LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR SUPER-QUADRATIC SCHRÖDINGER-POISSON SYSTEMS IN $ \mathbb{R}^{3} $

Sofiane Khoutir

doi:10.11948/20200274

2021 Volume 11 Issue 3

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Sofiane Khoutir. LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR SUPER-QUADRATIC SCHRÖDINGER-POISSON SYSTEMS IN $ \mathbb{R}^{3} $[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1520-1534. doi: 10.11948/20200274

Citation:

Sofiane Khoutir. LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR SUPER-QUADRATIC SCHRÖDINGER-POISSON SYSTEMS IN $ \mathbb{R}^{3} $[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1520-1534. doi: 10.11948/20200274

LEAST ENERGY SIGN-CHANGING SOLUTIONS FOR SUPER-QUADRATIC SCHRÖDINGER-POISSON SYSTEMS IN $ \mathbb{R}^{3} $

Sofiane Khoutir

Faculty of Mathematics, University of Science and Technology Houari Boumediene, PB 32 El-Alia, Bab Ezzouar 16111 Algiers, Algeria

Author Bio: Email: skhoutir@usthb.dz

Abstract

Abstract

In this paper, we study the following Schrödinger-Poisson systems

$ \begin{equation*} \begin{cases} -\Delta u+Vu+\lambda \phi u = f(u), &\quad x \in \mathbb{R}^{3},\\ -\Delta \phi = u^{2}, &\quad x \in \mathbb{R}^{3}, \end{cases} \end{equation*} $

where $ V, \: \lambda>0 $ and $ f \in C\left(\mathbb{R}, \mathbb{R}\right) $. Under some relaxed assumptions on $ f $, using variational methods in combination with the Pohozǎev identity, we prove that the above system possesses a least energy sign-changing solution and a ground state solution provided that $ \lambda $ is sufficiently small. Moreover, we prove that the energy of a sign-changing solution is strictly larger than that of the ground state solution. Our results generalize and extend some recent results in the literature.
- Schrödinger-Poisson system /
- sing-changing solution /
- ground state solution
MSC: 35J47, 35J50, 35J60

