Normalized solutions for the Sobolev critical Schrödinger equation with trapping potential

Abstract

We study the existence and multiplicity of positive normalized solutions with prescribed $L^2$-norm for the Sobolev critical Schrödinger equation $-\hat{I}”U+V(x)U=\hat{I}»U+|U{|}^{2^{\ast }-2}U$ in ${\mathbb{R}}^N$, ${\int }_{{\mathbb{R}}^N}{U}^{2}dx=\ddot{I}{}^2$, where $N\ge 3$, $V\ge 0$ is a trapping potential, $\hat{I}»\in \mathbb{R}$ and $2^{\ast }=\frac{2N}{N-2}$. Our first result is that the existence of local minimum solutions for $\ddot{I}\in (0,\ddot{I}{}^{\ast })$, for some suitable $\ddot{I}{}^{\ast }>0$, under appropriate assumptions on the potential. These solutions correspond to ground states. Our second result concerns the existence of mountain pass solutions, under the same assumptions.