CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE
CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE
CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE
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<strong>CONCENTRATING</strong> <strong>SOLUTIONS</strong> <strong>FOR</strong> A <strong>PLANAR</strong><br />
ELLIPTIC PROBLEM INVOLVING NONLINEARITIES<br />
WITH LARGE EXPONENT<br />
PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Abstract. We consider the boundary value problem ∆u + u p = 0 in a<br />
bounded, smooth domain Ω in R 2 with homogeneous Dirichlet boundary<br />
condition and p a large exponent. We find topological conditions on Ω<br />
which ensure the existence of a positive solution u p concentrating at<br />
exactly m points as p → ∞. In particular, for a non simply connected<br />
domain such a solution exists for any given m ≥ 1.<br />
1. Introduction and statement of main results<br />
This paper is concerned with analysis of solutions to the boundary value<br />
problem:<br />
⎧<br />
⎨<br />
⎩<br />
∆u + u p = 0 in Ω,<br />
u > 0 in Ω,<br />
u = 0 on ∂Ω,<br />
where Ω is a smooth bounded domain in R 2 and p is a large exponent.<br />
Let us consider the Rayleigh quotient:<br />
∫<br />
Ω<br />
I p (u) =<br />
|∇u|2<br />
( ∫ , u ∈ H<br />
Ω |u|p+1 ) 2<br />
0 1 (Ω) \ {0},<br />
p+1<br />
and set<br />
S p =<br />
inf I p (u).<br />
u∈H0 1(Ω)\{0} (1.1)<br />
Since H 1 0 (Ω) is compactly embedded in Lp+1 (Ω) for any p > 0, standard variational<br />
methods show that S p is achieved by a positive function u p which<br />
solves problem (1.1). The function u p is known as least energy solution.<br />
In [29]-[30] the authors show that the least energy solution has L ∞ -norm<br />
1991 Mathematics Subject Classification. 35J60, 35B33, 35J25, 35J20, 35B40.<br />
Key words and phrases. Large exponent, Concentrating solutions, Green’s function,<br />
Finite dimensional reduction.<br />
First author is supported by M.U.R.S.T., project “Variational methods and nonlinear<br />
differential equations”. Second author has been partially supported by Fondecyt 1040936<br />
(Chile). Second and third authors are supported by M.U.R.S.T., project “Metodi variazionali<br />
e topologici nello studio di fenomeni non lineari”.<br />
1
2 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
bounded and bounded away from zero uniformly in p, for p large. Furthermore,<br />
up to subsequence, the renormalized energy density p|∇u p | 2 concentrates<br />
as a Dirac delta around a critical point of the Robin’s function<br />
H(x, x), where H is the regular part of Green function of the laplacian in<br />
Ω with homogeneous Dirichlet boundary condition. Namely, the Green’s<br />
function G(x, y) is the solution of the problem<br />
{ −∆x G(x, y) = δ y (x) x ∈ Ω,<br />
G(x, y) = 0 x ∈ ∂Ω,<br />
and H(x, y) is the regular part defined as<br />
H(x, y) = G(x, y) − 1<br />
2π log 1<br />
|x − y| .<br />
In [1] and [16] the authors give a further description of the asymptotic behaviour<br />
of u p , as p → ∞, by identifying a limit profile problem of Liouvilletype:<br />
{ ∆u<br />
∫ + e u = 0 in R 2 ,<br />
R 2 e u (1.2)<br />
< ∞,<br />
and showing that ‖u p ‖ ∞ → √ e as p → +∞.<br />
Problem (1.2) possesses exactly a three-parameters family of solutions<br />
U δ,ξ (x) = log<br />
where δ is a positive number and ξ ∈ R 2 (see [6]).<br />
8δ 2<br />
(δ 2 + |x − ξ| 2 ) 2 , (1.3)<br />
The aim of this paper is to build solutions for problem (1.1) that, up to<br />
a suitable normalization, look like a sum of concentrated solutions for the<br />
limit profile problem (1.2) centered at several points ξ 1 , . . . , ξ m , as p → ∞.<br />
In this case, when m is possibly greater than 1, the function responsible to<br />
locate the concentration points ξ 1 , . . . , ξ m is more involved than the Robin’s<br />
function. In fact, location of such points is related to critical points of the<br />
function<br />
m∑<br />
m∑<br />
ϕ m (ξ 1 , . . . , ξ m ) = H(ξ j , ξ j ) + G(ξ i , ξ j ).<br />
j=1<br />
Let us mention that the same function ϕ m is responsible for the location<br />
of the points of concentration for solutions to the mean field equation in<br />
bounded domains Ω ⊂ R 2 (see [3, 12, 14]).<br />
Our main result reads as follows.<br />
Theorem 1.1. Assume that Ω is not simply connected. Then given any<br />
m ≥ 1 there exists p m > 0 such that for any p ≥ p m problem (1.1) has a<br />
solution u p which concentrates at m different points in Ω, according to (1.6),<br />
(1.7) and (1.8), as p goes to +∞.<br />
i,j=1<br />
i≠j
This result is consequence of a more general theorem, which we state below,<br />
that ensures the existence of solutions to problem (1.1) which concentrates<br />
at m different points of Ω, under the assumption that the function ϕ m has<br />
a non-trivial critical value.<br />
Let Ω m be Ω × Ω × . . . × Ω m times. We define<br />
ϕ m (x 1 , . . . , x m ) = +∞ if x i = x j for some i ≠ j.<br />
Let D be an open set compactly contained in Ω m with smooth boundary. We<br />
recall that ϕ m links in D at critical level C relative to B and B 0 if B and B 0<br />
are closed subsets of ¯D with B connected and B 0 ⊂ B such that the following<br />
conditions hold: Let us set Γ to be the class of all maps Φ ∈ C(B, D) with<br />
the property that there exists a function Ψ ∈ C([0, 1] × B, D) such that:<br />
We assume<br />
Ψ(0, ·) = Id B , Ψ(1, ·) = Φ, Ψ(t, ·)| B0 = Id B0 for all t ∈ [0, 1].<br />
sup ϕ m (y) < C ≡ inf<br />
y∈B 0<br />
sup<br />
Φ∈Γ y∈B<br />
ϕ m (Φ(y)) , (1.4)<br />
and for all y ∈ ∂D such that ϕ m (y) = C, there exists a vector τ y tangent to<br />
∂D at y such that<br />
∇ϕ m (y) · τ y ≠ 0 . (1.5)<br />
Under these conditions a critical point ȳ ∈ D of ϕ m with ϕ m (ȳ) = C exists,<br />
as a standard deformation argument involving the negative gradient flow of<br />
ϕ m shows. It is easy to check that the above conditions hold if<br />
inf ϕ m(x) < inf<br />
x∈D<br />
x∈∂D ϕ m(x), or sup<br />
x∈D<br />
ϕ m (x) > sup ϕ m (x) ,<br />
x∈∂D<br />
namely the case of (possibly degenerate) local minimum or maximum points<br />
of ϕ m . We call C a non-trivial critical level of ϕ m in D.<br />
Theorem 1.2. Let m ≥ 1 and assume that there is an open set D compactly<br />
contained in Ω m where ϕ m has a non-trivial critical level C. Then, there<br />
exists p m > 0 such that for any p ≥ p m problem (1.1) has a solution u p<br />
which concentrates at m different points of Ω, i.e. as p goes to +∞<br />
pu p+1<br />
p<br />
m∑<br />
⇀ 8πe δ ξj weakly in the sense of measure in Ω (1.6)<br />
i=1<br />
for some ξ ∈ D such that ϕ m (ξ 1 , . . . , ξ m ) = C and ∇ϕ m (ξ 1 , . . . , ξ m ) = 0.<br />
More precisely, there is an m-tuple ξ p = (ξ p 1 , . . . , ξp m) ∈ D converging (up to<br />
subsequence) to ξ such that, for any δ > 0, as p goes to +∞<br />
and<br />
u p → 0 uniformly in Ω \ ∪ m j=1 B δ(ξ p i ) (1.7)<br />
sup<br />
x∈B δ (ξ p i ) u p (x) → √ e. (1.8)<br />
3
4 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
The detailed proof of how Theorem 1.2 implies the result contained in Theorem<br />
1.1 can be found in [12].<br />
As already mentioned, the case of (possibly degenerate) local maximum or<br />
minimum for ϕ m is included. This simple fact allows us to obtain an existence<br />
result for solutions to problem (1.1) also when Ω is simply connected.<br />
Indeed, we can construct simply connected domains of dumbell-type where<br />
a large number of concentrating solutions can be found.<br />
Let h be an integer. By h-dumbell domain with thin handles we mean the<br />
following: let Ω 0 = Ω 1 ∪ . . . ∪ Ω h , with Ω 1 , . . . , Ω h smooth bounded domains<br />
in R 2 such that ¯Ω i ∩ ¯Ω j = ∅ if i ≠ j. Assume that<br />
Ω i ⊂ {(x 1 , x 2 ) ∈ R 2 : a i ≤ x 1 ≤ b i }, Ω i ∩ {x 2 = 0} ̸= ∅,<br />
for some b i < a i+1 and i = 1, . . . , h. Let<br />
C ε = {(x 1 , x 2 ) ∈ R 2 : |x 2 | ≤ ε, x 1 ∈ (a 1 , b h )}, for some ε > 0.<br />
We say that Ω ε is a h-dumbell with thin handles if Ω ε is a smooth simply<br />
connected domain such that Ω 0 ⊂ Ω ε ⊂ Ω 0 ∪ C ε , for some ε > 0.<br />
The following result holds true.<br />
Theorem 1.3. There exist ε h > 0 and p h > 0 such that for any ε ∈ (0, ε h )<br />
and p ≥ p h problem (1.1) in Ω ε has at least 2 h −1 families of solutions which<br />
concentrate at different points of Ω ε , according to (1.6), (1.7) and (1.8), as<br />
( ) h<br />
p goes to +∞. More precisely, for any integer 1 ≤ m ≤ h there exist<br />
m<br />
families of solutions of (1.1) which concentrate at m different points of Ω ε .<br />
The detailed proof of how Theorem 1.2 implies the result contained in Theorem<br />
1.3 can be found in [14]. We also refer the reader to [4] and [7], where<br />
domains like dumbells with thin handles are considered.<br />
The proof of all our results relies on a Lyapunov-Schmidt procedure, based<br />
on a proper choice of the ansatz for the solution we are looking for. Usually,<br />
in other related problems of asymptotic analysis, the ansatz for the solution<br />
is built as the sum of a main term, which is a solution (properly modified or<br />
projected) of the associated limit problem, and a lower order term, which<br />
can be determined by a fixed point argument. In our problem, this is not<br />
enough. Indeed, in order to perform the fix point argument to find the lower<br />
order term in the ansatz (see Lemma 4.1), we need to improve substantially<br />
the main term in the ansatz, adding two other terms in the expansion of the<br />
solution (see section 2). This fact is basically due to estimate (4.7).<br />
By performing a finite dimensional reduction, we find an actual solution to<br />
our problem adjusting points ξ inside Ω to be critical points of a certain<br />
function F (ξ) (see (5.2)). It is quite standard to show that this function<br />
F (ξ) is a perturbation of ϕ m (ξ) in a C 0 −sense. On the other hand, it is not<br />
at all trivial to show the C 1 closeness between F and ϕ m . This difficulty is<br />
related to the difference between the exponential decay of the concentration
paremeters δ ∼ e − p 4 (see (2.3)) and the polinomial decay 1 p 4 of the error term<br />
‖∆U ξ + U p ξ ‖ ∗ of our approximating function U ξ (see Proposition 2.1). We<br />
are able to overcome this difficulty (see Lemma 5.3) using a Pohozaev-type<br />
identity.<br />
Now, we would like to compare problem (1.1) with some widely studied<br />
problems which have some analogies with it.<br />
In higher dimension the problem equivalent to Problem (1.1) is the slightly<br />
sub-critical problem<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
∆u + u N+2<br />
N−2 −ε = 0 in Ω,<br />
u > 0 in Ω,<br />
u = 0 on ∂Ω,<br />
5<br />
(1.9)<br />
where Ω is a smooth bounded domain in R N , N ≥ 3 and ε is a positive<br />
parameter. Indeed, in dimension N ≥ 3, the embedding of H0 1(Ω) in Lp+1 (Ω)<br />
is compact for every p < N+2<br />
N−2<br />
. Hence the minimum of the Rayleigh quotient<br />
corresponding to problem (1.9) is achieved by a positive function u ε , called<br />
least energy solution, which, after a multiplication by a suitable positive<br />
constant, is a solution to (1.9).<br />
It is well known that, as ε goes to 0, the least energy solution u ε concentrates<br />
around a point, which is a critical point of the Robin’s function of the<br />
corresponding Green’s function (see [2, 17, 20, 27]). Also the converse is<br />
true: around any stable critical point of the Robin’s function one can build<br />
a family of solutions for (1.9) concentrating precisely there (see [24, 27, 28]).<br />
In [2] and [22] the authors showed that also for problem (1.9) there exist<br />
solutions with concentration in multiple points and, as in the problem that<br />
we are considering in the present paper, the points of concentration are given<br />
by critical points of a certain function defined in terms of both the Green’s<br />
function and Robin’s function.<br />
The analogies between problems (1.1) and (1.9) break down here. Indeed,<br />
while for (1.1) one can find solutions with an arbitrarily large number of<br />
condensation points in any given not simply connected domain Ω, in [2] the<br />
authors proved that solutions to (1.9) can have at most a finite number of<br />
peaks which depends on Ω (see, also, [15, 22]).<br />
The property of problem (1.1) to have solution with an arbitrarily large<br />
number of points of condensation is what one expects to happen in the<br />
slightly super-critical version of problem (1.9), namely<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
∆u + u N+2<br />
N−2 +ε = 0 in Ω,<br />
u > 0 in Ω,<br />
u = 0 on ∂Ω.<br />
(1.10)<br />
Indeed, a conjecture for (1.10) is that, given any domain Ω with a hole, one<br />
can see solutions with an arbitrarily large number of peaks (see [9, 10, 25]).
