| Abstract: |
| In this talk, I would like to present some recent results on the existence and multiplicity of positive solutions in $H^1(\mathbb{R}^N)$, $N\ge3$, with prescribed $L^2$-norm, for the Sobolev critical Schr\{o}dinger equation with trapping potential.
Under suitable assumptions on the potential, the associated energy functional exhibits a mountain pass geometry on the $L^2$-sphere, which yields the existence of both a local minimizer and a mountain pass solution. In particular, for small values of the prescribed mass, we prove the existence of two distinct solutions: a ground state and a mountain pass solution. |
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