SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH CRITICAL GROWTH*

doi:10.1007/s10473-023-0521-9

Abstract

Abstract: In this paper, we study the following Schrödinger-Poisson system with critical growth: \begin{equation*} \begin{cases}-\Delta u+V(x)u+ \phi(x)u =f(u)+|u|^4u, \ & x\in\mathbb{R}^3, \\ -\Delta \phi=u^2, \ & x\in\mathbb{R}^3. \end{cases} \end{equation*} We establish the existence of a positive ground state solution and a least energy sign-changing solution, providing that the nonlinearity $f$ is super-cubic, subcritical and that the potential $V(x)$ has a potential well.

Key words: Schrödinger-Poisson system, ground state solution, sign-changing solution, critical growth

CLC Number:

35A01

Yinbin Deng, Wei Shuai, Xiaolong Yang. SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH CRITICAL GROWTH^*[J].Acta mathematica scientia,Series B, 2023, 43(5): 2291-2308.

Trendmd

References

[1] Alves C O, Souto M A. Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z Angew Math Phys, 2014, 65(6): 1153-1166
[2] Alves C O, Souto M A S, Soares S H M. Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition. J Math Anal Appl, 2011, 377(2): 584-592
[3] Alves C O, Souto M A S, Soares S H M. A sign-changing solution for the Schrödinger-Poisson equation in $\mathbb{R}^3$. Rocky Mountain J Math, 2017, 47(1): 1-25
[4] Ambrosetti A. On Schrödinger-Poisson systems. Milan J Math, 2008, 76: 257-274
[5] Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson problem. Commun Contemp Math, 2008, 10(3): 391-404
[6] Azzollini A, d'Avenia P, Pomponio A. On the Schrödinger-Maxwell equations under the effect of a general nonlinear term. Ann Inst H Poincaré Anal Non Linéaire, 2010, 27(2): 779-791
[7] Bartsch T, Willem M. Infinitely many radial solutions of a semilinear elliptic problem on $\mathbb{R}^N$. Arch Ration Mech Anal, 1993, 124(3): 261-276
[8] Bellazzini J, Siciliano G. Stable standing waves for a class of nonlinear Schrödinger-Poisson equations. Z Angew Math Phys, 2011, 62(2): 267-280
[9] Benguria R, Brézis H, Lieb E. The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun Math Phys, 1981, 79(2): 167-180
[10] Cao D, Zhu X. On the existence and nodal character of solutions of semilinear elliptic equations. Acta Math Sci, 1988, 8B(3): 345-359
[11] Catto I, Lions P. Binding of atoms and stability of molecules in Hartree and Thomas-Fermi type theories. Commun Partial Differ Equ, 1993, 18(7/8): 1149-1159
[12] D'Aprile T, Wei J. Standing waves in the Maxwell-Schrödinger equation and an optimal configuration problem. Calc Var Partial Differ Equ, 2006, 25(1): 105-137
[13] He X, Zou W. Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth. J Math Phys, 2012, 53(2): 023702
[14] Ianni I. Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem. Topol Methods Nonlinear Anal, 2013, 41(2): 365-385
[15] Jiang Y, Zhou H. Bound states for a stationary nonlinear Schrödinger-Poisson system with sign-changing potential in $\mathbb{R}^3$. Acta Math Sci, 2009, 29B(4): 1095-1104
[16] Kim S, Seok J. On nodal solutions of the nonlinear Schrödinger-Poisson equations. Commun Contemp Math, 2012, 14(6): 1250041
[17] Li G, Peng S, Yan S. Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system. Commun Contemp Math, 2010, 12(6): 1069-1092
[18] Lieb E. Thomas-Fermi and related theories of atoms and molecules. Rev Mod Phys, 1981, 53(4): 263-301
[19] Lions P. The concentration-compactness principle in the calculus of variations. The locally compact case, part 1. Ann Inst H Poincaré Anal Non Linéaire, 1984, 1(2): 109-145
[20] Lions P. Solutions of Hartree-Fock equations for Coulomb systems. Commun Math Phys, 1987, 109(1): 33-97
[21] Liu X, Liu J, Wang Z. Ground states for quasilinear Schrödinger equations with critical growth. Calc Var Partial Differ Equ, 2013, 46(3/4): 641-669
[22] Liu Z, Wang Z, Zhang J. Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system. Ann Mat Pura Appl, 2016, 195(3): 775-794
[23] Markowich P, Ringhofer C, Schmeiser C. Semiconductor Equations.Vienna: Springer-Verlag, 1990
[24] Mauser N J. The Schrödinger-Poisson-$X_\alpha$ equation. Appl Math Lett, 2001, 14(6): 759-763
[25] Miranda C. Un'osservazione su un teorema di Brouwer. Bol Un Mat Ital, 1940, 3: 5-7
[26] Ruiz D. The Schrödinger-Poissom equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2): 655-674
[27] Ruiz D. On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases. Arch Rational Mech Anal, 2010, 198(1): 349-368
[28] S$\acute{a}$nchez O, Soler J. Long-time dynamics of the Schrödinger-Poisson-Slater system. J Statistical Physics, 2004, 114(1/2): 179-204
[29] Shuai W, Wang Q. Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$. Z Angew Math Phys, 2015, 66(6): 3267-3282
[30] Wang J, Tian L, Xu J, Zhang F. Existence and concentration of positive solutions for semilinear Schröinger-Poisson systems in $\mathbb{R}^3$. Calc Var Partial Differ Equ, 2013, 48(1/2): 243-273
[31] Wang J, Tian L, Xu J, Zhang F. Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth. Z Angew Math Phys, 2015, 66(5): 2441-2471
[32] Wang Z, Zhou H. Positive solution for a nonlinear stationary Schrödinger Poisson system in $\mathbb{R}^3$. Discrete Contin Dyn Syst, 2007, 18(4): 809-816
[33] Wang Z, Zhou H. Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$. Calc Var Partial Differ Equ, 2015, 52(3/4): 927-943