6 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
For this fact, despite of being compact and hence sub-critical in dimension<br />
2, problem (1.1) shares patterns similar to the ones associated to slighlty<br />
super-critical problem (1.10) in higher dimension.<br />
However, again the analogies between (1.1) and (1.10) in higher dimension<br />
break down here. Indeed, the dilation invariance, which is crucial in the<br />
study of problem (1.10), does not play a role in finding solutions to problem<br />
(1.1), as already observed for a similar two-dimensional problem in [12] and<br />
[14] (see, also, [3]): only translation invariance is concerned in the study of<br />
(1.1).<br />
The only translation invariance is the crucial key which allows to find solutions<br />
to the sub-critical problem (see [11, 19, 21, 23, 31])<br />
⎧<br />
⎨<br />
⎩<br />
−ε 2 ∆u + u = u p in Ω,<br />
u > 0 in Ω,<br />
u = 0 on ∂Ω,<br />
(1.11)<br />
where Ω is a bounded open domain in R N , p > 1 if N = 2 and 1 < p < N+2<br />
N−2<br />
if N ≥ 3, ε is a positive parameter.<br />
Moreover, the property of problem (1.1) to have solution with an arbitrarily<br />
large number of points of condensation is what happens in the sub-critical<br />
problem (1.11). In fact, in [8] the authors prove that if the reduced cohomology<br />
of Ω is not trivial, then for any integer k such a problem has at least<br />
one k−peaks solution, provided the parameter ε is small enough.<br />
However, again the analogies between (1.1) and (1.11) break down here. Indeed,<br />
problem (1.1) is somehow almost critical in R 2 , since the limit problem<br />
as p goes to +∞ is (1.2) which is critical in R 2 , while the limit problem of<br />
(1.11) as ε goes to 0 is the sub-critical problem<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
−∆u + u = u p in R N ,<br />
u > 0 in R N ,<br />
u ∈ H 1 (R N ).<br />
The paper is organized as follows. In Section 2 we describe exactly the<br />
ansatz for the solution we are searching for. We rewrite the problem in term<br />
of a linear operator L for which a solvability theory is performed in Section<br />
3. In Section 4 we solve an auxiliary non linear problem. We reduce (1.1) to<br />
solve a finite system c ij = 0, as we will see in Section 5. Section 5 contains<br />
also the proof of Theorem 1.2.<br />
2. A first approximation of the solution<br />
In this Section we will provide an ansatz for solutions of problem (1.1).<br />
A useful observation is that u satisfies equation (1.1) if and only if<br />
v(y) = δ 2<br />
p−1 u(δy + ξ),<br />
y ∈ Ω ξ,δ
satisfies<br />
⎧<br />
⎨<br />
⎩<br />
∆v + v p = 0, in Ω ξ,δ ,<br />
v > 0, in Ω ξ,δ , v = 0, on ∂Ω ξ,δ ,<br />
7<br />
(2.1)<br />
where ξ is a given point in Ω, δ is a positive number with δ → 0, and Ω ξ,δ<br />
is the expanding domain defined by Ω−ξ<br />
δ<br />
.<br />
In this section we will show that the basic elements for the construction<br />
of an approximate solution to problem (1.1) which exhibits one point of<br />
concentration (or equivalently of problem (2.1)) are the radially symmetric<br />
solutions of the problem (1.2) given by U δ,ξ defined in (1.3).<br />
For U δ,ξ (x) defined in (1.3), we denote by P U δ,ξ (x) its projection on the<br />
space H0 1(Ω), namely P U δ,ξ(x) is the unique solution of<br />
{ ∆P Uδ,ξ = ∆U δ,ξ in Ω,<br />
P U δ,ξ (x) = 0 on ∂Ω.<br />
Since P U δ,ξ (x)−U δ,ξ (x)+log(8δ 2 )+4 log<br />
1<br />
|x−ξ| = O(δ2 ) uniformly on x ∈ ∂Ω<br />
as δ → 0 (together with any boundary derivatives), by harmonicity we get<br />
P U δ,ξ (x) = U δ,ξ (x) − log(8δ 2 ) + 8πH(x, ξ) + O(δ 2 ) in C 1 (¯Ω) (2.2)<br />
P U δ,ξ (x) = 8πG(x, ξ) + O(δ 2 ) in C 1 loc (¯Ω \ {ξ}),<br />
provided ξ is bounded away from ∂Ω.<br />
Assume now that<br />
δ = µe − p 1<br />
4 ,<br />
C<br />
and define<br />
u(x) = e p<br />
2(p−1)<br />
Observe that, as p → ∞,<br />
p p<br />
p−1 µ 2<br />
p−1<br />
≤ µ ≤ C, (2.3)<br />
P U δ,ξ (x) x ∈ Ω. (2.4)<br />
u(ξ) → √ e and u(x) = O( 1 ) for x ≠ ξ.<br />
p<br />
Furthermore, under the extra assumption that the parameter µ is defined<br />
by the relation<br />
log(8µ 4 ) = 8πH(ξ, ξ),<br />
a direct computation shows that a good first approximation for a solution<br />
to problem (1.1) exhibiting only one point of concentration is given by a<br />
perturbation of the function u defined in (2.4).<br />
Indeed, in the expanded variable y = x−ξ<br />
δ<br />
∈ Ω−ξ<br />
δ<br />
, if we define v(y) =<br />
δ 2<br />
p−1 u(δy + ξ), then our first approximation (2.4) looks like<br />
1<br />
p p<br />
p−1<br />
(p + U 1,0 (y) + O(e − p 4 |y| + e<br />
− p 4 )<br />
)<br />
(2.5)
8 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
and hence<br />
∆v + v p ∼ 1<br />
p p<br />
p−1<br />
[<br />
(<br />
−e U1,0(y) +<br />
1 + U 1,0(y)<br />
p<br />
) p ]<br />
,<br />
which, roughly speaking, implies that the error for u to be a solution of (1.1)<br />
exhibiting one points of concentration, or equivalently for v to be a solution<br />
of (2.1), is of order 1 p 2 .<br />
However, as we will see below, this is not enough to built an actual solution<br />
to (1.1) starting from u(x). We need to refine this first approximation, or<br />
equivalently, according to (2.5), we need to go further in the expansion of<br />
¯v(y) = p + U 1,0 (y) + o(1), by identifying first and second order terms in<br />
¯v − p − U 1,0 .<br />
Let us call<br />
and consider<br />
v ∞ (y) = U 1,0 (y)<br />
ˆv(y) = v ∞ (y) + 1 p w 0(y) + 1 p 2 w 1(y),<br />
where w 0 , w 1 solve<br />
where<br />
∆w i +<br />
f 0 = 4v 2 ∞ , f 1 = 8<br />
8<br />
(1 + |y| 2 ) 2 w i =<br />
1<br />
(1 + |y| 2 ) 2 f i(y) in R 2 , i = 1, 2,<br />
(<br />
w 0 v ∞ − 1 3 v3 ∞ − 1 2 w2 0 − 1 8 v4 ∞ + 1 )<br />
2 w 0v∞<br />
2 . (2.6)<br />
According to [5], for a radial function f(y) = f(|y|) there exists a radial<br />
solution<br />
w(r) = 1 − (∫ r2 r<br />
)<br />
φ f (s) − φ f (1)<br />
r<br />
1 + r 2 (s − 1) 2 ds + φ f (1)<br />
1 − r<br />
0<br />
for the equation<br />
8<br />
∆w +<br />
(1 + |y| 2 ) 2 w = f(y),<br />
where φ f (s) = ( s2 +1<br />
s 2 −1 )2 (s−1) 2 ∫ s<br />
s 0 t 1−t2 f(t)dt for s ≠ 1 and φ<br />
1+t 2 f (1) = lim s→1 φ f (s).<br />
It is a straightforward computation to show that<br />
w(r) = C f log r + D f + O(<br />
∫ +∞<br />
r<br />
s| log s||f|(s)ds +<br />
| log r|<br />
r 2 ) as r → +∞,<br />
where C f = ∫ +∞<br />
0<br />
t t2 −1<br />
t 2 +1 f(t)dt, provided ∫ +∞<br />
0<br />
t|f|(t)dt < +∞.<br />
r<br />
Therefore, up to replacing w(r) with w(r) − D 2 −1 f , we have shown<br />
r 2 +1<br />
Lemma 2.1. Let f ∈ C 1 ([0, +∞)) such that ∫ +∞<br />
0<br />
t| log t||f|(t)dt < +∞.<br />
There exists a C 2 radial solution w(r) of equation<br />
∆w +<br />
8<br />
(1 + |y| 2 w = f(|y|) in R2<br />
) 2
such that as r → +∞<br />
( ∫ )<br />
+∞<br />
w(r) = t t2 − 1<br />
t 2 + 1 f(t)dt log r + O(<br />
and<br />
0<br />
∂ r w(r) =<br />
( ∫ +∞<br />
0<br />
∫ +∞<br />
r<br />
s| log s||f|(s)ds +<br />
| log r|<br />
r 2 )<br />
) ∫<br />
t t2 − 1 1 +∞<br />
t 2 + 1 f(t)dt r + O(1 | log r|<br />
s|f|(s)ds +<br />
r r<br />
r 3 ).<br />
By means of Lemma 2.1, since f 0 has at most logarithmic growth at infinity<br />
(see (2.6)), we can define w 0 (r) as a radial function satisfying<br />
w 0 (y) = C 0 log |y| + O( 1 ) as |y| → +∞, (2.7)<br />
|y|<br />
where C 0 = 4 ∫ +∞<br />
0<br />
t t2 −1<br />
(t 2 +1) 3 log 2 (<br />
8<br />
(1+t 2 ) 2 ) = 12 − 4 log 8.<br />
More precisely, since we will need the exact expression of w 0 , we have that<br />
w 0 (y) = 1 2 v2 ∞(y) + 6 log(|y| 2 2 log 8 − 10<br />
+ 1) +<br />
|y| 2 + 1<br />
(<br />
+ |y|2 − 1<br />
|y| 2 2 log 2 (|y| 2 + 1) − 1 + 1<br />
2 log2 8 + 4<br />
)<br />
−8 log |y| log(|y| 2 + 1) ,<br />
∫ +∞<br />
|y| 2<br />
9<br />
(2.8)<br />
ds<br />
s + 1 log s + 1<br />
s<br />
as we can see by direct inspection. From (2.7) we get that also f 1 growths at<br />
most logarithmically at infinity and w 1 can be defined as a radial function<br />
satisfying<br />
for a suitable constant C 1 .<br />
w 1 (y) = C 1 log |y| + O( 1 ) as |y| → +∞, (2.9)<br />
|y|<br />
1<br />
We will see now that the profile p<br />
p p−1<br />
a better approximation for a solution to the equation<br />
∆v + v p = 0<br />
(<br />
)<br />
p + v ∞ (y) + 1 p w 0(y) + 1 w<br />
p 2 1 (y)<br />
in the region |y| ≤ Ce p 8 . Indeed, by Taylor expansions of exponential and<br />
logarithmic function, we have that, for |y| ≤ Ce p 8 ,<br />
(<br />
1 + a p + b<br />
p 2 + c ) [<br />
p<br />
p 3 = e a 1 + 1 a2<br />
(b −<br />
p 2 ) + 1 (<br />
p 2 c − ab + a3<br />
(2.10)<br />
3<br />
+ b2 2 + a4<br />
8 − a2 b<br />
2<br />
)<br />
+ O<br />
(<br />
log 6 )]<br />
(|y| + 2)<br />
p 3<br />
provided −4 log(|y| + 2) ≤ a(y) ≤ C and |b(y)| + |c(y)| ≤ C log(|y| + 2).<br />
Observe that in our case 2C ≥ a + b p + c<br />
p 2 ≥ − 3 4 p for |y| ≤ Ce p 8 .<br />
is
10 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Hence, by the choice of v ∞ , w 0 and w 1 and expansion (2.10), we obtain that<br />
∆v + v p = O( 1 p 4 log 6 (|y| + 2)<br />
(1 + |y| 2 ) 2 ) in |y| ≤ Ce p 8 .<br />
As before, we will see now that a proper choice of the parameter µ will automatically<br />
imply that this approximation for v is also good for the boundary<br />
condition to be satisfied. Indeed, observe that by (2.7), (2.9) and Lemma<br />
2.1 we get: for i = 1, 2<br />
(<br />
P w i ( x − ξ )<br />
) = w i ( x − ξ ) − 2πC i H(x, ξ) + C i log δ + O(δ) in C 1 (¯Ω)(2.11)<br />
δ<br />
δ<br />
(<br />
P w i ( x − ξ )<br />
) = −2πC i G(x, ξ) + O(δ) in C 1<br />
δ<br />
loc (¯Ω \ {ξ}),<br />
provided ξ is bounded away from ∂Ω. If we take µ as a solution of<br />
(<br />
log(8µ 4 ) = 8πH(ξ, ξ) 1 − C 0<br />
4p − C )<br />
1<br />
4p 2 + log δ (<br />
C 0 + C )<br />
1<br />
,<br />
p p<br />
we get that<br />
u(x) = e 2(p−1)<br />
p<br />
p p<br />
p−1 µ 2<br />
p−1<br />
[P U δ,ξ (x) + 1 p P (<br />
w 0 ( x − ξ<br />
δ<br />
)<br />
