Normalized solutions for the Sobolev critical Schr\"{o}dinger equation with trapping potential

Abstract

We study the existence and multiplicity of positive normalized solutions with prescribed $L^{2}$-norm for the Sobolev critical Schr\"odinger equation $-\Delta U + V(x) U = \lambda U + |U|^{2^*-2} U$ in $\mathbb{R}^N$, $\int_{\mathbb{R}^N} U^2\,dx = \rho^2$, where $N \ge 3$, $V\ge 0$ is a trapping potential, $\lambda \in \mathbb{R}$ and $2^*=\frac{2N}{N-2}$. Our first result is that the existence of local minimum solutions for $\rho \in (0, \rho^*)$, for some suitable $\rho^* > 0$, 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.