Existence of solutions for Schrödinger-Poisson system with asymptotically periodic terms

Wang, Da-Bin ¹ ; Ma, Lu-Ping ¹ ; Guan, Wen ¹ ; Wu, Hong-Mei ¹

¹ Department of Applied Mathematics, Lanzhou University of Technology, 730050 Lanzhou, People’s Republic of China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 5, p. 591-601.

Résumé

In this paper, we consider the following nonlinear Schrödinger-Poisson system

$ \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Delta u + V(x)u+K(x)\phi u= f(x,u), x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^{2}, x\in \mathbb{R}^3, \end{array} \right. $

where~$V, K\in L^{\infty}(\mathbb{R}^3)$ and $f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous. We prove that the problem has a nontrivial solution under asymptotically periodic case of $V, K$, and $f$ at infinity. Moreover, the nonlinear term $f$ does not satisfy any monotone condition.

Détail

DOI : 10.22436/jnsa.011.05.01

Classification : 34C25, 58E50
Keywords: Schrödinger-Poisson system, asymptotically periodic, variational method

Bibliographie

[1] Alves, C. O.; M. A. S. Souto Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains , Z. Angew. Math. Phys., Volume 65 (2014), pp. 1153-1166 | DOI | Zbl

[2] Alves, C. O.; Souto, M. A. S.; Soares, S. H. M. Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., Volume 377 (2011), pp. 584-592 | DOI | Zbl

[3] Ambrosetti, A.; Badiale, M.; Cingolani, S. Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., Volume 140 (1997), pp. 285-300 | DOI

[4] Ambrosetti, A.; P. H. Rabinowitz Dual variational methods in critical point theory and applications, J. Functional Analysis, Volume 14 (1973), pp. 349-381 | DOI

[5] Ambrosetti, A.; D. Ruiz Multiple bound states for the Schrödinger-Poisson equation, Commun. Contemp. Math., Volume 10 (2008), pp. 391-404 | DOI | Zbl

[6] A. Azzollini Concentration and compactness in nonlinear Schrödinger-Poisson systemwith a general nonlinearity, J. Differential Equations, Volume 249 (2010), pp. 1746-1763 | DOI

[7] Benci, V.; Fortunato, D. An eigenvalue problem for the Schrödinger-Maxwell equations, Topol. Methods Nonlinear Anal., Volume 11 (1998), pp. 283-293 | DOI | Zbl

[8] Berestycki, H.; Lions, P.-L. Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal., Volume 82 (1983), pp. 347-375 | DOI | Zbl

[9] Cerami, G.; Vaira, G. Positive solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, Volume 248 (2010), pp. 521-543

[10] G. M. Coclite A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., Volume 7 (2003), pp. 417-423 | Zbl

[11] D’Aprile, T.; Mugnai, D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. London Ser. A, Volume 134 (2004), pp. 893-906 | DOI | Zbl

[12] D’Aprile, T.; Mugnai, D. Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., Volume 4 (2004), pp. 307-322 | DOI

[13] D’Aprile, T.; Wei, J. On bound states concentrating on spheres for the Maxwell-Schrödinger equation, SIAM J. Math. Anal., Volume 37 (2005), pp. 321-342 | DOI | Zbl

[14] R. de Marchi Schrödinger equations with asymptotically periodic terms, Proc. Roy. Soc. Edinburgh Sect. A, Volume 145 (2015), pp. 745-757 | Zbl | DOI

[15] Pino, M. del; Felmer, P. Semi-classical states of nonlinear Schrödinger equations: A variational reduction method, Math. Ann., Volume 324 (2002), pp. 1-32 | DOI | Zbl

[16] Ding, Y.; F. Lin Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, Volume 30 (2007), pp. 231-249 | DOI

[17] He, X.; W. Zou Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., Volume 2012 (2012 ), pp. 1-19 | Zbl | DOI

[18] Huang, L.; Rocha, E. M.; Chen, J. Two positive solutions of a class of Schrödinger-Poisson system with indefinite nonlinearity, J. Differential Equations, Volume 255 (2013), pp. 2463-2483 | Zbl | DOI

[19] I. Ianni Sign-changing radial solutions for the Schrödinger-Poisson-Slater problem, Topol. Methods Nonlinear Anal., Volume 41 (2013), pp. 365-385 | Zbl

