## Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential.(English)Zbl 1383.35213

The purpose of this paper is to study several variants of the NLS equation. This study is concentrated on the two ground states, i.e. the least energy solitons. The main result states that the solution of a NLS equation with a potential with a single negative eigenvalue either blows up in finite time or scatters as $$t\to\pm\infty$$ to the first ground state.
Then modulation and linearized equations around the ground state are analyzed. The proofs use implicit function theorem, Ascoli-Arzela theorem, Lagrange multipliers, Duhamel formula, Strichart estimate, Gagliardi-Nirenberg, Hölder, radial Sobolev, Sobolev, Cauchy-Schwarz and Hölder inequalities. A table of notation is given in the end.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35C08 Soliton solutions 35P25 Scattering theory for PDEs 35B44 Blow-up in context of PDEs
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