PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS

Abstract

AbstractWe 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.