Multiple positive solutions for semilinear elliptic systems with nonlinear boundary condition

Abstract

In this paper, we study the nonlinear boundary value problem - Δ u + u = f u ( u , v ) in Ω , - Δ v + v = f v ( u , v ) in Ω , ∂ u ∂ n = g u ( u , v ) , ∂ v ∂ n = g v ( u , v ) on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary and the pairs of ( f u , f v ) and ( g u , g v ) are subcritical nonlinearities. With the help of the Nehari manifold, we show existence of at least two positive solutions when one of the pairs of ( f u , f v ) and ( g u , g v ) is sublinear and the other one superlinear.