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 $\hat{I}\text{c◯}\subset {\mathbf{R}}^n$ ($n\ge 3$) be a smooth bounded domain and let $1<q<2\le p<\frac{2n}{n-2}$.Then, for every $r,\hat{I}½>0$, there exists $\hat{I}{»}^{\ast }>0$ with the following property: for every $\hat{I}»\in ]0,\hat{I}{»}^{\ast }[$, $\hat{I}¼\in ]-\hat{I}{»}^{\ast },\hat{I}{»}^{\ast }[$, and for every convex dense set $S\subset H^{-1}(\hat{I}\text{c◯})$, there exists $\tilde{\ddot{I}}†\in S$, with $\Vert \tilde{\ddot{I}}†{\Vert }_{H^{-1}(\hat{I}\text{c◯})}<r$, such that the problem $$\cases{-Δu=λ(|u|^{{{4}\over {n-2}}}u+ν|u|^{q-2}u+μ|u|^{p-2}u+\tildeφ) & in $Ω$\cr & \cr u=0 & on $\partialΩ$\cr}$$ has at least two solutions whose norms in $H_0^1(\hat{I}\text{c◯})$ are less than or equal to $r$.

Accepted in Optimization