Infinite sharp conditions by Nehari manifolds for nonlinear Schrödinger equations. (English) Zbl 1439.35441
Summary: We study the Cauchy problem of the nonlinear Schrödinger equation \(i\varphi_t+\Delta \varphi +|\varphi |^{p-1}\varphi =0\). By constructing infinite Nehari manifolds with geometric features, we not only obtain infinite invariant sets of solutions, but also give infinite sharp conditions for global existence and finite time blow up of solutions.
MSC:
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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