On ground states and compactly supported solutions of elliptic equations with non-Lipschitz nonlinearities

In a bounded domain $\Omega\subset\mathbb{R}^N$, we consider the Dirichlet boundary-value problem for an elliptic equation with a non-Lipschitz nonlinearity of the form
\begin{equation*} \Delta u = \lambda u-|u|^{\alpha-1}u, \quad \lambda \in \mathbb{R}, \quad 0<\alpha<1. \end{equation*}
The problem of the existence of a solution of the ground-state-type with compact support is examined.