Multiple critical points in closed sets via minimax theorems
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929673355657216 |
|---|---|
| author | Ricceri, Biagio |
| author_facet | Ricceri, Biagio |
| contents | 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 $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03703 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multiple critical points in closed sets via minimax theorems Ricceri, Biagio Analysis of PDEs 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 $r$. |
| title | Multiple critical points in closed sets via minimax theorems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.03703 |