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

The aim of this paper is to prove the nonexistence of positive radial solutions to the problem $-{\mathrm{\Delta }}_{\phi }u=\lambda f(u)$, $x\in B_1(0)$, $u(x)=0$ on $\mathrm{\mid }x\mathrm{\mid }=1$, for $\lambda >0$ sufficiently large. Here, $\phi$ is a continuous function, ${\mathrm{\Delta }}_{\phi }$ denotes the $\phi$-Laplacian operator which is defined by ${\mathrm{\Delta }}_{\phi }(u):=div(\phi (\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })\mathrm{\nabla }u)$, and $B_1(0)$ is the unit ball in ${\mathbb{R}}^N$, with $N>1$. Furthermore, $f$ is a continuous, nondecreasing function such that $f(0)<0$, and its behavior at infinity is intimately related to $\phi$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis.