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Fractional p-Laplacian problems with negative powers in a ball or an exterior domain

Abstract

This paper is concerned with the qualitative properties of positive solutions to fractional p-Laplacian problems

$$\begin{aligned}(-\Delta )^s_p u + |x|^{-\alpha }u^{-q} = 0\end{aligned}$$

in \(B_1 \setminus \{0\}\) and \(\mathbb {R}^n \setminus \overline{B_1}\), where \(0<s<1\), \(p\ge 2\), \(q>0\), \(\alpha \ge 0\) and \(B_1\) is the unit ball centered at the origin. By deriving a decay at infinity principle and exploiting the direct method of moving planes for the fractional p-Laplacian, we prove the symmetry or monotonicity of positive solutions to the above problems.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable suggestions and comments which lead to the improvement of the paper.

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Correspondence to Vu Ho.

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Le, P., Ho, V. Fractional p-Laplacian problems with negative powers in a ball or an exterior domain. J. Pseudo-Differ. Oper. Appl. 11, 789–803 (2020). https://doi.org/10.1007/s11868-019-00307-0

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  • DOI: https://doi.org/10.1007/s11868-019-00307-0

Keywords

  • Fractional p-Laplacian
  • Negative powers
  • Symmetry of solutions
  • Method of moving planes

Mathematics Subject Classification

  • 35R11
  • 35J92
  • 35B06
  • 35J75