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Hardy-Sobolev type inequality and supercritical extremal problem
Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
$ \begin{equation} -\varepsilon^{2}\Delta w+V(x)w = \varepsilon^{-\theta}W(x)(I_\theta*(W|w|^p))|w|^{p-2}w,\quad x\in\mathbb{R}^N, ~~~~~~~~~~~~(*)\end{equation} $ |
$ \varepsilon>0,\ N>2,\ I_\theta $ |
$ \theta\in(0,N),\ p\in\big[2,\frac{N+\theta}{N-2}\big),\ \min V>0 $ |
$ \inf W>0 $ |
$ (\ast) $ |
References:
[1] |
N. Ackermann,
A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
C. O. Alves, D. Cassani, C. Tarsi and M. Yang,
Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $\mathbb{R}^2$, J. Differ. Equations, 261 (2016), 1933-1972.
doi: 10.1016/j.jde.2016.04.021. |
[3] |
C. O. Alves, F. Gao, M. Squassina and M. Yang,
Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943-3988.
doi: 10.1016/j.jde.2017.05.009. |
[4] |
M. Bahrami, A. Großardt, S. Donadi and A. Bassi,
The Schrödinger-Newton equation and its foundations, New J. Phys., 16 (2014), 115007, 17pp.
doi: 10.1088/1367-2630/16/11/115007. |
[5] |
V. Benci,
On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.
doi: 10.1090/S0002-9947-1982-0675067-X. |
[6] |
D. Bonheure, S. Cingolani and J. Van Schaftingen,
The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.
doi: 10.1016/j.jfa.2017.02.026. |
[7] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[8] |
M. Clapp and D. Salazar,
Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.
doi: 10.1016/j.jmaa.2013.04.081. |
[9] |
Y. Ding and J. Wei,
Multiplicity of semiclassical solutions to nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 987-1010.
doi: 10.1007/s11784-017-0410-8. |
[10] |
M. Ghimenti and J. Van Schaftingen,
Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.
doi: 10.1016/j.jfa.2016.04.019. |
[11] |
M. Ghimenti, V. Moroz and J. Van Schaftingen,
Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747.
doi: 10.1090/proc/13247. |
[12] |
D. Giulini and A. Großardt,
The Schrödinger equation as a nonrelativistic limit of self-gravitating Klein-Gordon and Dirac fields, Class. Quantum Gravity, 25 (2012), 215010.
|
[13] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University press, 1952.
![]() ![]() |
[14] |
N. S. Landkof, Foundations of Modern Potential Theory-Translated from the Russian by A.P. Doohovskoy, Springer-Verlag Berlin Heidelberg New York, 1972. |
[15] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
|
[16] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[17] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear Anal., Theory Methods Appl., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[18] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[19] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[20] |
V. Moroz and J. Van Schaftingen,
Ground states 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. |
[21] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[22] |
V. Moroz and J. Van Schaftingen,
Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.
doi: 10.1090/S0002-9947-2014-06289-2. |
[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. Pekar, Untersuchung über die Elektronnentheorie der Kristalle, Akademie Verlag, Berlin, 1954. |
[25] |
D. Ruiz and J. Van Schaftingen,
Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differ. Equations, 264 (2018), 1231-1262.
doi: 10.1016/j.jde.2017.09.034. |
[26] |
T. Wang, Existence and nonexistence of nontrivial solutions for Choquard type equations, Electron. J. Differ. Equ., 2016 (2016), Paper No. 3, 17 pp. |
[27] |
J. Wei and M. Winter,
Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp.
doi: 10.1063/1.3060169. |
[28] |
M. Willem, Minimax Theorems, Birckhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
M. Yang,
Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^2$ with critical exponential growth, ESAIM, Control Optim. Calc. Var., 24 (2018), 177-209.
doi: 10.1051/cocv/2017007. |
show all references
References:
[1] |
N. Ackermann,
A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 277-320.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
C. O. Alves, D. Cassani, C. Tarsi and M. Yang,
Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in $\mathbb{R}^2$, J. Differ. Equations, 261 (2016), 1933-1972.
