New criteria for the existence of non-trivial fixed points in cones

Abstract We give new criteria for the existence of nontrivial fixed points on cones assuming some monotonicity of the operator on a suitable conical shell. Moreover, we give an application to the existence of multiple solutions for a nonlocal boundary value problem that models the displacement of a beam subject to some feedback controllers. MSC:47H10, 34B10, 34B18

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

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 $0\leq c\in L^{\infty}\left(\Omega\right)$. We study existence of strictly positive solutions for elliptic problems of the form $-\left(\left\| u^{\prime}\right\|^{p-2}u^{\prime}\right) ^{\prime}+c\left(x\right) u^{p-1}=m\left(x\right) u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$. We mention that our results are new even in the case $c\equiv0$.Comment: To appear in Bulletin of the Australian Mathematical Societ