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doi: 10.3934/dcdsb.2022112
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## Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well

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

*Corresponding author: Chun-Lei Tang

Received  November 2021 Revised  April 2022 Early access June 2022

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

In the present paper, we investigate a class of nonlinear Schrödinger-Poisson system
 $\begin{equation*} \begin{cases} -\Delta u+V_\lambda(x)u+\mu \phi u = f(u) \quad \quad \ {\rm{in}} \ {\mathbb{R}}^{3},\\ -\Delta \phi = u^2 \quad \quad \quad \quad \ \ \quad \quad \quad \quad \quad \quad {\rm{in}} \ {\mathbb{R}}^{3}, \end{cases} \end{equation*}$
where
 $\mu>0$
and
 $V_\lambda(x) = \lambda V(x)+1$
with
 $\lambda>0$
. Under some mild assumptions on
 $V$
and
 $f$
, we prove the existence of ground state sign-changing solution for
 $\lambda>0$
large enough by adopting the deformation lemma and constrained minimization arguments. Then, the least energy of sign-changing solutions is strictly large than two times the ground state energy. Additionally, the phenomenon of concentration for ground state sign-changing solutions is also analysed as
 $\lambda\rightarrow \infty$
and
 $\mu\rightarrow0$
.
Citation: Jin-Cai Kang, Xiao-Qi Liu, Chun-Lei Tang. Ground state sign-changing solution for Schrödinger-Poisson system with steep potential well. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022112
##### References:
 [1] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\mathbb{ R}}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149. [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] 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. [5] S.-J. Chen and C.-L. Tang, High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050. [6] S. Chen and X. Tang, Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446-458.  doi: 10.1016/j.camwa.2017.04.031. [7] S. Chen, X. Tang and J. Peng, Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential, Studia Sci. Math. Hungar., 55 (2018), 53-93.  doi: 10.1556/012.2018.55.1.1388. [8] X. Chen and C. Tang, Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth, Commun. Pure Appl. Anal., 20 (2021), 2291-2312.  doi: 10.3934/cpaa.2021077. [9] M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502, 19 pp. doi: 10.1063/1.4941036. [10] L. Gu, H. Jin and J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897, 16 pp. doi: 10.1016/j.na.2020.111897. [11] H. Guo and T. Wang, A note on sign-changing solutions for the schrödinger poisson system, Electronic Research Archive, 28 (2020), 195-203.  doi: 10.3934/era.2020013. [12] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453. [13] I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. [14] C. Ji, F. Fang and B. Zhang, Least energy sign-changing solutions for the nonlinear Schrödinger-Poisson system, Electron. J. Differential Equations, 2017 (2017), 282, 13 pp. [15] Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006. [16] S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14 (2012), 1250041, 16 pp. doi: 10.1142/S0219199712500411. [17] G. Li, X. Luo and W. Shuai, Sign-changing solutions to a gauged nonlinear Schrödinger equation, J. Math. Anal. Appl., 455 (2017), 1559-1578.  doi: 10.1016/j.jmaa.2017.06.048. [18] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2. [19] S. Liu and S. Mosconi, On the Schrödinger-Poisson system with indefinite potential and 3-sublinear nonlinearity, J. Differential Equations, 269 (2020), 689-712.  doi: 10.1016/j.jde.2019.12.023. [20] W. Liu and Z. Wang, Least energy nodal solution for nonlinear Schrödinger equation without (AR) condition, J. Math. Anal. Appl., 462 (2018), 285-297.  doi: 10.1016/j.jmaa.2018.02.005. [21] Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8. [22] Q.-J. Lou and Z.-Q. Han, The Nehari manifold for the Schrödinger-Poisson systems with steep well potential, Complex Var. Elliptic Equ., 64 (2019), 586-605.  doi: 10.1080/17476933.2018.1471070. [23] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. [24] E. G. Murcia and G. Siciliano, Least energy radial sign-changing solution for the Schrödinger-Poisson system in ${\mathbb{R}}^3$ under an asymptotically cubic nonlinearity, J. Math. Anal. Appl., 474 (2019), 544-571.  doi: 10.1016/j.jmaa.2019.01.063. [25] 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. [26] W. Shuai and Q. 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. [27] J. Sun and T.-f. Wu, Bound state nodal solutions for the non-autonomous Schrödinger-Poisson system in ${\mathbb{R}}^3$, J. Differential Equations, 268 (2020), 7121-7163.  doi: 10.1016/j.jde.2019.11.070. [28] J. Sun and T.-f. Wu, On Schrödinger-Poisson systems under the effect of steep potential well $(2 < p < 4)$, J. Math. Phys., 61 (2020), 071506, 13 pp. doi: 10.1063/1.5114672. [29] J. Sun, T.-F. Wu and Y. Wu, Existence of nontrivial solution for Schrödinger-Poisson systems with indefinite steep potential well, Z. Angew. Math. Phys., 68 (2017), 22 pp. doi: 10.1007/s00033-017-0817-5. [30] Z. Wang and H.-S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5. [31] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982), 567-576. [32] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3. [33] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhäuser, Boston, Mass., 1996. doi: 10.1007/978-1-4612-4146-1. [34] Y. Ye and C.-L. Tang, Existence and multiplicity of solutions for Schrödinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differential Equations, 53 (2015), 383-411.  doi: 10.1007/s00526-014-0753-6. [35] 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, 12 pp. doi: 10.1016/j.amc.2020.125035. [36] W. Zhang, X. Tang and J. Zhang, Existence and concentration of solutions for Schrödinger-Poisson system with steep potential well, Math. Methods Appl. Sci., 39 (2016), 2549-2557.  doi: 10.1002/mma.3712. [37] 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. [38] X.-J. Zhong and C.-L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a 3-linear growth nonlinearity, J. Math. Anal. Appl., 455 (2017), 1956-1974.  doi: 10.1016/j.jmaa.2017.04.010. [39] 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.

