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June  2021, 20(6): 2291-2312. doi: 10.3934/cpaa.2021077

Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth

School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author

Received  January 2021 Revised  March 2021 Published  May 2021

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11971393)

In this paper, we investigate the existence and asymptotic behavior of least energy sign-changing solutions for the following Schrödinger-Poisson system
$ \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*} $
where
$ \lambda>0 $
is a parameter. Under some suitable conditions on
$ f $
and
$ V $
, we get a least energy sign-changing solution
$ u_\lambda $
via variational method and its energy is strictly larger than twice that of least energy solutions. Moreover, the asymptotic behavior of
$ u_\lambda $
as
$ \lambda\rightarrow 0^+ $
is also analyzed.
Citation: Xiaoping Chen, Chunlei Tang. Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2291-2312. doi: 10.3934/cpaa.2021077
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G. CeramiS. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453.  Google Scholar

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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.  Google Scholar

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E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev inequality and related inequalities, Ann. Math., 118 (1983), 349-374.  doi: 10.2307/2007032.  Google Scholar

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J. LiuJ. 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.  Google Scholar

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C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, inG. Prodi (Ed.), Eigenvalues of Nonlinear Problems, CIME, (1974), 141–195.  Google Scholar

[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.  Google Scholar

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W. A. Strauss, Existence of solitary waves in higher dimensions., Commun. Math. Phys., 55, (1977), 149–162.  Google Scholar

[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.  Google Scholar

[22]

D. B. WangH. 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.  Google Scholar

[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.  Google Scholar

[24]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. M. Zou and M. Schechter, Critical Point Theory and its Applications, Springer, New York, 2006.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

G. CeramiS. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. LiuJ. 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.  Google Scholar

[15]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Un. Mat. Ital., 3 (1940), 5-7.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

P. H. Rabinowitz, Variational methods for nonlinear eigenvalue problems, inG. Prodi (Ed.), Eigenvalues of Nonlinear Problems, CIME, (1974), 141–195.  Google Scholar

[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.  Google Scholar

[20]

W. A. Strauss, Existence of solitary waves in higher dimensions., Commun. Math. Phys., 55, (1977), 149–162.  Google Scholar

[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.  Google Scholar

[22]

D. B. WangH. 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.  Google Scholar

[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.  Google Scholar

[24]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. M. Zou and M. Schechter, Critical Point Theory and its Applications, Springer, New York, 2006.  Google Scholar

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