Positive Solutions of the One-Dimensional p -Laplacian with Nonlinearity Defined on a Finite Interval

Abstract

We use the quadrature method to show the existence and multiplicity of positive solutions of the boundary value problems involving one-dimensional $p$-Laplacian ${\left(u^{\prime }\left(t\right){|}^{p-2}{u}^{\prime }\left(t\right)\right)}^{\prime }+\lambda f\left(u\left(t\right)\right)=0$, $t\in \left(\mathrm{0,1}\right)$, $u(0)=u(1)=0$, where $p\in (\mathrm{1,2}]$, $\lambda \in (0,\mathrm{\infty })$ is a parameter, $f\in C^1([0,r),[0,\mathrm{\infty }))$ for some constant $r>0$, $f(s)>0$ in $(0,r)$, and ${\lim }_{s\to r^{-}}(r-s{)}^{p-\mathrm{1}}f(s)=+\infty$.