A critical fractional Laplace equation in the resonant case

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities (-Δ)ˢ u-λ u=|u|2^*-2u & in Ω,\\ u=0 & in Rⁿ Ω, started in the recent papers {13}, {17}-{19}. Here s∈ (0,1) is a fixed parameter, (-Δ )ˢ is the fractional Laplace operator, λ is a positive constant, 2^*=2n/(n-2s) is the fractional critical Sobolev exponent and Ω is an open bounded subset of ⁿ, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when n=4s and λ is an eigenvalue of the operator (-Δ)ˢ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in {13}, {17}-{19}, we are able to show that this critical problem admits a nontrivial solution provided "•" $n> 4s$ and λ> 0, "•" $n=4s$ and λ> 0 is different from the eigenvalues of (-Δ)ˢ, "•" $2s< n< 4s$ and λ> 0 is sufficiently large. In this way we extend completely the famous result of Brezis and Nirenberg (see {4}, {5}, {9}, {23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.