CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

CONCENTRATING SOLUTIONS FOR A PLANAR 

ELLIPTIC PROBLEM INVOLVING NONLINEARITIES 

WITH LARGE EXPONENT 

PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA 

Abstract. We consider the boundary value problem ∆u + u p = 0 in a 

bounded, smooth domain Ω in R 2 with homogeneous Dirichlet boundary 

condition and p a large exponent. We find topological conditions on Ω 

which ensure the existence of a positive solution u p concentrating at 

exactly m points as p → ∞. In particular, for a non simply connected 

domain such a solution exists for any given m ≥ 1. 

1. Introduction and statement of main results 

This paper is concerned with analysis of solutions to the boundary value 

problem: 

⎧ 

⎨ 

⎩ 

∆u + u p = 0 in Ω, 

u > 0 in Ω, 

u = 0 on ∂Ω, 

where Ω is a smooth bounded domain in R 2 and p is a large exponent. 

Let us consider the Rayleigh quotient: 

∫ 

Ω 

I p (u) = 

|∇u|2 

( ∫ , u ∈ H 

Ω |u|p+1 ) 2 

0 1 (Ω) \ {0}, 

p+1 

and set 

S p = 

inf I p (u). 

u∈H0 1(Ω)\{0} (1.1) 

Since H 1 0 (Ω) is compactly embedded in Lp+1 (Ω) for any p > 0, standard variational 

methods show that S p is achieved by a positive function u p which 

solves problem (1.1). The function u p is known as least energy solution. 

In [29]-[30] the authors show that the least energy solution has L ∞ -norm 

1991 Mathematics Subject Classification. 35J60, 35B33, 35J25, 35J20, 35B40. 

Key words and phrases. Large exponent, Concentrating solutions, Green’s function, 

Finite dimensional reduction. 

First author is supported by M.U.R.S.T., project “Variational methods and nonlinear 

differential equations”. Second author has been partially supported by Fondecyt 1040936 

(Chile). Second and third authors are supported by M.U.R.S.T., project “Metodi variazionali 

e topologici nello studio di fenomeni non lineari”. 

1

2 PIERPAOLO ESPOSITO, MONICA MUSSO, AND ANGELA PISTOIA 

bounded and bounded away from zero uniformly in p, for p large. Furthermore, 

up to subsequence, the renormalized energy density p|∇u p | 2 concentrates 

as a Dirac delta around a critical point of the Robin’s function 

H(x, x), where H is the regular part of Green function of the laplacian in 

Ω with homogeneous Dirichlet boundary condition. Namely, the Green’s 

function G(x, y) is the solution of the problem 

{ −∆x G(x, y) = δ y (x) x ∈ Ω, 

G(x, y) = 0 x ∈ ∂Ω, 

and H(x, y) is the regular part defined as 

H(x, y) = G(x, y) − 1 

2π log 1 

|x − y| . 

In [1] and [16] the authors give a further description of the asymptotic behaviour 

of u p , as p → ∞, by identifying a limit profile problem of Liouvilletype: 

{ ∆u 

∫ + e u = 0 in R 2 , 

R 2 e u (1.2) 

< ∞, 

and showing that ‖u p ‖ ∞ → √ e as p → +∞. 

Problem (1.2) possesses exactly a three-parameters family of solutions 

U δ,ξ (x) = log 

where δ is a positive number and ξ ∈ R 2 (see [6]). 

8δ 2 

(δ 2 + |x − ξ| 2 ) 2 , (1.3) 

The aim of this paper is to build solutions for problem (1.1) that, up to 

a suitable normalization, look like a sum of concentrated solutions for the 

limit profile problem (1.2) centered at several points ξ 1 , . . . , ξ m , as p → ∞. 

In this case, when m is possibly greater than 1, the function responsible to 

locate the concentration points ξ 1 , . . . , ξ m is more involved than the Robin’s 

function. In fact, location of such points is related to critical points of the 

function 

m∑ 

m∑ 

ϕ m (ξ 1 , . . . , ξ m ) = H(ξ j , ξ j ) + G(ξ i , ξ j ). 

j=1 

Let us mention that the same function ϕ m is responsible for the location 

of the points of concentration for solutions to the mean field equation in 

bounded domains Ω ⊂ R 2 (see [3, 12, 14]). 

