Blowing-up Solutions for 2nd-Order Critical Elliptic Equations: The Impact of the Scalar Curvature
Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M...
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| Published in: | International mathematics research notices Vol. 2023; no. 2; pp. 901 - 931 |
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| Abstract | Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$. |
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| AbstractList | Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation$$\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0\hbox{ in }M$$is that $h_0\in C^1(M)$ touches the Scalar curvature somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet's condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in\{4,5\}$, our proof requires the introduction of a suitable mass that is defined only where Druet's condition holds. This mass carries global information both on $h_0$ and $(M,g)$. Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to perturbations of the equation $$ \begin{align*} &\Delta_gu+h_0u=u^{\frac{n+2}{n-2}},\ u>0 \ \textrm{in }M\end{align*}$$is that $h_0\in C^1(M)$ touches the Yamabe potential somewhere when $n\geq 4$ (the condition is different for $n=6$). In this paper, we prove that Druet’s condition is also sufficient provided we add its natural differentiable version. For $n\geq 6$, our arguments are local. For the low dimensions $n\in \{4,5\}$, our proof requires to introduce a suitable mass that is defined only where Druet’s condition holds. This mass carries global information both on $h_0$ and $(M,g)$. |
| Author | Vétois, Jérôme Robert, Frédéric |
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| Cites_doi | 10.1155/S1073792804133278 10.4171/134 10.1090/S0273-0979-1987-15514-5 10.1007/978-3-0348-0373-1_6 10.1007/s00208-013-0971-9 10.4310/jdg/1090426771 10.4310/jdg/1231856261 10.1007/s00526-018-1433-8 10.1007/s11854-019-0008-8 10.1080/03605302.2012.745552 10.1007/s11118-016-9608-4 10.1007/s00526-014-0802-1 10.1016/j.jde.2011.09.032 10.1142/S021919979900002X 10.4310/jdg/1406552253 10.21711/217504322010/em191 10.1090/S0002-9947-04-03681-5 10.1090/S0894-0347-07-00575-9 10.4310/MRL.2012.v19.n4.a18 10.4310/jdg/1228400630 10.4310/jdg/1143651772 10.1016/j.jfa.2006.11.010 |
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| References | Schoen (2023012014102883600_ref27) 1988 Marques (2023012014102883600_ref18) 2005; 71 Brendle (2023012014102883600_ref2) 2009; 81 Druet (2023012014102883600_ref8) 2005; 357 Premoselli (2023012014102883600_ref20) 2018; 57 Chen (2023012014102883600_ref3) 2012; 252 Druet (2023012014102883600_ref6) 2004; 23 Li (2023012014102883600_ref17) 1999; 1 Robert (2023012014102883600_ref23) 2013 Druet (2023012014102883600_ref5) 2003; 63 Robert (2023012014102883600_ref26) 2019 Lee (2023012014102883600_ref14) 1987; 17 Khuri (2023012014102883600_ref13) 2009; 81 Druet (2023012014102883600_ref9) 2004 Hebey (2023012014102883600_ref11) 2014 Vétois (2023012014102883600_ref29) 2019; 1 Pistoia (2023012014102883600_ref21) 2019 Robert (2023012014102883600_ref24) 2014; 98 Brendle (2023012014102883600_ref1) 2008; 21 Schoen (2023012014102883600_ref28) 1991 Morabito (2023012014102883600_ref19) 2017; 47 Druet (2023012014102883600_ref7) 2010 Esposito (2023012014102883600_ref10) 2014; 358 Robert (2023012014102883600_ref25) 2015; 54 Robert (2023012014102883600_ref22) 2013; 38 del Pino (2023012014102883600_ref4) 2019; 137 Li (2023012014102883600_ref15) 2005; 24 Hebey (2023012014102883600_ref12) 2012; 19 Li (2023012014102883600_ref16) 2007; 245 |
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| Snippet | Given a closed manifold $(M^n,g)$, $n\geq 3$, Druet [5, 7] proved that a necessary condition for the existence of energy-bounded blowing-up solutions to... Given a closed manifold $(M^n,g)$, $n\geq 3$, Olivier Druet proved that a necessary condition for the existence of energy-bounded blowing-up solutions to... |
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