Find in Library

In this article, we consider the global existence and stability issues of the nonlinear Schrödinger equation with partial confinement. First, by establishing some new cross-invariant manifolds and variational problems, a new sharp criterion of global existence is derived in the $ L^{2} $-critical and $ L^{2} $-supercritical cases. Then, the existence of orbitally stable standing waves is obtained in the $ L^{2} $-subcritical and $ L^{2} $-critical cases by taking advantage of the profile decomposition technique. Our work extends and complements some earlier results.