show all references

##### References:
 [1] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${\mathbb{ R}}^N$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149. [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] 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. [5] S.-J. Chen and C.-L. Tang, High energy solutions for the superlinear Schrödinger-Maxwell equations, Nonlinear Anal., 71 (2009), 4927-4934.  doi: 10.1016/j.na.2009.03.050. [6] S. Chen and X. Tang, Ground state sign-changing solutions for asymptotically cubic or super-cubic Schrödinger-Poisson systems without compact condition, Comput. Math. Appl., 74 (2017), 446-458.  doi: 10.1016/j.camwa.2017.04.031. [7] S. Chen, X. Tang and J. Peng, Existence and concentration of positive solutions for Schrödinger-Poisson systems with steep well potential, Studia Sci. Math. Hungar., 55 (2018), 53-93.  doi: 10.1556/012.2018.55.1.1388. [8] X. Chen and C. Tang, Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth, Commun. Pure Appl. Anal., 20 (2021), 2291-2312.  doi: 10.3934/cpaa.2021077. [9] M. Du, L. Tian, J. Wang and F. Zhang, Existence and asymptotic behavior of solutions for nonlinear Schrödinger-Poisson systems with steep potential well, J. Math. Phys., 57 (2016), 031502, 19 pp. doi: 10.1063/1.4941036. [10] L. Gu, H. Jin and J. Zhang, Sign-changing solutions for nonlinear Schrödinger-Poisson systems with subquadratic or quadratic growth at infinity, Nonlinear Anal., 198 (2020), 111897, 16 pp. doi: 10.1016/j.na.2020.111897. [11] H. Guo and T. Wang, A note on sign-changing solutions for the schrödinger poisson system, Electronic Research Archive, 28 (2020), 195-203.  doi: 10.3934/era.2020013. [12] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann., 261 (1982), 493-514.  doi: 10.1007/BF01457453. [13] I. Ianni, Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. [14] C. Ji, F. Fang and B. Zhang, Least energy sign-changing solutions for the nonlinear Schrödinger-Poisson system, Electron. J. Differential Equations, 2017 (2017), 282, 13 pp. [15] Y. Jiang and H.-S. Zhou, Schrödinger-Poisson system with steep potential well, J. Differential Equations, 251 (2011), 582-608.  doi: 10.1016/j.jde.2011.05.006. [16] S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., 14 (2012), 1250041, 16 pp. doi: 10.1142/S0219199712500411. [17] G. Li, X. Luo and W. Shuai, Sign-changing solutions to a gauged nonlinear Schrödinger equation, J. Math. Anal. Appl., 455 (2017), 1559-1578.  doi: 10.1016/j.jmaa.2017.06.048. [18] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2. [19] S. Liu and S. Mosconi, On the Schrödinger-Poisson system with indefinite potential and 3-sublinear nonlinearity, J. Differential Equations, 269 (2020), 689-712.  doi: 10.1016/j.jde.2019.12.023. [20] W. Liu and Z. Wang, Least energy nodal solution for nonlinear Schrödinger equation without (AR) condition, J. Math. Anal. Appl., 462 (2018), 285-297.  