Existence and concentration properties of ground state solutions for elliptic systems

References

• Lions JL. Optimal control of systems governed by partial differential equations. Berlin: Springer-Verlag; 1971.
   Google Scholar
• Nagasawa M. Schrödinger equations and diffusion theory. Basel: Birkhäuser; 1993.
   Google Scholar
• Ávila AI, Yang J. On the existence and shape of least energy solutions for some elliptic systems. J Differ Equ. 2003;191:348–376. doi: 10.1016/S0022-0396(03)00017-2
   Web of Science ®Google Scholar
• Ávila AI, Yang J. Multiple solutions of nonlinear elliptic systems. Nonlinear Differ Equ Appl. 2005;12:459–479. doi: 10.1007/s00030-005-0022-7
   Web of Science ®Google Scholar
• Bartsch T, De Figueiredo DG. Infinitely many solutions of nonlinear elliptic systems. In: Escher J, Simonett G, editors. Progress in nonlinear differential equations and their applications Vol. 35. Birkhäuser: Basel/Switzerland; 1999. p. 51–67.
   Google Scholar
• Clapp M, Faya J. Concentration with a single sign-changing layer at the higher critical exponents. Adv Nonlinear Anal. 2018;7:271–283. doi: 10.1515/anona-2016-0056
   Web of Science ®Google Scholar
• De Figueiredo DG, Yang J. Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal. 1998;33:211–234. doi: 10.1016/S0362-546X(97)00548-8
   Web of Science ®Google Scholar
• Ding YH, Lee C, Zhao FK. Semiclassic limits of ground state solutions to Schrödinger systems. Calc Var. 2014;51:725–760. doi: 10.1007/s00526-013-0693-6
   Web of Science ®Google Scholar
• Fonda A, Garrione M, Gidoni P. Periodic perturbations of Hamiltonian systems. Adv Nonlinear Anal. 2016;5:367–382.
   Web of Science ®Google Scholar
• Li G, Yang J. Asymptotically linear elliptic systems. Commun Part Diff Equ. 2004;29:925–954. doi: 10.1081/PDE-120037337
   Web of Science ®Google Scholar
• Qin DD, Chen J, Tang XH. Existence and non-existence of nontrivial solutions for Schrödinger systems via Nehari-Pohozaev manifold. Comput Math Appl. 2017;74(12):3141–3160. doi: 10.1016/j.camwa.2017.08.010
   Web of Science ®Google Scholar
• Qin DD, Tang XH, Wu QF. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Commun Pure Appl Anal. 2019;18(3):1261–1280. doi: 10.3934/cpaa.2019061
   Web of Science ®Google Scholar
• Sirakov B. On the existence of solutions of Hamiltonian elliptic systems in RN. Adv Differ Equ. 2000;5:1445–1464.
   Google Scholar
• Tang XH, Lin XY. Existence of ground state solutions of Nehari–Pankov type to Schrödinger systems. Sci China Math. 2017;60. DOI:10.1007/s11425-016-5135-4
   PubMedGoogle Scholar
• Wang J, Xu JX, Zhang FB. Existence of semiclassical ground-state solutions for semilinear elliptic systems. Pro Roy Soc Edinburgh Sect A. 2012;142:867–895. doi: 10.1017/S0308210511000254
   Web of Science ®Google Scholar
• Wang XM, Lin T-C, Wang Z-Q. Existence and concentration of ground states for saturable nonlinear Schrödinger equations with intensity functions in R2. Nonlinear Anal. 2018;173:19–36. doi: 10.1016/j.na.2018.03.005
   Google Scholar
• Zhao FK, Zhao LG, Ding YH. Multiple solutions for asymptotically linear elliptic systems. Nonlinear Differ Equ Appl. 2008;15:673–688. doi: 10.1007/s00030-008-7080-6
   Web of Science ®Google Scholar
• Zhao FK, Zhao LG, Ding YH. Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems. ESAIM Control Optim Calc Var. 2010;16:77–91. doi: 10.1051/cocv:2008064
   Web of Science ®Google Scholar
• Zhao LG, Zhao FK. On ground state solutions for superlinear Hamiltonian elliptic systems. Z Angew Math Phys. 2013;64:403–418. doi: 10.1007/s00033-012-0258-0
   Web of Science ®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
• Yang MB, Chen WX, Ding YH. Solutions of a class of Hamiltonian elliptic systems in RN. J Math Anal Appl. 2010;362:338–349. doi: 10.1016/j.jmaa.2009.07.052
   Web of Science ®Google Scholar
• Zhang J, Tang XH, Zhang W. On semiclassical ground state solutions for Hamiltonian elliptic systems. Appl Anal. 2015;94:1380–1396. doi: 10.1080/00036811.2014.931940
   Web of Science ®Google Scholar
• Zhang J, Tang XH, Zhang W. Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms. Nonlinear Anal. 2014;95:1–10. doi: 10.1016/j.na.2013.07.027
   Web of Science ®Google Scholar
• Zhang J, Zhang W, Xie X. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Commun Pure Appl Anal. 2016;15:599–622. doi: 10.3934/cpaa.2016.15.599
