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Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

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Abstract

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities

$$ - {\Delta _{\mathbb{B}}}u + \lambda u = a\left( z \right)f\left( u \right),\,\,\,\,\,\,u \ge 0\,\,{\rm{in}}\,\,_ + ^N,$$

where N = n + 1 ≥ 3, λ > 0, z = (t, x1, ⋯, xn), and \({\Delta _{\mathbb{B}}} = {\left( {t{\partial _t}} \right)^2} + \partial _{{x_1}}^2 + \cdots + \partial _{{x_n}}^2\). Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Hua Chen

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Peng Luo

  3. Department of Mathematics, School of Science, Wuhan University of Technology, Wuhan, 430070, China

    Shuying Tian

Authors
  1. Hua Chen
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  2. Peng Luo
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  3. Shuying Tian
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Corresponding authors

Correspondence to Hua Chen, Peng Luo or Shuying Tian.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11631011, 11601402, 12171183, 11831009, 12071364, 11871387).

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Chen, H., Luo, P. & Tian, S. Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations. Chin. Ann. Math. Ser. B 43, 685–718 (2022). https://doi.org/10.1007/s11401-022-0353-2

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  • DOI: https://doi.org/10.1007/s11401-022-0353-2

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