) + 1 (<br />
p 2 P<br />
w 1 ( x − ξ<br />
δ<br />
)]<br />
)<br />
is a good first approximation in order to construct a solution for (1.1) with<br />
just one concentration point.<br />
Let us remark that µ bifurcates, as p gets large, by ¯µ = e − 3 4 e 2πH(ξ,ξ) , solution<br />
of equation<br />
More precisely,<br />
log(8µ 4 ) = 8πH(ξ, ξ) − C 0<br />
4<br />
= 8πH(ξ, ξ) − 3 + log 8.<br />
µ = e − 3 4 e 2πH(ξ,ξ) (1 + O( 1 p )).<br />
Let us see now how things generalize if we want to construct a solution to<br />
problem (1.1) which exhibits m points of concentration. Let ε > 0 fixed and<br />
take an m−uple ξ = (ξ 1 , . . . , ξ m ) ∈ O ε , where<br />
O ε = {ξ = (ξ 1 , . . . , ξ m ) ∈ Ω m : dist (ξ i , ∂Ω) ≥ 2ε, |ξ i − ξ j | ≥ 2ε, i ≠ j} .<br />
Define<br />
U ξ (x) =<br />
where<br />
m∑<br />
j=1<br />
γµ<br />
1<br />
γ = p p<br />
2<br />
p−1<br />
j<br />
[P U δj ,ξ j<br />
(x) + 1 (<br />
p P w 0 ( x − ξ )<br />
j<br />
) + 1 (<br />
δ j p 2 P w 1 ( x − ξ ) ]<br />
j<br />
) ,<br />
δ j<br />
p−1 e − p<br />
2(p−1)<br />
and δ j = µ j e − p 1<br />
4 ,<br />
C ≤ µ j ≤ C. (2.12)
The parameters µ j will be chosen later. Observe that for any j ≠ i and<br />
x = δ i y + ξ i<br />
1<br />
[P U δj ,ξ j<br />
(x) + 1 (<br />
p P w 0 ( x − ξ )<br />
j<br />
) + 1 (<br />
δ j p 2 P w 1 ( x − ξ )]<br />
j<br />
)<br />
δ j<br />
γµ<br />
2<br />
p−1<br />
j<br />
= 8π<br />
γµ<br />
2<br />
p−1<br />
j<br />
[<br />
G(ξ i , ξ j ) 1 − C 0<br />
4p − C ]<br />
1<br />
4p 2<br />
p + O( e− 4 + |x − ξ i |<br />
).<br />
γ<br />
Hence, we get that the function U ξ (x) is a good approximation for a solution<br />
to problem (1.1) exhibiting m points of concentration provided<br />
log(8µ 4 i ) =<br />
(<br />
8πH(ξ i , ξ i ) 1 − C 0<br />
4p − C )<br />
1<br />
4p 2 + log δ (<br />
i<br />
C 0 + C )<br />
1<br />
p p<br />
+8π ∑ 2<br />
p−1 [<br />
µ i<br />
2<br />
G(ξ j , ξ i ) 1 − C 0<br />
p−1<br />
4p − C ]<br />
1<br />
4p<br />
j≠i µ<br />
.<br />
A direct computation shows that, for p large, µ satisfies<br />
j=1<br />
γµ<br />
2<br />
p−1<br />
j<br />
j<br />
∑<br />
µ i = e − 3 2πH(ξ<br />
4 e i ,ξ i )+2π<br />
j≠i G(ξ j,ξ i ) 1 (1 + O( )). (2.13)<br />
p<br />
Indeed, with this choice of the parameters µ i , we have that<br />
m∑ 1<br />
[P U δj ,ξ j<br />
(x) + 1 (<br />
p P w 0 ( x − ξ )<br />
j<br />
) + 1 (<br />
δ j p 2 P<br />
= 1<br />
γµ<br />
2<br />
p−1<br />
i<br />
for x = δ i y + ξ i .<br />
w 1 ( x − ξ j<br />
δ j<br />
)<br />
(p + v ∞ (y) + 1 p w 0(y) + 1 p 2 w 1(y) + O(e − p 4 |y| + e<br />
− p 4 )<br />
)<br />
(2.14)<br />
Remark 2.1. Let us remark that U ξ is a positive function. Since |v ∞ +<br />
1<br />
p w 0 + 1 w<br />
p 2 1 | ≤ C in |y| ≤ ɛ δ i<br />
, by (2.14) we get that U ξ is positive in B(ξ i , ɛ)<br />
for any i = 1, . . . , m. Moreover, by (2.11) we get that<br />
(<br />
P w i ( x − ξ )<br />
j<br />
) → −2πC i G(·, ξ j )<br />
δ j<br />
in C 1 -norm on |x − ξ j | ≥ ɛ, i = 0, 1, and then<br />
P U δj ,ξ j<br />
+ 1 (<br />
p P w 0 ( x − ξ )<br />
j<br />
) + 1 (<br />
δ j p 2 P w 1 ( x − ξ )<br />
j<br />
)<br />
δ j<br />
→ 8πG(·, ξ j )<br />
in C 1 -norm on |x−ξ j | ≥ ɛ. Hence, since ∂G<br />
∂ n (·, ξ j) < 0 on ∂Ω, U ξ is a positive<br />
function in Ω.<br />
)]<br />
11
12 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
We will look for solutions u of problem (1.1) in the form u = U ξ + φ, where<br />
φ will represent an higher order term in the expansion of u. Let us set<br />
W ξ (x) = pU p−1<br />
ξ<br />
(x).<br />
In terms of φ, problem (1.1) becomes<br />
⎧<br />
⎨ L(φ) = −[R + N(φ)] in Ω,<br />
where<br />
and<br />
⎩<br />
φ = 0<br />
on ∂Ω,<br />
(2.15)<br />
L(φ) := ∆φ + W ξ φ (2.16)<br />
R ξ := ∆U ξ + U p ξ , N(φ) = [(U ξ + φ) p − U p ξ<br />
p−1<br />
− pUξ<br />
φ].<br />
(2.17)<br />
The main step in solving problem (2.15) for small φ, under a suitable choice<br />
of the points ξ i , is that of a solvability theory for the linear operator L.<br />
In developing this theory, we will take into account the invariance, under<br />
translations and dilations, of the problem ∆v + e v = 0 in R 2 . We will<br />
perform the solvability theory for the linear operator L in weighted L ∞<br />
spaces, following [12]. For any h ∈ L ∞ (Ω), define<br />
⎛<br />
⎞<br />
‖h‖ ∗ = sup ∣<br />
x∈Ω<br />
−1<br />
∣ m∑ δ<br />
⎝<br />
j ⎠<br />
j=1 (δj 2 + |x − ξ h(x) ∣∣ . (2.18)<br />
j| 2 ) 3 2<br />
We conclude this section by proving an estimate of R in ‖ · ‖ ∗ .<br />
Proposition 2.1. For fixed ε > 0, there exist C > 0 and p 0 > 0 such that<br />
for any ξ ∈ O ε and p ≥ p 0<br />
‖∆U ξ + U p ξ ‖ ∗ ≤ C p 4 . (2.19)<br />
Proof Observe that<br />
m∑ 1<br />
∆U ξ =<br />
j=1 γµ j<br />
m∑ 1<br />
=<br />
j=1 γµ<br />
2<br />
p−1<br />
2<br />
p−1<br />
j<br />
(<br />
(<br />
−e U j<br />
+ 1<br />
pδj<br />
2 ∆w 0 ( x − ξ j<br />
) + 1<br />
δ j p 2 δj<br />
2 ∆w 1 ( x − ξ )<br />
j<br />
)<br />
δ j<br />
−e U j<br />
+ 1<br />
pδ 2 j<br />
˜f 0 ( x − ξ j<br />
) + 1<br />
δ j p 2 δj<br />
2<br />
)<br />
− 1 p eU j<br />
w 0 ( x − ξ j<br />
) − 1 δ j p 2 eU j<br />
w 1 ( x − ξ j<br />
)<br />
δ j<br />
˜f 1 ( x − ξ j<br />
δ j<br />
)<br />
, (2.20)<br />
where, for j = 1, . . . , m, U j = U δj ,ξ j<br />
and, for i = 0, 1, ˜fi (y) =<br />
1<br />
(1+|y| 2 ) 2 f i (y)<br />
with f 0 , f 1 given in (2.6). By (2.2) and (2.11), formula (2.20) gives that, if
|x − ξ j | ≥ ε for any j = 1, . . . , m,<br />
⎛<br />
⎞−1 m∑<br />
∣<br />
δ (<br />
⎝<br />
j ⎠<br />
(δj 2 + |x − ξ ∆U<br />
j| 2 ) 3 ξ + U p ξ<br />
2<br />
j=1<br />
)<br />
(x) ∣ (<br />
≤ Ce p 4 ( C )<br />
p )p + pe − p 2<br />
13<br />
= O(pe − p 4 ), (2.21)<br />
and, if |x − ξ i | ≤ ε for some i = 1, . . . , m,<br />
|∆U ξ + U p ξ | = ∣ (<br />
1<br />
8<br />
γδi 2 2<br />
−<br />
µ<br />
p−1 (1 + |y| 2 ) 2 + 1 p ˜f 0 (y) + 1 p ˜f 2 1 (y) (2.22)<br />
i<br />
− 1 8<br />
p (1 + |y| 2 ) 2 w 0(y) − 1 )<br />
8<br />
p 2 (1 + |y| 2 ) 2 w 1(y)<br />
+U p ξ (δ iy + ξ i ) + O(pe − p 2 )<br />
∣ ∣ ,<br />
where we denote y = x−ξ i<br />
δ i<br />
. By (2.14) we deduce that, for x = δ i y + ξ i<br />
(<br />
U p ξ (x) = ( p<br />
2<br />
) p 1 + 1<br />
p−1 p v ∞(y) + 1 p<br />
γµ<br />
2 w 0(y) + 1 p p ) p<br />
p 3 w 1(y) + O( e− 4<br />
p |y| + e− 4<br />
p ) .<br />
Since (<br />
i<br />
p<br />
2<br />
p−1<br />
γµ i<br />
U p ξ (x) = 1<br />
) p =<br />
1<br />
2 , by (2.10) we get for |x − ξ i | ≤ ε √ δ i<br />
γδi 2µ p−1<br />
i<br />
(<br />
γδ 2 i µ 2<br />
p−1<br />
i<br />
8<br />
(1 + |y| 2 ) 2 [1 + 1 p<br />
w 0 (y) − 1 )<br />
8<br />
2 log2 (<br />
(1 + |y| 2 ) 2 )<br />
+ 1 8<br />
(w<br />
p 2 1 − log(<br />
(1 + |y| 2 ) 2 )w 0 + 1 8<br />
3 log3 (<br />
(1 + |y| 2 ) 2 )<br />
+ w2 0<br />
2 + 1 8<br />
8 log4 (<br />
(1 + |y| 2 ) 2 ) − w )<br />
0 8<br />
2 log2 (<br />
(1 + |y| 2 ) 2 )<br />
(<br />
log 6 )]<br />
(|y| + 2)<br />
+O<br />
p 3 + p 2 e − p 4 y + p 2 e − p 4 , y = x − ξ i<br />
.<br />
δ i<br />
Hence, in this region we obtain that<br />
⎛<br />
⎞−1 m∑<br />
∣<br />
δ (<br />
⎝<br />
j ⎠<br />
(δj 2 + |x − ξ ∆U<br />
j| 2 ) 3 ξ + U p ξ<br />
2<br />
j=1<br />
≤ ∣ (δ2 i + |x − ξ i| 2 ) 3 2<br />
(∆U ξ + U p )<br />
ξ<br />
(x) ∣ δ i<br />
)<br />
(x) ∣ ∣ (2.23)<br />
≤ C (<br />
γ (1 + |y|2 ) 3 1 log 6 )<br />
(|y| + 2)<br />
2 O<br />
p 3 (1 + |y| 2 ) 2 ≤ C p 4 , y = x − ξ i<br />
.<br />
δ i<br />
On the other hand, if ε √ δ i ≤ |x − ξ i | ≤ ε we have that<br />
( p<br />
)<br />
U p e<br />
ξ (x) = O 2 1<br />
γ (1 + |y| 2 ) 2 , y = x − ξ i<br />
,<br />
δ i
14 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
since (1 + s p )p ≤ e s . Thus, in this region<br />
⎛<br />
⎞−1 m∑<br />
∣<br />
δ ( )<br />
⎝<br />
j ⎠<br />
j=1 (δj 2 + |x − ξ ∆U<br />
j| 2 ) 3 ξ + U p ξ<br />
(x) ∣ (2.24)<br />
2<br />
(<br />
)<br />
p<br />
= O<br />
≤ Cpe − p x − ξ i<br />
(1 + |y| 2 ) 1 8 , y = .<br />
2<br />
δ i<br />
By (2.21), (2.23) and (2.24) we obtain the desired result.<br />
3. Analysis of the linearized operator<br />
In this Section, we prove bounded invertibility of the operator L, uniformly<br />
on ξ ∈ O ε , by using L ∞ -norms introduced in (2.18). Let us recall that<br />
L(φ) = ∆φ + W ξ φ, where W ξ (x) = pU p−1<br />
ξ<br />
(x). For simplicity of notation, we<br />
will omit the dependence of W ξ on ξ.<br />
As in Proposition 2.1, we have for the potential W (x) the following expansions.<br />
If |x − ξ i | ≤ ε for some i = 1, . . . , m<br />
p<br />
W (x) = p(<br />
γµ<br />
2<br />
p−1<br />
i<br />
+O( e− p 4<br />
p |y| + e− p 4<br />
p ) ) p−1<br />
= δ −2<br />
i<br />
(<br />
) p−1 (1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) (3.1)<br />
1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4<br />
p |y| + e− p 4<br />
p ) ) p−1<br />
where again we use the notation y = x−ξ i<br />
δ i<br />
. In this region, we have that<br />
W (x) ≤ C δi<br />
2 e v∞(y) e − 1 p v∞(y) ( )<br />
= O e U i(x)<br />
,<br />
since v ∞ (y) ≥ −2p. Indeed, by Taylor expansions of exponential and logarithmic<br />
functions as in (2.10), we obtain that, if |x − ξ i | ≤ ε √ δ i (and<br />
|y| ≤ √ ε<br />
δi<br />
),<br />
W (x) = δ −2<br />
i<br />
=<br />
(<br />
1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4<br />
p |y| + e− p 4<br />
p ) ) p−1<br />
(<br />
8<br />
δi 2(1 + |y|2 ) 2 1 + 1 p (w 0 − v ∞ − 1 2 v2 ∞) + O( log4 (|y| + 2)<br />
p 2 )<br />
If |x − ξ i | ≥ ε for any i = 1, . . . , m:<br />
Summing up, we have that<br />
W (x) = O(p( C p )p−1 ).<br />
)<br />
.<br />
□
Lemma 3.1. Let ε > 0 be fixed. There exist D 0 > 0 and p 0 > 0 such that<br />
W (x) ≤ D 0 m ∑<br />
e U j(x)<br />
j=1<br />
for any ξ ∈ O ε and p ≥ p 0 . Furthermore,<br />
(<br />
8<br />
W (x) =<br />
δi 2(1 + |y|2 ) 2 1 + 1 p (w 0 − v ∞ − 1 )<br />
2 v2 ∞) + O( log4 (|y| + 2)<br />
p 2 )<br />
for any |x − ξ i | ≤ ε √ δ i , where y = x−ξ i<br />
δ i<br />
.<br />
Remark 3.1. As for W , let us point out that, if |x − ξ i | ≤ ε for some<br />
i = 1, . . . , m, there holds<br />
p(U ξ + O( 1 p<br />
