Sublinear elliptic equations with a sharp change of sign in the nonlinearity

We study the semilinear indefinite elliptic problem $$-\mathrm{\Delta }u=Q_{\mathrm{\Omega }}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u\text{in }{\mathbb{R}}^N,$$ where $Q_{\mathrm{\Omega }}={\chi }_{\mathrm{\Omega }}-{\chi }_{{\mathbb{R}}^N\setminus \mathrm{\Omega }}$, $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth subset, $N\ge 3$, and $1\le p<2$, with $p=1$ corresponding to the sign nonlinearity. Using a variational approach, we investigate the uniqueness or multiplicity of nonnegative solutions depending on the shape of $\mathrm{\Omega }$ and the existence of different types of nodal solutions. We also show that all solutions have compact support and analyze how the support of the ground state depends on $p$, proving convergence to the whole space as $p\to 2^{-}$ and identifying some qualitative features such as starshapedness and Lipschitz regularity of the support. We also establish a link between these problems and a two-phase Serrin-type torsion overdetermined problem.