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References

[1]	C. O. Alves, M. A. S. Souto and S. H. M. Soares, A sign-changing solution for the Schrödinger-Poisson equation, Rocky Mountain J. Math., 2017, 47(1), 1-25. Google Scholar
[2]	A. Ambrosetti and D. Ruiz, Multiple bound states for the Schrödinger-Poisson problem. Commun. Contemp. Math., 2008, 10, 391-404. doi: 10.1142/S021919970800282X CrossRef Google Scholar
[3]	A. Azzollini, P. d'Avenia and A. Pomponio, On the Schrödinger-Maxwell equations under the effect of a general nonlinear term, Ann. Inst. H. Poincaré Anal. Non Linéair, 2010, 27, 779-791. doi: 10.1016/j.anihpc.2009.11.012 CrossRef Google Scholar
[4]	V. Benci and F. Donato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., 1998, 11, 283-293. doi: 10.12775/TMNA.1998.019 CrossRef Google Scholar
[5]	H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, existence of a ground state, Arch. Ration. Mech. Anal., 1983, 82, 313-346. doi: 10.1007/BF00250555 CrossRef Google Scholar
[6]	G. Cerami and G. Vaira, Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Diff. Eqs., 2010, 248, 521-543. doi: 10.1016/j.jde.2009.06.017 CrossRef Google Scholar
[7]	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., 2016, 67(4), 102-120. doi: 10.1007/s00033-016-0695-2 CrossRef Google Scholar
[8]	T. D'Aprile and D. Mugnai, Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect. A, 2004, 134, 893-906. doi: 10.1017/S030821050000353X CrossRef Google Scholar
[9]	D. G. De Figueiredo, Lectures on the Ekeland variational principle with applications and detours. Vol. 81. Berlin: Springer, 1989. Google Scholar
[10]	X. He and W. Zou, Existence and concentration behavior of positive solutions of Kirchhoff type equation in $\mathbb{R}^3$, J. Diff. Eqs., 2012, 252(2), 1813-1834. doi: 10.1016/j.jde.2011.08.035 CrossRef Google Scholar
[11]	W. Huang and X. Tang, The existence of infinitely many solutions for the nonlinear Schrödinger-Maxwell equations, Results Math., 2014, 65, 223-234. doi: 10.1007/s00025-013-0342-6 CrossRef Google Scholar
[12]	W. Huang and X. Tang, Ground-state solutions for asymptotically cubic Schrödinger-Maxwell equations, Mediterr. J. Math., 2016, 13, 3469-3481. doi: 10.1007/s00009-016-0697-5 CrossRef Google Scholar
[13]	L. Jeanjean and S. Le Coz, An existence and stability result for standing waves of nonlinear Schr{ö}dinger equations, Adv. Diff. Eqs., 2006, 11, 813-840. Google Scholar
[14]	H. Kikuchi, Existence and stability of standing waves for Schr{\"O}dinger-Poisson-Slater equation, Adv. Nonlinear Stud., 2007, 7, 403-437. Google Scholar
[15]	S. Khoutir and H. Chen, Multiple nontrivial solutions for a nonhomogeneous Schrödinger-Poisson system in $\mathbb{R}^3$, Electron. J. Qual. Theo. Diff. Eqs., 2017, 28, 1-17. Google Scholar
[16]	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., 2016, 435, 783-799. doi: 10.1016/j.jmaa.2015.10.076 CrossRef Google Scholar
[17]	H. Liu, H. Chen and G. Wang, Multiplicity for a 4-sublinear Schrödinger-Poisson system with sign-changing potential via Morse theory, Compt. Rendus Math., 2016, 354, 75-80. doi: 10.1016/j.crma.2015.10.018 CrossRef Google Scholar
[18]	H. Liu and H. Chen, Multiple solutions for a nonlinear Schrödinger-Poisson system with sign-changing potential, Comput. Math. Appl., 2016, 71(7), 1405-1416. doi: 10.1016/j.camwa.2016.02.010 CrossRef Google Scholar
[19]	D. Ruiz, The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 2006, 237, 655-674. doi: 10.1016/j.jfa.2006.04.005 CrossRef Google Scholar
[20]	O. Sánchez and J. Soler, Long-time dynamics of the Schrödinger-Poisson-Slater system, J. Stat. Phys., 2004, 114, 179-204. doi: 10.1023/B:JOSS.0000003109.97208.53 CrossRef Google Scholar
[21]	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., 2015, 66(6), 3267-3282. doi: 10.1007/s00033-015-0571-5 CrossRef Google Scholar
[22]	J. Sun, H. Chen and L. Yang, Positive solutions of asymptotically linear Schrödinger-Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal., 2011, 74, 413-423. doi: 10.1016/j.na.2010.08.052 CrossRef Google Scholar
[23]	J. Sun, Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 2012, 390, 514-522. doi: 10.1016/j.jmaa.2012.01.057 CrossRef Google Scholar
[24]	J. Sun, T. Wu and Z. Feng, Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Diff. Eqs., 2016, 260, 586-627. doi: 10.1016/j.jde.2015.09.002 CrossRef Google Scholar
[25]	Z. Wang and H. Zhou, Sign-changing solutions for the nonlinear Schrödinger-Poisson system in $\mathbb{R}^3$, Calc. Var. Partial Differential Equations, 2015, 52, 927-943. doi: 10.1007/s00526-014-0738-5 CrossRef Google Scholar
[26]	M. Willem, Minimax Theorems, Birkhäuser Boston Inc, Boston, 1996. Google Scholar
[27]	L. Xu and H. Chen, Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger-Poisson systems, Appl. Math. Comput., 2014, 243, 817-824. Google Scholar
[28]	M. Yang and Z. Han, Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems, Nonlinear Anal. RWA, 2012, 13, 1093-1101. doi: 10.1016/j.nonrwa.2011.07.008 CrossRef Google Scholar
[29]	J. Zhang, On the Schrödinger-Poisson equations with a general nonlinearity in the critical growth, Nonlinear Anal., 2012, 75, 6391-6401. doi: 10.1016/j.na.2012.07.008 CrossRef Google Scholar
[30]	J. Zhang, On ground state and nodal solutions of Schrödinger-Poisson equations with critical growth, J. Math. Anal. Appl., 2015, 428, 387-404. doi: 10.1016/j.jmaa.2015.03.032 CrossRef Google Scholar