Normalized solutions for Schrödinger equations with negative potential and nonlocal perturbation

In this paper, we study normalized solutions for the following Schrödinger equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is an unknown Lagrange multiplier, [Formula: see text] is the Riesz potential with order [Formula: see text], and the potential [Formula: see text] satisfies some suitable assumptions. The simultaneous presence of the nonlocal Choquard term, the Sobolev critical term, and the negative potential makes the geometric structure of the corresponding functional considerably more delicate. To overcome these difficulties, we develop some variational techniques and establish sharp energy estimates to recover compactness. Our main results are as follows: i) The existence of two solutions, one being a local minimizer with negative energy, and the other of mountain-pass type with positive energy, under explicit conditions on [Formula: see text] when [Formula: see text]; ii) The existence of a ground state solution for all [Formula: see text] when [Formula: see text]. This work provides the first existence results for the case [Formula: see text] and [Formula: see text]. In particular, our results extend and improve those of [Ding-Zhong, J. Differ. Equ. 334 (2022) 194-215] and the related literature.