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Infinitely many segregated solutions for coupled nonlinear Schrödinger systems
Standing waves for Schrödinger-Poisson system with general nonlinearity
1. | School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
2. | School of Mathematics and Statistics, Hunan University of Technology and Business, Changsha, 410205 Hunan, China |
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*} $ |
$ \varepsilon>0 $ |
$ V $ |
$ f, $ |
$ V $ |
$ u\rightarrow\frac{f(u)}{u^3} $ |
$ f(u) = |u|^{p-2}u $ |
$ 3<p<6 $ |
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[3] |
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. |
[4] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[5] |
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. |
[6] |
J. Byeon and L. Jeanjean,
Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[7] |
G. Cerami and G. Vaira,
Positive solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differ. Equ., 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[8] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[9] |
S. T. Chen and X. H. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[10] |
S. T. Chen and X. H. Tang,
Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193.
doi: 10.1016/j.aml.2018.11.007. |
[11] |
S. T. Chen and X. H. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111.
doi: 10.1016/j.jmaa.2018.12.037. |
[12] |
T. D'Aprile and J. C. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[13] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Junior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[15] |
X. M. He,
Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.
doi: 10.1007/s00033-011-0120-9. |
[16] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[17] |
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. |
[18] |
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. |
[19] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[20] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[21] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[22] |
G. B. Li and S. S. Yan,
Eigenvalue problems for quasilinear elliptic equations on $\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314.
doi: 10.1080/03605308908820654. |
[23] |
P. L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[24] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire., 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[25] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[26] |
N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. |
[27] |
M. del Pino and P. Felmer,
Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[28] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[29] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[30] |
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. |
[31] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[32] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoamericana., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[33] |
J. Seok,
On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.
doi: 10.1016/j.jmaa.2012.12.054. |
[34] |
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differe. Equ., 260 (2016), 2119-2149.
doi: 10.1016/j.jde.2015.09.057. |
[35] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[36] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials, Calc. Var. PDE, 56 (2017), Art. 110, 25 pp.
doi: 10.1007/s00526-017-1214-9. |
[37] |
X. H. Tang, X. Y. Lin and J. S. Yu,
Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 31 (2018), 369-383.
doi: 10.1007/s10884-018-9662-2. |
[38] |
X. F. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[39] |
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. PDE., 48 (2013), 243-273.
doi: 10.1007/s00526-012-0548-6. |
[40] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471.
doi: 10.1007/s00033-015-0531-0. |
[41] |
M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[42] |
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.
doi: 10.3934/dcds.2017195. |
[43] |
X. Zhang and J. K. Xia,
Semi-classical solutions for Schrödinger-Poisson equations with a critical frequency, J. Differ. Equ., 265 (2018), 2121-2170.
doi: 10.1016/j.jde.2018.04.023. |
[44] |
L. G. Zhao and F. K. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
[45] |
L. G. Zhao and F. K. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
show all references
References:
[1] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
doi: 10.1142/S021919970800282X. |
[2] |
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. |
[3] |
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. |
[4] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[5] |
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. |
[6] |
J. Byeon and L. Jeanjean,
Standing waves with a critical frequency for nonlinear Schrödinger equations Ⅱ, Calc. Var. Partial Differ. Equ., 18 (2003), 207-219.
doi: 10.1007/s00526-002-0191-8. |
[7] |
G. Cerami and G. Vaira,
Positive solutions for some non-autonomous Schrödinger-Maxwell systems, J. Differ. Equ., 248 (2010), 521-543.
doi: 10.1016/j.jde.2009.06.017. |
[8] |
S. T. Chen and X. H. Tang, Ground state sign-changing solutions for a class of Schrödinger-Poisson type problems in $\mathbb{R}^3$, Z. Angew. Math. Phys., 67 (2016), Art. 102, 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[9] |
S. T. Chen and X. H. Tang,
Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete. Contin. Dyn. Syst., 38 (2018), 2333-2348.
doi: 10.3934/dcds.2018096. |
[10] |
S. T. Chen and X. H. Tang,
Geometrically distinct solutions for Klein-Gordon-Maxwell systems with superlinear nonlinearities, Appl. Math. Letters, 90 (2019), 188-193.
doi: 10.1016/j.aml.2018.11.007. |
[11] |
S. T. Chen and X. H. Tang,
Ground state solutions of Schrödinger-Poisson systems with variable potential and convolution nonlinearity, J. Math. Anal. Appl., 473 (2019), 87-111.
doi: 10.1016/j.jmaa.2018.12.037. |
[12] |
T. D'Aprile and J. C. Wei,
On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., 37 (2005), 321-342.
