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March  2020, 19(3): 1563-1579. doi: 10.3934/cpaa.2020078

Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent

Ting Guo , Xianhua Tang , , Qi Zhang and  Zu Gao

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author

Received  June 2019 Revised  August 2019 Published  November 2019

Fund Project: X. H. Tang was partially supported by the National Natural Science Foundation of China (No: 11571370).

This paper is dedicated to studying the Choquard equation
$ \begin{equation*} \left\{ \begin{array}{ll} -\Delta u+V(x)u = (I_{\alpha}\ast|u|^{p})|u|^{p-2}u+g(u),\; \; \; \; \; x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}) ,\\ \end{array} \right. \end{equation*} $
where
$ N\geq4 $
,
$ \alpha\in(0, N) $
,
$ V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R}) $
 is sign-changing and periodic,
$ I_{\alpha} $
 is the Riesz potential,
$ p = \frac{N+\alpha}{N-2} $
 and
$ g\in\mathcal{C}(\mathbb{R}, \mathbb{R}) $
. The equation is strongly indefinite, i.e., the operator
$ -\Delta+V $
 has infinite-dimensional negative and positive spaces. Moreover, the exponent
$ p = \frac{N+\alpha}{N-2} $
 is the upper critical exponent with respect to the Hardy-Littlewood-Sobolev inequality. Under some mild assumptions on
$ g $
, we obtain the existence of nontrivial solutions for this equation.
Keywords: Choquard equation, indefinite, critical growth, local nonlinear term.
Mathematics Subject Classification: Primary: 35B33; Secondary: 47J30.
Citation: Ting Guo, Xianhua Tang, Qi Zhang, Zu Gao. Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078
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]

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.

[3]

J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2002), 85-93.  doi: 10.1090/S0002-9939-01-06143-3.

[4]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Continuous Dynam. Systems - A, 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.

[5]

S. T. Chen and X. H. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, (2019), https://doi.org/10.1016/j.jde.2019.08.036. doi: 10.1016/j.jde.2019.08.036.

[6]

S. T. Chen, A. Fiscella, P. Pucci and X. H. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, (2019), https://doi.org/10.1016/j.jde.2019.09.041. doi: 10.1016/j.jde.2019.09.041.

[7]

P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model-an ODE approach, Differential Integral Equations, 21 (2008), 665-679. 

[8]

Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differ. Equ., 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.

[9] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. 
[10]

Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.

[11]

F. S. Gao and M. B. Yang, The Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[12]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472. 

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[14]

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

[15]

E. H. Lieb and M. Loss, Graduate Studies in Mathematics, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[16]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[17]

G. D. Li and C. L. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 76 (2018), 2625-2647.  doi: 10.1016/j.camwa.2018.08.052.

[18]

X. F. Li, S. W. Ma and G. Zhang, Existence and qualitative properties of solutions for Choquard equations with a local term, Nonlinear Anal. RWA., 45 (2019), 1-25.  doi: 10.1016/j.nonrwa.2018.06.007.

[19]

L. Mattner, Strict definiteness of integrals via complete monotonicity of derivatives, Trans. Amer. Math. Soc., 349 (1997), 3321-3342.  doi: 10.1090/S0002-9947-97-01966-1.

[20]

G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191.

[21]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[22]

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.

[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. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[25]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[26]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 1 (2018), 1-15.  doi: 10.1007/s10884-018-9662-2.

[27]

X. H. Tang, Non-Nehari-Manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.

[28]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.

[29]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110-134.  doi: 10.1007/s00526-017-1214-9.

[30]

X. H. Tang and S. T. Chen, Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9 (2020), 413-437.  doi: 10.1515/anona-2020-0007.

[31]

J. Van Schaftingen and J. K. Xia, Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464 (2018), 1184-1202.  doi: 10.1016/j.jmaa.2018.04.047.

[32]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 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]

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.

[3]

J. Chabrowski and A. Szulkin, On a semilinear Schrödinger equation with critical Sobolev exponent, Proc. Amer. Math. Soc., 130 (2002), 85-93.  doi: 10.1090/S0002-9939-01-06143-3.

[4]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Continuous Dynam. Systems - A, 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.

[5]

S. T. Chen and X. H. Tang, On the planar Schrödinger-Poisson system with the axially symmetric potential, J. Differential Equations, (2019), https://doi.org/10.1016/j.jde.2019.08.036. doi: 10.1016/j.jde.2019.08.036.

[6]

S. T. Chen, A. Fiscella, P. Pucci and X. H. Tang, Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations, J. Differential Equations, (2019), https://doi.org/10.1016/j.jde.2019.09.041. doi: 10.1016/j.jde.2019.09.041.

[7]

P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model-an ODE approach, Differential Integral Equations, 21 (2008), 665-679. 

[8]

Y. H. Ding and C. Lee, Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms, J. Differ. Equ., 222 (2006), 137-163.  doi: 10.1016/j.jde.2005.03.011.

[9] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. 
[10]

Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.

[11]

F. S. Gao and M. B. Yang, The Brézis-Nirenberg type critical problem for nonlinear Choquard equation, Sci. China Math., 61 (2018), 1219-1242.  doi: 10.1007/s11425-016-9067-5.

[12]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472. 

[13]

E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 57 (1976/1977), 93-105.  doi: 10.1002/sapm197757293.

[14]

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

[15]

E. H. Lieb and M. Loss, Graduate Studies in Mathematics, American Mathematical Society, Providence, 2001. doi: 10.1090/gsm/014.

[16]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[17]

G. D. Li and C. L. Tang, Existence of a ground state solution for Choquard equation with the upper critical exponent, Comput. Math. Appl., 76 (2018), 2625-2647.  doi: 10.1016/j.camwa.2018.08.052.

[18]

X. F. Li, S. W. Ma and G. Zhang, Existence and qualitative properties of solutions for Choquard equations with a local term, Nonlinear Anal. RWA., 45 (2019), 1-25.  doi: 10.1016/j.nonrwa.2018.06.007.

[19]

L. Mattner, Strict definiteness of integrals via complete monotonicity of derivatives, Trans. Amer. Math. Soc., 349 (1997), 3321-3342.  doi: 10.1090/S0002-9947-97-01966-1.

[20]

G. P. Menzala, On regular solutions of a nonlinear equation of Choquard's type, Proc. Roy. Soc. Edinburgh Sect. A, 86 (1980), 291-301.  doi: 10.1017/S0308210500012191.

[21]

I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schrödinger-Newton equations, Classical Quantum Gravity, 15 (1998), 2733-2742.  doi: 10.1088/0264-9381/15/9/019.

[22]

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.

[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. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954.

[25]

M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.  doi: 10.1007/BF01174186.

[26]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 1 (2018), 1-15.  doi: 10.1007/s10884-018-9662-2.

[27]

X. H. Tang, Non-Nehari-Manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116.  doi: 10.1017/S144678871400041X.

[28]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.

[29]

X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110-134.  doi: 10.1007/s00526-017-1214-9.

[30]

X. H. Tang and S. T. Chen, Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions, Adv. Nonlinear Anal., 9 (2020), 413-437.  doi: 10.1515/anona-2020-0007.

[31]

J. Van Schaftingen and J. K. Xia, Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent, J. Math. Anal. Appl., 464 (2018), 1184-1202.  doi: 10.1016/j.jmaa.2018.04.047.

[32]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.

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