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The semirelativistic Choquard equation with a local nonlinear term
1. | Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toruń, Poland |
2. | Università degli Studi di Milano-Bicocca, Via R. Cozzi 55, I-20125 Milano, Italy |
$ \mathbb{R}^N $ |
$ \begin{equation*} \sqrt{ -\Delta + m^2} u - mu + V(x)u = \left( \int_{ \mathbb{R} ^N} \frac{|u(y)|^p}{|x-y|^{N-\alpha}} \, dy \right) |u|^{p-2}u - \Gamma (x) |u|^{q-2}u, \end{equation*} $ |
$ m > 0 $ |
$ V $ |
$ \mathbb{Z}^N $ |
$ \mathbb{R}_{+}^{N+1} $ |
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] |
B. Bieganowski,
Solutions of the fractional Schródinger equation with a sign-changing nonlinearity, J. Math. Anal. Appl., 450 (2017), 461-479.
doi: 10.1016/j.jmaa.2017.01.037. |
[3] |
B. Bieganowski and J. Mederski,
Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 143-161.
doi: 10.3934/cpaa.2018009. |
[4] |
V. I. Bogachev, Measure Theory, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[5] |
X. Cabré and J. Solà-Morales,
Layers solutions in a half-space for boundary reactions, Comm. Pure Applied Math., 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[6] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[7] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
Y. H. Chen and C. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[9] |
Y. Cho and T. Ozawa,
On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.
doi: 10.1137/060653688. |
[10] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[11] |
S. Cingolani, M. Clapp and S. Secchi,
Intertwining semiclassical solutions to a Schródinger-Newton system, Discrete Continuous Dynmical Systems Series S, 6 (2013), 891-908.
doi: 10.3934/dcdss.2013.6.891. |
[12] |
S. Cingolani, S. Secchi and M. Squassina,
Semiclassical limit for Schródinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[13] |
S. Cingolani and S. Secchi,
Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 73-90.
doi: 10.1017/S0308210513000450. |
[14] |
V. Coti Zelati and P. Rabinowitz,
Homoclinic type solutions for a semilinear elliptic PDE on $ {\mathbb{R} ^n}$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
[15] |
V. Coti Zelati and M. Nolasco,
Existence of ground states for nonlinear, pseudorelativistic Schródinger equations, Red. Lincei Mat. Appl., 22 (2011), 51-72.
doi: 10.4171/RLM/587. |
[16] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[17] |
J. Fröhlich and E. Lenzmann, Mean-field Limit of Quantum Bose Gases and Nonlinear Hartree Equation, in Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, 26 pp., École Polytech., Palaiseau, 2004. |
[18] |
L. Jeanjean and K. Tanaka,
A positive solution for a nonlinear Schródinger equation of $ {\mathbb{R} ^N}$, Indiana Univ. Math. Journal, 54 (2005), 443-464.
doi: 10.1512/iumj.2005.54.2502. |
[19] |
E. H. Lieb and H.-T. Yau,
The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.
|
[20] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001.
doi: 10.1090/gsm/014. |
[21] |
E. H. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
|
[22] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. IHP, Analyse Non Linéaire, 1 (1984), 109-145.
|
[23] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[24] |
I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schródinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733–2742.
doi: 10.1088/0264-9381/15/9/019. |
[25] |
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. |
[26] |
R. Penrose,
Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[27] |
R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. |
[28] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ {\mathbb{R} ^N}$, Journal of Mathematical Physics, 54 (2013), 031501, 17 pp.
doi: 10.1063/1.4793990. |
[29] |
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3, Springer, Berlin; UMI, Bologna, 2007. |
[30] |
P. Tod,
The ground state energy of the Schródinger-Newton equation, Physics Letters A, 280 (2001), 173-176.
doi: 10.1016/S0375-9601(01)00059-7. |
[31] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp.
doi: 10.1063/1.3060169. |
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] |
B. Bieganowski,
Solutions of the fractional Schródinger equation with a sign-changing nonlinearity, J. Math. Anal. Appl., 450 (2017), 461-479.
doi: 10.1016/j.jmaa.2017.01.037. |
[3] |
B. Bieganowski and J. Mederski,
Nonlinear Schrödinger equations with sum of periodic and vanishing potentials and sign-changing nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 143-161.
