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On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $
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 |
$ \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*} $ |
$ N\geq4 $ |
$ \alpha\in(0, N) $ |
$ V\in\mathcal{C}(\mathbb{R}^{N}, \mathbb{R}) $ |
$ I_{\alpha} $ |
$ p = \frac{N+\alpha}{N-2} $ |
$ g\in\mathcal{C}(\mathbb{R}, \mathbb{R}) $ |
$ -\Delta+V $ |
$ p = \frac{N+\alpha}{N-2} $ |
$ g $ |
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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