Our main result reads as follows. 

Theorem 1.1. Assume that Ω is not simply connected. Then given any 

m ≥ 1 there exists p m > 0 such that for any p ≥ p m problem (1.1) has a 

solution u p which concentrates at m different points in Ω, according to (1.6), 

(1.7) and (1.8), as p goes to +∞. 

i,j=1 

i≠j

This result is consequence of a more general theorem, which we state below, 

that ensures the existence of solutions to problem (1.1) which concentrates 

at m different points of Ω, under the assumption that the function ϕ m has 

a non-trivial critical value. 

Let Ω m be Ω × Ω × . . . × Ω m times. We define 

ϕ m (x 1 , . . . , x m ) = +∞ if x i = x j for some i ≠ j. 

Let D be an open set compactly contained in Ω m with smooth boundary. We 

recall that ϕ m links in D at critical level C relative to B and B 0 if B and B 0 

are closed subsets of ¯D with B connected and B 0 ⊂ B such that the following 

conditions hold: Let us set Γ to be the class of all maps Φ ∈ C(B, D) with 

the property that there exists a function Ψ ∈ C([0, 1] × B, D) such that: 

We assume 

Ψ(0, ·) = Id B , Ψ(1, ·) = Φ, Ψ(t, ·)| B0 = Id B0 for all t ∈ [0, 1]. 

sup ϕ m (y) < C ≡ inf 

y∈B 0 

sup 

Φ∈Γ y∈B 

ϕ m (Φ(y)) , (1.4) 

and for all y ∈ ∂D such that ϕ m (y) = C, there exists a vector τ y tangent to 

∂D at y such that 

∇ϕ m (y) · τ y ≠ 0 . (1.5) 

Under these conditions a critical point ȳ ∈ D of ϕ m with ϕ m (ȳ) = C exists, 

as a standard deformation argument involving the negative gradient flow of 

ϕ m shows. It is easy to check that the above conditions hold if 

inf ϕ m(x) < inf 

x∈D 

x∈∂D ϕ m(x), or sup 

x∈D 

ϕ m (x) > sup ϕ m (x) , 

x∈∂D 

namely the case of (possibly degenerate) local minimum or maximum points 

of ϕ m . We call C a non-trivial critical level of ϕ m in D. 

Theorem 1.2. Let m ≥ 1 and assume that there is an open set D compactly 

contained in Ω m where ϕ m has a non-trivial critical level C. Then, there 

exists p m > 0 such that for any p ≥ p m problem (1.1) has a solution u p 

which concentrates at m different points of Ω, i.e. as p goes to +∞ 

pu p+1 

p 

m∑ 

⇀ 8πe δ ξj weakly in the sense of measure in Ω (1.6) 

i=1 

for some ξ ∈ D such that ϕ m (ξ 1 , . . . , ξ m ) = C and ∇ϕ m (ξ 1 , . . . , ξ m ) = 0. 

More precisely, there is an m-tuple ξ p = (ξ p 1 , . . . , ξp m) ∈ D converging (up to 

subsequence) to ξ such that, for any δ > 0, as p goes to +∞ 

and 

u p → 0 uniformly in Ω \ ∪ m j=1 B δ(ξ p i ) (1.7) 

sup 

x∈B δ (ξ p i ) u p (x) → √ e. (1.8) 

3


The detailed proof of how Theorem 1.2 implies the result contained in Theorem 

1.1 can be found in [12]. 

As already mentioned, the case of (possibly degenerate) local maximum or 

minimum for ϕ m is included. This simple fact allows us to obtain an existence 

result for solutions to problem (1.1) also when Ω is simply connected. 

Indeed, we can construct simply connected domains of dumbell-type where 

a large number of concentrating solutions can be found. 

