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

TL;DR

This paper establishes the existence of normalized solutions to a nonlinear Schrödinger equation with potentials, addressing nonradial cases and convergence to ground states as a parameter approaches zero.

Contribution

It introduces a variational approach to find normalized solutions with potentials, handling nonradial terms and proving convergence to ground states.

Findings

Existence of positive normalized solutions for the Schrödinger equation.

Construction of solutions via energy minimization under constraints.

Radial solutions exist and converge to ground states as a parameter tends to zero.

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 $\mu \to 0^+$.