p 3 ))p−2 ≤ Cp(<br />
2<br />
) p−2 e p−2<br />
p v ∞( x−ξ i ) δ i = O<br />
(e ) U i(x)<br />
.<br />
p−1<br />
γµ<br />
i<br />
Since this estimate is true if |x − ξ i | ≥ ε for any i = 1, . . . , m, we have that<br />
p(U ξ + O( 1 m∑<br />
p 3 ))p−2 ≤ C e Uj(x) . (3.2)<br />
j=1<br />
In an heuristic way, the operator L is close to ˜L defined by<br />
( m<br />
)<br />
∑<br />
˜L(φ) = ∆φ + φ.<br />
e U i<br />
i=1<br />
The operator ˜L is ”essentially” a superposition of linear operators which,<br />
after a dilation and translation, approach, as p → ∞, the linear operator in<br />
R 2 ,<br />
8<br />
φ → ∆φ +<br />
(1 + |y| 2 ) 2 φ,<br />
namely, equation ∆v+e v 8<br />
= 0 linearized around the radial solution log .<br />
(1+|y| 2 ) 2<br />
Set<br />
z 0 (y) = |y|2 − 1<br />
|y| 2 + 1 , z y i<br />
i(y) = 4<br />
1 + |y| 2 i = 1, 2.<br />
The first ingredient to develop the desired solvability theory for L is the well<br />
known fact that any bounded solution of L(φ) = 0 in R 2 is precisely a linear<br />
combination of the z i , i = 0, 1, 2, see [3] for a proof.<br />
The second ingredient is a detailed analysis of L − ˜L. It has been proved in<br />
[12] and [14] that the operator ˜L is invertible in the set of functions which,<br />
roughly speaking, are orthogonal to the functions z i for i = 1, 2, and the<br />
operatorial norm of ˜L −1 behaves like p as p → +∞. Since L is close to ˜L up<br />
to terms of order at least 1 p<br />
(see Lemma 3.1), the invertibility of L becomes<br />
delicate and non trivial.<br />
In [12] and [14] there were established a priori estimates respectively in<br />
weighted L ∞ −norms and in H0 1 (Ω)−norms. We will follow the approach in<br />
[12] since the estimates there are stronger and in this context very helpful.<br />
15
16 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Given h ∈ C(¯Ω), we consider the linear problem of finding a function<br />
φ ∈ W 2,2 (Ω) such that<br />
2∑ m∑<br />
L(φ) = h + c ij e U j<br />
Z ij , in Ω, (3.3)<br />
i=1 j=1<br />
∫<br />
φ = 0, on ∂Ω, (3.4)<br />
e U j<br />
Z ij φ = 0, for all i = 1, 2, j = 1, . . . , m, (3.5)<br />
Ω<br />
for some coefficients c ij , i = 1, 2 and j = 1, . . . , m. Here and in the sequel,<br />
for any i = 0, 1, 2 and j = 1, . . . , m we denote<br />
⎧<br />
⎪⎨<br />
Z ij (x) := z i ( x − ξ j<br />
δ j<br />
) =<br />
The main result of this Section is the following:<br />
⎪⎩<br />
|x−ξ j | 2 −δ 2 j<br />
δ 2 j +|x−ξ j| 2 if i = 0<br />
4δ j (x−ξ j ) i<br />
δ 2 j +|x−ξ j| 2 if i = 1, 2.<br />
Proposition 3.1. Let ε > 0 be fixed. There exist p 0 > 0 and C > 0 such<br />
that, for h ∈ C(¯Ω) there is a unique solution to problem (3.3)-(3.5), for any<br />
p > p 0 and ξ ∈ O ε , which satisfies<br />
‖φ‖ ∞ ≤ Cp‖h‖ ∗ . (3.6)<br />
Proof. The proof of this result consists of six steps.<br />
1 st Step. The operator L satisfies the maximum principle in ˜Ω := Ω \<br />
∪ m j=1 B(ξ j, Rδ j ) for R large, independent on p. Namely,<br />
if L(ψ) ≤ 0 in ˜Ω and ψ ≥ 0 on ∂ ˜Ω, then ψ ≥ 0 in ˜Ω.<br />
In order to prove this fact, we show the existence of a positive function Z<br />
in ˜Ω satisfying L(Z) < 0. We define Z to be<br />
( )<br />
m∑ a(x − ξj )<br />
Z(x) = z 0 , a > 0.<br />
δ j<br />
j=1<br />
First, observe that, if |x−ξ j | ≥ Rδ j for R > 1 a<br />
, then Z(x) > 0. On the other<br />
hand, we have:<br />
⎛<br />
m∑<br />
W (x)Z(x) ≤ D 0 ⎝<br />
e U j(x)<br />
j=1<br />
⎞<br />
⎠ Z(x) ≤ D 0 Z(x)<br />
m∑<br />
j=1<br />
8δ 2 j<br />
|x − ξ j | 4 ,<br />
where D 0 is the constant in Lemma 3.1. Further,by definition of z 0 ,<br />
m∑<br />
−∆Z(x) = a 2 8δ2 j (a2 |x − ξ j | 2 − δj 2)<br />
(a 2 |x − ξ j | 2 + δj 2)3<br />
≥ 1 3<br />
j=1<br />
m∑<br />
j=1<br />
8a 2 δ 2 j<br />
(a 2 |x − ξ j | 2 + δ 2 j )2 ≥ 4 27<br />
m∑<br />
j=1<br />
8δ 2 j<br />
a 2 |x − ξ j | 4
provided R ><br />
√<br />
2<br />
a . Hence<br />
LZ(x) ≤<br />
(<br />
− 4<br />
27<br />
)<br />
1<br />
a 2 + D ∑ m<br />
0<br />
j=1<br />
8δ 2 j<br />
|x − ξ j | 4 < 0<br />
provided that a is chosen sufficiently small, but independent of p.<br />
function Z(x) is what we are looking for.<br />
17<br />
The<br />
2 nd Step. Let R be as before. Let us define the “inner norm” of φ in the<br />
following way<br />
‖φ‖ i = sup<br />
x∈∪ m j=1 B(ξ j,Rδ j )<br />
|φ|(x).<br />
We claim that there is a constant C > 0 such that, if L(φ) = h in Ω,<br />
h ∈ C 0,α (¯Ω), then<br />
‖φ‖ ∞ ≤ C[‖φ‖ i + ‖h‖ ∗ ],<br />
for any h ∈ C 0,α (¯Ω). We will establish this estimate with the use of suitable<br />
barriers. Let M = 2 diam Ω. Consider the solution ψ j (x) of the problem:<br />
{<br />
−∆ψj = 2δ j<br />
|x−ξ j<br />
in Rδ<br />
| 3 j < |x − ξ j | < M<br />
ψ j (x) = 0 on |x − ξ j | = Rδ j and |x − ξ j | = M.<br />
Namely, the function ψ j (x) is the positive function defined by:<br />
where<br />
and<br />
ψ j (x) = −<br />
2δ j<br />
|x − ξ j | + A + B log |x − ξ j|,<br />
B = 2( δ j<br />
M − 1 R ) 1<br />
log( M Rδ j<br />
) < 0<br />
A = 2δ j<br />
− B log M.<br />
M<br />
Hence, the function ψ j (x) is uniformly bounded from above by a constant<br />
independent of p, since we have that, for Rδ j ≤ |x − ξ j | ≤ M,<br />
Define now the function<br />
ψ j (x) ≤ A + B log(Rδ j ) = 2δ j<br />
M − B log M Rδ j<br />
= 2 R .<br />
∑<br />
˜φ(x) m = 2‖φ‖ i Z(x) + ‖h‖ ∗ ψ j (x),<br />
where Z was defined in the previous Step. First of all, observe that by the<br />
definition of Z, choosing R larger if necessary,<br />
˜φ(x) ≥ 2‖φ‖ i Z(x) ≥ ‖φ‖ i ≥ |φ|(x) for |x − ξ j | = Rδ j , j = 1, . . . , m,<br />
and, by the positivity of Z(x) and ψ j (x),<br />
j=1<br />
˜φ(x) ≥ 0 = |φ|(x) for x ∈ ∂Ω.
18 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Since by definition of ‖ · ‖ ∗ we have that<br />
⎛<br />
⎞<br />
m∑ δ<br />
⎝<br />
j ⎠<br />
(δj 2 + |x − ξ ‖h‖<br />
j| 2 ) 3 ∗ ≥ |h(x)|, (3.7)<br />
2<br />
j=1<br />
finally we obtain that<br />
L ˜φ<br />
∑ m m (<br />
∑<br />
≤ ‖h‖ ∗ Lψ j (x) = ‖h‖ ∗ −<br />
2δ )<br />
j<br />
|x − ξ<br />
j=1<br />
j=1<br />
j | 3 + W (x)ψ j(x)<br />
m (<br />
∑<br />
≤ ‖h‖ ∗ −<br />
2δ j<br />
|x − ξ<br />
j=1<br />
j | 3 + 2mD )<br />
0<br />
R<br />
eU j(x)<br />
⎛<br />
⎞<br />
m∑ δ<br />
≤ −‖h‖ ∗ ⎝<br />
j ⎠<br />
(δj 2 + |x − ξ ≤ −|h(x)| ≤ |Lφ|(x)<br />
j| 2 ) 3 2<br />
j=1<br />
provided R > 16mD 0 and p large enough. Hence, by the maximum principle<br />
in Step 1 we obtain that<br />
|φ|(x) ≤ ˜φ(x) for x ∈ ˜Ω,<br />
and therefore, since Z(x) ≤ 1 and ψ j (x) ≤ 2 R ,<br />
‖φ‖ ∞ ≤ C[‖φ‖ i + ‖h‖ ∗ ].<br />
3 rd Step. We prove uniform a priori estimates for solutions φ of problem<br />
Lφ = h in Ω, φ = 0 on ∂Ω, when h ∈ C 0,α (¯Ω) and φ satisfies (3.5) and in<br />
addition the orthogonality conditions:<br />
∫<br />
e U j<br />
Z 0j φ = 0, for j = 1, . . . , m. (3.8)<br />
Ω<br />
Namely, we prove that there exists a positive constant C such that for any<br />
ξ ∈ O ε and h ∈ C 0,α (¯Ω)<br />
‖φ‖ ∞ ≤ C‖h‖ ∗ ,<br />
for p sufficiently large. By contradiction, assume the existence of sequences<br />
p n → ∞, points ξ n ∈ O ε , functions h n and associated solutions φ n such that<br />
‖h n ‖ ∗ → 0 and ‖φ n ‖ ∞ = 1.<br />
Since ‖φ n ‖ ∞ = 1, Step 2 shows that lim inf n→+∞ ‖φ n ‖ i > 0. Let us set<br />
ˆφ n j (y) = φ n(δj ny + ξn j ) for j = 1, . . . , m. By Lemma 3.1 and (3.7), elliptic<br />
estimates readily imply that ˆφ n j converges uniformly over compact sets to a<br />
bounded solution ˆφ ∞ j of the equation in R 2 :<br />
∆φ +<br />
8<br />
(1 + |y| 2 ) 2 φ = 0.<br />
This implies that ˆφ ∞ j is a linear combination of the functions z i , i = 0, 1, 2.<br />
Since ‖ ˆφ n j ‖ ∞ ≤ 1, by Lebesgue theorem the orthogonality conditions (3.5)<br />
and (3.8) on φ n pass to the limit and give ∫ R 2 8<br />
(1+|y| 2 ) 2 z i (y) ˆφ ∞ j = 0 for any i =
0, 1, 2. Hence, ˆφ ∞ j ≡ 0 for any j = 1, . . . , m contradicting lim inf n→+∞ ‖φ n ‖ i ><br />
0.<br />
4 th Step. We prove that there exists a positive constant C > 0 such that<br />
any solution φ of equation Lφ = h in Ω, φ = 0 on ∂Ω, satisfies<br />
‖φ‖ ∞ ≤ Cp‖h‖ ∗ ,<br />
when h ∈ C 0,α (¯Ω) and we assume on φ only the orthogonality conditions<br />
(3.5). Proceeding by contradiction as in Step 3, we can suppose further that<br />
p n ‖h n ‖ ∗ → 0 as n → +∞ (3.9)<br />
but we loss in the limit the condition ∫ 8<br />
R 2 z<br />
(1+|y| 2 ) 2 0 (y) ˆφ ∞ j<br />
have that<br />
19<br />
= 0. Hence, we<br />
ˆφ n j → C j<br />
|y| 2 − 1<br />
|y| 2 + 1 in C0 loc (R2 ) (3.10)<br />
for some constants C j . To reach a contradiction, we have to show that<br />
C j = 0 for any j = 1, . . . , m. We will obtain it from the stronger condition<br />
(3.9) on h n .<br />
To this end, we perform the following construction. By Lemma 2.1, we<br />
8<br />
find radial solutions w and t respectively of equations ∆w + w =<br />
(1+|y| 2 ) 2<br />
8<br />
8<br />
8<br />
z<br />
(1+|y| 2 ) 2 0 (y) and ∆t + t = in R<br />
(1+|y| 2 ) 2 (1+|y| 2 ) 2 , such that as |y| → +∞<br />
2<br />
w(y) = 4 3<br />
1<br />
1<br />
log |y| + O( ) , t(y) = O(<br />
|y| |y| ),<br />
since 8 ∫ +∞<br />
0<br />
t (t2 −1) 2<br />
dt = 4 (t 2 +1) 4 3 and 8 ∫ +∞<br />
0<br />
t t2 −1<br />
dt = 0.<br />
(t 2 +1) 3<br />
For simplicity, from now on we will omit the dependence on n. For j =<br />
1, . . . , m, define now<br />
u j (x) = w( x − ξ j<br />
) + 4 δ j 3 (log δ j)Z 0j (x) + 8π 3 H(ξ j, ξ j )t( x − ξ j<br />
)<br />
δ j<br />
and denote by P u j the projection of u j onto H0 1(Ω). Since u j − P u j +<br />
4<br />
3 log 1<br />
|·−ξ j |<br />
= O(δ j ) on ∂Ω (together with boundary derivatives), by harmonicity<br />
we get<br />
The function P u j solves<br />
P u j = u j − 8π 3 H(·, ξ j) + O(e − p 4 ) in C 1 (¯Ω),<br />
P u j = − 8π 3 G(·, ξ j) + O(e − p 4 ) in C 1 loc (¯Ω \ {ξ j }).<br />
∆P u j + W (x)P u j = e U j<br />
Z 0j +<br />
(3.11)<br />
(<br />
)<br />
W (x) − e U j<br />
P u j + R j , (3.12)<br />
where<br />
R j (x) =<br />
(<br />
P u j − u j + 8π )<br />
3 H(ξ j, ξ j ) e U j<br />
.
