On one-dimensional superlinear indefinite problems involving the $$\phi $$ ϕ -Laplacian

Abstract

Let $$\Omega := ( a,b ) \subset \mathbb {R}$$ , $$m\in L^{1} ( \Omega ) $$ and $$\phi :\mathbb {R\rightarrow R}$$ be an odd increasing homeomorphism. We consider the existence of positive solutions for problems of the form $$\begin{aligned} \left\{ \begin{array} [c]{ll} -\phi ( u^{\prime } ) ^{\prime }=m ( x ) f ( u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on } \partial \Omega , \end{array} \right. \end{aligned}$$ where $$f: [ 0,\infty ) \rightarrow [ 0,\infty ) $$ is a continuous function which is, roughly speaking, superlinear with respect to $$\phi $$ . Our approach combines the Guo-Krasnoselskiĭ fixed-point theorem with some estimates on related nonlinear problems. We mention that our results are new even in the case $$m\ge 0$$ .