An existence result for superlinear semipositone p-Laplacian systems on the exterior of a ball

We study the existence of positive radial solutions to the problem {equation*} \{ {aligned} -Δ_p u &= λ K_1(|x|) f(v) {.3in}in Ω_e,\\ -Δ_p v &= λ K_2(|x|) g(u) {.31in}in Ω_e, \ &= v=0 {.7in} if |x|=r_0, \(x)&→ 0,v(x)→ 0 {.4in} as |x |→∞, {aligned} . {equation*} where Δₚ w:=div(|∇ w|ᵖ⁻²∇ w), $1 < p < n$, λ is a positive parameter, r₀>0 and Ωₑ:=ⁿ|~|x|>r₀\. Here, Kᵢ:[r₀,∞)→ (0,∞), $i=1,2$ are continuous functions such that limr → ∞ Kᵢ(r)=0, and f, g:[0,∞)→ R are continuous functions which are negative at the origin and have a superlinear growth at infinity. We establish the existence of a positive radial solution for small values of λ via degree theory and rescaling arguments.