A note on sign-changing solutions to supercritical Yamabe-type equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866917570045542400 |
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| author | Julio-Batalla, Jurgen |
| author_facet | Julio-Batalla, Jurgen |
| contents | On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-Δ_g u + λu = λ|u|^{q-1}u$, where $λ\in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$ with focal submanifolds of positive dimension. If $k>0$ is the minimum of the dimensions of the focal submanifolds of $f$, we let $q^* =\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant sign-changing solutions to the equation when $1<q<q^*$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on sign-changing solutions to supercritical Yamabe-type equations Julio-Batalla, Jurgen Analysis of PDEs Differential Geometry On a closed Riemannian manifold $(M^n ,g)$, we consider the Yamabe-type equation $-Δ_g u + λu = λ|u|^{q-1}u$, where $λ\in \mathbb{R}_{+}$ and $q>1$. We assume that $M$ admits a proper isoparametric function $f$ with focal submanifolds of positive dimension. If $k>0$ is the minimum of the dimensions of the focal submanifolds of $f$, we let $q^* =\frac{n-k+2}{n-k-2}$. We prove the existence of infinite $f$-invariant sign-changing solutions to the equation when $1<q<q^*$. |
| title | A note on sign-changing solutions to supercritical Yamabe-type equations |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2401.09612 |