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April  2022, 21(4): 1157-1187. doi: 10.3934/cpaa.2022014

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 Published  April 2022 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 and Applied Analysis, 2022, 21 (4) : 1157-1187. doi: 10.3934/cpaa.2022014
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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

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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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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