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

Abstract

The aim of this paper is to prove the nonexistence of positive radial solutions to the problem $-{\Delta }_{\varphi }u=\lambda f(u)$, $x\in B_1(0)$, $u(x)=0$ on $|x|=1$, for $\lambda >0$ sufficiently large. Here, $\varphi$ is a continuous function, ${\Delta }_{\varphi }$ denotes the $\varphi$-Laplacian operator which is defined by ${\Delta }_{\varphi }(u):=div(\varphi (|\nabla u|)\nabla u)$, 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 $\varphi$. Our findings are presented in a combined format, employing both an indirect argument and an energy analysis.