Normalized solutions for the nonlinear Schrödinger equation with potentials
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arXiv
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| Format: | Preprint |
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2025
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| author | Rizzi, Matteo Peng, Xueqin |
| author_facet | Rizzi, Matteo Peng, Xueqin |
| contents | In this paper, we find normalized solutions to the following Schrödinger equation \begin{equation}\notag \begin{aligned} &-Δu-\fracμ{|x|^2}h(x)u+λu =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 $λ$ 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 $μ\to 0^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20961 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions for the nonlinear Schrödinger equation with potentials Rizzi, Matteo Peng, Xueqin Analysis of PDEs 35J10, 35J50, 35J60 In this paper, we find normalized solutions to the following Schrödinger equation \begin{equation}\notag \begin{aligned} &-Δu-\fracμ{|x|^2}h(x)u+λu =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 $λ$ 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 $μ\to 0^+$. |
| title | Normalized solutions for the nonlinear Schrödinger equation with potentials |
| topic | Analysis of PDEs 35J10, 35J50, 35J60 |
| url | https://arxiv.org/abs/2507.20961 |