Positive solutions for a p−q‐Laplacian system with critical nonlinearities. (7th August 2017) - Yin, Honghui; Han, Yefei
Abstract : 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 − p is the critical Sobolev exponent and △ s ·=div(|∇·| s −2 ∇·) is the s‐Laplacian operator. The main results are obtained by variational methods.