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Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics
Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth
School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China |
$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda\phi(x)u = |u|^4u+ f(u),\ \ \ &\ x \in \mathbb{R}^{3},\\ -\Delta \phi = u^2, \ \ \ &\ x \in \mathbb{R}^{3}, \end{cases} \end{equation*} $ |
$ \lambda>0 $ |
$ f $ |
$ V $ |
$ u_\lambda $ |
$ u_\lambda $ |
$ \lambda\rightarrow 0^+ $ |
References:
[1] |
T. Bartsch and T. Weth,
Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[2] |
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. |
[3] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[4] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[6] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[7] |
J. Q. Chen,
Multiple positive solutions of a class of non autonomous Schrödinger-Poisson systems, Nonlinear Anal. Real World Appl., 21 (2015), 13-26.
doi: 10.1016/j.nonrwa.2014.06.002. |
[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), 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[9] |
L. H. Gu, H. Jin and J. J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897.
doi: 10.1016/j.na.2020.111897. |
[10] |
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. |
[11] |
H. Hofer,
Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.
doi: 10.1007/BF01457453. |
[12] |
Q. F. Jin,
Multiple sign-changing solutions for nonlinear Schrödinger equations with potential well, Appl. Anal., 99 (2020), 2555-2570.
doi: 10.1080/00036811.2019.1572883. |
[13] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[14] |
J. Liu, J. F. Liao and C. L. Tang,
Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.
doi: 10.1088/1361-6544/aa5659. |
[15] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[16] |
J. J. Nie and Q. Q. Li, Multiplicity of sign-changing solutions for a supercritical nonlinear Schrödinger equation, Appl. Math. Lett., 109 (2020), 106569.
doi: 10.1016/j.aml.2020.106569. |
[17] |
A. X. Qian, J. M. Liu and A. M. Mao, Ground state and nodal solutions for a Schrödinger-Poisson equation with critical growth, J. Math. Phys., 59 (2018), 121509.
doi: 10.1063/1.5050856. |
[18] |
P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, inG. Prodi (Ed.), Eigenvalues of Nonlinear Problems, CIME, (1974), 141–195. |
[19] |
W. Shuai and Q. F. Wang,
Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3267-3282.
doi: 10.1007/s00033-015-0571-5. |
[20] |
W. A. Strauss, Existence of solitary waves in higher dimensions., Commun. Math. Phys., 55, (1977), 149–162. |
[21] |
K. M. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equ., 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[22] |
D. B. Wang, H. B. Huang and W. Guan,
Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301.
doi: 10.1016/j.jmaa.2019.07.052. |
[23] |
Z. P. Wang and H. S. Zhou,
Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 52 (2015), 927-943.
doi: 10.1007/s00526-014-0738-5. |
[24] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[25] |
Y. Z. Wu and Y. S. Huang,
Sign-changing solutions for Schrödinger equations with indefinite supperlinear nonlinearities, J. Math. Anal. Appl., 401 (2013), 850-860.
doi: 10.1016/j.jmaa.2013.01.006. |
[26] |
X. Y. Yang, X. H. Tang and Y. P. Zhang, Positive, negative, and sign-changing solutions to a quasilinear Schrödinger equation with a parameter, J. Math. Phys., 60 (2019), 121510.
doi: 10.1063/1.5116602. |
[27] |
L. F. Yin, X. P. Wu and C. L. Tang, Existence and concentration of ground state solutions for critical Schrödinger-Poisson system with steep potential well, Appl. Math. Comput., 374 (2020), 125035.
doi: 10.1016/j.amc.2020.125035. |
[28] |
X. J. Zhong and C. L. Tang,
Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 39 (2018), 166-184.
doi: 10.1016/j.nonrwa.2017.06.014. |
[29] |
W. M. Zou and M. Schechter, Critical Point Theory and its Applications, Springer, New York, 2006. |
show all references
References:
[1] |
T. Bartsch and T. Weth,
Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 259-281.
doi: 10.1016/j.anihpc.2004.07.005. |
[2] |
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. |
[3] |
V. Benci and D. Fortunato,
Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations, Rev. Math. Phys., 14 (2002), 409-420.
