The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

Abstract

We study the following fractional Yamabe-type equation:\left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.Here Ω is a regular bounded domain of{\mathbb{R}^{n}},{n\geq 2}, and{A_{s}},{s\in(0,1)}, represents the fractional Laplacian operator{(-\Delta)^{s}}in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

On the sharp constant in the Bianchi-Egnell stability inequality

“…It follows by the implicit function theorem (see [3] for a similar argument for s = 1, which remains valid for all s ∈ (0, d 2 ) as observed in [1]) that for ε > 0 small enough the minimum of the distance dist(U + ερ, M) is in fact achieved in U. Hence…”

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

“…It follows that ðÀDÞ s is given by: The critical points of J in R þ :¼ fu 2 R; u [ 0g correspond to the solutions of (1.2), see [3]. As mentioned previously, J fails to satisfy the Palais Smale condition, since the exponent 2n…”

“…Using the above decomposition and computation like the one of Proposition 1 of [3], we have the following expansion:…”

“…This leads to the possibility of existence of non-compact orbits of ðÀoJÞ along which J is bounded and its gradient tends to zero, the so-called critical points at infinity, see [7]. Although some results were established on (1.1), see for example [3,25], questions concerning existence of solutions, multiplicity or non-existence remains open for the critical problem (1.1).…”

“…It is interesting to note that although the existence of a solution has been proved for (1.2) in the case of a thin annuli-domain, see [3], the evaluation of the topological contribution of the critical points at infinity is interesting to prove multiplicity results as in [14,24]. Notice also that the blow-up approach of [1] is quite different to the approach of this paper which is based on the critical points theory at infinity of Bahri [7].…”