[20] Jeanjean, L.; Tanaka, K. Singularly perturbed elliptic problems with superlinear or asympotically linear nonlinearities, Calc. Var. Partial Differential Equations, Volume 21 (2004), pp. 287-318 | DOI

[21] Jiang, Y.; H.-S. Zhou Schrödinger-Poisson system with steep potential well , J. Differential Equations, Volume 251 (2011), pp. 582-608 | DOI

[22] Kim, S.; Seok, J. On nodal solutions of the nonlinear Schrödinger-Poisson equations, Commun. Contemp. Math., Volume 2012 (2012 ), pp. 1-16 | DOI

[23] Li, G.; Peng, S.; Wang, C. Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., Volume 2011 (2011 ), pp. 1-19 | DOI

[24] Li, G.; Peng, S.; Yan, S. Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system , Commun. Contemp. Math., Volume 12 (2010), pp. 1069-1092 | DOI

[25] Li, G.; A. Szulkin An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., Volume 4 (2002), pp. 763-776 | DOI | Zbl

[26] Li, Y.; Wang, Z.-Q.; J. Zeng Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 23 (2006), pp. 829-837 | DOI

[27] Lins, H. F.; E. A. B. Silva Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., Volume 71 (2009), pp. 2890-2905 | DOI

[28] Liu, Z.; Guo, S.; Fang, Y. Multiple semiclassical states for coupled Schrödinger-Poisson equations with critical exponential growth, J. Math. Phys., Volume 2015 (2015), pp. 1-22 | Zbl | DOI

[29] Liu, Z.; Wang, Z.-Q.; Zhang, J. Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., Volume 195 (2016), pp. 775-794 | DOI | Zbl

[30] Rabinowitz, P. H. Minimax theorems in critical point theory with applications to differential equations, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, 1986

[31] Rabinowitz, P. H. On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., Volume 43 (1992), pp. 270-291 | DOI

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

[33] M. Schechter A variation of the mountain pass lemma and applications, J. London Math. Soc., Volume 44 (1991), pp. 491-502 | DOI

[34] Shuai, W.; Q. Wang Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in \(R^3\), Z. Angew. Math. Phys., Volume 66 (2015), pp. 3267-3282 | Zbl | DOI

[35] Silva, E. A. B.; G. F. Vieira Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., Volume 72 (2010), pp. 2935-2949 | Zbl | DOI

[36] Silva, E. A. B.; G. F. Vieira Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, Volume 39 (2010), pp. 1-33 | DOI | Zbl

[37] Sun, J.; S. Ma Ground state solutions for some Schrödinger-Poisson systems with periodic potentials, J. Differential Equations, Volume 260 (2016), pp. 2119-2149 | Zbl | DOI

[38] Sun, J.; Wu, T.-F.; Z. Feng Multiplicity of positive solutions for a nonlinear Schrödinger-Poisson system, J. Differential Equations, Volume 260 (2016), pp. 586-627 | DOI

[39] Szulkin, A.; T. Weth The method of Nehari manifold, Handbook of nonconvex analysis and applications, 597–632, Int. Press, Somerville, MA, 2010

[40] Vaira, G. Gound states for Schrödinger-Poisson type systems, Ricerche di Matematica, Volume 60 (2011), pp. 263-297 | DOI

[41] Wang, J.; Xu, J.; Zhang, F.; Chen, X. Existence of multi-bump solutions for a semilinear Schrödinger-Poisson system, Nonlinearity, Volume 26 (2013), pp. 1377-1399 | Zbl | DOI

[42] Wang, Z.; H.-S. Zhou Sign-changing solutions for the nonlinear Schrödinger-Poisson system in, Calc. Var. Partial Differential Equations, Volume 52 (2015), pp. 927-943 | DOI

[43] M. Willem Minimax Theorems, Birkhäuser Boston, Bosten, 1996

[44] Yang, M.; Zhao, F.; Ding, Y. On the existence of solutions for Schrödinger-Maxwell systems in \(R^3\) , Rocky Mountain J. Math., Volume 42 (2012), pp. 1655-1674 | DOI

[45] Zhang, H.; Xu, J.; F. Zhang Positive ground states for asymptotically periodic Schrödinger-Poisson systems, Math. Methods Appl. Sci., Volume 36 (2013), pp. 427-439 | DOI | Zbl

[46] Zhao, L.; Zhao, F. On the existence of solutions for the Schrödinger-Poisson equations, J. Math. Anal. Appl., Volume 346 (2008), pp. 155-169 | DOI