doi: 10.3934/cpaa.2018009. |
[4] |
V. I. Bogachev, Measure Theory, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[5] |
X. Cabré and J. Solà-Morales,
Layers solutions in a half-space for boundary reactions, Comm. Pure Applied Math., 58 (2005), 1678-1732.
doi: 10.1002/cpa.20093. |
[6] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[7] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[8] |
Y. H. Chen and C. Liu,
Ground state solutions for non-autonomous fractional Choquard equations, Nonlinearity, 29 (2016), 1827-1842.
doi: 10.1088/0951-7715/29/6/1827. |
[9] |
Y. Cho and T. Ozawa,
On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060-1074.
doi: 10.1137/060653688. |
[10] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[11] |
S. Cingolani, M. Clapp and S. Secchi,
Intertwining semiclassical solutions to a Schródinger-Newton system, Discrete Continuous Dynmical Systems Series S, 6 (2013), 891-908.
doi: 10.3934/dcdss.2013.6.891. |
[12] |
S. Cingolani, S. Secchi and M. Squassina,
Semiclassical limit for Schródinger equations with magnetic field and Hartree-type nonlinearities, Proc. Roy. Soc. Edinburgh, 140 (2010), 973-1009.
doi: 10.1017/S0308210509000584. |
[13] |
S. Cingolani and S. Secchi,
Ground states for the pseudo-relativistic Hartree equation with external potential, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 73-90.
doi: 10.1017/S0308210513000450. |
[14] |
V. Coti Zelati and P. Rabinowitz,
Homoclinic type solutions for a semilinear elliptic PDE on $ {\mathbb{R} ^n}$, Comm. Pure Appl. Math., 45 (1992), 1217-1269.
doi: 10.1002/cpa.3160451002. |
[15] |
V. Coti Zelati and M. Nolasco,
Existence of ground states for nonlinear, pseudorelativistic Schródinger equations, Red. Lincei Mat. Appl., 22 (2011), 51-72.
doi: 10.4171/RLM/587. |
[16] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Comm. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[17] |
J. Fröhlich and E. Lenzmann, Mean-field Limit of Quantum Bose Gases and Nonlinear Hartree Equation, in Séminaire: Équations aux Dérivées Partielles 2003–2004, Exp. No. XIX, 26 pp., École Polytech., Palaiseau, 2004. |
[18] |
L. Jeanjean and K. Tanaka,
A positive solution for a nonlinear Schródinger equation of $ {\mathbb{R} ^N}$, Indiana Univ. Math. Journal, 54 (2005), 443-464.
doi: 10.1512/iumj.2005.54.2502. |
[19] |
E. H. Lieb and H.-T. Yau,
The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.
|
[20] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14, American Mathematical Society, 2001.
doi: 10.1090/gsm/014. |
[21] |
E. H. Lieb and B. Simon,
The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys., 53 (1977), 185-194.
|
[22] |
P.-L. Lions,
The concentration-compactness principle in the calculus of variations. The locally compact case. Part I, Ann. IHP, Analyse Non Linéaire, 1 (1984), 109-145.
|
[23] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[24] |
I. M. Moroz, R. Penrose and P. Tod, Spherically-symmetric solutions of the Schródinger-Newton equations, Topology of the Universe Conference (Cleveland, OH, 1997), Classical Quantum Gravity, 15 (1998), 2733–2742.
doi: 10.1088/0264-9381/15/9/019. |
[25] |
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. |
[26] |
R. Penrose,
Quantum computation, entanglement and state reduction, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 356 (1998), 1927-1939.
doi: 10.1098/rsta.1998.0256. |
[27] |
R. Penrose, The Road to Reality. A Complete Guide to the Laws of the Universe, Alfred A. Knopf Inc., New York, 2005. |
[28] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $ {\mathbb{R} ^N}$, Journal of Mathematical Physics, 54 (2013), 031501, 17 pp.
doi: 10.1063/1.4793990. |
[29] |
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces, Lecture Notes of the Unione Matematica Italiana, 3, Springer, Berlin; UMI, Bologna, 2007. |
[30] |
P. Tod,
The ground state energy of the Schródinger-Newton equation, Physics Letters A, 280 (2001), 173-176.
doi: 10.1016/S0375-9601(01)00059-7. |
[31] |
J. Wei and M. Winter, Strongly interacting bumps for the Schrödinger-Newton equation, J. Math. Phys., 50 (2009), 012905, 22 pp.
doi: 10.1063/1.3060169. |
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