Standing wave solutions for a generalized quasilinear Schrödinger equation with indefinite potential

References

• Brüll L, Lange H. Solitary waves for quasilinear Schrödinger equations. Exposition Math. 1986;4:279–288.
   Google Scholar
• Bass FG, Nasanov NN. Nonlinear electromagnetic-spin waves. Phys Rep. 1990;189:165–223.
   Web of Science ®Google Scholar
• Guliyev VS, Guliyev RV, Ragusa MA, et al. Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete Contin Dyn Syst B. 2020;25(2):671–690.
   Web of Science ®Google Scholar
• Hasse RW. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys B. 1980;37:83–87.
   Google Scholar
• Kosevich AM, Ivanov BA, Kovalev AS. Magnetic solitons. Phys Rep. 1990;194:117–238.
   Web of Science ®Google Scholar
• Lange H, Poppenberg M, Teismann H. Nash–Moser methods for the solution of quasilinear Schrödinger equations. Commun Partial Differ Equ. 1999;24:1399–1418.
   Web of Science ®Google Scholar
• Makhankov VG, Fedyanin VK. Nonlinear effects in quasi-one-dimensional models and condensed matter theory. Phys Rep. 1984;104:1–86.
   Web of Science ®Google Scholar
• Poppenberg M, Schmitt K, Wang Z-Q. On the existence of soliton solutions to quasilinear Schrödinger equations. Calc Var Partial Differ Equ. 2002;14:329–344.
   Web of Science ®Google Scholar
• Quispel GRW, Capel HW. Equation of motion for the Heisenberg spin chain. Physica A. 1982;110:41–80.
   Web of Science ®Google Scholar
• Xie W, Chen C. Sign-changing solutions for the nonlinear Chern–Simons–Schrödinger equations. Appl Anal. 2020;99(5):880–898.
   Web of Science ®Google Scholar
• Kurihura S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Jpn. 1981;50:3262–3267.
   Google Scholar
• Laedke E, Spatschek K, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys. 1983;24:2764–2769.
   Web of Science ®Google Scholar
• Brandi H, Manus C, Mainfray G, et al. Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys Rev B. 1993;5:3539–3550.
   Google Scholar
• Chen XL, Sudan RN. Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma. Phys Rev Lett. 1993;70:2082–2085.
   PubMed Web of Science ®Google Scholar
• De Bouard A, Hayashi N, Saut J. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun Math Phys. 1997;189:73–105.
   Web of Science ®Google Scholar
• Ritchie B. Relativistic self-focusing and channel formation in laser-plasma interactions. Phys Rev E. 1994;50:687–689.
   Web of Science ®Google Scholar
• Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal Theory Methods Appl. 2013;80:194–201.
   Web of Science ®Google Scholar
• Chen H, Liu S. Standing waves with large frequency for 4-superlinear Schröinger–Poisson systems. Ann Mat Pura Appl. 2015;194(4):43–53.
   Google Scholar
• Bartsch T, Wang ZQ. Existence and multiplicity results for some superlinear elliptic problems on RN. Commun Partial Differ Equ. 1995;20:1725–1741.
   Web of Science ®Google Scholar
• Ambrosetti A, Rabinowitz PH. Dual variational methods in critical point theory and applications. J Funct Anal. 1973;14:349–381.
   Google Scholar
• Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrodinger equations with critical growth. J Differ Equ. 2015;258:115–147.
   Web of Science ®Google Scholar
• Fang X-D, Szulkin A. Multiple solutions for a quasilinear Schrödinger equation. J Differ Equ. 2013;254:2015–2032.
   Web of Science ®Google Scholar
• Chang K-C. Infinite-dimensional Morse theory and multiple solution problems. Boston (MA): Birkhäuser Boston, Inc.; 1993. (Progress in nonlinear differential equations and their applications; vol. 6).
   Google Scholar
• Bartsch T, Li S. Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 1997;28:419–441.
   Web of Science ®Google Scholar
• Liu JQ. The Morse index of a saddle point. J Syst Sci Math Sci. 1989;2:32–39.
   Google Scholar
• Liu S. Existence of solutions to a superlinear p-Laplacian equation. Electron J Differ Equ. 2001;66:6.
   Google Scholar
• Wang ZQ. On a superlinear elliptic equation. Ann Inst H Poincaré Anal Non Linéaire. 1991;8:43–57.
   Web of Science ®Google Scholar
• Rabinowitz PH. Minimax methods and their application to partial differential equations. In: Seminar on nonlinear partial differential equations (Berkeley, California, 1983). New York (NY): Springer; 1984. p. 307–320. (Mathematical Sciences Research Institute Publications; vol. 2).
   Google Scholar
• Chen S-J, Tang C-L. High energy solutions for the superlinear Schrödinger–Maxwell equations. Nonlinear Anal. 2009;71:4927–4934.
   Web of Science ®Google Scholar