An existence result for a class of Kirchhoff type systems via sub and supersolutions method

This paper deals with the existence of positive solutions for the following Kirchhoff type systems { − M 1 ( ∫ Ω | ∇ u | p d x ) Δ p u = λ a ( x ) f ( u , v ) in  Ω , − M 2 ( ∫ Ω | ∇ v | q d x ) Δ q v = λ b ( x ) g ( u , v ) in  Ω , u = v = 0 on  ∂ Ω , where Ω is a bounded smooth domain of R N , p , q > 1 , M i : R 0 + → R + , i = 1 , 2 are two continuous and increasing functions, λ is a positive parameter, and a , b ∈ C ( Ω ¯ ) . We discuss the existence of a large positive solution for λ large when lim t → ∞ f ( t , M [ g ( t , t ) ] 1 q − 1 ) t p − 1 = 0 for every M > 0 , and lim t → ∞ g ( t , t ) t q − 1 = 0 . In particular, we do not assume any sign conditions on f ( 0 , 0 ) or g ( 0 , 0 ) .