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doi: 10.3934/cpaa.2022014
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Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods

Kaimin Teng 1,, and  Xian Wu 2,

1. 

Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
2. 

Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

* Corresponding author

Received  July 2020 Revised  October 2021 Early access December 2021

Fund Project: The work is supported by NSFC grant 11501403, by fund program for the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (2018) and by the Natural Science Foundation of Shanxi Province (No. 201901D111085)

In this paper, we study the following fractional Schrödinger-Poiss-on system
$ \begin{equation*} \begin{cases} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u = g(u) & \hbox{in $\mathbb{R}^3$,} \\ \varepsilon^{2t}(-\Delta)^t\phi = u^2,\,\, u>0& \hbox{in $\mathbb{R}^3$,} \end{cases} \end{equation*} $
where
$ s,t\in(0,1) $
,
$ \varepsilon>0 $
 is a small parameter. Under some local assumptions on
$ V(x) $
 and suitable assumptions on the nonlinearity
$ g $
, we construct a family of positive solutions
$ u_{\varepsilon}\in H_{\varepsilon} $
 which concentrate around the global minima of
$ V(x) $
 as
$ \varepsilon\rightarrow0 $
.
Keywords: Fractional Schrödinger-Poisson system, concentration, bound state, penalization method.
Mathematics Subject Classification: Primary: 35B38, Secondary: 35R11.
Citation: Kaimin Teng, Xian Wu. Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022014
References:
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L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

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J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.  Google Scholar

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show all references

References:
[1]

C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^N$ via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016), 1-19.  doi: 10.1007/s00526-016-0983-x.  Google Scholar

[2]

V. Ambrosi, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl., 196 (2017), 2043-2062.  doi: 10.1007/s10231-017-0652-5.  Google Scholar

[3]

J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200.  doi: 10.1007/s00205-006-0019-3.  Google Scholar

[4]

C. Bucur and E. Valdinoci, Nonlocal diffusion and applications, in Lecture Notes of the Unione Matemat-ica Italiana, Springer, International Publishing, 2016. doi: 10.1007/978-3-319-28739-3.  Google Scholar

[5]

J. Byeon and Z. Q. Wang, Standing waves witha criticak frequency for nonlinear Schrödinger equations II, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219.  doi: 10.1007/s00526-002-0191-8.  Google Scholar

[6]

S. Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.  doi: 10.1016/j.aim.2010.07.016.  Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[8]

R. Cont and P. Tankov, Financial Modeling with Jump Processes, Chapman Hall/CRC Financial Mathematics Series, Boca Raton, 2004.  Google Scholar

[9]

M. del Pino and P. L. Felmer, Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.  doi: 10.1007/BF01189950.  Google Scholar

[10]

P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Royal Soc. Edinb. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar

[11]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Commun. Pure Appl. Math., LXIX (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Equations of Second Order, Springer-Verlag, New York, 1998.  Google Scholar

[13]

X. M. He, Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys. 5 (2011), 869–889. doi: 10.1007/s00033-011-0120-9.  Google Scholar

[14]

Y. He and G. B. Li, Standing waves for a class of Schrödinger-Poisson equations in $\mathbb{R}^3$ involving critical Sobolev exponents, Ann. Acad. Sci. Fenn. Math., 40 (2015), 729-766.  doi: 10.5186/aasfm.2015.4041.  Google Scholar

[15]

X. M. He and W. M. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differ. Equ., 55 (2016), 1-39.  doi: 10.1007/s00526-016-1045-0.  Google Scholar

[16]

I. Ianni and G. Vaira, Solutions of the Schrödinger-Poisson problem concentrating on spheres, Part I: Necessary conditions, Math. Models Meth. Appl. Sci., 19 (2009), 707-720.  doi: 10.1142/S0218202509003589.  Google Scholar

[17]

I. Ianni and G. Vaira, On concentration of positive bound states for the Schrödinger-Poisson problem with potentials, Adv. Nonlinear Stud., 8 (2008), 573-595.  doi: 10.1515/ans-2008-0305.  Google Scholar

[18]

N. Laskin, Fractional Schrödinger equation, Phys. Rev., 66 (2002), 56-108.  doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[19]

Z. S. Liu and J. J. Zhang, Multiplicity and concentration of positive solutions for the fractional Schrödinger-Poisson systems with critical growth, ESAIM: Control, Optim. Calc. Var., 23 (2017), 1515-1542.  doi: 10.1051/cocv/2016063.  Google Scholar

[20]

R. Metzler and J. Klafter, The random walks guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar

[21]

P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[22]

E. G. Murcia and G. Siciliano, Positive semiclassical states for a fractional Schrödinger-Poisson system, Differ. Integral Equ., 30 (2017), 231-258.   Google Scholar

[23]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional sobolev spaces, Bullet. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[24]

D. Ruiz, Semiclassical states for coupled Schrödinger-Maxwell equations: Concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.  doi: 10.1142/S0218202505003939.  Google Scholar

[25]

D. Ruiz and G. Vaira, Cluster solutions for the Schrödinger-Poinsson-Slater problem around a local minimum of potential, Rev. Mat. Iberoam., 27 (2011), 253-271.  doi: 10.4171/RMI/635.  Google Scholar

[26]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[27]

S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), 031501.  doi: 10.1063/1.4793990.  Google Scholar

[28]

K. M. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equ., 261 (2016), 3061-3106.  doi: 10.1016/j.jde.2016.05.022.  Google Scholar

[29]

K. M. Teng, Ground state solutions for the nonlinear fractional Schrödinger-Poisson system, Appl. Anal., 98 (2019), 1959-1996.  doi: 10.1080/00036811.2018.1441998.  Google Scholar

[30]

K. M. Teng and R. P. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math. Meth. Appl. Sci., 41 (2018), 8258-8293.  doi: 10.1002/mma.5289.  Google Scholar

[31]

K. M. Teng and Yi qun Cheng, Multiplicity and concentration of nontrivial solutions for fractional Schrödinger-Poisson system involving critical growth, Nonlinear Anal., 202 (2021), 112144.  doi: 10.1016/j.na.2020.112144.  Google Scholar

[32]

M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[33]

J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang, Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 48 (2013), 243-273.  doi: 10.1007/s00526-012-0548-6.  Google Scholar

[34]

J. Zhang, J. M. DO Ó and M. Squassina, Fractional Schrödinger-Poisson system with a general subcritical or critical nonlinearity, Adv. Nonlinear Stud., 16 (2016), 15-30.  doi: 10.1515/ans-2015-5024.  Google Scholar

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