Global bifurcation of positive solutions for a class of superlinear elliptic systems

Abstract

We consider a system of semilinear equations of the form−Δu=λf(v)inΩ;−Δv=λg(u)inΩ;u=0=von∂Ω,} where λ∈R is the bifurcation parameter, Ω⊂RN; N≥2 is a bounded domain with smooth boundary ∂Ω. The nonlinearities f,g:R→(0,+∞) are nondecreasing continuous functions that have superlinear growth at infinity. We use bifurcation theory, combined with an approximation scheme, to establish the existence of an unbounded branch of positive solutions, emanating from the origin, which is bounded in positive λ-direction.If in addition, Ω is convex and f,g∈C1 satisfy certain subcriticality condition, we show that the branch must bifurcate from infinity at λ=0.