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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

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.

Citation: Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011
References:
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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.  Google Scholar

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A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger–Poisson problem, Commun. Contemp. Math., 10 (2008), 391-404.  doi: 10.1142/S021919970800282X.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[12]

C. Y. ChenY. 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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Z. LiangJ. 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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

C. LiuH. 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.  Google Scholar

[26]

Z. LiuZ. 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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

J. SunT. 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.  Google Scholar

[31]

J. SunT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

C. Y. ChenY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

Z. LiangJ. 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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

C. LiuH. 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.  Google Scholar

[26]

Z. LiuZ. 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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[30]

J. SunT. 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.  Google Scholar

[31]

J. SunT. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

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