Infinitely many solutions for nonlinear fourth-order Schrödinger equations with mixed dispersion

References

• Bonheure D, Casteras J, Dos Santos E, et al. Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation. SIAM J Math Anal. 2018;50:5027–5071.
   Web of Science ®Google Scholar
• Meystre P. Atom optics. New York: Springer; 2001.
   Google Scholar
• Mills D. Nonlinear optics. Berlin, Heidelberg: Springer-Verlag; 1998.
   Google Scholar
• Bahri A, Li Y. On a min–max procedure for the existence of a positive solution for certain scaler field equation in Rn. Rev Mat Iberoam. 1990;6:1–15.
   Google Scholar
• Bahri A, Lions P. On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann I H Poincaré–AN. 1997;14:365–413.
   Web of Science ®Google Scholar
• Cao D, Noussair E, Yan S. Multiscale bump standing waves with a critical frequency for nonlinear Schrödinger equations. Trans Am Math Soc. 2008;360:3813–3838.
   Web of Science ®Google Scholar
• Gidas B, Ni W, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Adv Math. 1981;7:369–402.
   Google Scholar
• Lions P. The concentration compactness principle in the calculus of variations, the locally compact case I. Ann I H Poincaré–AN. 1984;1:109–145.
   Web of Science ®Google Scholar
• Lions P. The concentration compactness principle in the calculus of variations, the locally compact case II. Ann I H Poincaré–AN. 1984;1:223–283.
   Web of Science ®Google Scholar
• Wei J, Yan S. Infinitely many positive solutions for the nonlinear Schrödinger equations in RN. Calc Var. 2010;37:423–439.
   Web of Science ®Google Scholar
• Chen W, Wei J, Yan S. Infinitely many solutions for the Schrödinger equations in RN with critical growth. J Differ Equ. 2012;252:2425–2447.
   Web of Science ®Google Scholar
• Li G, Peng S, Yan S. Infinitely many positive solutions for the nonlinear Schrödinger–Poisson system. Commun Contemp Math. 2010;12:1069–1092.
   Web of Science ®Google Scholar
• Li G, Peng S, Wang C. Infinitely many solutions for the nonlinear Schrödinger equations with electromagenetic fields. J Differ Equ. 2011;251:3500–3521.
   Web of Science ®Google Scholar
• Peng S, Wang Z. Segregated and synchronized vector solutions for nonlinear Schrödinger systems. Arch Rational Mech Anal. 2013;208:305–339.
   Web of Science ®Google Scholar
• Wang L, Zhao C. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete Contin Dyn Sys. 2017;37:1701–1731.
   Web of Science ®Google Scholar
• Ao W, Wei J. Infinitely many positive solutions for nonlinear equations with non-symmetric potential. Calc Var. 2014;51:761–798.
   Web of Science ®Google Scholar
• Cerami G, Passaseo D, Solimini S. Infinitely many positive solutions to some scalar field equations with nonsymmetric coefficients. Comm Pure Appl Math. 2013;66:372–413.
   Web of Science ®Google Scholar
• Devillanova G, Solimini S. Infinitely many positive solutions to some nonsymmetric scalar field equations: the planar case. Calc Var. 2015;52:857–898.
   Web of Science ®Google Scholar
• Karpman V, Shagalov A. Stability of solitons described by nonlinear Schrödinger-type equations with high-order dispersion. Phys D. 2000;144:194–210.
   Web of Science ®Google Scholar
• Fibich G, Ilan B, Papanicolaou G. Self-focusing with fourth-order dispersion. SIAM J Appl Math. 2002;62:1437–1462.
   Web of Science ®Google Scholar
• Bonheure D, Casteras J, Mandel R. On a fourth-order nonlinear Helmholtz equation. J Lond Math Soc. 2019;99:831–852.
   Web of Science ®Google Scholar
• Evequoz G, Weth T. Dual variational methods and nonvanishing for the nonlinear Helmholtz equation. Adv Math. 2015;280:690–728.
   Web of Science ®Google Scholar
• He Y, Luo X, Rădulescu VD. Nodal multi-peak standing waves of fourth-order Schrödinger equations with mixed dispersion. J Geom Ana. 2022;32:1–36. DOI:10.1007/s12220-021-00795-x
   Web of Science ®Google Scholar
• Luo X, Yang T. Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. Sci China Math. 2022;65:1–26. DOI:10.1007/s11425-022-1997-3
   Google Scholar
• Bonheure D, Nascimento R. Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion. Contributes to Nonlinear Elliptic Equations and Systems, Progr. in Nonlinear Differ Equ and Appl. 2015;86:31–53.
   Google Scholar