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On fair entropy of the tent family
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 |
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.
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. |
[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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