-004-1084-9. Google Scholar [14] M. Clapp and Y. Ding, Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Differential Integral Equations, 16 (2003), 981-992. Google Scholar [15] 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. Google Scholar [16] Y. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation, J. Differential Equations, 249 (2010), 1015-1034. doi: 10.1016/j.jde.2010.03.022. Google Scholar [17] Y. Ding and K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manus. Math., 112 (2003), 109-135. doi: 10.1007/s00229-003-0397-x. Google Scholar [18] F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math. Google Scholar [19] F. Gao and M. Yang, On nonlocal Choquard equations with Hardy-Littlewood-Sobolev critical exponents, J. Math. Anal. Appl., 448 (2017), 1006-1041. doi: 10.1016/j.jmaa.2016.11.015. Google Scholar [20] F. Gao and M. Yang, A strongly indefinite Choquard equation with critical exponent due to Hardy-Littlewood-Sobolev inequality, Commun. Contemp. Math., https://doi.org/10.1142/S0219199717500377. doi: 10.1142/S0219199717500377. Google Scholar [21] M. Ghimenti, V. Moroz and J. Van Schaftingen, Least Action nodal solutions for ghe quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747. doi: 10.1090/proc/13247. Google Scholar [22] 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. Google Scholar [23] Y. Guo and Z. Tang, Multi-bump solutions for Schrödinger equation involving critical growth and potential wells, Discrete Contin. Dyn. Syst., 35 (2015), 3393-3415. doi: 10.3934/dcds.2015.35.3393. Google Scholar [24] Y. Guo and Z. Tang, Sign changing bump solutions for Schrödinger equations involving critical growth and indefinite potential wells, J. Differential Equations, 259 (2015), 6038-6071. doi: 10.1016/j.jde.2015.07.015. Google Scholar [25] Y. S. Jiang and H. S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608. doi: 10.1016/j.jde.2011.05.006. Google Scholar [26] E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27. doi: 10.2140/apde.2009.2.1. Google Scholar [27] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976/77), 93-105. Google Scholar [28] E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014. Google Scholar [29] P.L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072. doi: 10.1016/0362-546X(80)90016-4. Google Scholar [30] 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. Google Scholar [31] V. Moroz and J. Van Schaftingen, Groundstates 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. Google Scholar [32] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equation: Hardy-Littlewood-Sobolev critical exponent, Commun. Contemp. Math., 17 (2015), 1550005, 12pp. doi: 10.1142/S0219199715500054. Google Scholar [33] S. Pekar, Untersuchungüber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar [34] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitat., 28 (1996), 581-600. doi: 10.1007/BF02105068. Google Scholar [35] 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 [36] Z. Tang, Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials, Commun. Pure Appl. Anal., 13 (2014), 237-248. doi: 10.3934/cpaa.2014.13.237. Google Scholar [37] J. C. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22 pp. doi: 10.1063/1.3060169. Google Scholar [38] M. 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