Existence of solutions for critical fractional p&q-Laplacian system.

In this article, we are concerned with the existence of solutions for the following critical fractional p&q-Laplacian system (1) (− Δ) p s 1 u + (− Δ) q s 2 u = λ Q (x) | u | r − 2 u + 2 α α + β | u | α − 2 u | v | β , i n Ω ; (− Δ) p s 1 v + (− Δ) q s 2 v = μ Q (x) | v | r − 2 v + 2 β α + β | u | α | v | β − 2 v i n Ω ; u = v = 0 , i n R n ∖ Ω , where Ω is a smooth bounded set in R n , λ , μ > 0 are two parameters, 0 < s 2 < s 1 < 1 , 1 < q < p < r < p s 1 ∗ , n > p s 1 , and α > 1 , β > 1 satisfy α + β = p s 1 ∗ with p s 1 ∗ = n p n − p s 1 is the fractional Sobolev critical exponent, and (− △) t s is the fractional t-Laplacian operator. By using variational methods, we show that there exists a nontrivial weak solution for system (1). [ABSTRACT FROM AUTHOR]