Positive solutions of the second-order differential systems in reactor dynamics

In this paper, we are concerned with the existence of positive solutions of the second-order cooperative system-u″=-λu+φu+g(t)f(u),t∈(0,1),-φ″=μu,t∈(0,1),u(0)=u(1)=0,φ(0)=φ(1)=0,where λ>-π2 is a constant, μ>0 is a parameter, g:[0,1]→[0,∞) is continuous and g≢0 on any subinterval of [0,1],f:[0,∞)→[0,∞) is continuous and f(s)>0 for s>0. Under some suitable conditions on the nonlinearity f, we show that above system has at least one positive solution for any μ∈(0,∞). The proof of our main results is based upon bifurcation techniques.