We consider the problem $$\begin{aligned} (P_\lambda )\quad -\Delta _{p}u=\lambda u^{p-1}+a(x)u^{q-1},\quad u\ge 0\quad \text{ in } \Omega \end{aligned}$$ under Dirichlet or Neumann boundary conditions. Here $$\Omega $$ is a smooth bounded domain of $${\mathbb {R}}^{N}$$ ( $$N\ge 1$$ ), $$\lambda \in {\mathbb {R}}$$ , $$1<q<p$$ and $$a\in C({\overline{\Omega }})$$ changes sign. These conditions...