Strictly positive solutions for one-dimensional nonlinear elliptic problems

We study existence and nonexistence of strictly positive solutions for the elliptic problems of the form $Lu=m\left( x\right) u^{p}$ in a bounded open interval, with zero boundary conditions, where $L$ is a strongly uniformly elliptic differential operator, $p\in\left( 0,1\right) $, and $m$ is a function that changes sign. We also characterize the set of values $p$ for which the problem admits a solution, and in addition an existence result for other nonlinearities is presented.