Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight


Let \(p>0\) and \((-\Delta )^{s}\) is the fractional Laplacian with \(0< s<1\). The purpose of this paper is to establish a classification result for positive stable solutions to a fractional singular elliptic equation with weight

$$ (-\Delta )^{s} u=-h(x)u^{-p}\text{ in }\mathbb{R}^{N}. $$

Here \(N>2s\) and \(h\) is a nonnegative, continuous function satisfying \(h(x)\geq C|x|^{a}\), \(a\geq 0\), when \(|x|\) large. We prove the nonexistence of positive stable solutions of this equation under the condition

$$ N< 2s+\frac{2(a+2s)}{p+1}\left (p+\sqrt{p^{2}+p}\right ) $$

or equivalently

$$ p>p_{c}(N,s,a), $$


$$ p_{c}(N,s,a)= \textstyle\begin{cases} \frac{(N-2s)^{2}-2(N+a)(a+2s)+2\sqrt{(a+2s)^{3}(2N-2s+a)}}{(N-2s)(10s+4a-N)}&\text{ if }N< 10s+4a \\ +\infty &\text{ if }N\geq 10s+4a \end{cases}\displaystyle . $$

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We would like to express our gratitude to the referee for the careful reading and helpful suggestions. This research is funded by VNU - University of Education under Research Project number QS.19.07.

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Correspondence to Anh Tuan Duong.

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Duong, A.T., Luong, V.T. & Nguyen, T.Q. Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight. Acta Appl Math 170, 579–591 (2020).

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  • Liouville type theorems
  • Stable solutions
  • Fractional singular elliptic equations
  • Negative exponent nonlinearity

Mathematics Subject Classification

  • 35B53
  • 35J60
  • 35B35