Normalized solutions for the nonlinear Schr\"{o}dinger equation with potentials

Abstract

In this paper, we find normalized solutions to the following Schr\"{o}dinger equation \begin{equation}\notag \begin{aligned} &-\Delta u-\frac{\mu}{|x|^2}h(x)u+\lambda u =f(u)\quad\text{in}\quad\mathbb{R}^{N},\\ & u>0,\quad \int_{\mathbb{R}^{N}}u^2dx=a^2, \end{aligned} \end{equation} where $N\geq3$, $a>0$ is fixed, $f$ satisfies mass-subcritical growth conditions and $h$ is a given bounded function with $||h||_\infty\le 1$. The $L^2(\mathbb{R}^N)$-norm of $u$ is fixed and $\lambda$ 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 $\mu \to 0^+$.