A class of equations with three solutions

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$beasmoothboundeddomain,let$q>1$,with$q<{{n+2}\over {n-2}}$if$n\geq 3$andlet$\lambda_1$bethefirsteigenvalueoftheproblem$\cases{-\Delta u=\lambda u & in\Omega\cr & \cr u=0 & on\partial\Omega$
.\cr}Then, for every $\lambda>\lambda_1$ and for every convex set $S\subseteq L^{\infty}(\Omega)$ dense in $L^2(\Omega)$, there exists $\alpha\in S$ such that the problem\cases{-\Delta u=\lambda(u^±(u^+)^q)+\alpha(x) & in $\mathrm{\Omega }$ \cr & \cr u=0 & on $\mathrm{\partial }\mathrm{\Omega }$\cr}has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)$ of the functionalu\to {{1}\over {2}}\int_{\Omega}|\nabla u(x)|^2dx-\lambda\int_{\Omega}\left ({{1}\over {2}}|u^+(x)|^2-{{1}\over {q+1}}|u^+(x)|^{q+1}\right )dx-\int_{\Omega}\alpha(x)u(x)dx\ $$where$u^+=\max{u,0}$.