Multiple critical points in closed sets via minimax theorems

TL;DR

This paper develops a minimax framework to establish the existence of multiple critical points in closed sets, with applications to nonlinear elliptic boundary value problems demonstrating the existence of multiple solutions.

Contribution

It introduces a general scheme based on minimax theorems for proving multiplicity of critical points, extending previous theories with new applications to elliptic PDEs.

Findings

Multiple solutions exist for certain nonlinear elliptic boundary value problems.

The scheme guarantees at least two solutions under specified conditions.

The approach applies to convex dense sets in function spaces.

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 $r$.