{
  "id": "2002.01413",
  "version": "v1",
  "published": "2020-02-04T17:13:05.000Z",
  "updated": "2020-02-04T17:13:05.000Z",
  "title": "An alternative theorem for gradient systems",
  "authors": [
    "Biagio Ricceri"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Here is one of the result obtained in this paper: Let $\\Omega\\subset {\\bf R}^2$ be a smooth bounded domain and let $F, G : {\\bf R}\\to {\\bf R}$ be two $C^1$ functions satisfying the following conditions: $(i)$ for some $p>0$, one has $$\\limsup_{|\\xi|\\to +\\infty}{{|F'(\\xi)|+|G'(\\xi)|}\\over {|\\xi|^p}}<+\\infty\\ ;$$ $(ii)$ $F$ is non-negative, non-decreasing, $\\lim_{\\xi\\to +\\infty}{{F(\\xi)}\\over {\\xi^2}}=0$, $\\lim_{\\xi\\to 0^+}{{F(\\xi)}\\over {\\xi^2}}=+\\infty$ and the function $\\xi\\to {{F'(\\xi)}\\over {\\xi}}$ is strictly decreasing in $]0,+\\infty[$ ; $(iii)$ $G$ is positive and convex. Then, for every positive function $\\alpha\\in L^{\\infty}(\\Omega)$, the problem $$\\cases {-\\Delta u=\\alpha(x) G(v(x))F'(u) & in $\\Omega$ \\cr & \\cr -\\Delta v=-\\alpha(x) F(u(x))G'(v) & in $\\Omega$ \\cr & \\cr u=v=0 & on $\\partial\\Omega$\\cr} $$ has a non-zero weak solution belonging to $L^{\\infty}(\\Omega)\\times L^{\\infty}(\\Omega)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-04T17:13:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient systems",
      "alternative theorem",
      "non-zero weak solution belonging",
      "smooth bounded domain",
      "conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}