{
  "abstract": "A nonlinear scalar field equation is studied, which takes the form \\begin{equation*} - \\triangle{u}=g(u) \\ \\textrm{in} \\ R^{N}, \\quad \\quad u \\in E, \\end{equation*} where $E$ is a suitable Banach space.. The nonlinear term $g(s)$ has subcritical growth at infinity. Near $s=0$, the nonlinear term $g(s)$ grows like $s^q$ for any $q>0$ if $N=2$ and for any $q \\in (0,\\ (N+2)/(N-2))$ if $N\\ge 3$. Through the technique of adding one dimension of space, by mountain pass theorem and symmetric mountain theorem, we prove the multiplicity of radial solutions if $N\\ge 2$ and the multiplicity of nonradial solutions if $N\\ge 4$. Unlike [1] and [2], the proof presented here does not rely on any decomposition results. During the treatment for $N=2$, we provide a new proof for certain Gagliardo-Nirenberg type inequality on $R^2$, which is originally due to [3]. Based on this inequality we get a Trudinger-Moser type inequality involving gradient norm through Taylor expansion. Eventually we construct a special path to obtain the least energy level for solutions. [1] J. Mederski, Nonradial solutions of nonlinear scalar field equations. Nonlinearity 33 (2020), 6349--6380. [2] L. Jeanjean, S.S. Lu, Nonlinear scalar field equations with general nonlinearity. Nonlinear Anal. 190 (2020), 111604. [3] D.E. Edmunds, A.A.Ilyin, Asymptotically sharp multiplicative inequalities. Bull. London Math. Soc. 27 (1995) 71--74.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02524",
  "author_total": 2,
  "authors": [
    {
      "name": "Qiuping Lu"
    },
    {
      "name": "Shuolei Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2aq3a7rjes2kq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02524",
  "fulltext_url": null,
  "id": "2aq3a7rjes2kq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02524",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02524",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2aq3a7rjes2kq.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02524",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02524"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane",
  "title_text": "Radial and nonradial solutions for nonlinear scalar field equations and a new proof for certain Gagliardo-Nirenberg type inequality on the plane",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/2aq3a7rjes2kq",
  "venue": "arXiv",
  "versions": []
}