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We obtain threshold results for the existence, non-existence and multiplicityof normalized solutions for semi-linear elliptic equations in the exterior of aball. To the best of our knowledge, it is the first result in the literatureaddressing this problem. In particular, we show that the prescribed mass canaffect the number of normalized solutions and has a stabilizing effect in themass supercritical case. Furthermore, in the threshold we find a new exponent p= 6 when N = 2, which does not seem to have played a role for this equation inthe past. Moreover, our findings are "quite surprising" and completelydifferent from the results obtained on the entire space and on balls. We willalso show that the nature of the domain is crucial for the existence andstability of standing waves. As a foretaste, it is well-known that in thesupercritical case these waves are unstable in RN . In this paper, we will showthat in the exterior domain they are strongly stable.

Threshold for Existence, Non-existence and Multiplicity of positive  solutions with prescribed mass for an NLS with a pure power nonlinearity in  the exterior of a ball