doi: 10.1016/j.jmaa.2018.02.005. [21] Z. Liu, Z.-Q. Wang and J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775-794.  doi: 10.1007/s10231-015-0489-8. [22] Q.-J. Lou and Z.-Q. Han, The Nehari manifold for the Schrödinger-Poisson systems with steep well potential, Complex Var. Elliptic Equ., 64 (2019), 586-605.  doi: 10.1080/17476933.2018.1471070. [23] C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. [24] E. G. Murcia and G. Siciliano, Least energy radial sign-changing solution for the Schrödinger-Poisson system in ${\mathbb{R}}^3$ under an asymptotically cubic nonlinearity, J. Math. Anal. Appl., 474 (2019), 544-571.  doi: 10.1016/j.jmaa.2019.01.063. [25] 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. [26] W. Shuai and Q. 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. [27] J. Sun and T.-f. Wu, Bound state nodal solutions for the non-autonomous Schrödinger-Poisson system in ${\mathbb{R}}^3$, J. Differential Equations, 268 (2020), 7121-7163.  doi: 10.1016/j.jde.2019.11.070. [28] J. Sun and T.-f. Wu, On Schrödinger-Poisson systems under the effect of steep potential well $(2 < p < 4)$, J. Math. Phys., 61 (2020), 071506, 13 pp. doi: 10.1063/1.5114672. [29] J. Sun, T.-F. Wu and Y. Wu, Existence of nontrivial solution for Schrödinger-Poisson systems with indefinite steep potential well, Z. Angew. Math. Phys., 68 (2017), 22 pp. doi: 10.1007/s00033-017-0817-5. [30] Z. Wang and H.-S. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in ${\mathbb{R}}^3$, Calc. Var. Partial Differential Equations, 52 (2015), 927-943.  doi: 10.1007/s00526-014-0738-5. [31] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982), 567-576. [32] T. Weth, Energy bounds for entire nodal solutions of autonomous superlinear equations, Calc. Var. Partial Differ. Equ., 27 (2006), 421-437.  doi: 10.1007/s00526-006-0015-3. [33] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, vol. 24, Birkhäuser, Boston, Mass., 1996. doi: 10.1007/978-1-4612-4146-1. [34] Y. Ye and C.-L. Tang, Existence and multiplicity of solutions for Schrödinger-Poisson equations with sign-changing potential, Calc. Var. Partial Differential Equations, 53 (2015), 383-411.  doi: 10.1007/s00526-014-0753-6. [35] 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, 12 pp. doi: 10.1016/j.amc.2020.125035. [36] W. Zhang, X. Tang and J. Zhang, Existence and concentration of solutions for Schrödinger-Poisson system with steep potential well, Math. Methods Appl. Sci., 39 (2016), 2549-2557.  doi: 10.1002/mma.3712. [37] 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. [38] X.-J. Zhong and C.-L. Tang, Ground state sign-changing solutions for a Schrödinger-Poisson system with a 3-linear growth nonlinearity, J. Math. Anal. Appl., 455 (2017), 1956-1974.  doi: 10.1016/j.jmaa.2017.04.010. [39] 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.
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