Let h be an integer. By h-dumbell domain with thin handles we mean the 

following: let Ω 0 = Ω 1 ∪ . . . ∪ Ω h , with Ω 1 , . . . , Ω h smooth bounded domains 

in R 2 such that ¯Ω i ∩ ¯Ω j = ∅ if i ≠ j. Assume that 

Ω i ⊂ {(x 1 , x 2 ) ∈ R 2 : a i ≤ x 1 ≤ b i }, Ω i ∩ {x 2 = 0} ̸= ∅, 

for some b i < a i+1 and i = 1, . . . , h. Let 

C ε = {(x 1 , x 2 ) ∈ R 2 : |x 2 | ≤ ε, x 1 ∈ (a 1 , b h )}, for some ε > 0. 

We say that Ω ε is a h-dumbell with thin handles if Ω ε is a smooth simply 

connected domain such that Ω 0 ⊂ Ω ε ⊂ Ω 0 ∪ C ε , for some ε > 0. 

The following result holds true. 

Theorem 1.3. There exist ε h > 0 and p h > 0 such that for any ε ∈ (0, ε h ) 

and p ≥ p h problem (1.1) in Ω ε has at least 2 h −1 families of solutions which 

concentrate at different points of Ω ε , according to (1.6), (1.7) and (1.8), as 

( ) h 

p goes to +∞. More precisely, for any integer 1 ≤ m ≤ h there exist 

m 

families of solutions of (1.1) which concentrate at m different points of Ω ε . 

The detailed proof of how Theorem 1.2 implies the result contained in Theorem 

1.3 can be found in [14]. We also refer the reader to [4] and [7], where 

domains like dumbells with thin handles are considered. 

The proof of all our results relies on a Lyapunov-Schmidt procedure, based 

on a proper choice of the ansatz for the solution we are looking for. Usually, 

in other related problems of asymptotic analysis, the ansatz for the solution 

is built as the sum of a main term, which is a solution (properly modified or 

projected) of the associated limit problem, and a lower order term, which 

can be determined by a fixed point argument. In our problem, this is not 

enough. Indeed, in order to perform the fix point argument to find the lower 

order term in the ansatz (see Lemma 4.1), we need to improve substantially 

the main term in the ansatz, adding two other terms in the expansion of the 

solution (see section 2). This fact is basically due to estimate (4.7). 

By performing a finite dimensional reduction, we find an actual solution to 

our problem adjusting points ξ inside Ω to be critical points of a certain 

function F (ξ) (see (5.2)). It is quite standard to show that this function 

F (ξ) is a perturbation of ϕ m (ξ) in a C 0 −sense. On the other hand, it is not 

at all trivial to show the C 1 closeness between F and ϕ m . This difficulty is 

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 

‖∆U ξ + U p ξ ‖ ∗ of our approximating function U ξ (see Proposition 2.1). We 

are able to overcome this difficulty (see Lemma 5.3) using a Pohozaev-type 

identity. 

Now, we would like to compare problem (1.1) with some widely studied 

problems which have some analogies with it. 

In higher dimension the problem equivalent to Problem (1.1) is the slightly 

sub-critical problem 

⎧ 

⎪⎨ 

⎪⎩ 

∆u + u N+2 

N−2 −ε = 0 in Ω, 

u > 0 in Ω, 

u = 0 on ∂Ω, 

5 

(1.9) 

where Ω is a smooth bounded domain in R N , N ≥ 3 and ε is a positive 

parameter. Indeed, in dimension N ≥ 3, the embedding of H0 1(Ω) in Lp+1 (Ω) 

is compact for every p < N+2 

N−2 

. Hence the minimum of the Rayleigh quotient 

corresponding to problem (1.9) is achieved by a positive function u ε , called 

least energy solution, which, after a multiplication by a suitable positive 

constant, is a solution to (1.9). 

It is well known that, as ε goes to 0, the least energy solution u ε concentrates 

around a point, which is a critical point of the Robin’s function of the 

corresponding Green’s function (see [2, 17, 20, 27]). Also the converse is 

true: around any stable critical point of the Robin’s function one can build 

a family of solutions for (1.9) concentrating precisely there (see [24, 27, 28]). 

In [2] and [22] the authors showed that also for problem (1.9) there exist 

solutions with concentration in multiple points and, as in the problem that 

we are considering in the present paper, the points of concentration are given 

by critical points of a certain function defined in terms of both the Green’s 

function and Robin’s function. 

