A class of equations with three solutions

Abstract

Here is one of the results obtained in this paper: Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain, let $q>1$, with $q<{{n+2}\over {n-2}}$ if $n\geq 3$ and let $\lambda_1$ be the first eigenvalue of the problem $$\cases{-\Delta u=\lambda u & in $\Omega$ \cr & \cr u=0 & on $\partial\Omega$\ .\cr}$$ 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 $$\cases{-\Delta u=\lambda(u^+-(u^+)^q)+\alpha(x) & in $\Omega$ \cr & \cr u=0 & on $\partial\Omega$\cr}$$ has at least three weak solutions, two of which are global minima in $H^1_0(\Omega)$ of the functional $$u\to {{1}\over {2}}\int_{\Omega}|\nabla u(x)|^2dx-\lambda\int_{\Omega}\left ({{1}\over {2}}|u^+(x)|^2-{{1}\over {q+1}}|u^+(x)|^{q+1}\right )dx-\int_{\Omega}\alpha(x)u(x)dx\ $$ where $u^+=\max\{u,0\}$.