Compactly Supported Solutions near a Nehari Critical Value

We study non-negative solutions of the indefinite sublinear Dirichlet problem $$\{\begin{array}{ll}-\mathrm{\Delta }u=\lambda u+m(x)\mathrm{\mid }u{\mathrm{\mid }}^{\alpha -1}u& \text{ in }\mathrm{\Omega },\\ u=0& \text{ on }\mathrm{\partial }\mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $0<\alpha <1$, $m\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ is a sign-changing or non-positive weight, and $\lambda \in \mathbb{R}$.