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In this paper, we consider the following Brézis-Nirenberg problem involving the fractional Laplacian operator:where s ∈ (0, 1), Ω is a bounded smooth domain of R N (N > 6s) and 2 * s = 2NN −2s 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.MSC 2010 : Primary 35J60; Secondary 47J30, 58E05

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In this paper, we consider the following nonlinear Kirchhoff type problem:where a, b > 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u| p−2 u with p ∈ (2, 4].