Positive solutions for the one-dimensional Sturm-Liouville superlinear p-Laplacian problem

Abstract

We prove the existence of positive classical solutions for the p-Laplacian problem $$\displaylines{ -(r(t)\phi (u'))'=f(t,u),\quad t\in (0,1), \cr au(0)-b\phi ^{-1}(r(0))u'(0)=0,\ cu(1)+d\phi ^{-1}(r(1))u'(1)=0, }$$ where $\phi (s)=|s|^{p-2}s$, $p>1$, $f:(0,1)\times [ 0,\infty )\to\mathbb{R}$ is a Caratheodory function satisfying $$ \limsup_{z\to 0^{+}} \frac{f(t,z)}{z^{p-1}}<\lambda_1 <\liminf_{z\to \infty }\frac{f(t,z)}{z^{p-1}} $$ uniformly for a.e. $t \in (0,1)$, where $\lambda _1$ denotes the principal eigenvalue of $-(r(t)\phi (u'))'$ with Sturm-Liouville boundary conditions. Our result extends a previous work by Manasevich, Njoku, and Zanolin to the Sturm-Liouville boundary conditions with more general operator.

Keywords

• p-Laplacian
• superlinear
• positive solutions