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The Asymptotic Behavior and Symmetry of Positive Solutions to p-Laplacian Equations in a Half-Space


We study a nonlinear equation in the half-space with a Hardy potential, specifically,

$$ - {\Delta _p}u = \lambda {{{u^{p - 1}}} \over {x_1^p}} - x_1^\theta f\left(u \right)\,\,\,\,{\rm{in}}\,\,T,$$

where Δp stands for the p-Laplacian operator defined by Δpu = div(∣Δup−2Δu), p > 1, θ > −p, and T is a half-space {x1 > 0}. When λ > Θ (where Θ is the Hardy constant), we show that under suitable conditions on f and θ, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as x1 → 0+, and the symmetric property of the positive solution are obtained.

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Corresponding author

Correspondence to Yimin Zhang.

Additional information

L. Wei was supported by NSFC (11871250). Y.M. Zhang was supported by NSFC (11771127, 12171379) and the Fundamental Research Funds for the Central Universities (WUT: 2020IB011, 2020IB017, 2020IB019).

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Chen, Y., Wei, L. & Zhang, Y. The Asymptotic Behavior and Symmetry of Positive Solutions to p-Laplacian Equations in a Half-Space. Acta Math Sci 42, 2149–2164 (2022).

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Key words

  • p-Lapacian
  • Hardy potential
  • symmetry
  • uniqueness
  • asymptotic behavior

2010 MR Subject Classification

  • 35J20
  • 35J60