doi: 10.1142/S0129055X02001168. |
[4] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[5] |
K. J. Brown and Y. Zhang,
The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differ. Equ., 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[6] |
G. Cerami, S. Solimini and M. Struwe,
Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[7] |
J. Q. Chen,
Multiple positive solutions of a class of non autonomous Schrödinger-Poisson systems, Nonlinear Anal. Real World Appl., 21 (2015), 13-26.
doi: 10.1016/j.nonrwa.2014.06.002. |
[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), 18 pp.
doi: 10.1007/s00033-016-0695-2. |
[9] |
L. H. Gu, H. Jin and J. J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897.
doi: 10.1016/j.na.2020.111897. |
[10] |
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. |
[11] |
H. Hofer,
Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.
doi: 10.1007/BF01457453. |
[12] |
Q. F. Jin,
Multiple sign-changing solutions for nonlinear Schrödinger equations with potential well, Appl. Anal., 99 (2020), 2555-2570.
doi: 10.1080/00036811.2019.1572883. |
[13] |
E. H. Lieb,
Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. Math., 118 (1983), 349-374.
doi: 10.2307/2007032. |
[14] |
J. Liu, J. F. Liao and C. L. Tang,
Ground state solution for a class of Schrödinger equations involving general critical growth term, Nonlinearity, 30 (2017), 899-911.
doi: 10.1088/1361-6544/aa5659. |
[15] |
C. Miranda,
Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.
|
[16] |
J. J. Nie and Q. Q. Li, Multiplicity of sign-changing solutions for a supercritical nonlinear Schrödinger equation, Appl. Math. Lett., 109 (2020), 106569.
doi: 10.1016/j.aml.2020.106569. |
[17] |
A. X. Qian, J. M. Liu and A. M. Mao, Ground state and nodal solutions for a Schrödinger-Poisson equation with critical growth, J. Math. Phys., 59 (2018), 121509.
doi: 10.1063/1.5050856. |
[18] |
P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, inG. Prodi (Ed.), Eigenvalues of Nonlinear Problems, CIME, (1974), 141–195. |
[19] |
W. Shuai and Q. F. Wang,
Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Z. Angew. Math. Phys., 66 (2015), 3267-3282.
doi: 10.1007/s00033-015-0571-5. |
[20] |
W. A. Strauss, Existence of solitary waves in higher dimensions., Commun. Math. Phys., 55, (1977), 149–162. |
[21] |
K. M. Teng,
Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Differ. Equ., 261 (2016), 3061-3106.
doi: 10.1016/j.jde.2016.05.022. |
[22] |
D. B. Wang, H. B. Huang and W. Guan,
Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301.
doi: 10.1016/j.jmaa.2019.07.052. |
[23] |
Z. P. Wang and H. S. Zhou,
Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Calc. Var. Partial Differ. Equ., 52 (2015), 927-943.
doi: 10.1007/s00526-014-0738-5. |
[24] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
[25] |
Y. Z. Wu and Y. S. Huang,
Sign-changing solutions for Schrödinger equations with indefinite supperlinear nonlinearities, J. Math. Anal. Appl., 401 (2013), 850-860.
doi: 10.1016/j.jmaa.2013.01.006. |
[26] |
X. Y. Yang, X. H. Tang and Y. P. Zhang, Positive, negative, and sign-changing solutions to a quasilinear Schrödinger equation with a parameter, J. Math. Phys., 60 (2019), 121510.
doi: 10.1063/1.5116602. |
[27] |
L. F. Yin, X. P. Wu and C. L. Tang, Existence and concentration of ground state solutions for critical Schrödinger-Poisson system with steep potential well, Appl. Math. Comput., 374 (2020), 125035.
doi: 10.1016/j.amc.2020.125035. |
[28] |
X. J. Zhong and C. L. Tang,
Ground state sign-changing solutions for a Schrödinger-Poisson system with a critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl., 39 (2018), 166-184.
doi: 10.1016/j.nonrwa.2017.06.014. |
[29] |
W. M. Zou and M. Schechter, Critical Point Theory and its Applications, Springer, New York, 2006. |
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