20 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Multiply (3.12) by φ and integrate by parts to obtain:<br />
∫<br />
∫ (<br />
) ∫ ∫<br />
e U j<br />
Z 0j φ + W (x) − e U j<br />
P u j φ = P u j h −<br />
Ω<br />
Ω<br />
First of all, by Lebesgue theorem and (3.10) we get that<br />
∫<br />
∫<br />
e U j<br />
8(|y| 2 − 1) 2<br />
Z 0j φ → C j<br />
Ω<br />
R 2<br />
The more delicate term is ∫ Ω<br />
we have that<br />
∫<br />
Ω<br />
− 8π 3<br />
(<br />
W (x) − e U j<br />
(<br />
) ∫<br />
W (x) − e U j<br />
P u j φ =<br />
∑<br />
k≠j<br />
= 4 log δ j<br />
3 p<br />
− 8π 3<br />
∑<br />
k≠j<br />
= − C j<br />
3<br />
Ω<br />
Ω<br />
R j φ. (3.13)<br />
(1 + |y| 2 ) 4 = 8π 3 C j. (3.14)<br />
)<br />
P u j φ. By Lemma 3.1 and (3.11)<br />
B(ξ j ,ε √ δ j )<br />
∫<br />
G(ξ k , ξ j )<br />
B(ξ k ,ε √ W (x)φ + O(e − p 8 )<br />
δ k )<br />
∫<br />
B(0, √ ε )<br />
δ j<br />
∫<br />
G(ξ k , ξ j )<br />
(<br />
)<br />
W (x) − e U j<br />
P u j φ<br />
(<br />
8<br />
(1 + |y| 2 ) 2 w 0 − v ∞ − 1 2 v2 ∞<br />
ε<br />
B(0, √ ) δk<br />
8<br />
(1 + |y| 2 ) 2 ˆφ k + O( 1 p )<br />
)<br />
z 0 (y) ˆφ j<br />
8(|y|<br />
∫R 2 − 1) 2 (<br />
2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 )<br />
2 v2 ∞ (y) + o(1)<br />
since Lebesgue theorem and (3.10) imply:<br />
∫<br />
and<br />
∫<br />
B(0, √ ε )<br />
δ j<br />
ε<br />
B(0, √ ) δk<br />
(<br />
8<br />
(1 + |y| 2 ) 2 w 0 − v ∞ − 1 2 v2 ∞<br />
)<br />
z 0 (y) ˆφ j →<br />
∫<br />
8(|y| 2 − 1) 2 (<br />
C j<br />
R 2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 )<br />
2 v2 ∞<br />
8<br />
(1 + |y| 2 ) ˆφ<br />
∫<br />
2 k → C k<br />
R 2 8<br />
(1 + |y| 2 ) 2 |y| 2 − 1<br />
|y| 2 + 1 = 0.<br />
In a straightforward but tedious way, by (2.8) we can compute:<br />
8(|y|<br />
∫R 2 − 1) 2 (<br />
2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 )<br />
2 v2 ∞ (y) = −8π,<br />
so that we obtain ∫ (<br />
)<br />
W (x) − e U j<br />
P u j φ = 8π<br />
Ω<br />
3 C j + o(1). (3.15)<br />
As far as the R.H.S. in (3.13), we have that by (3.11)<br />
∫<br />
∫<br />
)<br />
m∑ δ k<br />
| P u j h| = O<br />
(‖h‖ ∗ (<br />
Ω<br />
Ω (δk 2 + |x − ξ )|P u<br />
k| 2 ) 3 j | = O(p‖h‖ ∗ ) (3.16)<br />
2<br />
k=1
21<br />
since ∫ Ω |P u j| = O(| log δ j |) = O(p) and<br />
∫<br />
∫<br />
δ j<br />
1<br />
B(ξ j ,ε) (δj 2 + |x − ξ |u<br />
j| 2 ) 3 j | ≤<br />
|u<br />
2 R 2 (1 + |y| 2 ) 3 j |(δ j y + ξ j ) = O(p).<br />
2<br />
Finally, by (3.11)<br />
∫<br />
R j φ = O<br />
Ω<br />
(∫<br />
Ω<br />
e U j<br />
(|x − ξ j | + e − p 4 )<br />
)<br />
= O(e − p 4 ). (3.17)<br />
Hence, inserting (3.14)-(3.17) in (3.13) we obtain that<br />
16π<br />
3 C j = o(1)<br />
for any j = 1, . . . , m. Necessarily, C j = 0 and the claim is proved.<br />
5 th Step. We establish the validity of the a priori estimate:<br />
‖φ‖ ∞ ≤ Cp‖h‖ ∗ (3.18)<br />
for solutions of problem (3.3)-(3.5) and h ∈ C 0,α (¯Ω).<br />
gives<br />
⎛<br />
⎞<br />
2∑ m∑<br />
‖φ‖ ∞ ≤ Cp ⎝‖h‖ ∗ + |c ij | ⎠<br />
since<br />
i=1 j=1<br />
‖e U j<br />
Z ij ‖ ∗ ≤ 2‖e U j<br />
‖ ∗ ≤ 16.<br />
The previous Step<br />
Hence, arguing by contradiction of (3.18), we can proceed as in Step 3 and<br />
suppose further that<br />
2∑ m∑<br />
p n ‖h n ‖ ∗ → 0 , p n |c n ij| ≥ δ > 0 as n → +∞.<br />
i=1 j=1<br />
We omit the dependence on n. It suffices to estimate the values of the<br />
constants c ij . For i = 1, 2 and j = 1, . . . , m, multiply (3.3) by P Z ij and,<br />
integrating by parts, get:<br />
2∑<br />
m∑<br />
l=1 h=1<br />
∫<br />
( )<br />
c lh P Zlh , P Z ij + hP Z<br />
H 1 ij =<br />
0 Ω<br />
−<br />
∫<br />
Ω<br />
∫<br />
Ω<br />
W (x)φP Z ij (3.19)<br />
e U j<br />
Z ij φ,<br />
since ∆P Z ij = ∆Z ij = −e U j<br />
Z ij . For i = 1, 2 and j = 1, . . . , m we have the<br />
following expansions:<br />
( )<br />
∂H<br />
P Z ij = Z ij − 8πδ j ∂(ξ j ) i<br />
(·, ξ j ) + O δj<br />
3 ( ) (3.20)<br />
P Z 0j = Z 0j − 1 + O<br />
δ 2 j
22 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
in C 1 (¯Ω) and<br />
( )<br />
∂G<br />
P Z ij = −8πδ j ∂(ξ j ) i<br />
(·, ξ j ) + O δj<br />
3 ( )<br />
P Z 0j = O δj<br />
2 (3.21)<br />
in C 1 loc (¯Ω \ {ξ j }). By (3.20)-(3.21) we deduce the following “orthogonality”<br />
relations: for i, l = 1, 2 and j, h = 1, . . . , m with j ≠ h,<br />
(<br />
)<br />
(P Z ij , P Z lj = 64 ∫ )<br />
|y| 2<br />
δ<br />
H 1 0 (Ω) (1+|y| 2 ) 4 il + O(δj 2)<br />
)<br />
R (P 2 Z ij , P Z lh = O(δ jδ h ) (3.22)<br />
H 1 0 (Ω)<br />
and<br />
(P Z 0j , P Z lj<br />
)<br />
H 1 0 (Ω) = O(δ2 j )<br />
(P Z 0j , P Z lh<br />
)<br />
H 1 0 (Ω) = O(δ jδ h )<br />
(3.23)<br />
uniformly on ξ ∈ O ε , where δ il denotes the Kronecker’s symbol. In fact, we<br />
have that<br />
∫<br />
(P Z ij , P Z lj<br />
)H = e U j<br />
Z ij P Z lj<br />
1<br />
0 (Ω) Ω<br />
∫<br />
(<br />
)<br />
= e U j<br />
∂H<br />
Z ij Z lj − 8πδ j (ξ j , ξ j ) + O(δ j |x − ξ j | + δj 3 ) + O(δj 4 )<br />
∂(ξ j ) l<br />
B(ξ j ,ε)<br />
∫<br />
= 128<br />
R 2<br />
B(ξ j ,ε)<br />
y i y l<br />
(1 + |y| 2 ) 4 + O(δ2 j ) =<br />
⎛<br />
∫<br />
|y|<br />
⎝64<br />
2<br />
R 2<br />
(1 + |y| 2 ) 4 ⎞<br />
⎠ δ il + O(δ 2 j ),<br />
∫<br />
(P Z ij , P Z lh<br />
)H = e U j<br />
Z ij P Z lh<br />
1<br />
0 (Ω) Ω<br />
∫<br />
(<br />
)<br />
= e U j<br />
∂G<br />
Z ij −8πδ h (ξ j , ξ h ) + O(δ h |x − ξ j | + δ 3<br />
∂(ξ h )<br />
h) + O(δj 3 )<br />
l<br />
= O(δ j δ h ),<br />
∫<br />
(P Z 0j , P Z lj<br />
)H = e U j<br />
Z 0j P Z lj<br />
1<br />
0 (Ω) Ω<br />
∫<br />
(<br />
)<br />
= e U j<br />
∂H<br />
Z 0j Z lj − 8πδ j (ξ j , ξ j ) + O(δ j |x − ξ j | + δj 3 ) + O(δj 3 )<br />
∂(ξ j ) l<br />
B(ξ j ,ε)<br />
= O(δ 2 j )
23<br />
and<br />
∫<br />
(P Z 0j , P Z lh<br />
)H = e U j<br />
Z 0j P Z lh<br />
1<br />
0 (Ω) Ω<br />
∫<br />
(<br />
)<br />
= e U j<br />
∂G<br />
Z 0j −8πδ h (ξ j , ξ h ) + O(δ h |x − ξ j | + δ 3<br />
∂(ξ h )<br />
h) + O(δj 2 )<br />
l<br />
B(ξ j ,ε)<br />
= O(δ j δ h ).<br />
Now, since<br />
∫<br />
|<br />
∫<br />
hP Z ij | ≤ C ′ |h| ≤ C‖h‖ ∗ ,<br />
Ω<br />
Ω<br />
by (3.22) the L.H.S. of (3.19) is estimated as follows:<br />
L.H.S. = Dc ij + O ( e − p 2<br />
2∑ m∑<br />
|c lh | ) + O(‖h‖ ∗ ), (3.24)<br />
l=1 h=1<br />
where D = 64 ∫ R 2 |y| 2<br />
(1+|y| 2 ) 4 . Moreover, by Lemma 3.1 the R.H.S. of (3.19)<br />
takes the form:<br />
R.H.S. =<br />
=<br />
∫<br />
√<br />
B(ξ j ,ε δ j )<br />
∫<br />
= 1 p<br />
B(0, √ ε )<br />
δ j<br />
∫<br />
B(ξ j ,ε √ δ j )<br />
(<br />
W (x) − e U j<br />
∫<br />
√<br />
W (x)φP Z ij − e U j<br />
φZ ij + O( δ j ‖φ‖ ∞ ) (3.25)<br />
Ω<br />
) ∫<br />
√<br />
φP Z ij + e U j<br />
φ (P Z ij − Z ij ) + O( δ j ‖φ‖ ∞ )<br />
Ω<br />
(<br />
)<br />
32y i<br />
(1 + |y| 2 ) 3 w 0 − v ∞ − v2 ∞ ˆφ j + O( 1 2 p 2 ‖φ‖ ∞)<br />
in view of (3.20), where ˆφ j (y) = φ(δ j y + ξ j ). Inserting the estimates (3.24)<br />
and (3.25) into (3.19), we deduce that<br />
Dc ij + O ( e − p 2<br />
Hence, we obtain that<br />
2∑ m∑<br />
|c lh | ) = O(‖h‖ ∗ + 1 p ‖φ‖ ∞).<br />
l=1 h=1<br />
2∑<br />
l=1 h=1<br />
m∑<br />
|c lh | = O(‖h‖ ∗ + 1 p ‖φ‖ ∞). (3.26)<br />
Since ∑ 2<br />
l=1<br />
∑ mh=1<br />
|c lh | = o(1), as in Step 4 we have that<br />
ˆφ j → C j<br />
|y| 2 − 1<br />
|y| 2 + 1 in C0 loc (R2 )
24 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
for some constant C j , j = 1, . . . , m. Hence, in (3.25) we have a better<br />
estimate since by Lebesgue theorem the term<br />
∫<br />
(<br />
32y i<br />
(1 + |y| 2 ) 3 w 0 − v ∞ − 1 )<br />
2 v2 ∞ (y) ˆφ j (y)<br />
converges to<br />
B(0, √ ε )<br />
δ j<br />
∫<br />
32y i (|y| 2 (<br />
− 1)<br />
C j<br />
(1 + |y| 2 ) 4 w 0 − v ∞ − 1 )<br />
2 v2 ∞ (y) = 0.<br />
R 2<br />
Therefore, we get that the R.H.S. in (3.19) satisfies: R.H.S. = o( 1 p<br />
), and in<br />
turn, ∑ 2 ∑ mh=1<br />
l=1 |c lh | = O(‖h‖ ∗ ) + o( 1 p<br />
). This contradicts<br />
and the claim is established.<br />
2∑ m∑<br />
p |c ij | ≥ δ > 0,<br />
i=1 j=1<br />
6 th Step. We prove the solvability of (3.3)–(3.5). To this purpose, we<br />
consider the spaces:<br />
and<br />
2∑ m∑<br />
K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m}<br />
i=1 j=1<br />
∫<br />
Kξ ⊥ = {φ ∈ L 2 (Ω) :<br />
Let Π ξ : L 2 (Ω) → K ξ defined as:<br />
Ω<br />
e U j<br />
Z ij φ = 0 for i = 1, 2, j = 1, . . . , m}.<br />
2∑ m∑<br />
Π ξ φ = c ij P Z ij ,<br />
i=1 j=1<br />
where c ij are uniquely determined (as it follows by (3.22)-(3.23)) by the<br />
system:<br />
∫<br />
Ω<br />
e U h<br />
Z lh<br />
⎛<br />
⎝φ −<br />
2∑<br />
m∑<br />
i=1 j=1<br />
c ij P Z ij<br />
⎞<br />
⎠ = 0 for any l = 1, 2 , h = 1, . . . , m.<br />
