Positive solutions for nonlinear problems involving the one-dimensional ϕ-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 $\mathcal{L}u:=-\varphi {\left(u^{\prime }\right)}^{\prime }+r\left(x\right)\varphi \left(u\right)$, where $\varphi :\mathbb{R}\to \mathbb{R}$ is an odd increasing homeomorphism and $0\le r\in L^1\left(\Omega \right)$. 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 $f:\left[0,\infty \right)\to \left[0,\infty \right)$ is a continuos function which is, roughly speaking, sublinear with respect to $\varphi$. 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 $r\equiv 0$ and/or $m\ge 0$.

To appear in Journal of Mathematical Analysis and Applications