Superlinear elliptic systems with reaction terms involving product of powers

We consider a system of the form −Δu=λg1(x,u,v)inΩ;−Δv=λg2(x,u,v)inΩ;u=0=von∂Ω,where λ>0 is a parameter, Ω⊂RN(N≥2) is a bounded domain with sufficiently smooth boundary ∂Ω (a bounded open interval if N=1). Here gi(x,s,t):Ω×[0,+∞)×[0,+∞)→R (i=1,2) are Carathéodory functions that exhibit superlinear growth at infinity involving product of powers of u and v. Using re-scaling argument combined with Leray-Schauder degree theory and a version of Leray-Schauder continuation theorem, we show that the system has a connected set of positive solutions for λ small.