Let Π ⊥ ξ = Id − Π ξ : L 2 (Ω) → Kξ ⊥ . Problem (3.3)–(3.5), expressed in a weak<br />
form, is equivalent to find φ ∈ Kξ<br />
⊥ ∩ H1 0 (Ω) such that<br />
∫<br />
(φ, ψ) H 1<br />
0 (Ω) = (W φ − h) ψ dx, for all ψ ∈ Kξ ⊥ ∩ H0 1 (Ω).<br />
Ω<br />
With the aid of Riesz’s representation theorem, this equation gets rewritten<br />
in Kξ<br />
⊥ ∩ H1 0 (Ω) in the operatorial form<br />
(Id − K)φ = ˜h, (3.27)
where ˜h = Π ⊥ ξ ∆−1 h and K(φ) = −Π ⊥ ξ ∆−1 (W φ) is a linear compact operator<br />
in Kξ ⊥ ∩ H1 0 (Ω). The homogeneous equation φ = K(φ) in K⊥ ξ ∩ H1 0 (Ω),<br />
which is equivalent to (3.3)-(3.5) with h ≡ 0, has only the trivial solution in<br />
view of the a priori estimate (3.18). Now, Fredholm’s alternative guarantees<br />
unique solvability of (3.27) for any ˜h ∈ Kξ ⊥ . Moreover, by elliptic regularity<br />
theory this solution is in W 2,2 (Ω).<br />
At p > p 0 fixed, by density of C 0,α (¯Ω) in (C(¯Ω), ‖ · ‖ ∞ ), we can approximate<br />
h ∈ C(¯Ω) by smooth functions and, by (3.18) and elliptic regularity theory,<br />
we can show that (3.6) holds for any h ∈ C(¯Ω). The proof is complete. □<br />
Remark 3.2. Given h ∈ C(¯Ω), let φ be the solution of (3.3)-(3.5) given by<br />
Proposition 3.1. Multiplying (3.3) by φ and integrating by parts, we get<br />
∫ ∫<br />
‖φ‖ 2 H 1 0 (Ω) = W φ 2 − hφ.<br />
By Lemma 3.1 we get<br />
Ω<br />
‖φ‖ H 1<br />
0 (Ω) ≤ C (‖φ‖ ∞ + ‖h‖ ∗ ) .<br />
4. The nonlinear problem<br />
We want to solve the nonlinear auxiliary problem<br />
∆(U ξ + φ) + (U ξ + φ) p =<br />
2∑ m∑<br />
c ij e U j<br />
Z ij , in Ω, (4.1)<br />
i=1 j=1<br />
U ξ + φ > 0, in Ω (4.2)<br />
∫<br />
Ω<br />
Ω<br />
φ = 0, on ∂Ω, (4.3)<br />
e U j<br />
Z ij φ = 0, for all i = 1, 2, j = 1, . . . , m, (4.4)<br />
for some coefficients c ij , i = 1, 2 and j = 1, . . . , m, which depend on ξ.<br />
Recalling that<br />
N(φ) = |U ξ + φ| p − U p ξ<br />
we can rewrite (4.1) in the form<br />
− pU<br />
p−1<br />
ξ<br />
φ , R = ∆U ξ + U p ξ ,<br />
2∑ m∑<br />
L(φ) = − (R + N(φ)) + c ij e U j<br />
Z ij .<br />
i=1 j=1<br />
Using the theory developed in the previous Section for the linear operator<br />
L, we prove the following result:<br />
Lemma 4.1. Let ε > 0 be fixed. There exist C > 0 and p 0 > 0 such that,<br />
for any p > p 0 and ξ ∈ O ε , problem (4.1)-(4.4) has a unique solution φ ξ<br />
which satisfies<br />
25<br />
‖φ ξ ‖ ∞ ≤ C p 3 . (4.5)
26 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Further, there holds:<br />
2∑ m∑<br />
|c ij (ξ)| ≤ C p<br />
i=1 j=1<br />
4 , ‖φ ξ‖ H 1<br />
0 (Ω) ≤ C p 3 . (4.6)<br />
Proof. Let us denote by C ∗ the function space C(¯Ω) endowed with the<br />
norm ‖ · ‖ ∗ . Proposition 3.1 implies that the unique solution φ = T (h) of<br />
(3.3)-(3.5) defines a continuous linear map from the Banach space C ∗ into<br />
C 0 (¯Ω), with norm bounded by a multiple of p. Problem (4.1)-(4.4) becomes<br />
φ = A(φ) := −T (R + N(φ)).<br />
For a given number γ > 0, let us consider the region<br />
We have the following estimates:<br />
F γ := {φ ∈ C 0 (¯Ω) : ||φ|| ∞ ≤ γ p 3 }.<br />
‖N(φ)‖ ∗ ≤ Cp‖φ‖ 2 ∞<br />
‖N(φ 1 ) − N(φ 2 )‖ ∗ ≤ Cp (max i=1,2 ‖φ i ‖ ∞ ) ‖φ 1 − φ 2 ‖ ∞ ,<br />
for any φ, φ 1 , φ 2 ∈ F γ . In fact, by Lagrange theorem we have that:<br />
(<br />
|N(φ)| ≤ p(p − 1) U ξ + O( 1 ) p−2<br />
p 3 ) φ 2<br />
|N(φ 1 ) − N(φ 2 )| ≤ p(p − 1)<br />
(U ξ + O( 1 ) p−2 ( )<br />
p 3 ) max |φ i| |φ 1 − φ 2 |<br />
i=1,2<br />
(4.7)<br />
for any x ∈ Ω, and hence, by (3.2) we get (4.7) since ‖ ∑ m<br />
j=1 e U j<br />
‖ ∗ = O(1).<br />
By (4.7), Propositions 2.1 and 3.1 imply that<br />
and<br />
‖A(φ)‖ ∞ ≤ D ′ p (‖N(φ)‖ ∗ + ‖R‖ ∗ ) ≤ O(p 2 ‖φ‖ 2 ∞) + D p 3<br />
)<br />
‖A(φ 1 )−A(φ 2 )‖ ∞ ≤ C ′ p ‖N(φ 1 )−N(φ 2 )‖ ∗ ≤ Cp<br />
(max<br />
2 ‖φ i‖ ∞ ‖φ 1 −φ 2 ‖ ∞<br />
i=1,2<br />
for any φ, φ 1 , φ 2 ∈ F γ , where D is independent of γ. Hence, if ‖φ‖ ∞ ≤ 2D<br />
p 3 ,<br />
we have that<br />
‖A(φ)‖ ∞ = O( 1 p ‖φ‖ ∞) + D p 3 ≤ 2D<br />
p 3 .<br />
Choose γ = 2D. Then, A is a contraction mapping of F γ since<br />
‖A(φ 1 ) − A(φ 2 )‖ ∞ ≤ 1 2 ‖φ 1 − φ 2 ‖ ∞<br />
for any φ 1 , φ 2 ∈ F γ . Therefore, a unique fixed point φ ξ of A exists in F γ .<br />
By (3.26), we get that<br />
2∑ m∑<br />
|c ij (ξ)| = O<br />
(‖N(φ ξ )‖ ∗ + ‖R‖ ∗ + 1 )<br />
p ‖φ ξ‖ ∞ ≤ C p 4<br />
i=1 j=1
27<br />
and by Remark 3.2 we deduce that<br />
‖φ ξ ‖ H 1<br />
0 (Ω) = O (‖φ ξ‖ ∞ + ‖N(φ ξ )‖ ∗ + ‖R‖ ∗ ) ≤ C p 3 .<br />
The proof is now complete since φ ξ solves (4.1)-(4.4): in order to show<br />
the validity of (4.2), let us remark that p|φ ξ | → 0 in C(¯Ω) and by elliptic<br />
regularity theory p|φ ξ | → 0 in C 1 (¯Ω \ ∪ m j=1 B(ξ j, ɛ)) and so, we can proceed<br />
as in Remark 2.1 to show that U ξ + φ ξ > 0 in Ω.<br />
□<br />
Let ξ 1 , ξ 2 ∈ O ε . Since<br />
∆(φ ξ1 − φ ξ2 ) + pU p−1<br />
ξ 1<br />
(φ ξ1 − φ ξ2 ) = ((U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p )<br />
(<br />
+ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1<br />
)<br />
ξ 1<br />
(φ ξ2 − φ ξ1 ) + ∆(U ξ2 − U ξ1 )<br />
2∑ m∑<br />
+ (c ij (ξ 1 ) − c ij (ξ 2 )) e U j<br />
(ξ 1 )Z ij (ξ 1 )<br />
+<br />
i=1 j=1<br />
2∑ m∑<br />
i=1 j=1<br />
and by (3.2)<br />
(<br />
)<br />
c ij (ξ 2 ) e U j<br />
(ξ 1 )Z ij (ξ 1 ) − e U j<br />
(ξ 2 )Z ij (ξ 2 )<br />
(<br />
)<br />
‖ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1<br />
ξ 1<br />
(φ ξ2 − φ ξ1 ) ‖ ∗<br />
≤ C p 2 ‖φ ξ 1<br />
− φ ξ2 ‖ ∞ ‖p(U ξ1 + O( 1 p 3 ))p−2 ‖ ∗<br />
= o( 1 p ‖φ ξ 1<br />
− φ ξ2 ‖ ∞ )<br />
uniformly in O ε , by Proposition 3.1 and (4.6) we get<br />
‖φ ξ1 − φ ξ2 ‖ ∞ ≤ Cp‖(U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p ‖ ∗<br />
+ C 2∑ m∑<br />
p 3 ‖e U j<br />
(ξ 1 )Z ij (ξ 1 ) − e U j<br />
(ξ 2 )Z ij (ξ 2 )‖ ∗<br />
i=1 j=1<br />
+ Cp‖∆(U ξ2 − U ξ1 )‖ ∗ ,<br />
for any p ≥ p 0 and ξ 1 , ξ 2 ∈ O ε (here, ‖ · ‖ ∗ is considered with respect to ξ 1 ).<br />
Hence, for fixed p ≥ p 0 , the map ξ → φ ξ is continuous in C 0 (¯Ω) and, in view<br />
of Remark 3.2, in H0 1(Ω). Further, this map is a C1 -function in C 0 (¯Ω) as it<br />
follows by the Implicit Function Theorem applied to the equation:<br />
[<br />
( ) p ]<br />
G(ξ, φ) := Π ⊥ ξ U ξ + Π ⊥ ξ φ + ∆ −1 U ξ + Π ⊥ ξ φ + Π ξ φ = 0, φ ∈ C 0 (¯Ω),<br />
where Π ξ , Π ⊥ ξ<br />
are the maps from L2 (Ω) respectively onto<br />
2∑ m∑<br />
K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m}<br />
i=1 j=1
28 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
and<br />
∫<br />
Kξ ⊥ = {φ ∈ L 2 (Ω) :<br />
Ω<br />
e U j<br />
Z ij φ = 0 for i = 1, 2, j = 1, . . . , m}<br />
(see the notations in Step 6 in the proof of Proposition 3.1). Let us remark<br />
that Π ξ φ ∈ C0 k(¯Ω), for any k ≥ 0. Indeed, G(ξ, φ ξ ) = 0 and the linearized<br />
operator:<br />
∂G<br />
∂φ (ξ, φ ξ) = Π ⊥ ξ<br />
[<br />
( )]<br />
Id + p∆ −1 (U ξ + φ ξ ) p−1 Π ⊥ ξ + Π ξ<br />
is invertible for p large. In fact, easily we reduce the invertibility property to<br />
uniquely solve the equation ∂G<br />
∂φ (ξ, φ ξ)[φ] = h in Kξ<br />
⊥ for any h ∈ K⊥ ξ ∩C 0(¯Ω).<br />
By Fredholm’s alternative, we need to show that in Kξ ⊥ ∩C 0(¯Ω) there is only<br />
the trivial solution for the equation ∂G<br />
∂φ (ξ, φ ξ)[φ] = 0, or equivalently for<br />
(<br />
Lφ = p U p−1<br />
ξ<br />
− (U ξ + φ ξ ) p−1) 2∑ m∑<br />
φ + c ij e U j<br />
Z ij ,<br />
i=1 j=1<br />
for any choice of the coefficients c ij , since by elliptic regularity theory φ ∈<br />
C0 2(¯Ω). By Proposition 3.1 and (3.2), we derive that<br />
(<br />
‖φ‖ ∞ ≤ C ′ p‖p U p−1<br />
ξ<br />
− (U ξ + φ ξ ) p−1) φ‖ ∗<br />
≤ C ′ p 2 ‖φ‖ ∞ ‖φ ξ ‖ ∞ ‖p(U ξ + O( 1 p 3 ))p−2 ‖ ∗ < ‖φ‖ ∞<br />
and hence, φ = 0. Similarly, we have also that ξ → φ ξ is a C 1 -function in<br />
H 1 0 (Ω).<br />
5. Variational reduction<br />
After problem (4.1)-(4.4) has been solved, we find a solution of (2.15) (and<br />
hence for (1.1)) if ξ is such that<br />
c ij (ξ) = 0 for all i = 1, 2 , j = 1, . . . , m, (5.1)<br />
where c ij (ξ) are the coefficients in (4.1). Problem (5.1) has a variational<br />
structure. Associated to (1.1), let us consider the energy functional J p given<br />
by<br />
J p (u) = 1 ∫<br />
|∇u| 2 dx − 1 ∫<br />
|u| p+1 dx , u ∈ H0 1 (Ω),<br />
2 Ω<br />
p + 1 Ω<br />
and the finite dimensional restriction<br />
F (ξ) := J p (U ξ + φ ξ ) (5.2)<br />
where φ ξ is the unique solution to problem (4.1)-(4.4) given by Lemma 4.1.<br />
Critical points of F correspond to solutions of (5.1) for large p, as the following<br />
result states:<br />
Lemma 5.1. The functional F (ξ) is of class C 1 . Moreover, for all p sufficiently<br />
large, if D ξ F (ξ) = 0 then ξ satisfies (5.1).
