Strictly positive solutions for one-dimensional nonlinear problems involving the p-Laplacian

Abstract

Let $\Omega$ be a bounded open interval, and let $p>1$ and $q\in \left(0,p-1\right)$. Let $m\in L^{p^{\prime }}\left(\Omega \right)$ and $0\le c\in L^{\infty }\left(\Omega \right)$. We study existence of strictly positive solutions for elliptic problems of the form $-{\left({\Vert u^{\prime }\Vert }^{p-2}{u}^{\prime }\right)}^{\prime }+c\left(x\right)u^{p-1}=m\left(x\right)u^q$ in $\Omega$, $u=0$ on $\partial \Omega$. We mention that our results are new even in the case $c\equiv 0$.

To appear in Bulletin of the Australian Mathematical Society