doi: 10.1137/S0036141004442793. |
[13] |
G. M. Figueiredo, N. Ikoma and J. R. Santos Junior,
Existence and concentration result for the Kirchhoff type equations with general nonlinearities, Arch. Ration. Mech. Anal., 213 (2014), 931-979.
doi: 10.1007/s00205-014-0747-8. |
[14] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss., vol. 224, Springer, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[15] |
X. M. He,
Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 62 (2011), 869-889.
doi: 10.1007/s00033-011-0120-9. |
[16] |
X. M. He and W. M. Zou, Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 53 (2012), 023702, 19 pp.
doi: 10.1063/1.3683156. |
[17] |
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. |
[18] |
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. |
[19] |
I. Ianni,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅱ: Existence, Math. Models Methods Appl. Sci., 19 (2009), 877-910.
doi: 10.1142/S0218202509003656. |
[20] |
I. Ianni and G. Vaira,
Solutions of the Schrödinger-Poisson problem concentrating on spheres, part Ⅰ: necessary condition, Math. Models Methods Appl. Sci., 19 (2009), 707-720.
doi: 10.1142/S0218202509003589. |
[21] |
L. Jeanjean,
Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.
doi: 10.1016/S0362-546X(96)00021-1. |
[22] |
G. B. Li and S. S. Yan,
Eigenvalue problems for quasilinear elliptic equations on $\mathbb{R}^N$, Commun. Partial Differ. Equ., 14 (1989), 1291-1314.
doi: 10.1080/03605308908820654. |
[23] |
P. L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[24] |
P. L. Lions,
The concentration-compactness principle in the calculus of variations, The locally compact case, part 2, Ann. Inst. H. Poincaré Anal. Non. Linéaire., 1 (1984), 223-283.
doi: 10.1016/S0294-1449(16)30422-X. |
[25] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2. |
[26] |
N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovs, Nonlinear Analysis-Theory and Methods,, Springer Monographs in Mathematics, Springer, BerlinCham, 2019. |
[27] |
M. del Pino and P. Felmer,
Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[28] |
P. Pucci and J. Serrin,
A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[29] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[30] |
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. |
[31] |
D. Ruiz,
Semiclassical states for coupled Schrödinger-Maxwell concentration around a sphere, Math. Models Methods Appl. Sci., 15 (2005), 141-164.
doi: 10.1142/S0218202505003939. |
[32] |
D. Ruiz and G. Vaira,
Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of potential, Rev. Mat. Iberoamericana., 27 (2011), 253-271.
doi: 10.4171/RMI/635. |
[33] |
J. Seok,
On nonlinear Schrödinger-Poisson equations with general potentials, J. Math. Anal. Appl., 401 (2013), 672-681.
doi: 10.1016/j.jmaa.2012.12.054. |
[34] |
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differe. Equ., 260 (2016), 2119-2149.
doi: 10.1016/j.jde.2015.09.057. |
[35] |
X. H. Tang and S. T. Chen,
Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials, Discrete. Contin. Dyn. Syst., 37 (2017), 4973-5002.
doi: 10.3934/dcds.2017214. |
[36] |
X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozǎev type for Kirchhoff-type problems with general potentials, Calc. Var. PDE, 56 (2017), Art. 110, 25 pp.
doi: 10.1007/s00526-017-1214-9. |
[37] |
X. H. Tang, X. Y. Lin and J. S. Yu,
Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ., 31 (2018), 369-383.
doi: 10.1007/s10884-018-9662-2. |
[38] |
X. F. Wang,
On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153 (1993), 229-244.
doi: 10.1007/BF02096642. |
[39] |
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. PDE., 48 (2013), 243-273.
doi: 10.1007/s00526-012-0548-6. |
[40] |
J. Wang, L. X. Tian, J. X. Xu and F. B. Zhang,
Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth, Z. Angew. Math. Phys., 66 (2015), 2441-2471.
doi: 10.1007/s00033-015-0531-0. |
[41] |
M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[42] |
J. Zhang, W. Zhang and X. H. Tang,
Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.
doi: 10.3934/dcds.2017195. |
[43] |
X. Zhang and J. K. Xia,
Semi-classical solutions for Schrödinger-Poisson equations with a critical frequency, J. Differ. Equ., 265 (2018), 2121-2170.
doi: 10.1016/j.jde.2018.04.023. |
[44] |
L. G. Zhao and F. K. Zhao,
On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., 346 (2008), 155-169.
doi: 10.1016/j.jmaa.2008.04.053. |
[45] |
L. G. Zhao and F. K. Zhao,
Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear Anal., 70 (2009), 2150-2164.
doi: 10.1016/j.na.2008.02.116. |
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