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On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. (English) Zbl 0396.35028


MSC:

35J10 Schrödinger operator, Schrödinger equation
35J60 Nonlinear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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[1] Dunford, N.; Schwartz, J., (Linear Operators, Vol. I (1958), Interscience: Interscience New York) · Zbl 0084.10402
[2] de Gennes, P. G., Superconductivity of Metals and Alloys (1966), Benjamin: Benjamin New York, Chap. 6 · Zbl 0138.22801
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[4] Scott, A. C.; Chu, F. Y.F.; McLaughlin, D. W., The soliton: A new concept in applied science, (Proc. IEEE, 61 (1973)), 1143-1483
[5] Strauss, W. A., Nonlinear scattering theory, (Lavita, J. A.; Marchand, J.-P., Scattering Theory in Mathematical Physics (1974), Reidel: Reidel Dordrecht, Holland), 53-78 · Zbl 0297.35062
[6] Volevic, L. R.; Paneyakh, B. P., Certain spaces of generalized functions and embedding theorems, Russian Math. Surveys, 20, 1-73 (1965)
[7] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Physics JETP, 34, 62-69 (1972)
[8] Baillon, J. B.; Cazenave, T.; Figueira, M., C. R. Acad. Sci. Paris, 284, 869-872 (1977) · Zbl 0349.35048
[9] Glassey, R. T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., 18, 1794-1797 (1977) · Zbl 0372.35009
[10] J. E. Lin and W. Strauss; J. E. Lin and W. Strauss
[11] Lions, J. L.; Magenes, E., (Problèmes aux limites non homogènes et applications, Vol. 1 (1968), Dunod: Dunod Paris) · Zbl 0165.10801
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