Nonexistence of positive radial solutions for semipositone $ϕ$-Laplacian problems with superlinear reaction term

The aim of this paper is to prove the nonexistence of positive radial solutions to the problem -Δ_ϕu=λf(u), x∈ B₁(0), $u(x)=0$ on $|x|=1$, for $λ>0$ sufficiently large. Here, $ϕ$ is a continuous function, Δ_ϕ denotes the $ϕ$-Laplacian operator which is defined by Δ_ϕ(u):=div (ϕ(|∇ u|) ∇ u), and B₁(0) is the unit ball in RN, with $N>1$. Furthermore, f is a continuous, nondecreasing function such that $f(0)<0$, and its behavior at infinity is intimately related to $ϕ$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis.