{
  "id": "2505.03381",
  "version": "v1",
  "published": "2025-05-06T10:00:18.000Z",
  "updated": "2025-05-06T10:00:18.000Z",
  "title": "On the continuity of solutions to the anisotropic $N$-Laplacian with $L^1$ lower order term",
  "authors": [
    "Mariia Savchenko",
    "Igor Skrypnik",
    "Yevgeniia Yevgenieva"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We establish the continuity of bounded solutions to the anisotropic elliptic equation $$-\\sum\\limits_{i=1}^N\\Big(|u_{x_i}|^{p_i-2} u_{x_i}\\Big)_{x_i}=f(x),\\quad x\\in \\Omega,\\quad f(x)\\in L^1(\\Omega)$$ under the conditions $$\\min\\limits_{1\\leqslant i\\leqslant N} p_i >1,\\quad \\sum\\limits_{i=1}^N \\frac{1}{p_i}=1$$ and $$\\lim\\limits_{\\rho\\rightarrow 0}\\,\\sup\\limits_{x\\in \\Omega}\\int\\limits^{\\rho}_0\\Big(\\int\\limits_{B_r(x)}|f(y)|\\,dy\\Big)^{\\frac{1}{N-1}}\\frac{dr}{r}=0.$$ In the standard case $p_1=...=p_N=N$, these conditions recover the known results for the $N$-Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-06T10:00:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B65",
      "35D30",
      "35J92"
    ],
    "keywords": [
      "lower order term",
      "continuity",
      "anisotropic elliptic equation",
      "conditions",
      "standard case"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}