Existence results for classes of Laplacian systems with sign-changing weight

Abstract Consider the system { − Δ u = λ F ( x , u , v ) , in  Ω , − Δ v = λ H ( x , u , v ) , in  Ω , u = 0 = v , on  ∂ Ω , where F ( x , u , v ) = [ g ( x ) a ( u ) + f ( v ) ] , H ( x , u , v ) = [ g ( x ) b ( v ) + h ( u ) ] , λ > 0 is a parameter, Ω is a bounded domain in R N ; N ≥ 1 , with smooth boundary ∂ Ω and Δ is the Laplacian operator. Here g is a C 1 sign-changing function that may be negative near the boundary and f , h , a , b are C 1 nondecreasing functions satisfying a ( 0 ) ≥ 0 , b ( 0 ) ≥ 0 , lim s → ∞ a ( s ) s = 0 , lim s → ∞ b ( s ) s = 0 , lim s → ∞ f ( s ) = ∞ , lim s → ∞ h ( s ) = ∞ and lim s → ∞ f ( M h ( s ) ) s = 0 , ∀ M > 0 . We discuss the existence of positive solutions when f , h , a , b and g satisfy certain additional conditions. We employ the method of sub–super-solutions to obtain our results. Note that we do not require any sign-changing conditions on f ( 0 ) or h ( 0 ) . We also note that while a and b are assumed to be sublinear at ∞ , we only assume a combined sublinear effect of f and h at ∞ .