no. 1
Partial differential equations
Multiplicity for a 4-sublinear Schrödinger–Poisson system with sign-changing potential via Morse theory

Présenté par : the Editorial Board

Liu, Hongliang1 ; Chen, Haibo1 ; Wang, Gangwei2, 3
1 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
2 Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada
3 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China
Comptes Rendus. Mathématique, Tome 354 (2016) no. 1, pp. 75-80.

In this paper, we study a 4-sublinear Schrödinger–Poisson system with sign-changing potential. Under some suitable assumptions, the existence of two nontrivial solutions are obtained by using the Morse theory. Our result improves the recent ones of Chen and Zhang (2014) [6].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.018
Mots clés : Schrödinger–Poisson system, Morse theory, Critical groups, Sign-changing potential
Liu, Hongliang 1 ; Chen, Haibo 1 ; Wang, Gangwei 2, 3

1 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
2 Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada
3 School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China 
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Liu, Hongliang; Chen, Haibo; Wang, Gangwei. Multiplicity for a 4-sublinear Schrödinger–Poisson system with sign-changing potential via Morse theory. Comptes Rendus. Mathématique, Tome 354 (2016) no. 1, pp. 75-80. doi : 10.1016/j.crma.2015.10.018. http://www.numdam.org/articles/10.1016/j.crma.2015.10.018/

[1] Alama, S.; Tarantello, G. On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differ. Equ., Volume 1 (1993), pp. 439-475

[2] Cerami, G.; Vaira, G. Positive solutions for some non-autonomous Schrödinger–Poisson systems, J. Differ. Equ., Volume 248 (2010), pp. 521-543

[3] Chang, K. Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhäuser, Boston, Basel, Berlin, 1993

[4] Chen, J. Multiple positive solutions of a class of non autonomous Schrödinger–Poisson systems, Nonlinear Anal., Real World Appl., Volume 21 (2015), pp. 13-26

[5] Chen, P.; Tian, C. Infinitely many solutions for Schrödinger–Maxwell equations with indefinite sign subquadratic potentials, Appl. Math. Comput., Volume 226 (2014), pp. 492-502

[6] Chen, S.; Zhang, D. Existence of nontrivial solution for a 4-sublinear Schrödinger–Poisson system, Appl. Math. Lett., Volume 38 (2014), pp. 135-139

[7] Huang, W.; Tang, X. Semiclassical solutions for the nonlinear Schrödinger–Maxwell equations, J. Math. Anal. Appl., Volume 415 (2014), pp. 791-802

[8] Huang, W.; Tang, X. The existence of infinitely many solutions for the nonlinear Schrödinger–Maxwell equations, Results Math., Volume 65 (2014), pp. 223-234

[9] Huang, L.; Rocha, E.M.; Chen, J. Two positive solutions of a class of Schrödinger–Poisson system with indefinite nonlinearity, J. Differ. Equ., Volume 255 (2013), pp. 2463-2483

[10] Jiang, Y.; Zhou, H. Schrödinger–Poisson system with steep potential well, J. Differ. Equ., Volume 251 (2011), pp. 582-608

[11] Jiang, Y.; Zhou, H. Schrödinger–Poisson system with singular potential, J. Math. Anal. Appl., Volume 417 (2014), pp. 411-438

[12] Liu, H.; Chen, H.; Yang, X. Multiple solutions for superlinear Schrödinger–Poisson system with sign-changing potential and nonlinearity, Comput. Math. Appl., Volume 68 (2014), pp. 1982-1990

[13] Liu, Z.; Guo, S.; Zhang, Z. Existence of ground state solutions for the Schrödinger–Poisson systems, Appl. Math. Comput., Volume 244 (2014), pp. 312-323

[14] Mawhin, J.; Willem, M. Critical Point Theory and Hamiltonian Systems, Springer, Berlin, 1989

[15] Reed, M.; Simon, B. Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New York, London, 1978

[16] Ruiz, D. The Schrödinger–Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., Volume 237 (2006), pp. 655-674

[17] Sun, J.; Chen, H.; Yang, L. Positive solutions of asymptotically linear Schrödinger–Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal., Volume 74 (2011), pp. 413-423

[18] Sun, J.; Chen, H.; Nieto, J.J. On ground state solutions for some non autonomous Schrödinger–Poisson systems, J. Differ. Equ., Volume 252 (2012), pp. 3365-3380

[19] Xu, L.; Chen, H. Multiplicity of small negative-energy solutions for a class of nonlinear Schrödinger–Poisson systems, Appl. Math. Comput., Volume 243 (2014), pp. 817-824

[20] Xu, L.; Chen, H. Existence of infinitely many solutions for generalized Schrödinger–Poisson system, Bound. Value Probl., Volume 2014 (2014)

[21] Yang, M.; Han, Z. Existence and multiplicity results of the nonlinear Schrödinger–Poisson systems, Nonlinear Anal., Real World Appl., Volume 13 (2012), pp. 1093-1101

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☆ Research supported by Hunan Provincial Foundation For Postgraduate CX2014B044, National Natural Science Foundation of China 11271372 and Hunan Provincial Natural Science Foundation of China 12JJ2004.