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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 & Applied Analysis, 2020, 19 (3) : 1563-1579. doi: 10.3934/cpaa.2020078
References:
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P. Choquard, J. Stubbe and M. Vuffray, Stationary solutions of the Schrödinger-Newton model-an ODE approach, Differential Integral Equations, 21 (2008), 665-679.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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

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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

S. I. Pekar, Untersuchung über die Elektronentheorie der Kristalle, Akademie Verlag, Berlin, 1954. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

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