Two solutions for an elliptic problem with two singular terms

We study the Dirichlet boundary value problem with 0-boundary data for the semilinear elliptic equation $−\Delta u = (\lambda u^{s−1}−u^{r−1})χ_{u>0}$ in a bounded domain ,where $0 < r < s < 1$ and $\lambda\in (0,\infty)$. In particular, for $\lambda$ large enough, we prove the existence of at least two nonnegative solutions, one of which is positive, satisfies the Hopf’s boundary condition and corresponds to a local minimum of the energy functional. This paper is motivated by a recent result of the authors where the same conclusion was obtained for the case $0 < r ≤ 1 < s < 2.$