Multiple critical points in closed sets via minimax theorems

Abstract

In this paper, we apply our minimax theory ([4], [5], [6]) with the one developed by A. Moameni in [2] to formalize a general scheme giving the multiplicity of critical points. Here is a sample of application of the scheme to a critical elliptic problem: Let $\Omega\subset {\bf R}^n$ ($n\geq 3$) be a smooth bounded domain and let $1<q<2\leq p<{{2n}\over {n-2}}$.Then, for every $r, \nu>0$, there exists $\lambda^*>0$ with the following property: for every $\lambda\in ]0,\lambda^*[$, $\mu\in ]-\lambda^*,\lambda^*[$, and for every convex dense set $S\subset H^{-1}(\Omega)$, there exists $\tilde\varphi\in S$, with $\|\tilde\varphi\|_{H^{-1}(\Omega)}<r$, such that the problem $$\cases{-\Delta u=\lambda(|u|^{{{4}\over {n-2}}}u+\nu |u|^{q-2}u+\mu|u|^{p-2}u+\tilde\varphi) & in $\Omega$\cr & \cr u=0 & on $\partial\Omega$\cr}$$ has at least two solutions whose norms in $H^1_0(\Omega)$ are less than or equal to $r$.