Genev, Hristo; Venkov, George Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. (English) Zbl 1247.35143 Discrete Contin. Dyn. Syst., Ser. S 5, No. 5, 903-923 (2012). Summary: We study the Cauchy problem for the focusing time-dependent Schrödinger-Hartree equation \[ i \partial_t \psi + \triangle \psi = -({|x|^{-(n-2)}}\ast |\psi|^{\alpha})|\psi|^{\alpha - 2} \psi, \quad \alpha\geq 2, \] for space dimension \(n \geq 3\). We prove the existence of solitary wave solutions and give conditions for formation of singularities in dependence of the values of \(\alpha\geq 2\) and the initial data \(\psi(0,x)=\psi_0(x)\). Cited in 41 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A15 Variational methods applied to PDEs 35Q51 Soliton equations 35C08 Soliton solutions Keywords:nonlinear Schrödinger equation; solitary waves; blow-up solutions; variational methods PDFBibTeX XMLCite \textit{H. Genev} and \textit{G. Venkov}, Discrete Contin. Dyn. Syst., Ser. S 5, No. 5, 903--923 (2012; Zbl 1247.35143) Full Text: DOI