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A direct method of moving planes for fully nonlinear nonlocal operators and applications
Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space
School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China |
$ \begin{equation*} \begin{cases} (-\Delta)^su+\phi u = |u|^{s^*-2}u+\mu|u|^{p-2}u \,\,\,\rm{in}\ \mathbb{R}^3,\\ -\Delta \phi = 4\pi u^2\ \rm{in}\ \mathbb{R}^3, \end{cases} \end{equation*} $ |
$ s\in(\frac{3}{4},1) $ |
$ \mu>0 $ |
$ p\in(3,s^*) $ |
$ s^* = \frac{6}{3-2s} $ |
References:
[1] |
C. Bardos, L. Erdös, F. Golse, N. Mauser and H.-T. Yau,
Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Math. Acad. Sci. Paris, 334 (2002), 515-520.
doi: 10.1016/S1631-073X(02)02253-7. |
[2] |
J. Bellazzini, M. Ghimenti, C. Mercuri, V. Moroz and J. Van Schaftingen,
Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc., 370 (2018), 8285-8310.
doi: 10.1090/tran/7426. |
[3] |
H. Brezis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[4] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
X. J. Chang and Z. Q. Wang,
Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.
doi: 10.1016/j.jde.2014.01.027. |
[6] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^N$, Lecture Notes. Vol. 15 Scuola Normale Superiore di Pisa (New Series). Pisa: Edizioni della Normale; 2017.
doi: 10.1007/978-88-7642-601-8. |
[7] |
M. M. Fall and E. Valdinoci,
Uniqueness and nondegeneracy of positve solutions of $(-\Delta)^su+u = u^p$ in $\mathbb{R}^N$ when $s$ is close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[8] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[9] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A: Math., 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[10] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[12] |
I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger-Poisson-Slater problem, Comm. Contemp. Math., 14 (2012), 1250003, 22pp.
doi: 10.1142/S0219199712500034. |
[13] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[14] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[15] |
K. X. Li,
Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations, Appl. Math. Lett., 72 (2017), 1-9.
doi: 10.1016/j.aml.2017.03.023. |
[16] |
Y. Li, D. Zhao and Q. X. Wang, Ground state solution and nodal solution for fractional nonlinear Schrödinger equation with indefinite potential, J. Math. Phys., 60 (2019), 041501, 15pp.
doi: 10.1063/1.5067377. |
[17] |
S. B. Liu and S. Mosconi,
On the Schrödinger-Poisson system with indefinite potential and $3$-sublinear nonlinearity, J. Differential Equations, 269 (2020), 689-712.
doi: 10.1016/j.jde.2019.12.023. |
[18] |
Z. Liu, Z. Zhang and S. Huang,
Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations, 266 (2019), 5912-5941.
doi: 10.1016/j.jde.2018.10.048. |
[19] |
H. X. Luo and X. H. Tang,
Ground state and multiple solutions for the fractional Schrödinger-Poisson system with critical Sobolev exponent, Nonlinear Anal. Real World Appl., 42 (2018), 24-52.
doi: 10.1016/j.nonrwa.2017.12.003. |
[20] |
C, Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency,, Calc. Var. Partial Differential Equations, 55 (2016), Art. 146, 58 pp.
doi: 10.1007/s00526-016-1079-3. |
[21] |
E. G. Murcia and G. Siciliano,
Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258.
|
[22] |
D. Ruiz and G. Siciliano,
A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2018), 179-190.
doi: 10.1515/ans-2008-0106. |
[23] |
D. Ruiz,
On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[24] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimun of potential, Rev. Mat. Iberoam., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[25] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$,, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[26] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[27] |
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. |
[28] |
X. D. Shang and J. H. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[29] |
K. M. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[30] |
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. |
[31] |
M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[32] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[33] |
J. Zhang, J. M. do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity, Commun. Contemp. Math., 19 (2017), 1650028, 16pp. |
show all references
References:
[1] |
C. Bardos, L. Erdös, F. Golse, N. Mauser and H.-T. Yau,
Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Math. Acad. Sci. Paris, 334 (2002), 515-520.
doi: 10.1016/S1631-073X(02)02253-7. |
[2] |
J. Bellazzini, M. Ghimenti, C. Mercuri, V. Moroz and J. Van Schaftingen,
Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces, Trans. Amer. Math. Soc., 370 (2018), 8285-8310.
