Soliton solutions for quasilinear Schrödinger equations. II. (English) Zbl 1229.35268

Liu, Jia-quan; Wang, Ya-qi; Wang, Zhi-Qiang

J. Differ. Equations 187, No. 2, 473-493 (2003).

Summary: For a class of quasilinear Schrödinger equations, we establish the existence of ground states of soliton-type solutions by a variational method.
For Part I see [J. Liu and Z. Wang, Proc. Am. Math. Soc. 131, No. 2, 441–448 (2003; Zbl 1229.35269)].

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35Q51	Soliton equations

Keywords:

standing waves; quasilinear Schrödinger equations; Orlicz spaces; minimax methods

Citations:

Zbl 1229.35269

References:

[1]	Ambrosetti, A.; Rabinowitz, P., Dual variational methods in critical point theory, J. funct. anal., 14, 349-381, (1973) · Zbl 0273.49063
[2]	Arcoya, D.; Boccardo, L., Critical points for multiple integrals of the calculus of variations, Arch. rational mech. anal., 134, 249-274, (1996) · Zbl 0884.58023
[3]	Arcoya, D.; Boccardo, L., Some remarks on critical point theory for nondifferentiable functionals, Nonlinear differential equations appl., 6, 79-100, (1999) · Zbl 0923.35049
[4]	Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on \( R\^{}\{n\}\), Comm. partial differential equations, 20, 1725-1741, (1995) · Zbl 0837.35043
[5]	Bass, F.G.; Nasanov, N.N., Nonlinear electromagnetic spin waves, Phys. rep., 189, 165-223, (1990)
[6]	Berestycki, H.; Lions, P.L., Nonlinear scalar field equations, iexistence of a ground state, Arch. rational mech. anal., 82, 313-346, (1983)
[7]	Borovskii, A.V.; Galkin, A.L., Dynamical modulation of an ultrashort high-intensity laser pulse in matter, Jetp, 77, 562-573, (1993)
[8]	De Bouard, A.; Hayashi, N.; Saut, J.-C., Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. math. phys., 189, 73-105, (1997) · Zbl 0948.81025
[9]	Brandi, H.S.; Manus, C.; Mainfray, G.; Lehner, T.; Bonnaud, G., Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. fluids B, 5, 3539-3550, (1993)
[10]	Browder, F.E., Variational methods for nonlinear elliptic eigenvalue problems, Bull. amer. math. soc., 71, 176-183, (1965) · Zbl 0135.15802
[11]	Canino, A., Multiplicity of solutions for quasilinear elliptic equations, Topol. methods nonlinear anal., 6, 357-370, (1995) · Zbl 0863.35038
[12]	Chen, X.L.; Sudan, R.N., Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. rev. lett., 70, 2082-2085, (1993)
[13]	Corvellec, J.; Degiovanni, M.; Marzocchi, M., Deformation properties for continuous functionals and critical point theory, Topol. methods nonlinear anal., 1, 151-171, (1993) · Zbl 0789.58021
[14]	Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. funct. anal., 69, 397-408, (1986) · Zbl 0613.35076
[15]	Hasse, R.W., A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. phys. B, 37, 83-87, (1980)
[16]	Kondrat’ev, V.; Shubin, M., Discreteness of spectrum for the Schrödinger operator on manifolds of bounded geometry, Operator theory: adv. appl., 110, 185-226, (1999) · Zbl 0985.58012
[17]	Kosevich, A.M.; Ivanov, B.A.; Kovalev, A.S., Magnetic solitons, Phys. rep., 194, 117-238, (1990)
[18]	Kurihura, S., Large-amplitude quasi-solitons in superfluid films, J. phys. soc. jpn, 50, 3262-3267, (1981)
[19]	Laedke, E.W.; Spatschek, K.H.; Stenflo, L., Evolution theorem for a class of perturbed envelope soliton solutions, J. math. phys., 24, 2764-2769, (1983) · Zbl 0548.35101
[20]	Lange, H.; Toomire, B.; Zweifel, P.F., Time-dependent dissipation in nonlinear Schrödinger systems, J. math. phys., 36, 1274-1283, (1995) · Zbl 0823.35159
[21]	Litvak, A.G.; Sergeev, A.M., One dimensional collapse of plasma waves, JETP lett., 27, 517-520, (1978)
[22]	Liu, J.-Q.; Wang, Z.-Q., Soliton solutions for quasilinear Schrödinger equations, Proc. amer. math. soc., 131, 441-448, (2003) · Zbl 1229.35269
[23]	Makhankov, V.G.; Fedyanin, V.K., Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. rep., 104, 1-86, (1984)
[24]	Nakamura, A., Damping and modification of exciton solitary waves, J. phys. soc. jpn, 42, 1824-1835, (1977)
[25]	Poppenberg, M.; Schmitt, K.; Wang, Z.Q., On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. var. partial differential equations, 14, 329-344, (2002) · Zbl 1052.35060
[26]	Porkolab, M.; Goldman, M.V., Upper hybrid solitons and oscillating two-stream instabilities, Phys. fluids, 19, 872-881, (1976)
[27]	Quispel, G.R.W.; Capel, H.W., Equation of motion for the Heisenberg spin chain, Physica A, 110, 41-80, (1982)
[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. · Zbl 0609.58002
[29]	Rao, M.M.; Ren, Z.D., Theory of Orlicz spaces, (1991), Marcel Dekker New York · Zbl 0724.46032
[30]	Ritchie, B., Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. rev. E, 50, 687-689, (1994)
[31]	Strauss, W.A., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028
[32]	Takeno, S.; Homma, S., Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations, Progr. theoret. phys., 65, 172-189, (1981)
[33]	Lions, P.-L., Concentration compactness principle in the calculus of variations. the limit case. part 1, Rev. mat. iberoamericana, 1.1, 145-201, (1985) · Zbl 0704.49005
[34]	Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001