   Web of Science ®Google Scholar
• Zhao FK, Ding YH. On Hamiltonian elliptic systems with periodic or non-periodic potentials. J Differ Equ. 2010;249:2964–2985. doi: 10.1016/j.jde.2010.09.014
   Web of Science ®Google Scholar
• Bartsch T, Ding YH. Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nach. 2006;279:1–22. doi: 10.1002/mana.200410420
   Web of Science ®Google Scholar
• Ackermann N. A nonlinear superposition principle and multibump solution of periodic Schrödinger equations. J Funct Anal. 2006;234:277–320. doi: 10.1016/j.jfa.2005.11.010
   Web of Science ®Google Scholar
• Buffoni B, Jeanjean L, Stuart CA. Existence of a nontrivial solution to a strongly indefinite semilinear equation. Proc Amer Math Soc. 1993;119:179–186. doi: 10.1090/S0002-9939-1993-1145940-X
   Web of Science ®Google Scholar
• Zhang J, Zhang W, Tang XH. Semiclassical limits of ground states for Hamiltonian elliptic system with gradient term. Nonlinear Anal-Real World Appl. 2018;40:377–402. doi: 10.1016/j.nonrwa.2017.08.010
   Web of Science ®Google Scholar
• Zhang J, Tang XH, Zhang W. On semiclassical ground states for Hamiltonian elliptic system with critical growth. Topol Methods Nonl Anal. 2017;49:245–272.
   Web of Science ®Google Scholar
• Ding YH, Lee C, Ruf B. On semiclassical states of a nonlinear Dirac equation. Proc Roy Soc Edinburgh A. 2013;143:765–790. doi: 10.1017/S0308210511001752
   Google Scholar
• Ding YH, Xu T. Localized concentration of semi-classical states for nonlinear Dirac equations. Arch Ration Mech Anal. 2015;216:415–447. doi: 10.1007/s00205-014-0811-4
   Web of Science ®Google Scholar
• Tang XH. Non-Nehari manifold method for superlinear Schrödinger equation. Taiwanese J Math. 2014;18:1957–1979. doi: 10.11650/tjm.18.2014.3541
   Web of Science ®Google Scholar
• del Pino M, Felmer P. Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math Ann. 2002;324:1–32. doi: 10.1007/s002080200327
   Web of Science ®Google Scholar
• Ding YH, Liu XY. Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities. Manuscript Math. 2013;140:51–82. doi: 10.1007/s00229-011-0530-1
   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
• Ramos M, Soares SHM. On the concentration of solutions of singularly perturbed Hamiltonian systems in RN. Port Math (N.S.). 2006;63:160–176.
   Google Scholar
• Montefusco E, Pellacci B, Squassina M. Semiclassical states for weakly coupled nonlinear Schrödinger systems. J Eur Math Soc (JEMS). 2008;10:47–71. doi: 10.4171/JEMS/103
   Web of Science ®Google Scholar
• Willem M. Minimax theorems. Boston: Birkhäuser; 1996.
   Google Scholar
• Tang XH. New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum. J Math Anal Appl. 2014;413(1):392–410. doi: 10.1016/j.jmaa.2013.11.062
   Web of Science ®Google Scholar
• Kryszewski W, Szulkin A. Generalized linking theorem with an application to a semilinear Schrödinger equations. Adv Differ Equ. 1998;3:441–472.
   Google Scholar
• Li G, Szulkin A. An asymptotically periodic equation with indefinite linear part. Commun Contemp Math. 2002;4:763–776. doi: 10.1142/S0219199702000853
   Web of Science ®Google Scholar
• Pankov A. Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J Math. 2005;73:259–287. doi: 10.1007/s00032-005-0047-8
   Google Scholar
• Qin DD, Tang XH. Solutions on asymptotically periodic elliptic system with new conditions. Results Math. 2016;70:539–565. doi: 10.1007/s00025-015-0491-x
   Web of Science ®Google Scholar
• Qin DD, Tang XH. Time-harmonic Maxwell equations with asymptotically linear polarization. Z Angew Math Phys. 2016;67(3). DOI:10.1007/s00033-016-0626-2 doi: 10.1007/s00033-016-0626-2
   PubMed Web of Science ®Google Scholar
• Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal. 2009;257(12):3802–3822. doi: 10.1016/j.jfa.2009.09.013
   Web of Science ®Google Scholar
• Tang XH, Qin DD. Ground state solutions for semilinear time-harmonic Maxwell equations. J Math Phys. 2016;57:041505. DOI:10.1063/1.4947179 doi: 10.1063/1.4947179
   Web of Science ®Google Scholar
• Gilbarg D, Trudinger NS. Elliptic partial differential equations of second order. Berlin: Springer; 1983.
   Google Scholar
• Simon B. Schrödinger semigroups. Bull Am Math Soc (N.S.). 1982;7:447–526. doi: 10.1090/S0273-0979-1982-15041-8
   Web of Science ®Google Scholar