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August  2021, 41(8): 3651-3682. doi: 10.3934/dcds.2021011

The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function

Juntao Sun 1,2, and  Tsung-fang Wu 3,,

1. 

School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
2. 

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
3. 

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

* Corresponding author: Tsung-fang Wu

Received  August 2019 Revised  November 2020 Published  August 2021 Early access  January 2021

Fund Project: J. Sun is supported by the National Natural Science Foundation of China (Grant No. 11671236). T.F. Wu is supported in part by the Ministry of Science and Technology, Taiwan (Grant 108-2115-M-390-007-MY2)

In this paper, we study the multiplicity of two spikes nodal solutions for a nonautonomous Schrödinger–Poisson system with the nonlinearity $ f(x)\vert u\vert ^{p-2}u(2<p<6) $ in $ \mathbb{R}^{3} $. By assuming that the weight function $ f\in C(\mathbb{R}^{3},\mathbb{R}^{+}) $ has $ m $ maximum points in $ \mathbb{R}^{3} $, we conclude that such system admits $ m^{2} $ distinct nodal solutions, each of which has exactly two nodal domains. The proof is based on a natural constraint approach developed by us as well as the generalized barycenter map.

Keywords: Nodal solution, Schrödinger-Poisson system, Variational method, Concentration-compactness principle.
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J61, 35Q40.
Citation: Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011
References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.  doi: 10.1007/s00033-013-0376-3.

[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]

A. Bahri and H. Berestycki, Points critiques de perturbations de fonctionnelles paries et applications, C. R. Acad. Sci. Paris Sér A-B, 291 (1980), A189–A192.

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Lineairé, 22 (2005), 259-281.  doi: 10.1016/j.anihpc.2004.07.005.

[6]

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.

[7]

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.

[8]

H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[9]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^{N}$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.

[10]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[11]

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.

[12]

C. Y. Chen, Y. C. Kuo and T. F. Wu, Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.  doi: 10.1017/S0308210511000692.

[13]

S. Chen and X. 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.

[14]

M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric doamin, Calc. Var. Partial Differential Equations, 21 (2004), 1-14.  doi: 10.1007/s00526-003-0241-x.

[15]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[16]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.

[17]

I. Ianni, Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. 

[18]

I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger–Poisson–Slater problem, Commun. Contemp. Math., 14 (2012), 1250003, 22pp. doi: 10.1142/S0219199712500034.

[19]

S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger–Poisson equations, Commun. Contemp. Math., 14 (2012), 1250041, 16pp. doi: 10.1142/S0219199712500411.

[20]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^{p} = 0$ in $\mathbb{R}^{3}, $, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[21]

Y. Li, F. Li and J. Shi, Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 134, 17 pp. doi: 10.1007/s00526-017-1229-2.

[22]

Z. Liang, J. Xu and X. Zhu, Revisit to sign-changing solutions for the nonlinear Schrödinger–Poisson system in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 435 (2016), 783-799.  doi: 10.1016/j.jmaa.2015.10.076.

[23]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, AMS, 2001. doi: 10.1090/gsm/014.

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[25]

C. Liu, H. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179.  doi: 10.1016/j.jmaa.2008.06.042.

[26]

Z. Liu, Z. 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.

[27]

S. I. Pohozaev, On an approach to nonlinear equations, Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331. 

[28]

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.

[29]

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.

[30]

J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differential Equations, 260 (2016), 586-627.  doi: 10.1016/j.jde.2015.09.002.

[31]

J. Sun, T. F. Wu and Z. Feng, Non-autonomous Schrödinger–Poisson problems in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.

[32]

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.

[33]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.

[34]

H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain, Nonlinear Differ. Equ. Appl., 11 (2004), 361-377.  doi: 10.1007/s00030-004-2008-2.

[35]

Z. Wang and H. 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.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Fixed-point Theorems, Springer, New York, 1986.

[37]

L. Zhao and F. 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.

show all references

References:
[1]

C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger–Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166.  doi: 10.1007/s00033-013-0376-3.

[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]

A. Bahri and H. Berestycki, Points critiques de perturbations de fonctionnelles paries et applications, C. R. Acad. Sci. Paris Sér A-B, 291 (1980), A189–A192.

[5]

T. Bartsch and T. Weth, Three nodal solutions of singularly perturbed elliptic equations on domains without topology, Ann. Inst. H. Poincaré Anal. Non Lineairé, 22 (2005), 259-281.  doi: 10.1016/j.anihpc.2004.07.005.

