# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021147
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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

Received  February 2021 Revised  July 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.
Citation: Masataka Shibata. Multiplicity of positive solutions to semi-linear elliptic problems on metric graphs. Communications on Pure &amp; Applied Analysis, 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 [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

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
 [1] Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [2] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [3] Min Liu, Zhongwei Tang. Multiplicity and concentration of solutions for Choquard equation via Nehari method and pseudo-index theory. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3365-3398. doi: 10.3934/dcds.2019139 [4] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [5] Said Taarabti. Positive solutions for the $p(x)-$Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021029 [6] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [7] Roy H. Goodman. NLS bifurcations on the bowtie combinatorial graph and the dumbbell metric graph. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2203-2232. doi: 10.3934/dcds.2019093 [8] Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325 [9] A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419 [10] Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003 [11] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 [12] Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4593-4608. doi: 10.3934/dcds.2021050 [13] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [14] Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044 [15] Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012 [16] Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911 [17] Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080 [18] Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166 [19] Lynnyngs Kelly Arruda, Francisco Odair de Paiva, Ilma Marques. A remark on multiplicity of positive solutions for a class of quasilinear elliptic systems. Conference Publications, 2011, 2011 (Special) : 112-116. doi: 10.3934/proc.2011.2011.112 [20] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201

2020 Impact Factor: 1.916