Global structure of positive solutions for a singular quasilinear elliptic problem

We are concerned with the existence of positive solutions for the quasilinear problem (P) −Δpu=λK (|x|) f (u), x∈RN∖B (r0), g~1 (u) u−a1∇u⋅x|x|→0, |x|→∞, g~2 (u) u+a2∂u∂n=0, x∈∂B (r0), (P) where Δps=div (|∇s|p−2⋅∇s), 10 is a parameter, ai, r0>0 are constants for i = 1, 2, B (r0): ={x∈RN: |x|<r0, ∂u∂n is the outward normal derivative of u on ∂B (r0), K: (r0, ∞) → (0, ∞) is a continuous function, f: (0, ∞) →R is a continuous function which satisfies lims→∞f (s) /φp (s) =∞, g~i: [0, ∞) → (0, ∞) are continuous functions. We investigate the global structure of positive solution for (P). The proof of main result is based upon bifurcation theory.