Soliton solutions for quasilinear Schrödinger equations, II

Journal of Differential Equations

Volumn 187, Issue 2, 2003, Pages 473-493

  Liu, Jia Quan ^a   Wang, Ya Qi ^a   Wang, Zhi Qiang ^b  

^a PEKING UNIVERSITY   (China)

^b UTAH STATE UNIVERSITY   (United States)

Author keywords

Minimax methods; Orlicz spaces; Quasilinear Schr dinger equations; Standing waves

Indexed keywords

EID: 0037455313     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-0396(02)00064-5     Document Type: Article

References (34)

1. A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory, J. Funct. Anal. 14 (1973) 349-381.
2. D. Arcoya, L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal. 134 (1996) 249-274.
3. D. Arcoya, L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, Nonlinear Differential Equations Appl. 6 (1999) 79-100.
4. n , Comm. Partial Differential Equations 20 (1995) 1725-1741.
5. F.G. Bass, N.N. Nasanov, Nonlinear electromagnetic spin waves, Phys. Rep. 189 (1990) 165-223.
6. H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I: existence of a ground state, Arch. Rational Mech. Anal. 82 (1983) 313-346.
7. A.V. Borovskii, A.L. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP 77 (1993) 562-573.
8. A. De Bouard, N. Hayashi, J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys. 189 (1997) 73-105.
9. H.S. Brandi, C. Manus, G. Mainfray, T. Lehner, G. Bormaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B 5 (1993) 3539-3550.
10. F.E. Browder, Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc. 71 (1965) 176-183.
11. A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal. 6 (1995) 357-370.
12. X.L. Chen, R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett. 70 (1993) 2082 2085.
13. J. Corvellec, M. Degiovanni, M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1993) 151-171.
14. A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986) 397-408.
15. R.W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B 37 (1980) 83-87.
16. V. Kondrat'ev, M. Shubin, Discreteness of spectrum for the Schrödinger operator on manifolds of bounded geometry, Operator theory: Adv. Appl. 110 (1999) 185-226.
17. A.M. Kosevich, B.A. Ivanov, A.S. Kovalev, Magnetic solitons, Phys. Rep. 194 (1990) 117-238.
18. S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Jpn 50 (1981) 3262-3267.
19. E.W. Laedke, K.H. Spatschek, L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys. 24 (1983) 2764-2769.
20. H. Lange, B. Toomire, P.F. Zweifel, Time-dependent dissipation in nonlinear Schrödinger systems, J. Math. Phys. 36 (1995) 1274 1283.
21. A.G. Litvak, A.M. Sergeev, One dimensional collapse of plasma waves, JETP Lett. 27 (1978) 517-520.
22. J.-Q. Liu, Z.-Q., Wang, Soliton solutions for quasilinear Schrödinger equations, Proc. Amer. Math. Soc. 131 (2003) 441-448.
23. V.G. Makhankov, V.K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104 (1984) 1-86.
24. A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc. Jpn 42 (1977) 1824-1835.
25. M. Poppenberg, K. Schmitt, Z.Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002) 329-344.
26. M. Porkolab, M.V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids 19 (1976) 872-881.
27. G.R.W. Quispel, H.W. Capel, Equation of motion for the Heisenberg spin chain, Physica A 110 (1982) 41-80.
28. P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Conference Series in Mathematics, Vol. 65, American Mathematical Society, 1986.
29. M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, New York, 1991.
30. B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E 50 (1994) 687-689.
31. W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977) 149-162.
32. S. Takeno, S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations, Progr. Theoret. Phys. 65 (1981) 172-189.
33. P.-L. Lions, Concentration compactness principle in the calculus of variations. The limit case. Part 1, Rev. Mat. Iberoamericana 1.1 (1985) 145-201.
34. M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.