On a geometric equation involving the Sobolev trace critical exponent
Journal of Inequalities and Applications
On a geometric equation involving the Sobolev trace critical exponent
Mohamed Ali Al-Ghamdi 0
Hichem Chtioui 0
Khadija Sharaf 0
0 Department of Mathematics, King Abdulaziz University , Jeddah , Saudi Arabia
In this paper we consider the problem of prescribing the mean curvature on the boundary of the unit ball of Rn, n ≥ 4. Under the assumption that the prescribed function is flat near its critical point, we give precise estimates on the losses of the compactness, and we provide a new existence result of Bahri-Coron type. Moreover, we establish, under generic boundary condition, a Morse inequality at infinity, which gives a lower bound on the number of solutions to the above problem. MSC: 58E05; 35J65; 53C21; 35B40
boundary mean curvature; variational problem; loss of compactness; β-flatness condition; critical point at infinity
1 Introduction
and hg is the mean curvature of g on Sn–.
which solves the following nonlinear boundary value equation:
u = in Bn,
n
⎩ ∂∂uν + n– u = n– Hu n–
on Sn–,
∇H(y) = .
(–)ind(H,y) = .
H(x) = H() +
where bi = bi(y) ∈ R \ {}, ∀i = , . . . , n – , in=– bi = and [sβ=] |∇sR(x)||x|s–β = o() as x
tends to zero. Here ∇s denotes all possible derivatives of order s and [β] is an integer part
of β. Let
we have already mentioned, another obstruction to solving the problem, the so-called
Kazdan-Warner obstruction. There have been many papers on the problem and related
ones, please see [–] and the references therein.
One group of existence results has been obtained under hypotheses involving the
Laplacian H at the critical points y of H; see [, ] for n = , and [–] for n ≥ . For
example, in [] and [], it is assumed that H is a Morse function and
The result has been extended to any dimension n ≥ in []. Roughly, it is assumed that
there exists β, n – < β < n – , such that in some geodesic normal coordinate system
centered at y, we have
K = y ∈ Sn–, ∇H(y) = ,
K+ = y ∈ K,
y∈K+
(–)n––˜i(y) = ;
see [].
Let us observe that a condition like (.) appeared first in [] concerning the scalar
curvature problem; see also [].
In this work we restrict our attention to problem (.) under condition that H is a
C-function satisfying (f)β condition with n – ≤ β < n – . This leads to an interesting
new phenomenon, that is, the presence of multiple blow-up points. In fact, when looking
to the possible formations of blow-up points, it comes out that the strong interaction of
the bubbles in the case where n – < β < n – forces all blow-up points to be single, while
in the case where β = n – , we have a balance phenomenon, that is, any interaction of
two bubbles is of the same order with respect to the self-interaction. We denote by the
operator which associates to H the solution v of (.), and we extend the definition of
to the case of weak solutions of (.). Let
Kn– = y ∈ K, β = β(y) = n – .
⎧ – kn=– bk(yij )
⎪⎪⎨ Mjj = nn–– c˜ H(ylj )n– , j ∈ {, . . . , N },
⎪⎪⎩ Mlj = – n– c [H(yGil )(Hyil(y,yijij))] n– , l, j ∈ {, . . . , N }, l = j,
Here G(q, ·) denotes Green’s function for the operator with point q.
Let ρ = ρ(yi , . . . , yiN ) be the least eigenvalue of M. We assume the following:
We now introduce the following set:
We then have the following theorem.
(–)n––˜i(y) +
y∈K+\Kn–
(yi ,...,yiN )∈Cn+–
(–)N–+ jN= n–+˜i(yij) = ,
then (.) has at least one solution. Moreover, for generic H, we have
S ≥ –
(–)n––˜i(y) –
y∈K+\Kn–
(yi ,...,yiN )∈Cn+–
(–)N–+ jN= n–+˜i(yij) ,
where S denotes the set of solutions of (.).
tion of a suitable pseudo-gradient, for which the Palais-Smale condition is satisfied along
the decreasing flow lines as long as these flow lines do not enter the neighborhood of some
specific critical points of H.
A similar Morse lemma at infinity has been established for problem (.) under the
hypothesis that the function H is of class C and the order of flatness at critical points of H
is β ∈ ]n – , n – [; see [].
The rest of this paper is organized as follows. In Section , we set up the variational
problem and we recall the expansion of the gradient of the associated Euler-Lagrange
functional near infinity. In Section , we construct a suitable pseudo-gradient and we
characterize the critical points at infinity. Lastly, in Section , we prove our main result.
2 General framework and some known facts
2.1 Variational problem
First, we recall the functional setting and the variational problem and its main features.
Problem (.) has a variational structure. The Euler-Lagrange functional is
J(u) =
defined on H(Bn) equipped with the norm
u =
Bn |∇u| dvg +
where dvg and dσg denote the Riemannian measure on Bn and Sn– induced by the
metric g. We denote by the unit sphere of H(Bn), and we set
+ = {u ∈
/u ≥ }.
u |x | +(xxn + ) , |x|x||++(xxnn+–) ,
Bn |∇u| +
u =
In the sequel, we identify the function H a (...truncated)