On singular $p$-Laplacian boundary value problems involving integral boundary conditions

Abstract

We prove the existence of positive solutions for the $p$-Laplacian equations \[-(\phi (u^{\prime }))^{\prime }=\lambda f(t,u),\qquad t\in (0,1) \] with integral boundary conditions. Here $\lambda $ is a positive parameter, $\phi (s)=|s|^{p-2}s,p>1,\ f:(0,1)\times (0,\infty )\rightarrow \mathbb{R\ }$ is $p$-superlinear or $p$-sublinear at $\infty $ and is allowed be singular at $t=0,1$ and $u=0.$

Keywords

• $p$-laplacian
• integral boundary conditions
• positive solutions