Polyharmonic boundary value problems

1.5 Paneitz-Branson type equations 19
In analogy to the second order Yamabe problem (for an overview see [381, Section
III.4]), obvious questions here concern the existence of conformal metrics with constant
or prescribed Q-curvature. Huge work has so far been done by research groups
around Chang-Yang-Gursky and Hebey, as well as many others. For a survey and
references see the books by Chang [89] and by Druet-Hebey-Robert [149]. Difficult
problems arise from ensuring the positivity requirement of the conformal factor
u > 0 and from the necessity to know about the kernel of the Paneitz operator. These
problems have only been solved partly yet.
In order to explain the geometrical importance of the Q-curvature, we assume
now for a moment that the manifold (M ,g) is four-dimensional. Then, the Paneitz
operator is defined by
P 4 4 := ∆ 2 −
4
∑
i, j=1
∇ i
2
3 Rgi
j − 2Ri j ∇ j
in such a way that under the conformal change of metrics gu = e 2u g one has
(P 4 4 )u(ϕ) = e −4u P 4 4 (ϕ).
In order to achieve a prescribed Q-curvature on the four-dimensional manifold
(M ,gu), one has to find u solving
where Q4 4 is the curvature invariant
P 4 4 u + 2Q 4 4 = 2Qe 4u ,
12Q 4 4 = −∆R + R 2 − 3|(Ri j)| 2 .
In this situation, one has the following Gauss-Bonnet-formula
Q + 1
8 |W|2
dS = 4π 2 χ(M ),
M
where W is the Weyl tensor and χ(M ) is the Euler characteristic. Since χ(M ) is a
topological and |W| 2 dS is a pointwise conformal invariant, this shows that
M QdS
is a conformal invariant, which governs e.g. the existence of conformal Ricci positive
metrics (see e.g. Chang-Gursky-Yang [90, 91]) and eigenvalue estimates for
Dirac operators (see Guofang Wang [407]). All these facts show that the Q-curvature
in the context of fourth order conformally covariant operators takes a role quite analogous
to the scalar curvature with respect to second order operators.
Getting back to the general case n > 4, let us outline what we are going to prove
in the present book. We do not aim at giving an overview – not even of parts – of
the theory of Paneitz operators but at giving a spot on some aspects of this issue.
Namely, in Section 7.9 we address the question whether in specific bounded smooth
domains Ω ⊂ R n (n > 4) there exists a metric gu = u 4/(n−4) (δi j) being conformal
to the flat euclidean metric and subject to certain homogeneous boundary condi-