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December  2021, 20(12): 4107-4126. doi: 10.3934/cpaa.2021147

Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs

Masataka Shibata

Department of Mathematics, Meijo University, 1-501 Shiogamaguchi, Tempaku-ku, Nagoya 468-8502, Japan

Received  February 2021 Revised  July 2021 Published  December 2021 Early access  September 2021

Fund Project: The author was supported by JSPS KAKENHI Grant Numbers 18K03356, 18K03362

We consider positive solutions of semi-linear elliptic equations
$ - \epsilon^2 u'' +u = u^p $
on compact metric graphs, where
$ p \in (1,\infty) $
 is a given constant and
$ \epsilon $
 is a positive parameter. We focus on the multiplicity of positive solutions for sufficiently small
$ \epsilon $
. For each edge of the graph, we construct a positive solution which concentrates some point on the edge if
$ \epsilon $
 is sufficiently small. Moreover, we give the existence result of solutions which concentrate inner vertices of the graph.
Keywords: metric graph, variational method, Nehari manifold, positive solution, multiplicity.
Mathematics Subject Classification: Primary: 34B45; Secondary: 35J20.
Citation: Masataka Shibata. Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs. Communications on Pure & Applied Analysis, 2021, 20 (12) : 4107-4126. doi: 10.3934/cpaa.2021147
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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