doi: 10.1090/tran/7426. |
[3] |
H. Brezis and E. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[4] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[5] |
X. J. Chang and Z. Q. Wang,
Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992.
doi: 10.1016/j.jde.2014.01.027. |
[6] |
S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb R^N$, Lecture Notes. Vol. 15 Scuola Normale Superiore di Pisa (New Series). Pisa: Edizioni della Normale; 2017.
doi: 10.1007/978-88-7642-601-8. |
[7] |
M. M. Fall and E. Valdinoci,
Uniqueness and nondegeneracy of positve solutions of $(-\Delta)^su+u = u^p$ in $\mathbb{R}^N$ when $s$ is close to 1, Comm. Math. Phys., 329 (2014), 383-404.
doi: 10.1007/s00220-014-1919-y. |
[8] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[9] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A: Math., 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[10] |
R. L. Frank and E. Lenzmann,
Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.
doi: 10.1007/s11511-013-0095-9. |
[11] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[12] |
I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger-Poisson-Slater problem, Comm. Contemp. Math., 14 (2012), 1250003, 22pp.
doi: 10.1142/S0219199712500034. |
[13] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[14] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[15] |
K. X. Li,
Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations, Appl. Math. Lett., 72 (2017), 1-9.
doi: 10.1016/j.aml.2017.03.023. |
[16] |
Y. Li, D. Zhao and Q. X. Wang, Ground state solution and nodal solution for fractional nonlinear Schrödinger equation with indefinite potential, J. Math. Phys., 60 (2019), 041501, 15pp.
doi: 10.1063/1.5067377. |
[17] |
S. B. Liu and S. Mosconi,
On the Schrödinger-Poisson system with indefinite potential and $3$-sublinear nonlinearity, J. Differential Equations, 269 (2020), 689-712.
doi: 10.1016/j.jde.2019.12.023. |
[18] |
Z. Liu, Z. Zhang and S. Huang,
Existence and nonexistence of positive solutions for a static Schrödinger-Poisson-Slater equation, J. Differential Equations, 266 (2019), 5912-5941.
doi: 10.1016/j.jde.2018.10.048. |
[19] |
H. X. Luo and X. H. Tang,
Ground state and multiple solutions for the fractional Schrödinger-Poisson system with critical Sobolev exponent, Nonlinear Anal. Real World Appl., 42 (2018), 24-52.
doi: 10.1016/j.nonrwa.2017.12.003. |
[20] |
C, Mercuri, V. Moroz and J. Van Schaftingen, Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency,, Calc. Var. Partial Differential Equations, 55 (2016), Art. 146, 58 pp.
doi: 10.1007/s00526-016-1079-3. |
[21] |
E. G. Murcia and G. Siciliano,
Positive semiclassical states for a fractional Schrödinger-Poisson system, Differential Integral Equations, 30 (2017), 231-258.
|
[22] |
D. Ruiz and G. Siciliano,
A note on the Schrödinger-Poisson-Slater equation on bounded domains, Adv. Nonlinear Stud., 8 (2018), 179-190.
doi: 10.1515/ans-2008-0106. |
[23] |
D. Ruiz,
On the Schrödinger-Poisson-Slater system: Behavior of minimizers, radial and nonradial cases, Arch. Ration. Mech. Anal., 198 (2010), 349-368.
doi: 10.1007/s00205-010-0299-5. |
[24] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimun of potential, Rev. Mat. Iberoam., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[25] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$,, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[26] |
R. Servadei and E. Valdinoci,
The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.
doi: 10.1090/S0002-9947-2014-05884-4. |
[27] |
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. |
[28] |
X. D. Shang and J. H. Zhang,
Ground states for fractional Schrödinger equations with critical growth, Nonlinearity, 27 (2014), 187-207.
doi: 10.1088/0951-7715/27/2/187. |
[29] |
K. M. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differential Equations, 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[30] |
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. |
[31] |
M. Willem, Minimax Theorems, Birkhauser, Basel, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[32] |
J. Zhang,
On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 75 (2012), 6391-6401.
doi: 10.1016/j.na.2012.07.008. |
[33] |
J. Zhang, J. M. do Ó and M. Squassina, Schrödinger-Poisson systems with a general critical nonlinearity, Commun. Contemp. Math., 19 (2017), 1650028, 16pp. |
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