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

TL;DR

This paper proves that for large positive parameters, there are no positive radial solutions to certain semipositone $ ext{phi}$-Laplacian boundary value problems, using energy methods and indirect arguments.

Contribution

It establishes the nonexistence of positive radial solutions for a class of $ ext{phi}$-Laplacian problems with superlinear reaction terms, extending previous results to more general operators.

Findings

No positive radial solutions for large $ ext{lambda}$

Energy analysis confirms nonexistence

Results apply to a broad class of $ ext{phi}$-Laplacian operators

Abstract

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