In this paper, we find normalized solutions to the following Schrödinger equation equation aligned &-Δu-μ|x|²h (x) u+λu =f (u) ^N, \\ & u>0, ₑ^₍u²dx=a², aligned equation where N3, a>0 is fixed, f satisfies mass-subcritical growth conditions and h is a given bounded function with ||h||_ 1. The L² (RN) -norm of u is fixed and λ appears as a Lagrange multiplier. Our solutions are constructed by minimizing the corresponding energy functional on a suitable constraint. Due to the presence of a possibly nonradial term h, establishing compactness becomes challenging. To address this difficulty, we employ the splitting lemma to exclude both the vanishing and the dichotomy of a given any minimizing sequence for appropriate a > 0. Furthermore, we show that if h is radial, then radial solutions can be obtained for any a>0. In this case, the radial symmetry allows us to prove that such solutions converge to a ground state solution of the limit problem as μ 0^+.
Rizzi et al. (Mon,) studied this question.