Local minimizers over the Nehari manifold for a class of concave-convex problems with sign changing nonlinearity

We study a $p$-Laplacian equation involving a parameter $\lambda$ and a concave-convex nonlinearity containing a weight which can change sign. By using the Nehari manifold and the fibering method, we show the existence of two positive solutions on some interval $(0,\lambda^*+\varepsilon)$, where $\lambda^*$ can be characterized variationally. We also study the asymptotic behavior of solutions when $\lambda\downarrow 0$.