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

## TL;DR

This paper establishes the existence of positive solutions for a class of one-dimensional concave-convex problems involving the $\, ext{phi}$-Laplacian, using weaker assumptions and combining fixed-point and sub-supersolutions methods.

## Contribution

It introduces a new approach that weakens the conditions on $\, ext{phi}$, $m$, and $n$, expanding the class of problems where positive solutions can be guaranteed.

## Key findings

- Existence of positive solutions under weaker assumptions.
- Application of combined fixed-point and sub-supersolutions methods.
- Extension to $\, ext{phi}$-Laplacian problems with concave-convex nonlinearities.

## Abstract

Let $\Omega=(a,b)\subset\mathbb{R}$, $0\leq m,n\in L^{1}(\Omega)$, $\lambda,\mu>0$ be real parameters, and $\phi:\mathbb{R}\rightarrow\mathbb{R}$ be an odd increasing homeomorphism. In this paper we consider the existence of positive solutions for problems of the form \[ \begin{cases} -\phi\left( u^{\prime}\right) ^{\prime}=\lambda m(x)f(u)+\mu n(x)g(u) & \text{ in }\Omega,\\ u=0 & \text{ on }\partial\Omega, \end{cases} \] where $f,g:[0,\infty)\rightarrow\lbrack0,\infty)$ are continuous functions which are, roughly speaking, sublinear and superlinear with respect to $\phi$, respectively. Our assumptions on $\phi$, $m$ and $n$ are substantially weaker than the ones imposed in previous works. The approach used here combines the Guo-Krasnoselski\u{\i}\ fixed-point theorem and the sub-supersolutions method with some estimates on related nonlinear problems.