Positive solutions for a p − q ‐Laplacian system with critical nonlinearities

In this paper, our main purpose is to establish the existence results of positive solutions for a p − q ‐Laplacian system involving concave‐convex nonlinearities:− △ p u − △ q u = λ V ( x ) | u | r − 2 u + 2 α α + β | u | α − 2 u | v | β , x ∈ Ω ,− △ p v − △ q v = θ V ( x ) | v | r − 2 v + 2 β α + β | u | α | v | β − 2 v , x ∈ Ω ,u = v = 0 , x ∈ ∂ Ω ,where Ω is a bounded domain in R N , λ , θ >0 and 1< r < q < p < N . We assume 1< α , β and α + β = p ∗ = N p N − pis the critical Sobolev exponent and △ s ·=div(|∇·| s −2 ∇·) is the s‐Laplacian operator. The main results are obtained by variational methods.