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Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials
School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
$\left\{ \begin{array}{ll} -\triangle u+V(x)u+φ u=f(u), \ \ \ \ x∈ \mathbb{R}^{3},\\ -\triangle φ=u^2, \ \ \ \ x∈ \mathbb{R}^{3}, \end{array} \right. $ |
$V(x)$ |
$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$ |
$V$ |
$f$ |
References:
[1] |
N. Ackermann and T. Weth,
Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math., 7 (2005), 269-298.
doi: 10.1142/S0219199705001763. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
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] |
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. |
[6] |
R. Benguria, H. Brezis and E. H. Lieb,
The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.
doi: 10.1007/BF01942059. |
[7] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[8] |
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. |
[9] |
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. |
[10] |
S. T. Chen and X. H. Tang,
Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[11] |
G. M. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[12] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. -A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[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] |
Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007.
doi: 10.1142/9789812709639.![]() ![]() ![]() |
[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, 19pp.
doi: 10.1063/1.3683156. |
[17] |
L. R. Huang, E. M. Rocha and J. Q. Chen,
Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.
doi: 10.1016/j.jde.2013.06.022. |
[18] |
W. N. Huang and X. H. Tang,
Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 415 (2014), 791-802.
doi: 10.1016/j.jmaa.2014.02.015. |
[19] |
W. N. Huang and X. H. Tang,
Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations with critical nonlinearity, Taiwan. J. Math., 18 (2014), 1203-1217.
doi: 10.11650/tjm.18.2014.3993. |
[20] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. -A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[21] |
L. Jeanjean and J. Toland,
Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 327 (1998), 23-28.
doi: 10.1016/S0764-4442(98)80097-9. |
[22] |
E. H. Lieb,
Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[23] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[24] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001.
doi: 10.1090/gsm/014. |
[25] |
P. L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[26] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2.![]() ![]() ![]() |
[27] |
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. |
[28] |
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. |
[29] |
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. |
[30] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[31] |
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.
doi: 10.1016/j.jde.2015.09.057. |
[32] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[33] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1.![]() ![]() ![]() |
[34] |
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. |
[35] |
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] |
N. Ackermann and T. Weth,
Multibump solutions of nonlinear periodic Schrödinger equations in a degenerate setting, Commun. Contemp. Math., 7 (2005), 269-298.
doi: 10.1142/S0219199705001763. |
[2] |
A. Ambrosetti and D. Ruiz,
Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.
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] |
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. |
[6] |
R. Benguria, H. Brezis and E. H. Lieb,
The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys., 79 (1981), 167-180.
doi: 10.1007/BF01942059. |
[7] |
I. Catto and P. L. Lions,
Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories, Part 1: A necessary and sufficient condition for the stability of general molecular system, Comm. Partial Differential Equations, 17 (1992), 1051-1110.
doi: 10.1080/03605309208820878. |
[8] |
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. |
[9] |
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. |
[10] |
S. T. Chen and X. H. Tang,
Nehari type ground state solutions for asymptotically periodic Schrödinger-Poisson systems, Taiwan. J. Math., 21 (2017), 363-383.
doi: 10.11650/tjm/7784. |
[11] |
G. M. Coclite,
A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 7 (2003), 417-423.
|
[12] |
T. D'Aprile and D. Mugnai,
Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. -A, 134 (2004), 893-906.
doi: 10.1017/S030821050000353X. |
[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] |
Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific, Singapore, 2007.
doi: 10.1142/9789812709639.![]() ![]() ![]() |
[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, 19pp.
doi: 10.1063/1.3683156. |
[17] |
L. R. Huang, E. M. Rocha and J. Q. Chen,
Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, 255 (2013), 2463-2483.
doi: 10.1016/j.jde.2013.06.022. |
[18] |
W. N. Huang and X. H. Tang,
Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 415 (2014), 791-802.
doi: 10.1016/j.jmaa.2014.02.015. |
[19] |
W. N. Huang and X. H. Tang,
Semiclassical solutions for the nonlinear Schrödinger-Maxwell equations with critical nonlinearity, Taiwan. J. Math., 18 (2014), 1203-1217.
doi: 10.11650/tjm.18.2014.3993. |
[20] |
L. Jeanjean,
On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer type problem set on $\mathbb{R}^N$, Proc. Roy. Soc. Edinburgh Sect. -A, 129 (1999), 787-809.
doi: 10.1017/S0308210500013147. |
[21] |
L. Jeanjean and J. Toland,
Bounded Palais-Smale mountain-pass sequences, C. R. Acad. Sci. Paris Sér. Ⅰ Math., 327 (1998), 23-28.
doi: 10.1016/S0764-4442(98)80097-9. |
[22] |
E. H. Lieb,
Thomas-Fermi and related theories and molecules, Rev. Modern Phys., 53 (1981), 603-641.
doi: 10.1103/RevModPhys.53.603. |
[23] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. of Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[24] |
E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, 2001.
doi: 10.1090/gsm/014. |
[25] |
P. L. Lions,
Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys., 109 (1987), 33-97.
doi: 10.1007/BF01205672. |
[26] |
P. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, New York, 1990.
doi: 10.1007/978-3-7091-6961-2.![]() ![]() ![]() |
[27] |
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. |
[28] |
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. |
[29] |
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. |
[30] |
M. Struwe,
A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., 187 (1984), 511-517.
doi: 10.1007/BF01174186. |
[31] |
J. J. Sun and S. W. Ma,
Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, 260 (2016), 2119-2149.
doi: 10.1016/j.jde.2015.09.057. |
[32] |
X. H. Tang and B. T. Cheng,
Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402.
doi: 10.1016/j.jde.2016.04.032. |
[33] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1.![]() ![]() ![]() |
[34] |
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. |
[35] |
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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