Ground states for asymptotically periodic Schrödinger-Poisson systems with critical growth

Central European Journal of Mathematics - Tập 12 - Trang 1484-1499 - 2014
Hui Zhang1, Junxiang Xu2, Fubao Zhang2, Miao Du2
1Department of Mathematics, Jinling Institute of Technology, Nanjing, China
2Department of Mathematics, Southeast University, Nanjing, China

Tóm tắt

For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.

Tài liệu tham khảo

Alves Claudianor O., Souto Marco A.S., Soares Sérgio H.M., Schrödinger-Poisson equations without Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 2011, 377(2), 584–592 Ambrosetti A., On Schrödinger-Poisson systems, Milan J. Math., 2008, 76(1), 257–274 Ambrosetti A., Ruiz D., Multiple bound states for the Schrödinger-Poisson problem, Commun. Contemp. Math., 2008, 10(3), 391–404 Azzollini A., Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Differential Equations, 2010, 249(7), 1746–1763 Azzollini A., Pomponio A., Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 2008, 345(1), 90–108 Benci V., Fortunato D., An eigenvalue problem for the Schrödinger-Maxwell equations, Top. Meth. Nonlinear Anal., 1998, 11(2), 283–293 Cerami G., Vaira G., Positive solutions for some nonautonomous Schrödinger-Poisson systems, J. Differential Equations, 2010, 248(3), 521–543 Coclite G.M., A multiplicity result for the nonlinear Schrödinger-Maxwell equations, Commun. Appl. Anal., 2003, 7(2–3), 417–423 D’Aprile T., Mugnai D., Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations, Proc. Roy. Soc. Edinburgh Sect., 2004, 134(5), 893–906 D’Aprile T., Mugnai D., Non-existence results for the coupled Klein-Gordon-Maxwell equations, Adv. Nonlinear Stud., 2004, 4(3), 307–322 D’Avenia P., Non-radially symmetric solutions of nonlinear Schrödinger equation coupled with Maxwell equations, Adv. Nonlinear Stud., 2002, 2(2), 177–192 D’Avenia P., Pomponio A., Vaira G., Infinitely many positive solutions for a Schrödinger-Poisson system, Appl. Math. Lett., 2011, 24(5), 661–664 He X.M., Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations, Z. Angew. Math. Phys., 2011, 62(5), 869–889 He X.M., Zou Z.W., Existence and concentration of ground states for Schrödinger-Poisson equations with critical growth, J. Math. Phys., 2012, 53(2), #023702 Ianni I., Solutions of the Schrödinger-Poisson problem concentrating on spheres. II. Existence, Math. Models Methods Appl. Sci., 2009, 19(6), 877–910 Ianni I., Vaira G., Solutions of the Schrödinger-Poisson problem concentrating on spheres. I. Necessary conditions, Math. Models Methods Appl. Sci., 2009, 19(5), 707–720 Jiang Y.S., Zhou H.S., Schrödinger-Poisson system with steep potential well, J. Differential Equations, 2011, 251(3), 582–608 Kikuchi H., On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations, Nonlinear Anal., 2007, 67(5), 1445–1456 Li G.B., Peng S., Wang C.H., Multi-bump solutions for the nonlinear Schrödinger-Poisson system, J. Math. Phys., 2011, 52(5), #053505 Li G.B., Peng S., Yan S., Infinitely many positive solutions for the nonlinear Schrödinger-Poisson system, Commun. Contemp. Math., 2010, 12(6), 1069–1092 Lins Haendel F., Silva Elves A.B., Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 2009, 71(7–8), 2890–2905 Mawhin J., Willem M., Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, 74, Springer-Verlag, New York, 1989 Ruiz D., The Schrödinger-Poisson equation under the effect of a nonlinear local term, J. Funct. Anal., 2006, 237(2), 655–674 Silva Elves A.B., Vieira Gilberto F., Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 2010, 39(1–2), 1–33 Sun J.T., Chen H.B., Nieto J., On ground state solutions for some non-autonomous Schrödinger-Poisson systems, J. Differential Equations, 2012, 252(5), 3365–3380 Sun J.T., Chen H.B., Yang L., Positive solutions of asymptotically linear Schrödinger-Poisson systems with a radial potential vanishing at infinity, Nonlinear Anal., 2011, 74(2), 413–423 Szulkin A., Weth T., The Method of Nehari Manifold, Gao D.Y. and Motreanu D. (Eds.), Handbook of Nonconvex Analysis and Applications, International Press, Boston, 2010, 597–632 Vaira G., Ground states for Schrödinger-Poisson type systems, Ricerche mat., 2011, 60(2), 263–297 Wang J., Tian L.X., Xu J.X., Zhang F.B., Existence and concentration of positive ground state solutions for Schrödinger-Poisson systems, Adv. Nonlinear Stud., 2013, 13(3), 553–582 Wang Z.P., Zhou H.S., Positive solution for a nonlinear stationary Schrödinger-Poisson system in ℝ3, Discrete Contin. Dyn. Syst., 2007, 18(4), 809–816 Willem M., Minimax Theorems, Progr. Nonlinear Differential Equations Appl., 24, Birkhäuser, Basel, 1996 Yang M.H., Han Z.Q., Existence and multiplicity results for the nonlinear Schrödinger-Poisson systems, Nonlinear Anal. Real World Appl., 2012, 13(3), 1093–1101 Zhang H., Xu J.X., Zhang F.B., Positive ground states for asymptotically periodic Schrödinger-Poisson systems, Math. Meth. Appl. Sci., 2013, 36(4), 427–439 Zhao L.G., Zhao F.K., Positive solutions for Schrödinger-Poisson equations with a critical exponent, Nonlinear. Anal., 2009, 70(6), 2150–2164