Proof. We have already shown that the map ξ → φ ξ is a C 1 −map into<br />
H 1 0 (Ω) and then, F (ξ) is a C1 -function of ξ.<br />
Since D ξ F (ξ) = 0, we have that<br />
∫<br />
0 = −<br />
= −<br />
= −<br />
Ω<br />
2∑ m∑<br />
i=1 j=1<br />
2∑ m∑<br />
i=1 j=1<br />
(∆(U ξ + φ ξ ) + (U ξ + φ ξ ) p ) (D ξ U ξ + D ξ φ ξ )<br />
∫<br />
c ij (ξ) e U j<br />
Z ij (D ξ U ξ + D ξ φ ξ )<br />
Ω<br />
∫<br />
c ij (ξ) e U j<br />
Z ij D ξ U ξ +<br />
Ω<br />
2∑<br />
m∑<br />
i=1 j=1<br />
∫<br />
c ij (ξ) D ξ (e U j<br />
Z ij )φ ξ<br />
Ω<br />
since ∫ Ω eU j<br />
Z ij φ ξ = 0. By the expression of U ξ , we have that:<br />
m∑<br />
[<br />
1<br />
∂ (ξj ) i<br />
U ξ = −<br />
2<br />
P<br />
p−1<br />
s=1 γµ s<br />
2<br />
+<br />
p 2 (p − 1) w 1 + 1 p ∇w 0 · y + 1 p 2 ∇w 1 · y<br />
(<br />
1<br />
+<br />
2<br />
P<br />
p−1<br />
γδ j µ j<br />
(<br />
1<br />
=<br />
2<br />
P<br />
p−1<br />
γδ j µ<br />
j<br />
2<br />
p − 1 U δ s ,ξ s<br />
− 2Z 0s +<br />
) ∣∣y=<br />
x−ξs<br />
δs<br />
Z ij − 1 p ∂ iw 0 ( x − ξ j<br />
) − 1 δ j p 2 ∂ iw 1 ( x − ξ j<br />
)<br />
δ j<br />
Z ij − 1 p ∂ iw 0 ( x − ξ j<br />
) − 1 δ j p 2 ∂ iw 1 ( x − ξ j<br />
)<br />
δ j<br />
29<br />
(<br />
2<br />
p(p − 1) w 0 (5.3)<br />
]<br />
∂ (ξj ) i<br />
log µ s<br />
)<br />
)<br />
+ O( 1 γ ),<br />
since P : L ∞ (Ω) → L ∞ 1<br />
(Ω) is a continuous operator (apply to<br />
p−1 U δ s ,ξ s<br />
,<br />
Z 0s , 1 p w j( x−ξs<br />
δ s<br />
), ∇w j ( x−ξs<br />
δ s<br />
) · ( x−ξs<br />
δ s<br />
) for j = 0, 1 which are bounded in Ω in<br />
view of Lemma 2.1), and<br />
∂ (ξj ) i<br />
(<br />
e U h<br />
Z lh<br />
)<br />
(<br />
= −4δ h e U δ<br />
h<br />
il<br />
δh 2 + |x − ξ h| 2 − 6(x − ξ )<br />
h) l (x − ξ j ) i<br />
(δh 2 + |x − ξ h| 2 ) 2 δ hj<br />
+3e U h<br />
Z 0h Z lh ∂ (ξj ) i<br />
log µ h , (5.4)<br />
where δ hj denotes the Kronecker’s symbol. Hence, by (5.3)-(5.4) for i = 1, 2<br />
and j = 1, . . . , m we get that<br />
1<br />
2∑ m∑<br />
)<br />
0 = ∂ (ξj ) i<br />
F (ξ) = −<br />
2<br />
c lh (ξ)<br />
(P Z ij , P Z lh<br />
p−1<br />
H<br />
γδ j µ l=1 h=1<br />
1 0 (Ω)<br />
j<br />
( )<br />
1<br />
+O + ‖φ ξ ‖ ∞ |∂<br />
pγδ (ξj ) i<br />
(e<br />
j<br />
∫Ω<br />
U ∑ 2 m∑<br />
h<br />
Z lh )| |c lh (ξ)|<br />
l=1 h=1
30 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
since ∂ i w j ( x−ξ s<br />
δ s<br />
) is bounded in Ω, j = 0, 1, in view of Lemma 2.1. Taking<br />
into account (3.22)-(3.23), (4.5) and (5.4) we get<br />
0 =<br />
64<br />
γδ j µ<br />
2<br />
p−1<br />
j<br />
( ∫<br />
R 2 |y| 2<br />
(1 + |y| 2 ) 4 dy )<br />
c ij (ξ) + O<br />
⎛<br />
⎝ 1<br />
pγδ j<br />
2∑<br />
m∑<br />
l=1 h,s=1<br />
⎞<br />
|c lh (ξ)| ⎠<br />
which implies for p large (independent of ξ ∈ O ε ) that c ij (ξ) = 0 for any<br />
i = 1, 2 and j = 1, . . . , m.<br />
□<br />
Next Lemma shows that the leading term of the function F (ξ) is given by<br />
ϕ m (ξ):<br />
Lemma 5.2. Let ε > 0. The following expansion holds:<br />
uniformly for ξ ∈ O ε .<br />
F (ξ) = 4πmp<br />
γ 2 − 32π2<br />
γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 4πm<br />
γ 2<br />
+ m ∫<br />
8<br />
2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )<br />
R 2 (<br />
Proof. First of all, multiply (4.1) by U ξ + φ ξ and integrate by parts to get:<br />
∫<br />
∫<br />
(U ξ + φ ξ ) p+1 =<br />
2∑ m∑<br />
∫<br />
|∇(U ξ + φ ξ )| 2 + c ij e U j<br />
Z ij (U ξ + φ ξ )<br />
Ω<br />
Ω<br />
i=1 j=1<br />
Ω<br />
∫<br />
=<br />
2∑ m∑<br />
∫<br />
|∇(U ξ + φ ξ )| 2 + c ij e U j<br />
Z ij U ξ<br />
Ω<br />
i=1 j=1<br />
Ω<br />
in view of (4.4). Since U ξ is a bounded function, by (4.6) we get that<br />
∫<br />
Ω<br />
∫<br />
(U ξ + φ ξ ) p+1 =<br />
uniformly for ξ ∈ O ε . We can write:<br />
F (ξ) = ( 1 2 − 1<br />
p + 1<br />
∫Ω<br />
)<br />
= ( 1 2 − 1<br />
∫<br />
p + 1<br />
(∫Ω<br />
) |∇U ξ | 2 + 2<br />
Ω<br />
|∇(U ξ + φ ξ )| 2 + O( 1 p 4 )<br />
|∇(U ξ + φ ξ )| 2 + O( 1 p 4 ) (5.5)<br />
Ω<br />
∫ )<br />
∇U ξ ∇φ ξ + |∇φ ξ | 2 + O( 1<br />
Ω<br />
p 4 ).
We expand the term ∫ Ω |∇U ξ| 2 : in view of (2.14) and (2.20) we have that<br />
∫<br />
m∑<br />
∫ (<br />
|∇U ξ | 2 1<br />
=<br />
2<br />
e U j<br />
− 1<br />
Ω<br />
p−1<br />
j=1<br />
B(ξ<br />
γµ<br />
j ,ε) pδ 2 ∆w 0 ( x − ξ j<br />
)<br />
j δ j<br />
j<br />
− 1<br />
p 2 δj<br />
2 ∆w 1 ( x − ξ )<br />
j<br />
) + O(p 2 e − p 2 ) U ξ + O(e − p 2 )<br />
δ j<br />
m∑<br />
∫ (<br />
1<br />
8<br />
=<br />
4<br />
j=1 γ 2 p−1 B(0,<br />
µ<br />
ε ) (1 + |y| 2 ) 2 − 1 p ∆w 0 − 1 )<br />
p 2 ∆w 1 + O(p 2 e −p ) ×<br />
δ<br />
j<br />
j<br />
×<br />
(p + v ∞ + 1 p w 0 + 1 )<br />
p 2 w 1 + O(e − p 4 |y| + e<br />
− p 4 ) + O(e − p 2 )<br />
=<br />
m∑<br />
j=1<br />
1<br />
γ 2 µ<br />
= 8πmp<br />
γ 2<br />
since µ − 4<br />
p−1<br />
j<br />
get that:<br />
4<br />
p−1<br />
j<br />
− 32π<br />
γ 2<br />
( ∫<br />
8πp +<br />
m ∑<br />
j=1<br />
R 2 (<br />
8<br />
(1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 )<br />
p )<br />
log µ j + m 8<br />
γ<br />
∫R 2 (<br />
2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )<br />
= 1 − 4 p log µ j + O( 1 p 2 ). Recalling property (2.13) of µ i , then we<br />
∫<br />
Ω<br />
|∇U ξ | 2 = 8πmp<br />
γ 2<br />
31<br />
− 64π2<br />
γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 24πm<br />
γ 2 (5.6)<br />
+ m 8<br />
γ<br />
∫R 2 (<br />
2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )<br />
uniformly for ξ ∈ O ε . Hence, using (4.6) and (5.6) we get that<br />
∫<br />
∇U ξ ∇φ ξ + 1 ∫<br />
|∇φ ξ | 2 = O( 1 ). (5.7)<br />
Ω 2 Ω<br />
Finally, inserting (5.6)-(5.7) in (5.5), we get that<br />
uniformly for ξ ∈ O ε .<br />
F (ξ) = 4πmp<br />
γ 2 − 32π2<br />
γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 4πm<br />
γ 2<br />
+ m ∫<br />
8<br />
2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 )<br />
R 2 (<br />
Now, we want to show that the expansion of F (ξ) in terms of ϕ m (ξ) holds<br />
in a C 1 -sense:<br />
Lemma 5.3. Let ε > 0. The following expansion holds:<br />
∇ (ξj ) i<br />
F (ξ) = − 32π2<br />
γ 2 ∇ (ξ j ) i<br />
ϕ m (ξ 1 , . . . , ξ m ) + o( 1 p 2 )<br />
uniformly for ξ ∈ O ε , for any j = 1, . . . , m and i = 1, 2.<br />
p 7 2
32 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Proof. Let j ∈ {1, . . . , m} and i ∈ {1, 2} be fixed. We want to expand the<br />
derivatives of F (ξ) in ξ:<br />
∫ ( )<br />
∂ (ξj ) i<br />
F (ξ) = − ∆u ξ + u p ξ<br />
∂ (ξj ) i<br />
u ξ ,<br />
Ω<br />
where u ξ = U ξ + φ ξ . Let us remark that it is very difficult to show directly<br />
that the expansion of F (ξ) holds in a C 1 −sense since there is a difference<br />
between the exponential decay of the concentration paremeters δ i = µ i e − p 4<br />
and the polinomial decay 1 of ‖R<br />
p 4 ξ ‖ ∗ (see Proposition 2.1). As usual in<br />
similar contexts (also in higher dimensions), we should be able to show that<br />
‖∂ ξ φ ξ ‖ ∞ is of order ‖φ ξ‖ ∞<br />
δ i<br />
. Unfortunately, since ‖φ ξ ‖ ∞ is only of order 1 ,<br />
p 3<br />
∂ ξ φ ξ is not a small function and, at a first glance, there is no hope for a<br />
C 1 −expansion of F (ξ). To overcome the problem, the idea is the following:<br />
first, we replace the term ∂ (ξj ) i<br />
u ξ with ∂ xi u ξ in the expression of ∂ (ξj ) i<br />
F (ξ)<br />
in a neighborhood of ξ j , up to higher order terms, and after, we use a<br />
Pohozaev-type identity based on integration by parts.<br />
To this purpose, let η be a radial smooth cut-off function such that 0 ≤ η ≤ 1,<br />
η ≡ 1 for |x| ≤ ɛ and η ≡ 0 for |x| ≥ 2ɛ. In view of (4.1) and (4.4), we can<br />
write:<br />
∫ ( )<br />
2∑ m∑<br />
∫<br />
∆u ξ + u p ξ<br />
∂ (ξj ) i<br />
φ ξ = c kl e U l<br />
Z kl ∂ (ξj ) i<br />
φ ξ (5.8)<br />
Ω<br />
= −<br />
=<br />
−<br />
2∑<br />
k=1 l=1<br />
2∑ m∑<br />
k=1 l=1<br />
2∑ m∑<br />
m∑<br />
∫<br />
c kl<br />
c kl<br />
∫<br />
c kl<br />
∫<br />
k=1 l=1<br />
2∑ m∑<br />
= −<br />
−<br />
k=1 l=1<br />
2∑ m∑<br />
k=1 l=1<br />
c kl<br />
∫<br />
Ω<br />
Ω<br />
c kl<br />
∫<br />
Ω<br />
Ω<br />
k=1 l=1<br />
( )<br />
∂ (ξj ) i<br />
e U l<br />
Z kl φ ξ<br />
Ω<br />
∂ xi<br />
(<br />
e U l<br />
Z kl<br />
)<br />
η(x − ξ j )φ ξ<br />
[∂ (ξj ) i<br />
(e U l<br />
Z kl<br />
)<br />
+ η(x − ξ j )∂ xi<br />
(e U l<br />
Z kl<br />
)]<br />
φ ξ<br />
Ω<br />
e U l<br />
Z kl ∂ xi (η(x − ξ j )φ ξ )<br />
[∂ (ξj ) i<br />
(e U l<br />
Z kl<br />
)<br />
