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PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 2 / October 2021
- Published online by Cambridge University Press:
- 09 March 2020, pp. 159-178
- Print publication:
- October 2021
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- Article
- Export citation
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We consider the fractional critical problem
$A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in 
$\unicode[STIX]{x1D6FA},u=0$ on 
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$, where 
$A_{s},s\in (0,1)$, is the fractional Laplace operator and 
$K$ is a given function on a bounded domain 
$\unicode[STIX]{x1D6FA}$ of 
$\mathbb{R}^{n},n\geq 2$. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when 
$K$ is close to 1.
