Stability and instability results for sign-changing solutions to second-order critical elliptic equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911781600886784 |
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| author | Premoselli, Bruno Vétois, Jérôme |
| author_facet | Premoselli, Bruno Vétois, Jérôme |
| contents | On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schrödinger equation $Δ_gu+h_0u=\left|u\right|^{2^*-2}u$, where $Δ_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^* :=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge7$ and $h_0\ne\frac{n-2}{4\left(n-1\right)}\text{Scal}_g$ at all points in $M$, where $\text{Scal}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge3$ showing the optimality of these assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_05679 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Stability and instability results for sign-changing solutions to second-order critical elliptic equations Premoselli, Bruno Vétois, Jérôme Analysis of PDEs On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\ge3$, we consider the stationary Schrödinger equation $Δ_gu+h_0u=\left|u\right|^{2^*-2}u$, where $Δ_g:=-\text{div}_g\nabla$, $h_0\in C^1\left(M\right)$ and $2^* :=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential function $h_0$ in $C^1\left(M\right)$, the sets of sign-changing solutions that are bounded in $H^1\left(M\right)$ are precompact in the $C^2$ topology. We obtain this result under the assumptions that $\left(M,g\right)$ is locally conformally flat, $n\ge7$ and $h_0\ne\frac{n-2}{4\left(n-1\right)}\text{Scal}_g$ at all points in $M$, where $\text{Scal}_g$ is the scalar curvature of the manifold. We then provide counterexamples in every dimension $n\ge3$ showing the optimality of these assumptions. |
| title | Stability and instability results for sign-changing solutions to second-order critical elliptic equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2201.05679 |