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Non-autonomous Schrödinger-Poisson system in $\mathbb{R}^{3}$
1. | School of Mathematics and Statistics, Shandong University of Technology Zibo 255049, China |
2. | School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China |
3. | Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan |
4. | School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, Texas 78539, USA |
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$ |
$\lambda >0$ |
$2 < p \le 4$ |
$K\left( x\right) $ |
$a\left( x\right) $ |
$\mathbb{R}^{3}$ |
$% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$ |
$\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$ |
$\lambda$ |
$p$ |
$3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$ |
References:
[1] |
A. Ambrosetti,
On the Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 39-404.
doi: 10.1142/S021919970800282X. |
[3] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[4] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[5] |
P. A. Binding, P. Drábek and Y. X. Huang,
On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.
|
[6] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490.
|
[7] |
K. J. Brown and T. F. Wu,
A fibrering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69 (2007), 1-9.
|
[8] |
K. J. Brown and T. F. Wu,
A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.
|
[9] |
K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[10] |
G. Cerami and G. Vaira,
Positive solutions for some non autonomous Schrödinger-Poisson Systems, J. Differential Equations, 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[11] |
C. Y. Chen, Y. C. Kuo and T. F. Wu,
Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.
doi: 10.1017/S0308210511000692. |
[12] |
G. M. Coclite and V. Georgiev,
Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.
|
[13] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[14] |
P. Drábek and S. I. Pohozaev,
Positive solutions for the $p$ -Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[15] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[16] |
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.
|
[17] |
I. Ianni and G. Vaira,
Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit, Nonlinear Differ. Equ. Appl., 22 (2015), 741-776.
doi: 10.1007/s00030-014-0303-0. |
[18] |
M. K. Kwong,
Uniqueness of positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
|
[19] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case Ⅰ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[20] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case Ⅱ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[21] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system, Applied Mathematics Letters, 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[22] |
Z. Nehari,
On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.
doi: 10.1090/S0002-9947-1960-0111898-8. |
[23] |
W. M. Ni and I. Takagi,
On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[24] |
D. Ruiz,
The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[25] |
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. |
[26] |
O. Sánchez and J. Soler,
Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.
doi: 10.1023/B:JOSS.0000003109.97208.53. |
[27] |
J. Sun, H. Chen and J. J. Nieto,
On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), 3365-3380.
doi: 10.1016/j.jde.2011.12.007. |
[28] |
J. Sun and T. F. Wu,
On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669.
doi: 10.1007/s00033-015-0494-1. |
[29] |
J. Sun, T. F. Wu and Z. Feng,
Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586-627.
doi: 10.1016/j.jde.2015.09.002. |
[30] |
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.
doi: 10.1016/S0294-1449(16)30238-4. |
[31] |
G. Vaira,
Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[32] |
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[33] |
L. Zhao, H. Liu and F. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.
doi: 10.1016/j.jde.2013.03.005. |
[34] |
L. Zhao and F. Zhao,
On the existence of solutions for the Schr ödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
show all references
References:
[1] |
A. Ambrosetti,
On the Schrödinger-Poisson systems, Milan J. Math., 76 (2008), 257-274.
doi: 10.1007/s00032-008-0094-z. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 39-404.
doi: 10.1142/S021919970800282X. |
[3] |
A. Azzollini and A. Pomponio,
Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008), 90-108.
doi: 10.1016/j.jmaa.2008.03.057. |
[4] |
V. Benci and D. Fortunato,
An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 11 (1998), 283-293.
doi: 10.12775/TMNA.1998.019. |
[5] |
P. A. Binding, P. Drábek and Y. X. Huang,
On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5 (1997), 1-11.
|
[6] |
H. Brézis and E. H. Lieb,
A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer Math. Soc., 88 (1983), 486-490.
|
[7] |
K. J. Brown and T. F. Wu,
A fibrering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69 (2007), 1-9.
|
[8] |
K. J. Brown and T. F. Wu,
A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22 (2009), 1097-1114.
|
[9] |
K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[10] |
G. Cerami and G. Vaira,
Positive solutions for some non autonomous Schrödinger-Poisson Systems, J. Differential Equations, 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[11] |
C. Y. Chen, Y. C. Kuo and T. F. Wu,
Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.
doi: 10.1017/S0308210511000692. |
[12] |
G. M. Coclite and V. Georgiev,
Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004), 1-31.
|
[13] |
T. D'Aprile and D. Mugnai,
Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.
|
[14] |
P. Drábek and S. I. Pohozaev,
Positive solutions for the $p$ -Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[15] |
I. Ekeland,
On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.
doi: 10.1016/0022-247X(74)90025-0. |
[16] |
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.
|
[17] |
I. Ianni and G. Vaira,
Non-radial sign-changing solutions for the Schrödinger-Poisson problem in the semiclassical limit, Nonlinear Differ. Equ. Appl., 22 (2015), 741-776.
doi: 10.1007/s00030-014-0303-0. |
[18] |
M. K. Kwong,
Uniqueness of positive solution of $Δ u-u+u^{p}=0$ in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.
|
[19] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case Ⅰ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 109-145.
doi: 10.1016/S0294-1449(16)30428-0. |
[20] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case Ⅱ, Ann. Inst. H. Poincar é Anal. Non Linéaire, 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[21] |
A. Mao, L. Yang, A. Qian and S. Luan,
Existence and concentration of solutions of Schrödinger-Poisson system, Applied Mathematics Letters, 68 (2017), 8-12.
doi: 10.1016/j.aml.2016.12.014. |
[22] |
Z. Nehari,
On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc., 95 (1960), 101-123.
doi: 10.1090/S0002-9947-1960-0111898-8. |
[23] |
W. M. Ni and I. Takagi,
On the shape of least energy solution to a Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.
doi: 10.1002/cpa.3160440705. |
[24] |
D. Ruiz,
The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 237 (2006), 655-674.
doi: 10.1016/j.jfa.2006.04.005. |
[25] |
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. |
[26] |
O. Sánchez and J. Soler,
Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Statist. Phys., 114 (2004), 179-204.
doi: 10.1023/B:JOSS.0000003109.97208.53. |
[27] |
J. Sun, H. Chen and J. J. Nieto,
On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 252 (2012), 3365-3380.
doi: 10.1016/j.jde.2011.12.007. |
[28] |
J. Sun and T. F. Wu,
On the nonlinear Schrödinger-Poisson systems with sign-changing potential, Z. Angew. Math. Phys., 66 (2015), 1649-1669.
doi: 10.1007/s00033-015-0494-1. |
[29] |
J. Sun, T. F. Wu and Z. Feng,
Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, 260 (2016), 586-627.
doi: 10.1016/j.jde.2015.09.002. |
[30] |
G. Tarantello,
On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.
doi: 10.1016/S0294-1449(16)30238-4. |
[31] |
G. Vaira,
Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011), 263-297.
doi: 10.1007/s11587-011-0109-x. |
[32] |
Z. Wang and H. Zhou,
Positive solution for a nonlinear stationary Schrödinger-Poisson system in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 18 (2007), 809-816.
doi: 10.3934/dcds.2007.18.809. |
[33] |
L. Zhao, H. Liu and F. Zhao,
Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential, J. Differential Equations, 255 (2013), 1-23.
doi: 10.1016/j.jde.2013.03.005. |
[34] |
L. Zhao and F. Zhao,
On the existence of solutions for the Schr ödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
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