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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.14719 |
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| _version_ | 1866911686405914624 |
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| author | Li, Jungang Wang, Zhiwei |
| author_facet | Li, Jungang Wang, Zhiwei |
| contents | We study the higher-order Schrödinger equation with critical Sobolev exponent on the hyperbolic space $\mathbb{H}^n$: $$P_m u + a(x)\,u = |u|^{q-2}u, \quad u \in D^{m,2}(\mathbb{H}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q = \frac{2n}{n-2m}$ is the critical exponent, and $a(x) \geq 0$ is a potential in $L^{n/2m}(\mathbb{H}^n)$. This problem simultaneously generalizes the classical work of Benci--Cerami from second-order to arbitrary order and from Euclidean space to hyperbolic space.
We establish a global compactness theorem (profile decomposition) for Palais--Smale sequences associated to this equation. The decomposition features two types of bubbles: concentrating bubbles arising from the conformal equivalence $\mathbb{H}^n \cong \mathbb{B}^n$, and isometry bubbles escaping to infinity. A key difficulty in the higher-order setting is that the classical positive/negative decomposition $u = u^+ + u^-$ fails in $W^{m,2}$ for $m \geq 2$. To overcome this, we employ the Moreau dual cone decomposition together with the positivity of the Green function of $P_m$ on $\mathbb{H}^n$, establishing an energy doubling inequality for sign-changing solutions: $I_\infty(u) \geq \frac{2m}{n}S^{n/2m}$.
As an application, under a concentration condition on the potential $a(x)$ of Passaseo type, we prove that the equation admits at least one positive solution, and a second positive solution under a smallness condition on $\|a\|_{L^{n/2m}}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_14719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global Compactness and Existence for Higher Order Critical Equations on Hyperbolic Spaces Li, Jungang Wang, Zhiwei Analysis of PDEs 35J91, 35B33, 58J05, 35A15 We study the higher-order Schrödinger equation with critical Sobolev exponent on the hyperbolic space $\mathbb{H}^n$: $$P_m u + a(x)\,u = |u|^{q-2}u, \quad u \in D^{m,2}(\mathbb{H}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q = \frac{2n}{n-2m}$ is the critical exponent, and $a(x) \geq 0$ is a potential in $L^{n/2m}(\mathbb{H}^n)$. This problem simultaneously generalizes the classical work of Benci--Cerami from second-order to arbitrary order and from Euclidean space to hyperbolic space. We establish a global compactness theorem (profile decomposition) for Palais--Smale sequences associated to this equation. The decomposition features two types of bubbles: concentrating bubbles arising from the conformal equivalence $\mathbb{H}^n \cong \mathbb{B}^n$, and isometry bubbles escaping to infinity. A key difficulty in the higher-order setting is that the classical positive/negative decomposition $u = u^+ + u^-$ fails in $W^{m,2}$ for $m \geq 2$. To overcome this, we employ the Moreau dual cone decomposition together with the positivity of the Green function of $P_m$ on $\mathbb{H}^n$, establishing an energy doubling inequality for sign-changing solutions: $I_\infty(u) \geq \frac{2m}{n}S^{n/2m}$. As an application, under a concentration condition on the potential $a(x)$ of Passaseo type, we prove that the equation admits at least one positive solution, and a second positive solution under a smallness condition on $\|a\|_{L^{n/2m}}$. |
| title | Global Compactness and Existence for Higher Order Critical Equations on Hyperbolic Spaces |
| topic | Analysis of PDEs 35J91, 35B33, 58J05, 35A15 |
| url | https://arxiv.org/abs/2411.14719 |