Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions

Abstract

We prove the existence of positive solutions for the $p$-Laplacian problem $\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}$ where $\phi (s)=|s|^{p-2}s$, $p \gt 1$, $H_{i}:\mathbb{R}\rightarrow\mathbb{R}$ can be nonlinear, $i=1,2$, $f:(0,\infty)\rightarrow \mathbb{R}$ is $p$-superlinear or $p$-sublinear at $\infty$ and is allowed be singular $(\pm\infty)$ at $0$, and $\lambda$ is a positive parameter.