Existence of positive solutions for a Schrödinger-Poisson system with critical growth

References

• D'Aprile T, Mugnai D. 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
   Google Scholar
• D'Aprile T, Mugnai D. Non-existence results for the coupled Klein-Gordon-Maxwell equations. Adv Nonlinear Stud. 2004;4:307–322.
   Web of Science ®Google Scholar
• Mugnai D. The Schrödinger-Poisson system with positive potential. Comm Partial Differ Equ. 2011;36:1099–1117. doi: 10.1080/03605302.2011.558551
   Web of Science ®Google Scholar
• Benci V, Fortunato D. Solitary waves of the nonlinear Klein-Gordon equation coupled with the Maxwell equations. Rev Math Phys. 2002;14:409–420. doi: 10.1142/S0129055X02001168
   Web of Science ®Google Scholar
• Benci V, Fortunato D. An eigenvalue problem for the Schrödinger-Maxwell equations. Topol Methods Nonlinear Anal. 1998;11:283–293. doi: 10.12775/TMNA.1998.019
   Google Scholar
• Vaira G. Ground states for Schrödinger-Poisson type systems. Ric Mat. 2011;60:263–297. doi: 10.1007/s11587-011-0109-x
   Google Scholar
• Ambrosetti A, Malchiodi A. Perturbation methods and semilinear elliptic problem on ${\mathbb{R}}^3$. Progr. Math.Boston, MA: Birkhäuser; 2005.
   Google Scholar
• He XM. Multiplicity and concentration of positive solutions for the Schrödinger-Poisson equations. Z Angew Math Phys. 2011;5:869–889. doi: 10.1007/s00033-011-0120-9
   Google Scholar
• Ianni I, Vaira G. On concentration of positive bound states for the Schrödinger-poisson problem with potentials. Adv Nonlinear Stud. 2008;8:573–595. doi: 10.1515/ans-2008-0305
   Web of Science ®Google Scholar
• Ruiz D, Vaira G. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local minimum of the potential. Rev Mat Iberoam. 2011;27:253–271. doi: 10.4171/RMI/635
   Web of Science ®Google Scholar
• Willem M. Minimax theorems. Progr. Nonlinear Differential Equations Appl., Vol. 24. Basel: Birkhäuser; 1996.
   Google Scholar
• Rabinowitz PH. On a class of nonlinear Schrödinger equations. Z Angew Math Phys. 1992;43:270–291. doi: 10.1007/BF00946631
   Web of Science ®Google Scholar
• Wang XF. On concentration of positive bound states of nonlinear Schrödinger equations. Comm Math Phys. 1993;153:229–244. doi: 10.1007/BF02096642
   Web of Science ®Google Scholar
• del Pino M, Felmer P. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc Var PDE. 1996;4:121–137. doi: 10.1007/BF01189950
   Web of Science ®Google Scholar
• Byeon J, Jeanjean L. Standing waves for nonlinear Schrödinger equations with a general nonlinearity. Arch Rational Mech Anal. 2007;185:185–200. doi: 10.1007/s00205-006-0019-3
   Web of Science ®Google Scholar
• del Pino M, Felmer P. Multi-peak bound states of nonlinear Schrödinger equations. Ann. IHP Anal Nonlineaire. 1998;15:127–149.
   Web of Science ®Google Scholar
• Jeanjean L, Tanaka K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc Var PDE. 2004;21:287–318. doi: 10.1007/s00526-003-0261-6
   Web of Science ®Google Scholar
• Ba N, Deng YY, Peng SJ. Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials. Ann mIH Poincaré-AN. 2010;27:1205–1226.
   Web of Science ®Google Scholar
• Cingolani S, Lazzo M. Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J Differ Equ. 2000;160:118–138. doi: 10.1006/jdeq.1999.3662
   Web of Science ®Google Scholar
• Wang XF, Zeng B. On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. SIAM J Math Anal. 1997;28:633–655. doi: 10.1137/S0036141095290240
   Web of Science ®Google Scholar
• Wang J, Tian LX, Xu JX, et al. Existence and concentration of positive ground state solutions for semilinear Schrödinger-Poisson systems. Adv Nonlinear Stud. 2013;13:553–582.
   Web of Science ®Google Scholar
• Wang J, Tian LX, Xu JX, et al. Existence and concentration of positive solutions for semilinear Schrödinger-Poisson systems in R3. Calc Var Partial Differ Equ. 2013;48:243–273. doi: 10.1007/s00526-012-0548-6
   Web of Science ®Google Scholar
• Wang J, Tian LX, Xu JX, et al. Existence of multiple positive solutions for Schrödinger-Poisson systems with critical growth. Z Angew Math Phys. 2015;66:2441–2471. doi: 10.1007/s00033-015-0531-0
   Web of Science ®Google Scholar
• Ambrosetti A, Malchiodi A, Ruiz D. Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J Anal Math. 2006;98:317–348. doi: 10.1007/BF02790279
   Web of Science ®Google Scholar
• Ambrosetti A, Wang ZQ. Nonlinear Schrödinger equations with vanishing and decaying potentials. Differ Integral Equ. 2005;18:1321–1332.
   Web of Science ®Google Scholar
• Ding YH, Liu X. Semi-classical limits of ground states of a nonlinear Dirac equation. J Differ Equ. 2012;252:4962–4987. doi: 10.1016/j.jde.2012.01.023
   Web of Science ®Google Scholar
• Ding YH, Liu X. Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities. Manuscripta Math. 2013;140:51–82. doi: 10.1007/s00229-011-0530-1
   Web of Science ®Google Scholar
• Szulkin A, Weth T. The method of Nehari manifold. In: Gao DY and Motreanu D, editors. Handbook of nonconvex analysis and applications. Boston, MA: International Press; 2010. p. 597–632.
   Google Scholar
• Mawhin J, Willen M. Critical point theory and Hamiltonian systems. New York: Springer-Verlag; 1989.
   Google Scholar
• Alves CO, Souto MA. On existence and concentration behavior of ground state solutions for a class of problems with critical growth. Comm Pure Appl Anal. 2002;1:417–431. doi: 10.3934/cpaa.2002.1.417
   Web of Science ®Google Scholar
• Brézis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Communs Pure Appl Math. 1983;36:437–477. doi: 10.1002/cpa.3160360405
   Web of Science ®Google Scholar
• Talenti G. Best constants in Sobolev inequality. Ann Mat Pura Appl. 1976;110:353–372. doi: 10.1007/BF02418013
   Google Scholar
• Miyagaki O-H. On a class of semilear elliptic problem in R3 with critical growth. Nonlinear Anlysis TMA. 1997;29:773–781. doi: 10.1016/S0362-546X(96)00087-9
   Web of Science ®Google Scholar
• Lions PL. The concentration compactness principle in the calculus of variations: the locally compact case. Parts 1, 2, Ann Inst H Poincaré Anal Non Linéaire. 1984;1:109–145, and 1984;2:223–283. doi: 10.1016/S0294-1449(16)30428-0
   Web of Science ®Google Scholar
• Wang J, Tian LX, Xu JX, et al. Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J Differ Equ. 2012;253:2314–2351. doi: 10.1016/j.jde.2012.05.023
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. Generalized linking theorem with an application to semilinear Schrödinger equations. Adv Diffr Equ. 1998;3:441–472.
   Google Scholar