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

On critical Choquard equation with potential well

Zifei Shen , Fashun Gao and  Minbo Yang ,

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
$λ, β∈\mathbb{R}^+$
,
$0<μ<N, N≥4, 2_{μ}^{*} = (2N-μ)/(N-2)$
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.
Keywords: Critical Choquard equation, Hardy-Littlewood-Sobolev inequality, potential well, Lusternik-Schnirelmann category.
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35A15.
Citation: Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[18]

F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[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. 

[28]

E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014.

[29]

P.L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.

[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.

[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.

[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.

[33]

S. Pekar, Untersuchungüber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[34]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitat., 28 (1996), 581-600.  doi: 10.1007/BF02105068.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[18]

F. Gao and M. Yang, On the Brezis-Nirenberg type critical problem for nonlinear Choquard equation, Sci China Math.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

E. Lenzmann, Uniqueness of ground states for pseudorelativistic Hartree equations, Anal. PDE, 2 (2009), 1-27.  doi: 10.2140/apde.2009.2.1.

[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. 

[28]

E. H. Lieb and M. Loss, Analysis, Gradute Studies in Mathematics, 1997. doi: 10.1090/gsm/014.

[29]

P.L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.

[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.

[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.

[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.

[33]

S. Pekar, Untersuchungüber die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[34]

R. Penrose, On gravity's role in quantum state reduction, Gen. Relativ. Gravitat., 28 (1996), 581-600.  doi: 10.1007/BF02105068.

[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.

[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.

[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.

[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.

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