+ η(x − ξ j )∂ xi<br />
(e U l<br />
Z kl<br />
)]<br />
φ ξ<br />
by an integration by parts of the derivative in x i . As for (5.4), we get that<br />
( )<br />
( )<br />
∂ (ξj ) i<br />
e U l<br />
Z kl + η(x − ξ j )∂ xi e U l<br />
Z kl = 4δ l e U η(x − ξ l<br />
j) − δ lj<br />
δl 2 + |x − ξ l| 2 δ ik<br />
+24δ l e U l (x − ξ l) k (x − ξ l ) i<br />
(δ 2 l + |x − ξ l| 2 ) 2 (δ lj − η(x − ξ j )) + 3e U l<br />
Z 0l Z kl ∂ (ξj ) i<br />
log µ l<br />
= 3e U l<br />
Z 0l Z kl ∂ (ξj ) i<br />
log µ l + O(e − 3 4 p ),
33<br />
where δ lj denotes the Kronecker’s symbol, and hence, we get that<br />
|<br />
2∑<br />
k=1 l=1<br />
m∑<br />
∫<br />
c kl<br />
Ω<br />
≤ C‖φ ξ ‖ ∞ max |c kl |<br />
k , l<br />
[∂ (ξj ) i<br />
(e U l<br />
Z kl<br />
)<br />
+ η(x − ξ j )∂ xi<br />
(e U l<br />
Z kl<br />
)]<br />
φ ξ |<br />
∫<br />
Ω<br />
(<br />
)<br />
e U l<br />
+ O(e − 3 4 p ) = O( 1 p 7 )<br />
in view of |Z 0l Z kl | ≤ 2, (4.5)-(4.6). Inserting in (5.8), we get that<br />
∫<br />
Ω<br />
(<br />
∆u ξ + u p )<br />
ξ<br />
∂ (ξj ) i<br />
φ ξ = −<br />
2∑<br />
k=1 l=1<br />
+O( 1 p 7 ).<br />
m∑<br />
∫<br />
c kl<br />
Ω<br />
e U l<br />
Z kl ∂ xi (η(x − ξ j )φ ξ )(5.9)<br />
Since pφ ξ → 0 in C 1 -norm away from ξ 1 , . . . , ξ m , (5.9) gives that<br />
∫<br />
Ω<br />
( )<br />
∆u ξ + u p ξ<br />
∂ (ξj ) i<br />
φ ξ = −<br />
= −<br />
2∑<br />
k=1 l=1<br />
∫<br />
m∑<br />
∫<br />
c kl<br />
B(ξ j ,ɛ)<br />
always in view of (4.1) and (4.5)-(4.6).<br />
B(ξ j ,ɛ)<br />
(<br />
∆u ξ + u p ξ<br />
e U l<br />
Z kl ∂ xi φ ξ + O( 1 p 7 )<br />
)<br />
∂ xi φ ξ + O( 1 p 7 ) (5.10)<br />
Now, by Lemma 2.1 we get that | 1 δ s<br />
∂ xi w l ( x−ξs<br />
δ s<br />
)| ≤ C for any l = 1, 2 and<br />
for x away from ξ s . Hence, by the expression of U ξ and (2.2), (2.11) we get<br />
that for |x − ξ j | ≤ 2ɛ:<br />
m∑<br />
(<br />
1<br />
∂ xi U ξ =<br />
2<br />
∂ xi P U δs ,ξ s<br />
+ 1 (<br />
p−1<br />
p ∂ x i<br />
P w 0 ( x − ξ )<br />
s<br />
)<br />
(5.11)<br />
δ<br />
s=1 γµ<br />
s<br />
s<br />
+ 1 (<br />
p 2 ∂ x i<br />
P w 1 ( x − ξ ))<br />
s<br />
)<br />
δ s<br />
m∑<br />
(<br />
1<br />
=<br />
2<br />
∂ xi U δs,ξs + 1 ∂ xi w 0 ( x − ξ s<br />
) + 1<br />
p−1<br />
pδ<br />
s=1 γµ<br />
s δ s p 2 ∂ xi w 1 ( x − ξ )<br />
s<br />
) + O( 1 δ s δ s γ )<br />
s<br />
(<br />
1<br />
= −<br />
2<br />
Z ij − 1<br />
p−1 p ∂ x i<br />
w 0 ( x − ξ j<br />
) − 1 δ<br />
γδ j µ<br />
j p 2 ∂ x i<br />
w 1 ( x − ξ )<br />
j<br />
) + O( 1 δ j γ ),<br />
j<br />
since ∂ xi U δs ,ξ s<br />
= − 1 δ s<br />
Z is . Since, as already observed, ∂ xi w l ( x−ξ j<br />
δ j<br />
) = O(δ j )<br />
uniformly away from ξ j , l = 1, 2, by (5.11) in particular we get:<br />
∂ xi U ξ = O( 1 ), (5.12)<br />
γ
34 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
for ɛ ≤ |x − ξ j | ≤ 2ɛ. Moreover, the maximum principle and Lemma 2.1<br />
imply that<br />
(<br />
P ∂ xi w l ( x − ξ )<br />
j<br />
) − ∂ xi w l ( x − ξ j<br />
) = O(δ j )<br />
δ j δ j<br />
in C(¯Ω) for any l = 1, 2, and hence, by (5.3) and (5.11) we get that:<br />
1<br />
∂ (ξj ) i<br />
U ξ + η(x − ξ j )∂ xi U ξ =<br />
γδ j µ<br />
2<br />
p−1<br />
j<br />
(P Z ij − Z ij ) + O( 1 γ ) = O( 1 γ ) (5.13)<br />
in view of (3.20). Now, by (5.12)-(5.13) we can write that:<br />
∫<br />
+<br />
Ω<br />
( )<br />
∆u ξ + u p ξ<br />
∂ (ξj ) i<br />
U ξ = −<br />
2∑<br />
m∑<br />
∫<br />
c kl<br />
k=1 l=1<br />
2∑ m∑<br />
= −<br />
= −<br />
k=1 l=1<br />
∫<br />
B(ξ j ,ɛ)<br />
Ω<br />
c kl<br />
∫<br />
2∑<br />
k=1 l=1<br />
m∑<br />
∫<br />
c kl<br />
e U l<br />
Z kl<br />
(<br />
∂ (ξj ) i<br />
U ξ + η(x − ξ j )∂ xi U ξ<br />
)<br />
B(ξ j ,ɛ)<br />
(<br />
∆u ξ + u p ξ<br />
Ω<br />
e U l<br />
Z kl ∂ xi U ξ + O( 1 p 5 )<br />
)<br />
∂ xi U ξ + O( 1 p 5 )<br />
e U l<br />
Z kl η(x − ξ j )∂ xi U ξ (5.14)<br />
in view of (4.1) and (4.6).<br />
Resuming, by (5.10) and (5.14) we have:<br />
∫ ( )<br />
∂ (ξj ) i<br />
F (ξ) = − ∆u ξ + u p ξ<br />
(∂ (ξj ) i<br />
U ξ + ∂ (ξj ) i<br />
φ ξ ) (5.15)<br />
Ω<br />
∫ (<br />
= ∆u ξ + u p )<br />
ξ<br />
(∂ xi U ξ + ∂ xi φ ξ ) + O( 1 p 5 )<br />
=<br />
B(ξ j ,ɛ)<br />
∫<br />
B(ξ j ,ɛ)<br />
( )<br />
∆u ξ + u p ξ<br />
∂ xi u ξ + O( 1 p 5 ).<br />
Now, the role of ∇ϕ m (ξ) becomes clear by means of the following Pohozaevtype<br />
identity (we follow some arguments of [13]): for any B ⊂ Ω and for<br />
any function u<br />
∫<br />
∫ (<br />
∆u ∇u = ∂ n u∇u − 1 )<br />
2 |∇u|2 n , (5.16)<br />
B<br />
∫<br />
B<br />
∂B<br />
u p ∇u = 1<br />
p + 1<br />
∫<br />
∂B<br />
u p+1 n,<br />
where n(x) is the unit outer normal vector of ∂B at x ∈ ∂B. Let<br />
ϕ j (x) = H(x, ξ j ) + ∑ l≠j<br />
G(x, ξ l )<br />
for any j = 1, . . . , m. Since, as already observed, pφ ξ → 0 in C 1 -norm away<br />
from ξ 1 , . . . , ξ m , for our function u ξ by (2.2) and (2.11) we have the following
asymptotic property:<br />
pu ξ (x) → 8π √ e<br />
m∑<br />
G(x, ξ l ) in C<br />
loc 1 (¯Ω \ {ξ 1 , . . . , ξ m }).<br />
l=1<br />
35<br />
(5.17)<br />
Apply now (5.16) on B = B(ξ j , ε), j = 1, . . . , m, and use (5.17) to obtain as<br />
p → +∞:<br />
∫<br />
∫<br />
B<br />
(∆u ξ + u p ξ )∇u ξ =<br />
∂B<br />
= 64π2 e<br />
p 2 ∫∂B<br />
− 1 2 | − 1<br />
2π<br />
(<br />
∂ n u ξ ∇u ξ − 1 2 |∇u ξ| 2 n + 1<br />
[<br />
(− 1<br />
2πɛ + ∂ nϕ j )(− 1<br />
2π<br />
]<br />
x − ξ j<br />
|x − ξ j | 2 + ∇ϕ j| 2 n<br />
= − 64π2 e<br />
p 2 ∇ϕ j (ξ j ) + o( 1 p 2 ),<br />
p + 1 up+1 ξ<br />
)<br />
n<br />
x − ξ j<br />
|x − ξ j | 2 + ∇ϕ j)<br />
+ o( 1 p 2 )<br />
since we decompose ∑ m<br />
l=1 G(x, ξ l ) = − 1<br />
2π ln |x − ξ j| + ϕ j (x) with ϕ j (x) an<br />
harmonic function near ξ j . In fact, we have used that<br />
1<br />
2πɛ<br />
∫<br />
∂B<br />
∇ϕ j = ∇ϕ j (ξ j )<br />
since ∇ϕ j is harmonic near ξ j , and by (5.16)<br />
∫ (<br />
∂ n ϕ j ∇ϕ j − 1 ) ∫<br />
2 |∇ϕ j| 2 n =<br />
∂B<br />
Combining with (5.15), finally we get:<br />
B<br />
∆ϕ j ∇ϕ j = 0.<br />
∂ (ξj ) i<br />
F (ξ) = − 32π2 e<br />
p 2 ∂ (ξj ) i<br />
ϕ m (ξ) + o( 1 p 2 ) = −32π2 γ 2 ∂ (ξ j ) i<br />
ϕ m (ξ) + o( 1 p 2 )<br />
since ∇ϕ j (ξ j ) = 1 2 ∇ ξ j<br />
ϕ m (ξ). The proof is now complete.<br />
Finally, we carry out the proof of our main result.<br />
Proof of Theorem 1.2. Let us consider the set D as in the statement of<br />
the theorem, C the associated critical value and ξ ∈ D. According to Lemma<br />
5.1, we have a solution of Problem (1.1) if we adjust ξ so that it is a critical<br />
point of F (ξ) defined by (5.2). This is equivalent to finding a critical point<br />
of<br />
[<br />
˜F (ξ) =<br />
γ2<br />
32π 2 F (ξ) − 4πmp<br />
γ 2 − 4πm<br />
γ 2 − m ∫<br />
]<br />
8<br />
2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) .<br />
On the other hand, from Lemmata 5.2 and 5.3, we have that for ξ ∈ D ∩ O ε ,<br />
R 2 (<br />
˜F (ξ) = ϕ m (ξ) + o(1)Θ p (ξ)<br />
where Θ p and ∇ ξ Θ p are uniformly bounded in the considered region as<br />
p → ∞.<br />
□
36 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA<br />
Let us observe that if M > C, then assumptions (1.4), (1.5) still hold for<br />
the function min{M, ϕ m (ξ)} as well as for min{M, ϕ m (ξ) + o(1)Θ p (ξ)}. It<br />
follows that the function min{M, ˜F (ξ)} satisfies for all p large assumptions<br />
(1.4),(1.5) in D and therefore has a critical value C p < M which is close to C<br />
in this region. If ξ p ∈ D is a critical point at this level for ˜F (ξ), then, since<br />
˜F (ξ p ) ≤ C p < M,<br />
we have that there exists ε > 0 such that |ξ p,j −ξ p,i | > 2ε, dist(ξ p,j , ∂Ω) > 2ε.<br />
This implies C 1 -closeness<br />
(<br />
of ˜F (ξ) and ϕ m (ξ)<br />
)<br />
at this level, hence ∇ϕ m (ξ p ) →<br />
0. The function u p (x) = U ξp (x) + φ ξp (x) is therefore a solution with the<br />
qualitative properties predicted by the theorem.<br />
□<br />
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P. Esposito - Dipartimento di Matematica, Università degli Studi “Roma<br />
Tre”, Largo S. Leonardo Murialdo, 1 – 00146 Roma, Italy.<br />
M. Musso - Departamento de Matematicas, Pontificia Universidad Catolica<br />
de Chile, Avenida Vicuna Mackenna 4860, Macul, Santiago, Chile and Dipartimento<br />
di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24<br />
– 10129 Torino, Italy.<br />
A. Pistoia - Dipartimento di Metodi e Modelli Matematici, Università di<br />
Roma “La Sapienza”, Via Scarpa, 16 – 00166 Roma, Italy.