The global Cauchy problem for the nonlinear Schrödinger equation revisited. (English) Zbl 0586.35042

For the Cauchy problem for the nonlinear Schrödinger equation \[ i(\partial /\partial t)\phi =-()\Delta \phi +f(\phi),\quad in\quad {\mathbb{R}}^ n, \] the authors prove the existence of weak global solutions \(\phi\) for the interactions f slowly increasing or repulsive rapidly increasing, and both the existence and uniqueness of a more regular solution \(\phi\) for f satisfying some additional condition. The basic condition on f is that \(f: {\mathbb{C}}\to {\mathbb{C}}\) is a continuous function such that \(| f(z)| \leq C(| z| +| z|^ p),\) \(z\in {\mathbb{C}}\), with some p, \(1\leq p<\infty\), and such that there exists a \(C^ 1\)-function \(V: {\mathbb{C}}\to {\mathbb{R}}\) with \(V(z)=V(| z|)\), \(z\in {\mathbb{C}}\), \(V(0)=0\), and \(f(z)=\partial V/\partial \bar z\). The other conditions are rather involved, dependent on the dimension n of space [cf. the authors, J. Funct. Anal. 32, 33-71 (1979; Zbl 0396.35029), and Ann. Inst. Henri Poincaré, N. Sér., Sect. A 28, 297-316 (1978; Zbl 0397.35012)]. The proof of existence uses the Galerkin method of approximation, and the proof of uniqueness is based on the method of contraction.
Reviewer: T.Ichinose


35J60 Nonlinear elliptic equations
35Q99 Partial differential equations of mathematical physics and other areas of application
35J10 Schrödinger operator, Schrödinger equation
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
49M15 Newton-type methods
Full Text: DOI Numdam EuDML


[1] Baillon, J. B.; Cazenave, T.; Figuera, M., C. R. acad. sci. Paris, t. 284, 869-872, (1977)
[2] Bergh, J.; Löfström, J., Interpolation spaces, (1976), Springer Berlin-Heidelberg-New York
[3] Cazenave, T., Proc. roy. soc. Edinburgh, t. 84, 327-346, (1979)
[4] Cazenave, T.; Haraux, A., Ann. fac. sc. Toulouse, t. 2, 21-25, (1980)
[5] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationnels, (1972), Dunod Paris
[6] Ginibre, J.; Velo, G., J. funct. anal., Ann. IHP, t. A28, 287-316, (1978), and
[7] Ginibre, J.; Velo, G., Ann. IHP, t. C1, 309-323, (1984)
[8] Ginibre, J.; Velo, G., C. R. acad. sci. Paris, t. 298, 137-140, (1984), and J. Math. Pur Appl., in press
[9] J. Ginibre, G. Velo, Math. Z., 1985, in press.
[10] Hartman, P., Ordinary differential equations, (1982), Birkhäuser Boston-Basel-Stuttgart
[11] Lin, J. E.; Strauss, W., J. funct. anal., t. 30, 245-263, (1978)
[12] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod and Gauthier-Villars Paris
[13] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications, Vol. 1, (1968), Dunod Paris
[14] Pecher, H., Math. Z., t. 185, 261-270, (1984)
[15] Pecher, H.; von Wahl, W., Manuscripta math., t. 27, 125-157, (1979)
[16] Reed, M., Abstract non-linear wave equations, (1976), Springer Berlin-Heidelberg-New York
[17] Segal, I. E., Bull. soc. math. France, t. 91, 129-135, (1963)
[18] Stein, E. M., Singular integrals and differentiability properties of functions, (1970), Princeton University Press Princeton
[19] Strauss, W., Anais acad. Brazil. ciencias, t. 42, 645-651, (1970)
[20] Strauss, W., J. funct. anal., J. Funct. Anal., t. 43, 281-293, (1981)
[21] Strichartz, R., Duke math. J., t. 44, 705-714, (1977)
[22] Treves, F., Basic linear differential equations, (1975), Academic Press New York-London
[23] Yosida, K., Functional analysis, (1978), Springer Berlin-Heidelberg-New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.