Positive solutions for concave-convex type problems for the one-dimensional $\varphi$-Laplacian

Abstract

Let $\Omega =(a,b)\subset \mathbb{R}$, $0\le m,n\in L^1(\Omega )$, $\lambda ,\mu >0$ be real parameters, and $\varphi :\mathbb{R}\to \mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -ϕ\left( u^{\prime}\right) ^{\prime}=λm(x)f(u)+μn(x)g(u) & \text{ in }Ω,\\ u=0 & \text{ on }\partialΩ, \end{cases} \] where $f,g:[0,\infty )\to [0,\infty )$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\varphi$, respectively. Our assumptions on $\varphi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\uı\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.

14 pages