The analogies between problems (1.1) and (1.9) break down here. Indeed, 

while for (1.1) one can find solutions with an arbitrarily large number of 

condensation points in any given not simply connected domain Ω, in [2] the 

authors proved that solutions to (1.9) can have at most a finite number of 

peaks which depends on Ω (see, also, [15, 22]). 

The property of problem (1.1) to have solution with an arbitrarily large 

number of points of condensation is what one expects to happen in the 

slightly super-critical version of problem (1.9), namely 

⎧ 

⎪⎨ 

⎪⎩ 

∆u + u N+2 

N−2 +ε = 0 in Ω, 

u > 0 in Ω, 

u = 0 on ∂Ω. 

(1.10) 

Indeed, a conjecture for (1.10) is that, given any domain Ω with a hole, one 

can see solutions with an arbitrarily large number of peaks (see [9, 10, 25]).


For this fact, despite of being compact and hence sub-critical in dimension 

2, problem (1.1) shares patterns similar to the ones associated to slighlty 

super-critical problem (1.10) in higher dimension. 

However, again the analogies between (1.1) and (1.10) in higher dimension 

break down here. Indeed, the dilation invariance, which is crucial in the 

study of problem (1.10), does not play a role in finding solutions to problem 

(1.1), as already observed for a similar two-dimensional problem in [12] and 

[14] (see, also, [3]): only translation invariance is concerned in the study of 

(1.1). 

The only translation invariance is the crucial key which allows to find solutions 

to the sub-critical problem (see [11, 19, 21, 23, 31]) 

⎧ 

⎨ 

⎩ 

−ε 2 ∆u + u = u p in Ω, 

u > 0 in Ω, 

u = 0 on ∂Ω, 

(1.11) 

where Ω is a bounded open domain in R N , p > 1 if N = 2 and 1 

N−2 

if N ≥ 3, ε is a positive parameter. 

Moreover, the property of problem (1.1) to have solution with an arbitrarily 

large number of points of condensation is what happens in the sub-critical 

problem (1.11). In fact, in [8] the authors prove that if the reduced cohomology 

of Ω is not trivial, then for any integer k such a problem has at least 

one k−peaks solution, provided the parameter ε is small enough. 

However, again the analogies between (1.1) and (1.11) break down here. Indeed, 

problem (1.1) is somehow almost critical in R 2 , since the limit problem 

as p goes to +∞ is (1.2) which is critical in R 2 , while the limit problem of 

(1.11) as ε goes to 0 is the sub-critical problem 

⎧ 

⎪⎨ 

⎪⎩ 

−∆u + u = u p in R N , 

u > 0 in R N , 

u ∈ H 1 (R N ). 

The paper is organized as follows. In Section 2 we describe exactly the 

ansatz for the solution we are searching for. We rewrite the problem in term 

of a linear operator L for which a solvability theory is performed in Section 

3. In Section 4 we solve an auxiliary non linear problem. We reduce (1.1) to 

solve a finite system c ij = 0, as we will see in Section 5. Section 5 contains 

also the proof of Theorem 1.2. 

2. A first approximation of the solution 

In this Section we will provide an ansatz for solutions of problem (1.1). 

A useful observation is that u satisfies equation (1.1) if and only if 

v(y) = δ 2 

p−1 u(δy + ξ), 

y ∈ Ω ξ,δ

satisfies 

⎧ 

⎨ 

⎩ 

∆v + v p = 0, in Ω ξ,δ , 

v > 0, in Ω ξ,δ , v = 0, on ∂Ω ξ,δ , 

7 

(2.1) 

where ξ is a given point in Ω, δ is a positive number with δ → 0, and Ω ξ,δ 

is the expanding domain defined by Ω−ξ 

δ 

. 

In this section we will show that the basic elements for the construction 

of an approximate solution to problem (1.1) which exhibits one point of 

concentration (or equivalently of problem (2.1)) are the radially symmetric 

solutions of the problem (1.2) given by U δ,ξ defined in (1.3). 

For U δ,ξ (x) defined in (1.3), we denote by P U δ,ξ (x) its projection on the 

space H0 1(Ω), namely P U δ,ξ(x) is the unique solution of 

{ ∆P Uδ,ξ = ∆U δ,ξ in Ω, 

P U δ,ξ (x) = 0 on ∂Ω. 

