Title:  Standing Waves of the Schrodinger Equation with Concentrated Nonlinearity

With Atanas Stefanov, we study the concentrated Nonlinear Schrodinger Equations in n-dimensions, with power non-linearities, driven by the fractional Laplacian.  We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. We also provided 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 $H^s(\mathbb{R}^n)\hookrightarrow L^\infty(\mathbb{R}^n)$, which provides an alternative calculation and justification of the sharp constants in these inequalities.