{
  "abstract": "We consider the functional $I_{δ,Ψ_1,Ψ_2}:X \\to \\mathbb{R}$ defined for any $z=(u,v) \\in X$ as \\begin{align*} I_{δ,Ψ_1,Ψ_2}(z) & = \\int_Ω Ψ_1(\\nabla u) \\, dx + \\int_Ω Ψ_2(\\nabla v ) \\, dx & - \\int_Ω H(δ,x,u(x),v(x)) \\,dx, \\end{align*} where $Ω$ is a smooth bounded domain of $\\mathbb{R}^N$, $Ψ_1, Ψ_2: \\mathbb{R}^N \\to \\mathbb{R}$ are convex functions satisfying suitable conditions for $1 < p, q < N$, and the nonlinearity $H$ is allowed to exhibit both subcritical and critical growth and may depend on a parameter $δ\\in I \\subseteq \\mathbb{R}$. Here, the functional space setting is given by the product space $X :=W_0^{1,p}(Ω)\\times W_0^{1,q}(Ω)$ equipped with the norm $\\|z\\|= \\|u\\|_{1,p} + \\|v\\|_{1,q}$ for any $z=(u,v)\\in X$, where $\\| \\cdot \\|_{1,s}$ is the usual norm in $W^{1,s}_0(Ω)$. In this paper we prove that $I_{δ,Ψ_1,Ψ_2}'$ is of class $(S)_+$ and we infer that each isolated critical point of $I_{δ,Ψ_1,Ψ_2}$ has critical groups of finite type and a Poincaré-Hopf formula holds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.40145",
  "author_total": 1,
  "authors": [
    {
      "name": "Natalino Borgia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctw1sfqmnscdw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.40145",
  "fulltext_url": null,
  "id": "ctw1sfqmnscdw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.40145",
  "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": "W7215084465",
  "pdf_url": "https://arxiv.org/pdf/2609.40145",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctw1sfqmnscdw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.40145",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.40145"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula",
  "title_text": "Topological invariants for anisotropic quasilinear elliptic systems: A Poincaré-Hopf formula",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/ctw1sfqmnscdw",
  "venue": "arXiv",
  "versions": []
}