Solutions with prescribed mass for critical Schrödinger–Poisson systems concentrating at a potential well

This paper is concerned with the existence and multiplicity of normalized solutions to the following Schrödinger–Poisson system: \{ array{ll - ² u + V (x) u - |u|^3 u = u + |u|^q-2 u + |u|^4 u, & ~ in R^3, \\ - ² = |u|^5, & ~ in R^3, array. } with prescribed mass aligned ₑ^{3} |u|^2 \, dx = a² ³, aligned where a > 0, > 0, q (2, 103), and > 0 is a small parameter. Here, R arises as a Lagrange multiplier, and the potential V: R^3 0, +) is a continuous function satisfying suitable conditions. By combining truncation techniques with some adequate estimates, we establish that, for sufficiently small > 0, normalized solutions do exist. Moreover, by employing Ljusternik-Schnirelmann theory, we find a relationship between the number of positive solutions and the topology of the set where the potential V attains its minimum. Our work extends and complements recent contributions of X. Feng [17, 18 (Z. Angew. Math. Phys. 2020), to the abstract setting of multiple normalized concentrating solutions. This study seems to be the first work dealing with the existence of multiple normalized semiclassical states for the Sobolev critical Schrödinger–Poisson system coupled with a nonlocal critical term in the whole space R³.