SIGN-CHANGING SOLUTIONS FOR THE NONLINEAR SCHRÖDINGER-POISSON SYSTEM WITH CRITICAL GROWTH^*

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Menghui WU, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SIGN-CHANGING SOLUTIONS FOR KIRCHHOFF-TYPE EQUATIONS WITH A STEEP POTENTIAL WELL^∗ [J]. Acta mathematica scientia,Series B, 2023, 43(4): 1781-1799.
[2]	Yuanyuan Luo, Dongmei Gao, Jun Wang. EXISTENCE OF A GROUND STATE SOLUTION FOR THE CHOQUARD EQUATION WITH NONPERIODIC POTENTIALS^* [J]. Acta mathematica scientia,Series B, 2023, 43(1): 303-323.
[3]	Jin DENG, Benniao LI. A GROUND STATE SOLUTION TO THE CHERN-SIMONS-SCHRÖDINGER SYSTEM [J]. Acta mathematica scientia,Series B, 2022, 42(5): 1743-1764.
[4]	Jianghao HAO, Yajing ZHANG. ESTIMATES FOR EXTREMAL VALUES FOR A CRITICAL FRACTIONAL EQUATION WITH CONCAVE-CONVEX NONLINEARITIES [J]. Acta mathematica scientia,Series B, 2022, 42(3): 903-918.
[5]	Jinlan TAN, Yongyong LI, Chunlei TANG. THE EXISTENCE AND CONCENTRATION OF GROUND STATE SOLUTIONS FOR CHERN-SIMONS-SCHRÖDINGER SYSTEMS WITH A STEEP WELL POTENTIAL [J]. Acta mathematica scientia,Series B, 2022, 42(3): 1125-1140.
[6]	Wenqing WANG, Anmin MAO. THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN [J]. Acta mathematica scientia,Series B, 2022, 42(2): 551-560.
[7]	Yongsheng JIANG, Na WEI, Yonghong WU. MULTIPLE SOLUTIONS FOR THE SCHRÖDINGER-POISSON EQUATION WITH A GENERAL NONLINEARITY [J]. Acta mathematica scientia,Series B, 2021, 41(3): 703-711.
[8]	Zhongyuan LIU. MULTIPLE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER EQUATIONS WITH SATURABLE NONLINEARITY [J]. Acta mathematica scientia,Series B, 2021, 41(2): 493-504.
[9]	Yaghoub JALILIAN. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR A COUPLED SYSTEM OF KIRCHHOFF TYPE EQUATIONS [J]. Acta mathematica scientia,Series B, 2020, 40(6): 1831-1848.
[10]	Jianhua CHEN, Xianjiu HUANG, Bitao CHENG, Xianhua TANG. EXISTENCE AND CONCENTRATION BEHAVIOR OF GROUND STATE SOLUTIONS FOR A CLASS OF GENERALIZED QUASILINEAR SCHRÖDINGER EQUATIONS IN $\mathbb{R}^N$ [J]. Acta mathematica scientia,Series B, 2020, 40(5): 1495-1524.
[11]	Xueliang DUAN, Gongming WEI, Haitao YANG. POSITIVE SOLUTIONS AND INFINITELY MANY SOLUTIONS FOR A WEAKLY COUPLED SYSTEM [J]. Acta mathematica scientia,Series B, 2020, 40(5): 1585-1601.
[12]	Yao DU, Chunlei TANG. GROUND STATE SOLUTIONS FOR A SCHRÖDINGER-POISSON SYSTEM WITH UNCONVENTIONAL POTENTIAL [J]. Acta mathematica scientia,Series B, 2020, 40(4): 934-944.
[13]	Wentao HUANG, Li WANG. GROUND STATE SOLUTIONS OF NEHARI-POHOZAEV TYPE FOR A FRACTIONAL SCHRÖ DINGER-POISSON SYSTEM WITH CRITICAL GROWTH [J]. Acta mathematica scientia,Series B, 2020, 40(4): 1064-1080.
[14]	Quanqing LI, Wenbo WANG, Kaimin TENG, Xian WU. GROUND STATES FOR FRACTIONAL SCHRÖDINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND CRITICAL GROWTH [J]. Acta mathematica scientia,Series B, 2020, 40(1): 59-74.
[15]	Lixia WANG, Xiaoming WANG, Luyu ZHANG. GROUND STATE SOLUTIONS FOR THE CRITICAL KLEIN-GORDON-MAXWELL SYSTEM [J]. Acta mathematica scientia,Series B, 2019, 39(5): 1451-1460.