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Properties of Solutions to a Harmonic-Mapping Type Equation with a Dirichlet Boundary Condition

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Abstract

In the present paper, we consider the problem

$$\left\{{\matrix{{- \Delta u = {u^{{\beta _1}}}|\nabla u{|^{{\beta _2}}},} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr {u = 0,} \hfill & {{\rm{on}}\,\,\partial \Omega ,} \hfill \cr {u > 0,} \hfill & {{\rm{in}}\,\,\Omega ,} \hfill \cr}} \right.$$

where β1, β2 > 0 and β1 + β2 < 1, and Ω is a convex domain in ℝn. The existence, uniqueness, regularity and \({{2 - {\beta _2}} \over {1 - {\beta _1} - {\beta _2}}}\)-concavity of the positive solutions of the problem (0.1) are proven.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China

    Bo Chen  (陈博)

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Zhengmao Chen  (陈正茂)

  3. School of Mathematics and Statistics, Hubei Minzu University, Enshi, 445000, China

    Junhui Xie  (谢君辉)

Authors
  1. Bo Chen  (陈博)
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  2. Zhengmao Chen  (陈正茂)
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  3. Junhui Xie  (谢君辉)
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Corresponding author

Correspondence to Junhui Xie  (谢君辉).

Additional information

The first author and the third author were supported by the National Natural Science Foundation of China (11761030) and the Cultivation Project for High-Level Scientific Research Achievements of Hubei Minzu University (PY20002). The second author was supported by the China Postdoctoral Science Foundation (2021M690773).

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Chen, B., Chen, Z. & Xie, J. Properties of Solutions to a Harmonic-Mapping Type Equation with a Dirichlet Boundary Condition. Acta Math Sci 43, 1161–1174 (2023). https://doi.org/10.1007/s10473-023-0310-5

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  • DOI: https://doi.org/10.1007/s10473-023-0310-5

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