[6]

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.

[7]

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.

[8]

H. Brezis and E. H. Lieb, A relation between pointwise convergence of functions and convergence functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.1090/S0002-9939-1983-0699419-3.

[9]

D. Cao and E. S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^{N}$, Ann. Inst. H. Poincaré Anal. Non Lineairé, 13 (1996), 567-588.  doi: 10.1016/S0294-1449(16)30115-9.

[10]

G. Cerami and D. Passaseo, The effect of concentrating potentials in some singularly perturbed problems, Calc. Var. Partial Differential Equations, 17 (2003), 257-281.  doi: 10.1007/s00526-002-0169-6.

[11]

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.

[12]

C. Y. Chen, Y. C. Kuo and T. F. Wu, Existence and multiplicity of positive solutions for the nonlinear Schrödinger-Poisson equations, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 745-764.  doi: 10.1017/S0308210511000692.

[13]

S. Chen and X. 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.

[14]

M. Clapp and T. Weth, Minimal nodal solutions of the pure critical exponent problem on a symmetric doamin, Calc. Var. Partial Differential Equations, 21 (2004), 1-14.  doi: 10.1007/s00526-003-0241-x.

[15]

T. D'Aprile and D. Mugnai, Non-existence results for the coupled Klein–Gordon–Maxwell equations, Adv. Nonlinear Stud., 4 (2004), 307-322.  doi: 10.1515/ans-2004-0305.

[16]

P. Drábek and S. I. Pohozaev, Positive solutions for the $p$-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.  doi: 10.1017/S0308210500023787.

[17]

I. Ianni, Sign-changing radial solutions for the Schrödinger–Poisson–Slater problem, Topol. Methods Nonlinear Anal., 41 (2013), 365-385. 

[18]

I. Ianni and D. Ruiz, Ground and bound states for a static Schrödinger–Poisson–Slater problem, Commun. Contemp. Math., 14 (2012), 1250003, 22pp. doi: 10.1142/S0219199712500034.

[19]

S. Kim and J. Seok, On nodal solutions of the nonlinear Schrödinger–Poisson equations, Commun. Contemp. Math., 14 (2012), 1250041, 16pp. doi: 10.1142/S0219199712500411.

[20]

M. K. Kwong, Uniqueness of positive solution of $\Delta u-u+u^{p} = 0$ in $\mathbb{R}^{3}, $, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[21]

Y. Li, F. Li and J. Shi, Existence and multiplicity of positive solutions to Schrödinger–Poisson type systems with critical nonlocal term, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 134, 17 pp. doi: 10.1007/s00526-017-1229-2.

[22]

Z. Liang, J. Xu and X. Zhu, Revisit to sign-changing solutions for the nonlinear Schrödinger–Poisson system in $\mathbb{R}^{3}$, J. Math. Anal. Appl., 435 (2016), 783-799.  doi: 10.1016/j.jmaa.2015.10.076.

[23]

E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, AMS, 2001. doi: 10.1090/gsm/014.

[24]

P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.  doi: 10.1016/S0294-1449(16)30422-X.

[25]

C. Liu, H. Wang and T. F. Wu, Multiplicity of 2-nodal solutions for semilinear elliptic problems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 348 (2008), 169-179.  doi: 10.1016/j.jmaa.2008.06.042.

[26]

Z. Liu, Z. 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.

[27]

S. I. Pohozaev, On an approach to nonlinear equations, Dokl. Akad. Nauk SSSR, 247 (1979), 1327-1331. 

[28]

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.

[29]

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.

[30]

J. Sun, T. F. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger–Poisson system, J. Differential Equations, 260 (2016), 586-627.  doi: 10.1016/j.jde.2015.09.002.

[31]

J. Sun, T. F. Wu and Z. Feng, Non-autonomous Schrödinger–Poisson problems in $\mathbb{R}^{3}$, Discrete Contin. Dyn. Syst., 38 (2018), 1889-1933.  doi: 10.3934/dcds.2018077.

[32]

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.

[33]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.  doi: 10.1016/S0294-1449(16)30238-4.

[34]

H. C. Wang and T. F. Wu, Symmetry breaking in a bounded symmetry domain, Nonlinear Differ. Equ. Appl., 11 (2004), 361-377.  doi: 10.1007/s00030-004-2008-2.

[35]

Z. Wang and H. 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.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications I, Fixed-point Theorems, Springer, New York, 1986.

[37]

L. Zhao and F. 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.

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