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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2411.03703 |
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- 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 $Ω\subset {\bf R}^n$ ($n\geq 3$) be a smooth bounded domain and let $1<q<2\leq p<{{2n}\over {n-2}}$.Then, for every $r, ν>0$, there exists $λ^*>0$ with the following property: for every $λ\in ]0,λ^*[$, $μ\in ]-λ^*,λ^*[$, and for every convex dense set $S\subset H^{-1}(Ω)$, there exists $\tildeφ\in S$, with $\|\tildeφ\|_{H^{-1}(Ω)}<r$, 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 $H^1_0(Ω)$ are less than or equal to $r$.