doi: 10.1016/j.jde.2016.04.021. |
[3] |
C. O. Alves, F. Gao, M. Squassina and M. Yang,
Singularly perturbed critical Choquard equations, J. Differ. Equations, 263 (2017), 3943-3988.
doi: 10.1016/j.jde.2017.05.009. |
[4] |
M. Bahrami, A. Großardt, S. Donadi and A. Bassi,
The Schrödinger-Newton equation and its foundations, New J. Phys., 16 (2014), 115007, 17pp.
doi: 10.1088/1367-2630/16/11/115007. |
[5] |
V. Benci,
On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274 (1982), 533-572.
doi: 10.1090/S0002-9947-1982-0675067-X. |
[6] |
D. Bonheure, S. Cingolani and J. Van Schaftingen,
The logarithmic Choquard equation: Sharp asymptotics and nondegeneracy of the groundstate, J. Funct. Anal., 272 (2017), 5255-5281.
doi: 10.1016/j.jfa.2017.02.026. |
[7] |
S. Cingolani, M. Clapp and S. Secchi,
Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63 (2012), 233-248.
doi: 10.1007/s00033-011-0166-8. |
[8] |
M. Clapp and D. Salazar,
Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 407 (2013), 1-15.
doi: 10.1016/j.jmaa.2013.04.081. |
[9] |
Y. Ding and J. Wei,
Multiplicity of semiclassical solutions to nonlinear Schrödinger equations, J. Fixed Point Theory Appl., 19 (2017), 987-1010.
doi: 10.1007/s11784-017-0410-8. |
[10] |
M. Ghimenti and J. Van Schaftingen,
Nodal solutions for the Choquard equation, J. Funct. Anal., 271 (2016), 107-135.
doi: 10.1016/j.jfa.2016.04.019. |
[11] |
M. Ghimenti, V. Moroz and J. Van Schaftingen,
Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc., 145 (2017), 737-747.
doi: 10.1090/proc/13247. |
[12] |
D. Giulini and A. Großardt,
The Schrödinger equation as a nonrelativistic limit of self-gravitating Klein-Gordon and Dirac fields, Class. Quantum Gravity, 25 (2012), 215010.
|
[13] |
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University press, 1952.
![]() ![]() |
[14] |
N. S. Landkof, Foundations of Modern Potential Theory-Translated from the Russian by A.P. Doohovskoy, Springer-Verlag Berlin Heidelberg New York, 1972. |
[15] |
E. H. Lieb,
Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1977), 93-105.
|
[16] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[17] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear Anal., Theory Methods Appl., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[18] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[19] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[20] |
V. Moroz and J. Van Schaftingen,
Ground states 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. |
[21] |
V. Moroz and J. Van Schaftingen,
Semi-classical states for the Choquard equation, Calc. Var. Partial Differ. Equ., 52 (2015), 199-235.
doi: 10.1007/s00526-014-0709-x. |
[22] |
V. Moroz and J. Van Schaftingen,
Existence of ground states for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc., 367 (2015), 6557-6579.
doi: 10.1090/S0002-9947-2014-06289-2. |
[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. Pekar, Untersuchung über die Elektronnentheorie der Kristalle, Akademie Verlag, Berlin, 1954. |
[25] |
D. Ruiz and J. Van Schaftingen,
Odd symmetry of least energy nodal solutions for the Choquard equation, J. Differ. Equations, 264 (2018), 1231-1262.
doi: 10.1016/j.jde.2017.09.034. |
[26] |
T. Wang, Existence and nonexistence of nontrivial solutions for Choquard type equations, Electron. J. Differ. Equ., 2016 (2016), Paper No. 3, 17 pp. |
[27] |
J. Wei and M. Winter,
Strongly interacting bumps for the Schrödinger-Newton equations, J. Math. Phys., 50 (2009), 012905, 22pp.
doi: 10.1063/1.3060169. |
[28] |
M. Willem, Minimax Theorems, Birckhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[29] |
M. Yang,
Semiclassical ground state solutions for a Choquard type equation in $\mathbb{R}^2$ with critical exponential growth, ESAIM, Control Optim. Calc. Var., 24 (2018), 177-209.
doi: 10.1051/cocv/2017007. |
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