Positive solutions for a p−q‐Laplacian system with critical nonlinearities. (7th August 2017)

• Record Type: Journal Article
• Authors: Yin, Honghui; Han, Yefei
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 18(2018)
• Page Start: 7110
• Page End: 7124
• Publication Date: 2017-08-07
• Subjects: critical exponent -- critical nonlinearities -- positive solutions -- p−q‐Laplacian; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4516
• Languages: English
• ISSNs: 0170-4214

Abstract

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.