Standing waves for a class of fractional p ‐Laplacian equations with a general critical nonlinearity

In this paper, we concern with the following fractional p ‐Laplacian equation with critical Sobolev exponentε p s− Δp s u + V ( x )up − 2u = λ f ( x )uq − 2 u +up s ∗ − 2 u in ℝ N ,u ∈ W s , pℝ N, u > 0 ,where ε  > 0 is a small parameter,   λ  > 0 , N is a positive integer, and N  >  p s with s  ∈ (0, 1) fixed, 1 < q ≤ p ,p s ∗ : = N p / N − p s . Since the nonlinearity h ( x , u ) : = λ f ( x )uq − 2 u +up s ∗ − 2 u does not satisfy the following Ambrosetti‐Rabinowitz condition:0 < μ H ( x , u ) : = μ ∫ 0 u h ( x , t ) d t ≤ h ( x , u ) u , x ∈ ℝ N , 0 ≠ u ∈ ℝ , with μ  >  p , it is difficult to obtain the boundedness of Palais‐Smale sequence, which is important to prove the existence of positive solutions. In order to overcome the above difficulty, we introduce a penalization method of fractional p ‐Laplacian type.