We consider the fractional elliptic problem: where B1 is the unit ball in ℝN, N ⩾ 3, s ∈ (0, 1) and p > (N + 2s)/(N − 2s). We prove that this problem has infinitely many solutions with slow decay O(|x|−2s/(p−1)) at infinity. In addition, for each s ∈ (0, 1) there exists Ps > (N + 2s)/(N − 2s), for any (N + 2s)/(N − 2s) < p < Ps, the above problem has a solution with fast decay O(|x|2s−N). This result is the extension of the work by Dávila, del Pino, Musso and Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) to the fractional case.

中文翻译：

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外域超临界分数椭圆问题的快慢衰减解

我们考虑分数椭圆问题：在哪里乙1是ℝ中的单位球ñ,ñ⩾ 3,s∈ (0, 1) 和p> (ñ+ 2s)/(ñ− 2s）。我们证明这个问题有无限多个缓慢衰减的解○(|X|-2s/(p-1)) 在无穷远处。此外，对于每个s∈ (0, 1) 存在磷s> (ñ+ 2s)/(ñ− 2s), 对于任何 (ñ+ 2s)/(ñ− 2s) <p<磷s, 上述问题有一个快速衰减的解○(|X|2s-ñ）。这个结果是 Dávila、del Pino、Musso 和 Wei (2008, Calc. Var. Partial Differ. Equ. 32, no. 4, 453–480) 对分数情况的扩展。