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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
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