The topological contribution of the critical points at infinity for critical fractional Yamabe-type equations

Abstract

In this paper, we study the critical fractional nonlinear PDE: $$(-\Delta )^{s}u= u^\frac{n+2s}{n-2s}$$ , $${u>0}$$ in $$\Omega $$ and $$u=0$$ on $$\partial \Omega $$ , where $$\Omega $$ is a thin annuli-domain of $${\mathbb{R}}^n, n\ge 2.$$ We compute the evaluation of the difference of topology induced by the critical points at infinity between the level sets of the associated variational function. Our Theorem can be seen as a nonlocal analog of the result of Ahmedou and El Mehdi (Duke Math J 94:215–229, 1998) on the classical Yamabe-type equation.