Positive solutions for nonlinear problems involving the one-dimensional {\phi}-Laplacian

Abstract

Let $\Omega:=\left( a,b\right) \subset\mathbb{R}$, $m\in L^{1}\left( \Omega\right) $ and $\lambda>0$ be a real parameter. Let $\mathcal{L}$ be the differential operator given by $\mathcal{L}u:=-\phi\left( u^{\prime}\right) ^{\prime}+r\left( x\right) \phi\left( u\right) $, where $\phi :\mathbb{R\rightarrow R}$ is an odd increasing homeomorphism and $0\leq r\in L^{1}\left( \Omega\right) $. We study the existence of positive solutions for problems of the form $\mathcal{L}u=\lambda m\left( x\right) f\left( u\right)$ in $\Omega,$ $u=0$ on $\partial\Omega$, where $f:\left[ 0,\infty\right) \rightarrow\left[ 0,\infty\right) $ is a continuos function which is, roughly speaking, sublinear with respect to $\phi$. Our approach combines the sub and supersolution method with some estimates on related nonlinear problems. We point out that our results are new even in the cases $r\equiv0$ and/or $m\geq0$.