{
  "abstract": "In this paper, we study normalized solutions to the following $L^2$-supercritical nonlinear Schrödinger equation with a periodic potential \\begin{equation*} \\begin{dcases} -Δu +V (x)u + λu=χ_{Ω}(x)f(u)\\quad \\text{in } \\mathbb{R}^N, u>0 \\quad \\text{in} \\ \\mathbb{R}^N, \\int_{\\mathbb{R}^N}\\abs{u}^2\\, dx =μ. \\end{dcases} \\end{equation*} Here $N \\geq 1$, $μ>0$ is prescribed, $λ\\in \\mathbb{R}$ is a Lagrange multiplier, $V\\in C(\\mathbb{R}^N)$ is $1$-periodic in $x_1,...,x_N$, $f \\in C^1(\\mathbb{R})$ is a nonlinearity having $L^2$-supercritical growth at infinity, $Ω\\subset \\mathbb{R}^N$ is either a (nonempty) bounded open set with smooth boundary $\\partial Ω$ or the whole space \\(\\mathbb{R}^N\\), and $χ_{Ω}$ is the characteristic function of $Ω$. In both cases, we prove the existence of normalized solutions of mountain pass type when $μ>0$ is small and $f$ behaves like a power function $|t|^{p-2}t$ with $2+4/N<p<2^*$ at infinity. Moreover, if $Ω$ is bounded, $1 \\leq N \\leq 4$ and $f$ has $L^2$-supercritical growth near the origin, then the existence of normalized solutions is obtained for all $μ>0$. On the other hand, when $Ω=\\mathbb{R}^N$, we prove the existence of solutions corresponding to local minimizers when $μ>0$ is small. When $Ω$ is bounded, the results are obtained via a monotonicity trick for mountain pass values and blow-up analysis with the Morse index estimates. In the case where $Ω= \\mathbb{R}^N$, we develop the concentration-compactness argument based on the Morse index for the existence of mountain pass type solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.02147",
  "author_total": 3,
  "authors": [
    {
      "name": "Zhentao He"
    },
    {
      "name": "Norihisa Ikoma"
    },
    {
      "name": "Chao Ji"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bd3je3dqxs901.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.02147",
  "fulltext_url": null,
  "id": "bd3je3dqxs901",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.02147",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7207750124",
  "pdf_url": "https://arxiv.org/pdf/2609.02147",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bd3je3dqxs901.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.02147",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.02147"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials",
  "title_text": "Normalized solutions of L^2-supercritical NLS equations with periodic potentials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/bd3je3dqxs901",
  "venue": "arXiv",
  "versions": []
}