We study the existence of positive solutions for the non-autonomous Schrödinger-Poisson system:
$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$
where $\lambda >0$, $2 < p \le 4$ and both $K\left( x\right) $ and $a\left( x\right) $ are nonnegative functions in $\mathbb{R}^{3}$, which satisfy the given conditions, but not require any symmetry property. Assuming that $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$ and $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$, we explore the existence of positive solutions, depending on the parameters $\lambda$ and $p$. More importantly, we establish the existence of ground state solutions in the case of $3.18 \approx \frac{{1 + \sqrt {73} }}{3} < P \le 4$.
Citation: |
A. Ambrosetti
, On the Schrödinger-Poisson systems, Milan J. Math., 76 (2008)
, 257-274.
doi: 10.1007/s00032-008-0094-z.![]() ![]() ![]() |
|
A. Ambrosetti
and D. Ruiz
, Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008)
, 39-404.
doi: 10.1142/S021919970800282X.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.
![]() ![]() |
|
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.
![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. M. Coclite
and V. Georgiev
, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 94 (2004)
, 1-31.
![]() ![]() |
|
T. D'Aprile
and D. Mugnai
, Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 4 (2004)
, 307-322.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
I. Ekeland
, On the variational principle, J. Math. Anal. Appl., 47 (1974)
, 324-353.
doi: 10.1016/0022-247X(74)90025-0.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. Vaira
, Ground states for Schrödinger-Poisson type systems, Ric. Mat., 60 (2011)
, 263-297.
doi: 10.1007/s11587-011-0109-x.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |