Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents

Abstract

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents: { − Δ u = | u | 2 ∗ − 2 u + α 2 ∗ | u | α − 2 | v | β u + λ | u | q − 2 u ,     in   Ω , − Δ v = | v | 2 ∗ − 2 v + β 2 ∗ | u | α | v | β − 2 v + μ | v | q − 2 v ,     in   Ω , u ,   v > 0 ,     in   Ω , u = v = 0 ,     on   ∂ Ω , where Ω ⊂ R N ( N ≥ 3 ) is a bounded domain with smooth boundary,~ λ ,   μ > 0 ,   1 < q < 2 ,   α > 1 ,   β > 1 ,   α + β = 2 ∗ = 2 N N − 2 . By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.