Multiple positive solutions of an elliptic equation with a convex–concave nonlinearity containing a sign-changing term

We study the existence of multiple positive solutions to a nonlinear Dirichlet problem for the $p$-Laplacian (in a bounded domain in $\mathbb R^N$) with a concave nonlinearity and with a nonlinear perturbation involving a function of the spatial variable whose sign can change the character of concavity. Under two different sets of conditions imposed on the perturbation, we prove the existence of two and three positive solutions, respectively.