Nodal solutions to Paneitz-type equations
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909135116697600 |
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| author | Julio-Batalla, Jurgen Petean, Jimmy |
| author_facet | Julio-Batalla, Jurgen Petean, Jimmy |
| contents | On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $Δ^2 u -αΔu +βu = u^q$, where $α$ and $β$ are positive constants satisfying that $α^2 \geq 4 β$. We let ${\bf m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f = \frac{n-{\bf m}+4}{n-{\bf m}-4}$, $q_f = \infty$ if $n\leq {\bf m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_08146 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nodal solutions to Paneitz-type equations Julio-Batalla, Jurgen Petean, Jimmy Analysis of PDEs Differential Geometry On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $Δ^2 u -αΔu +βu = u^q$, where $α$ and $β$ are positive constants satisfying that $α^2 \geq 4 β$. We let ${\bf m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f = \frac{n-{\bf m}+4}{n-{\bf m}-4}$, $q_f = \infty$ if $n\leq {\bf m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy. |
| title | Nodal solutions to Paneitz-type equations |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2403.08146 |