Multiple critical points in closed sets via minimax theorems

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 $\mathrm{\Omega }\subset {\mathbf{R}}^n$ ($n\ge 3$) be a smooth bounded domain and let $1<q<2\le p<\frac{2n}{n-2}$.Then, for every $r,\nu >0$, there exists ${\lambda }^{\ast }>0$ with the following property: for every $\lambda \in ]0,{\lambda }^{\ast }[$, $\mu \in ]-{\lambda }^{\ast },{\lambda }^{\ast }[$, and for every convex dense set $S\subset H^{-1}(\mathrm{\Omega })$, there exists $\tilde{\phi }\in S$, with $\mathrm{\parallel }\tilde{\phi }{\mathrm{\parallel }}_{H^{-1}(\mathrm{\Omega })}<r$, such that the problem \cases{-Δu=λ(|u|^{{{4}\over {n-2}}}u+ν|u|^{q-2}u+μ|u|^{p-2}u+\tilde\varphi) & in $Ω$\cr & \cr u=0 & on $\partialΩ$\cr} has at least two solutions whose norms in $H_0^1(\mathrm{\Omega })$ are less than or equal to $r$.