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

TL;DR

This paper investigates the existence and multiplicity of positive normalized solutions to a Sobolev critical Schrödinger equation with a trapping potential, identifying conditions for ground states and mountain pass solutions.

Contribution

It establishes the existence of ground state and mountain pass solutions for the Sobolev critical Schrödinger equation with prescribed $L^2$-norm, under specific potential assumptions.

Findings

Existence of local minimum solutions for small $\rho$

Existence of mountain pass solutions under the same conditions

Solutions correspond to ground states and excited states

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.