LAUSR

Abstract In this paper, we consider the following Brézis-Nirenberg problem involving the fractional Laplacian operator: (−Δ)su=λu+|u|2s∗−2uinΩ,u=0on∂Ω,$$\begin{array}{} \displaystyle\left\{\begin{array}{ll} (-\Delta)^s u=\lambda u+|u|^{2_s^{*}-2}u & \textrm{in}\ \, \Omega, \\ u=0 & \textrm{on}\ \, \partial\Omega, \end{array} \right. \end{array} $$ where s ∈ (0, 1), Ω is a bounded smooth domain of ℝN (N > 6s) and 2s∗=2NN−2s$\begin{array}{} \displaystyle 2_s^{*}=\frac{2N}{N-2s} \end{array}$ is the critical fractional Sobolev exponent. We show that, for each λ > 0, this problem has infinitely many sign-changing solutions by using a compactness result obtained in [34] and a combination of invariant sets method and Ljusternik-Schnirelman type minimax method.