On the standing waves of the Schroedinger equation with concentrated nonlinearity

Abstract

We study the concentrated NLS on R n , with power non-linearities, driven by the fractional Laplacian, (−∆) s , s > n 2 . We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space H s . Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable.Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding H s (R n ) → L ∞ (R n ), which provides an alternative calculation and justification of the sharp constants in these inequalities.