Normalized solutions for the nonlinear Schrödinger equation with potentials

In this paper, we find normalized solutions to the following Schrödinger equation {equation} {aligned} &-Δu-μ{|x|^2}h(x)u+λu =f(u)N,\\ & u>0, ∫_{RN}u^2dx=a^2, {aligned} {equation} where N≥3, $a>0$ 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⁺.