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Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs
Department of Mathematics, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, Nagoya 468-8502, Japan |
$ - \epsilon^2 u'' +u = u^p $ |
$ p \in (1,\infty) $ |
$ \epsilon $ |
$ \epsilon $ |
$ \epsilon $ |
References:
[1] |
S. Akduman and A. Pankov,
Nonlinear Schrödinger equation with growing potential on infinite metric graphs, Nonlinear Anal., 184 (2019), 258-272.
doi: 10.1016/j.na.2019.02.020. |
[2] |
R. Adami, E. Serra and P. Tilli,
NLS ground states on graphs, Calc. Var. Partial Differ. Equ., 54 (2015), 743-761.
doi: 10.1007/s00526-014-0804-z. |
[3] |
R. Adami, E. Serra and P. Tilli, Multiple positive bound states for the subcritical NLS equation on metric graphs, Calc. Var. Partial Differ. Equ., 58 (2019), 16 pp.
doi: 10.1007/s00526-018-1461-4. |
[4] |
A. Bahri and Y. Li,
On a min-max procedure for the existence of a positive solution for certain scalar field equations in ${\bf R}^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[5] |
C. Cacciapuoti, S. Dovetta and E. Serra,
Variational and stability properties of constant solutions to the NLS equation on compact metric graphs, Milan J. Math., 86 (2018), 305-327.
doi: 10.1007/s00032-018-0288-y. |
[6] |
M. del Pino and P. Felmer,
Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[7] |
S. Dovetta, M. Ghimenti, A. M. Micheletti and A. Pistoia,
Peaked and low action solutions of NLS equations on graphs with terminal edges, SIAM J. Math. Anal., 52 (2020), 2874-2894.
doi: 10.1137/19M127447X. |
[8] |
S. Dovetta, E. Serra and P. Tilli, Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs, Adv. Math., 374 (2020), 107352, 41 pp.
doi: 10.1016/j.aim.2020.107352. |
[9] |
S. Dovetta, E. Serra and P. Tilli,
NLS ground states on metric trees: existence results and open questions, J. Lond. Math. Soc., 102 (2020), 1223-1240.
doi: 10.1112/jlms.12361. |
[10] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in ${\bf R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[11] |
K. Kurata and M. Shibata, Least energy solutions to semi-linear elliptic problems on metric graphs, J. Math. Anal. Appl., 491 (2020), 124297, 22 pp.
doi: 10.1016/j.jmaa.2020.124297. |
[12] |
Y. Li, F. Li and J. Shi,
Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl., 459 (2018), 661-685.
doi: 10.1016/j.jmaa.2017.10.069. |
show all references
References:
[1] |
S. Akduman and A. Pankov,
Nonlinear Schrödinger equation with growing potential on infinite metric graphs, Nonlinear Anal., 184 (2019), 258-272.
doi: 10.1016/j.na.2019.02.020. |
[2] |
R. Adami, E. Serra and P. Tilli,
NLS ground states on graphs, Calc. Var. Partial Differ. Equ., 54 (2015), 743-761.
doi: 10.1007/s00526-014-0804-z. |
[3] |
R. Adami, E. Serra and P. Tilli, Multiple positive bound states for the subcritical NLS equation on metric graphs, Calc. Var. Partial Differ. Equ., 58 (2019), 16 pp.
doi: 10.1007/s00526-018-1461-4. |
[4] |
A. Bahri and Y. Li,
On a min-max procedure for the existence of a positive solution for certain scalar field equations in ${\bf R}^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.
doi: 10.4171/RMI/92. |
[5] |
C. Cacciapuoti, S. Dovetta and E. Serra,
Variational and stability properties of constant solutions to the NLS equation on compact metric graphs, Milan J. Math., 86 (2018), 305-327.
doi: 10.1007/s00032-018-0288-y. |
[6] |
M. del Pino and P. Felmer,
Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32.
doi: 10.1007/s002080200327. |
[7] |
S. Dovetta, M. Ghimenti, A. M. Micheletti and A. Pistoia,
Peaked and low action solutions of NLS equations on graphs with terminal edges, SIAM J. Math. Anal., 52 (2020), 2874-2894.
doi: 10.1137/19M127447X. |
[8] |
S. Dovetta, E. Serra and P. Tilli, Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs, Adv. Math., 374 (2020), 107352, 41 pp.
doi: 10.1016/j.aim.2020.107352. |
[9] |
S. Dovetta, E. Serra and P. Tilli,
NLS ground states on metric trees: existence results and open questions, J. Lond. Math. Soc., 102 (2020), 1223-1240.
doi: 10.1112/jlms.12361. |
[10] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in ${\bf R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[11] |
K. Kurata and M. Shibata, Least energy solutions to semi-linear elliptic problems on metric graphs, J. Math. Anal. Appl., 491 (2020), 124297, 22 pp.
doi: 10.1016/j.jmaa.2020.124297. |
[12] |
Y. Li, F. Li and J. Shi,
Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl., 459 (2018), 661-685.
doi: 10.1016/j.jmaa.2017.10.069. |
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