# American Institute of Mathematical Sciences

• Previous Article
Study of semi-linear $\sigma$-evolution equations with frictional and visco-elastic damping
• CPAA Home
• This Issue
• Next Article
On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $L_{2}(-\infty, \infty)$
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

 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.
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.
 [1] Bartosz Bieganowski, Simone Secchi. The semirelativistic Choquard equation with a local nonlinear term. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4279-4302. doi: 10.3934/dcds.2019173 [2] Gui-Dong Li, Chun-Lei Tang. Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term. Communications on Pure and Applied Analysis, 2019, 18 (1) : 285-300. doi: 10.3934/cpaa.2019015 [3] Hua Jin, Wenbin Liu, Huixing Zhang, Jianjun Zhang. Ground states of nonlinear fractional Choquard equations with Hardy-Littlewood-Sobolev critical growth. Communications on Pure and Applied Analysis, 2020, 19 (1) : 123-144. doi: 10.3934/cpaa.2020008 [4] Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure and Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026 [5] 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 [6] Youpei Zhang, Xianhua Tang, Vicenţiu D. Rădulescu. High and low perturbations of Choquard equations with critical reaction and variable growth. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1971-2003. doi: 10.3934/dcds.2021180 [7] Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459 [8] M. Grossi, P. Magrone, M. Matzeu. Linking type solutions for elliptic equations with indefinite nonlinearities up to the critical growth. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 703-718. doi: 10.3934/dcds.2001.7.703 [9] Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1319-1345. doi: 10.3934/cpaa.2021022 [10] Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131 [11] Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4887-4919. doi: 10.3934/dcds.2021061 [12] Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 [13] Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235 [14] Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168 [15] Sami Aouaoui. On some local-nonlocal elliptic equation involving nonlinear terms with exponential growth. Communications on Pure and Applied Analysis, 2017, 16 (5) : 1767-1784. doi: 10.3934/cpaa.2017086 [16] Nikolaos Bournaveas. Local well-posedness for a nonlinear dirac equation in spaces of almost critical dimension. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 605-616. doi: 10.3934/dcds.2008.20.605 [17] Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $\mathbb{T}^3$. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116 [18] Minbo Yang, Jianjun Zhang, Yimin Zhang. Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity. Communications on Pure and Applied Analysis, 2017, 16 (2) : 493-512. doi: 10.3934/cpaa.2017025 [19] Zhongwei Tang. Least energy solutions for semilinear Schrödinger equations involving critical growth and indefinite potentials. Communications on Pure and Applied Analysis, 2014, 13 (1) : 237-248. doi: 10.3934/cpaa.2014.13.237 [20] Yu Su, Zhaosheng Feng. Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022112

2020 Impact Factor: 1.916