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June  2021, 14(6): 1899-1916. doi: 10.3934/dcdss.2021064

Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space

Hangzhou Hu , Yuan Li and  Dun Zhao ,

School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

* Corresponding author: Dun Zhao

Received  December 2020 Revised  March 2021 Published  June 2021 Early access  May 2021

Fund Project: This work is supported by NSFC under Grant No. 12075102 and 11971212

We consider the following fractional Schrödinger-Poisson equation with combined nonlinearities
$ \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*} $
where
$ s\in(\frac{3}{4},1) $
,
$ \mu>0 $
,
$ p\in(3,s^*) $
 and
$ s^* = \frac{6}{3-2s} $
. By the perturbation approach we prove the existence of the ground state solution in fractional Coulomb-Sobolev space.
Keywords: Ground state, Fractional Schrödinger-Poisson equation, Coulomb-Sobolev space.
Mathematics Subject Classification: Primary: 35A15, 35Q55, 35R11; Secondary: 35Q40.
Citation: Hangzhou Hu, Yuan Li, Dun Zhao. Ground state for fractional Schrödinger-Poisson equation in Coulomb-Sobolev space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1899-1916. doi: 10.3934/dcdss.2021064
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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