MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRODINGER-POISSON EQUATIONS WITHÖSIGN-CHANGING POTENTIAL

doi:10.1016/S0252-9602(17)30021-8

Abstract

Abstract:

In this article,we study the following nonhomogeneous Schrödinger-Poisson equations

where λ > 0 is a parameter.Under some suitable assumptions on V,K,f and g,the existence of multiple solutions is proved by using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.In particular,the potential V is allowed to be signchanging.

Key words: Nonhomogeneous, sign-changing potential, Schrödinger-Poisson equations, Ekeland's variational principle, Mountain Pass Theorem

Lixia WANG, Shiwang MA, Na XU. MULTIPLE SOLUTIONS FOR NONHOMOGENEOUS SCHRODINGER-POISSON EQUATIONS WITHÖSIGN-CHANGING POTENTIAL[J].Acta mathematica scientia,Series B, 2017, 37(2): 555-572.

Trendmd

References

[1] Ackermann N. A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations. J Funct Anal, 2006, 234:277-320
[2] Alves C O, Carrião P C, Medeiros E S. Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions. Abstr Appl Anal, 2004, 3:251-268
[3] Ambrosetti A, Ruiz D. Multiple bound states for the Schrödinger-Poisson equation. Commun Contemp Math, 2008, 10:391-404
[4] Azzollini A, Pomponio A. Ground state solutions for the nonlinear Schrödinger-Maxwell equations. J Math Anal Appl, 2008, 345:90-108
[5] Bartsch T, Wang Z Q. Existence and multiplicity results for some superlinear elliptic problem on RN. Comm Partial Differ Equat, 1995, 20:1725-1741
[6] Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal, 1998, 11(2):283-293
[7] Brezis H, Lieb E H. A relation between pointwise convergence of functions and convergence functionals. Proc Amer Math Soc, 1983, 8:486-490
[8] Coclite G M. A multiplicity result for the nonlinear Schrödinger-Maxwell equations. Commun Appl Anal, 2003, 7(2/3):417-423
[9] Coclite G M. A multiplicity result for the Schrödinger-Maxwell equations with negative potential. Ann Polon Math, 2002, 79(1):21-30
[10] Chen S J, Tang C L. High energy solutions for the superlinear Schrödinger-Maxwell equations. Nonlinear Anal, 2009, 71:4927-4934
[11] Chen S J, Tang C L. Multiple solutions for a non-homogeneous Schrödinger-Maxwell and Klein-GordonMaxwell equations on R3. Nonlinear Differ Equat Appl, 2010, 17:559-574
[12] DAprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud, 2004, 4(3):307-322
[13] DAprile T, Mugnai D. Solitary waves for nonlinear Klein-Gordon-Maxwell and Schrödinger-Maxwell equations. Proc Roy Soc Edinburgh Sect A, 2004, 134:893-906
[14] Ding L, Li L, Zhang J L. Mulltiple solutions for nonhomogeneous Schrödinger-Poisson system with asymptotical nonlinearity in R3. Taiwanese Journal of Mathematics, 2013, 17(5):1627-1650
[15] Ding Y H, Szulkin A. Bound states for semilinear Schrödinger equations with sign-changing potential. Calc Var Partial Differ Equat, 2007, 29:397-419
[16] Du M, Zhang F B. Existence of positive solutions for a nonhomogeneous Schrödinger-Poisson system in R3. International Journal of Nonlinear Science, 2013, 16(2):185-192
[17] Jiang Y S, Wang Z P, Zhou H S. Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in R3. Nonlinear Anal, 2013, 83:50-57
[18] Kikuchi H. On the existence of a solution for elliptic system related to the Maxwell-Schrödinger equations. Nonlinear Anal, 2007, 67(5):1445-1456
[19] Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Berlin:Springer, 1989
[20] Mercuri C. Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity. (English summary) Atti Accad Naz Lincei Cl Sci Fis Mat Natur Rend Lincei (9) Mat Appl, 2008, 19(3):211-227
[21] Rabinowitz P H. Minimax Methods in Critical Point Theory with Applications to Differential Equations. Providence:Amer Math Soc, 1986
[22] Ruiz D. The Schrödinger-Poisson equation under the effect of a nonlinear local term. J Funct Anal, 2006, 237(2):655-674
[23] Salvatore A. Multiple solitary waves for a non-homogeneous Schrödinger-Maxwell system in R³. Adv Nonlinear Stud, 2006, 6(2):157-169
[24] Struwe M. Variational Methods:Applications to Nonlinear Partial Differential Equations and Hamiltonian systems. 3rd ed. Berlin:Springer, 2000
[25] Sun J T. Infinitely many solutions for a class of sublinear Schrödinger-Maxwell equations. J Math Anal Appl, 2012, 390:514-522
[26] Sun M Z, Su J B, Zhao L G. Infinitely many solutions for a Schrödinger-Poisson system with concave and convex nonlinearities. Discrete Contin Dyn Syst, 2015, 35:427-440
[27] Sun J T, Wu T F. Existence and multiplicity of positive solutions for a Schrödinger-Poisson system with a perturbation. Topological Methods in Nonlinear Analysis, 2015, 46:967-998
[28] Sun J T, Wu T F. On the nonlinear Schrödinger-Poisson system with sign-changing potential. Z Angew Math Phys, 2015, 66:1649-1669
[29] Wang Z P, Zhou H S. Positive solutions for a nonlinear stationary Schrödinger-Poisson system in R³. Discrete Contin Dyn Syst, 2007, 18:809-816
[30] Willem M. Minimax Theorems. Boston:Birkhäuser, 1996
[31] Willem M. Analyse Harmonique Réelle. Paris:Hermann, 1995
[32] Wu T F. Multiplicity results for a semi-linear elliptic equation involving sign-changing weigh function. Rocky Mountain J Mathematics, 2009, 39(3):995-1011
[33] Wu T F. Four positive solutions for a semilinear elliptic equation involving concave and convex nonlinearities. Nonlinear Anal, 2009, 70:1377-1392
[34] Wu T F. The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions. Nonlinear Anal, 2008, 68:1733-1745
[35] Ye Y W, Tang C L. Existence and multiplicity of solutions for Schrödinger-Poisson equations with signchanging potential. Calc Var, 2015, 53:383-411
[36] Zhao L G, Zhao F K. Positive solutions for Schrödinger-Poisson equations with a critical exponent. Nonlinear Anal, 2009, 70:2150-2164
[37] Zhao L G, Zhao F K. On the existence of solutions for the Schrödinger-Poisson equations. J Math Anal Appl, 2008, 346(1):155-169
[38] Zhao L G, Liu H D, Zhao F K. Existence and concentration of solutions for the Schrödinger-Poisson equations with steep well potential. J Differ Equat, 2013, 255:1-23