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Asymptotic analysis for fourth order Paneitz equations with critical growth. (English) Zbl 1251.58005

The fourth order Paneitz equations of critical growth is investigated in the case of $n$-dimensional closed conformally flat manifolds, $n\ge 5$. Such equations arise from conformal geometry, they are modelled on the Einstein case of a geometric equation describing the effects of conformal changes on the $Q$-curvature. Sharp asymptotics is obtained for arbitrary bounded energy sequences of solutions and stability and compactness are achieved, so establishing the criticality of the geometric equation with respect to the trace of its second order terms. For each positive numbers $b,c\in ℝ$, satisfying $c-\frac{{b}^{2}}{4}<0$, $\left(M,g\right)$ a smooth compact Riemannian manifold of dimension $n\ge 5$, the fourth order Paneitz equation of critical Sobolev growth is

${{\Delta }}_{g}^{2}u+b{{\Delta }}_{g}u+cu={u}^{{2}^{#}-1}\phantom{\rule{2.em}{0ex}}\left(1\right)$

where ${{\Delta }}_{g}$ is the Laplace-Beltrami operator and ${2}^{#}=\frac{2n}{n-4}$ is the critical Sobolev exponent. Let $R{c}_{g}$ and ${S}_{g}$ denote the Ricci and scalar curvature, respectively, of $g$ and consider ${A}_{g}$, the smooth $\left(2,0\right)$-tensor field given by

${A}_{g}=\frac{{\left(n-2\right)}^{2}+4}{2\left(n-1\right)\left(n-2\right)}{S}_{g}g-\frac{4}{n-2}R{c}_{g}·$

If $\stackrel{˜}{g}={u}^{4/\left(n-4\right)}g$ the conformal equation relating the $Q$-curvatures ${Q}_{g}$ and ${Q}_{\stackrel{˜}{g}}$ is

${{\Delta }}_{g}^{2}u-di{v}_{g}\left({A}_{g}du\right)+\frac{n-4}{2}{Q}_{g}u=\frac{n-4}{2}{Q}_{\stackrel{˜}{g}}{u}^{{2}^{#}-1}·$

When $g$ is Einstein, so that $R{c}_{g}=\lambda g$, for some $\lambda \in ℝ$, the last equation becomes

${{\Delta }}_{g}^{2}u+\frac{{b}_{n}\lambda }{n-1}{{\Delta }}_{g}u+\frac{{c}_{n}{\lambda }^{2}}{{\left(n-1\right)}^{2}}u=\frac{n-4}{2}{Q}_{\stackrel{˜}{g}}{u}^{{2}^{#}-1},\phantom{\rule{2.em}{0ex}}\left(2\right)$

where ${b}_{n}=\frac{{n}^{2}-2n-4}{2}$, ${c}_{n}=\frac{n\left(n-4\right)\left({n}^{2}-4\right)}{16}$ and ${c}_{n}-\frac{{b}_{n}^{2}}{4}=-1$. Given ${\Lambda }>0$, consider the set ${𝒮}_{b,c}^{{\Lambda }}=\left\{u\in {C}^{4}{\left(M\right),\phantom{\rule{4pt}{0ex}}u>0,\phantom{\rule{4.pt}{0ex}}\text{s.t.}\phantom{\rule{4.pt}{0ex}}\parallel u\parallel }_{{H}^{2}}\le {\Lambda }\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}u\phantom{\rule{4.pt}{0ex}}\text{solves}\phantom{\rule{4.pt}{0ex}}\text{(1)}\right\}$ (${H}^{2}$ is the Sobolev space of functions in ${L}^{2}$ with two derivatives in ${L}^{2}$). The equation (1) is said to be compact if for any ${\Lambda }>0$, ${𝒮}_{b,c}^{{\Lambda }}$ is compact in the ${C}^{4}$-topology, and it is stable if it is compact and for any $ϵ>0$, there exists $\delta >0$ such that for any ${b}^{\text{'}}$ and ${c}^{\text{'}}$ satisfying $|{b}^{\text{'}}-b|+|{c}^{\text{'}}-c|<\delta$, it holds that

${d}_{{C}^{4}}\left({𝒮}_{{b}^{\text{'}},{c}^{\text{'}}}^{{\Lambda }};{𝒮}_{b,c}^{{\Lambda }}\right)<ϵ·$

The main theorems proved in the paper are the following:

Theorem 1. Let $\left(M,g\right)$ be a smooth compact conformally flat Riemannian manifold of dimension $n=5,6,7$ and $b,c$ positive real numbers such that $c-\frac{{b}^{2}}{4}<0$. Let ${\left\{{b}_{\alpha }\right\}}_{\alpha \in ℕ}$ and ${\left\{{c}_{\alpha }\right\}}_{\alpha \in ℕ}$ be sequences of real numbers converging to $b$ and $c$, respectively, and ${\left\{{u}_{\alpha }\right\}}_{\alpha \in ℕ}$ be a bounded sequence in ${H}^{2}$ as $\alpha \to \infty$ of smooth positive solutions of (2) such that ${u}_{\alpha }\to {u}_{\infty }$ weakly in ${H}^{2}$ as $\alpha \to \infty$. Then either ${u}_{\alpha }\to {u}_{\infty }$ strongly in any ${C}^{k}$-topology, or ${u}_{\infty }\equiv 0$.

Theorem 2. Under the same hypothesis of the theorem above, assume that one of the following conditions holds true:

$n=5$ and ${\mu }_{x}\left(x\right)>0$ for al $x$, where ${\mu }_{x}\left(x\right)$ is the mass at $x$ of the operator ${P}_{g}={{\Delta }}_{g}^{2}+b{{\Delta }}_{g}+c$.

$n=6$ and $b\notin {𝒮}_{\omega }$, where if ${\lambda }_{1},\phantom{\rule{4pt}{0ex}}{\lambda }_{2}$ are the minimum and the maximum eigenvalues of ${A}_{g}$, resp., and ${𝒮}_{\omega }=\left[{\lambda }_{1},{\lambda }_{2}\right]$.

$n=8$ and $b<\frac{1}{8}{min}_{M}T{r}_{g}\left({A}_{g}\right)$.

$n=7$ or $n\ge 9$ and $b\ne \frac{1}{n}T{r}_{g}\left({A}_{g}\right)$ in M.

Then for any sequences ${\left\{{b}_{\alpha }\right\}}_{\alpha \in ℕ}$ and ${\left\{{c}_{\alpha }\right\}}_{\alpha \in ℕ}$, and any bounded sequence ${\left\{{u}_{\alpha }\right\}}_{\alpha \in ℕ}\subset {H}^{2}$ of positive solutions of (2) there holds that, up to a subsequence , ${u}_{\alpha }\to {u}_{\infty }$ in ${C}^{4}\left(M\right)$ for some smooth positive solution ${u}_{\infty }$ of (1). In particular, equation (1) is stable.

##### MSC:
 58J05 Elliptic equations on manifolds, general theory 35J30 Higher order elliptic equations, general 53A30 Conformal differential geometry