Existence of nonnegative solutions for a class of semilinear elliptic systems with indefinite weight

Abstract

We prove the existence of nonnegative solutions to the system − Δ u = λ a ( x ) f ( v ) in Ω , − Δ v = λ b ( x ) g ( u ) in Ω , u = 0 = v on ∂ Ω , where Ω is a bounded domain in R N with a smooth boundary ∂ Ω , λ is a positive parameter. By the method of monotone iteration and Schauder fixed point theorem, we prove the existence of a nonnegative solution to the above system for λ sufficiently small. In this study, we do not restrict the sign of a and b .