We obtain threshold results for the existence, non-existence and multiplicity of normalized solutions for semi-linear elliptic equations in the exterior of a ball. To the best of our knowledge, it is the first result in the literature addressing this problem for the \(L^2\) supercritical case. In particular, we show that the prescribed mass can affect the number of normalized solutions and has a stabilizing effect in the mass 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 in the past. Moreover, our findings are “quite surprising” and completely different from the results obtained on the entire space and on balls. We will also show that the nature of the domain is crucial for the existence and stability of standing waves. As a foretaste, it is well-known that in the supercritical case these waves are unstable in \(\mathbb {R}^N.\) In this paper, we will show that in the exterior domain they are strongly stable.
