| Abstract: |
| We study the existence and multiplicity of positive solutions with prescribed $L^2$-norm for the (stationary) nonlinear Schr\odinger equation with Sobolev critical power nonlinearity. In the free case on the full space, the associated energy functional has a mountain pass geometry on the $L^2$-sphere, which boils down, in higher dimension, to the existence of a mountain pass solution. We consider this problem, either in bounded domains (i.e., the normalized Brezis-Nirenberg problem) or in presence of a potential, wondering (i) whether a local minimum solution appears, thus providing an orbitally stable family of solitons, and (ii) if the existence of a mountain pass solution persists. This talk is based on joint works with Dario Pierotti and Junwei Yu (Politecnico di Milano). |
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