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Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type
Concentration of bound states for fractional Schrödinger-Poisson system via penalization methods
1. | Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China |
2. | Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China |
$ \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*} $ |
$ s,t\in(0,1) $ |
$ \varepsilon>0 $ |
$ V(x) $ |
$ g $ |
$ u_{\varepsilon}\in H_{\varepsilon} $ |
$ V(x) $ |
$ \varepsilon\rightarrow0 $ |
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. |
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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