Since P U δ,ξ (x)−U δ,ξ (x)+log(8δ 2 )+4 log 

1 

|x−ξ| = O(δ2 ) uniformly on x ∈ ∂Ω 

as δ → 0 (together with any boundary derivatives), by harmonicity we get 

P U δ,ξ (x) = U δ,ξ (x) − log(8δ 2 ) + 8πH(x, ξ) + O(δ 2 ) in C 1 (¯Ω) (2.2) 

P U δ,ξ (x) = 8πG(x, ξ) + O(δ 2 ) in C 1 loc (¯Ω \ {ξ}), 

provided ξ is bounded away from ∂Ω. 

Assume now that 

δ = µe − p 1 

4 , 

C 

and define 

u(x) = e p 

2(p−1) 

Observe that, as p → ∞, 

p p 

p−1 µ 2 

p−1 

≤ µ ≤ C, (2.3) 

P U δ,ξ (x) x ∈ Ω. (2.4) 

u(ξ) → √ e and u(x) = O( 1 ) for x ≠ ξ. 

p 

Furthermore, under the extra assumption that the parameter µ is defined 

by the relation 

log(8µ 4 ) = 8πH(ξ, ξ), 

a direct computation shows that a good first approximation for a solution 

to problem (1.1) exhibiting only one point of concentration is given by a 

perturbation of the function u defined in (2.4). 

Indeed, in the expanded variable y = x−ξ 

δ 

∈ Ω−ξ 

δ 

, if we define v(y) = 

δ 2 

p−1 u(δy + ξ), then our first approximation (2.4) looks like 

1 

p p 

p−1 

(p + U 1,0 (y) + O(e − p 4 |y| + e 

− p 4 ) 

) 

(2.5)


and hence 

∆v + v p ∼ 1 

p p 

p−1 

[ 

( 

−e U1,0(y) + 

1 + U 1,0(y) 

p 

) p ] 

, 

which, roughly speaking, implies that the error for u to be a solution of (1.1) 

exhibiting one points of concentration, or equivalently for v to be a solution 

of (2.1), is of order 1 p 2 . 

However, as we will see below, this is not enough to built an actual solution 

to (1.1) starting from u(x). We need to refine this first approximation, or 

equivalently, according to (2.5), we need to go further in the expansion of 

¯v(y) = p + U 1,0 (y) + o(1), by identifying first and second order terms in 

¯v − p − U 1,0 . 

Let us call 

and consider 

v ∞ (y) = U 1,0 (y) 

ˆv(y) = v ∞ (y) + 1 p w 0(y) + 1 p 2 w 1(y), 

where w 0 , w 1 solve 

where 

∆w i + 

f 0 = 4v 2 ∞ , f 1 = 8 

8 

(1 + |y| 2 ) 2 w i = 

1 

(1 + |y| 2 ) 2 f i(y) in R 2 , i = 1, 2, 

( 

w 0 v ∞ − 1 3 v3 ∞ − 1 2 w2 0 − 1 8 v4 ∞ + 1 ) 

2 w 0v∞ 

2 . (2.6) 

According to [5], for a radial function f(y) = f(|y|) there exists a radial 

solution 

w(r) = 1 − (∫ r2 r 

) 

φ f (s) − φ f (1) 

r 

1 + r 2 (s − 1) 2 ds + φ f (1) 

1 − r 

0 

for the equation 

8 

∆w + 

(1 + |y| 2 ) 2 w = f(y), 

where φ f (s) = ( s2 +1 

s 2 −1 )2 (s−1) 2 ∫ s 

s 0 t 1−t2 f(t)dt for s ≠ 1 and φ 

1+t 2 f (1) = lim s→1 φ f (s). 

It is a straightforward computation to show that 

w(r) = C f log r + D f + O( 

∫ +∞ 

r 

s| log s||f|(s)ds + 

| log r| 

r 2 ) as r → +∞, 

where C f = ∫ +∞ 

0 

t t2 −1 

t 2 +1 f(t)dt, provided ∫ +∞ 

0 

t|f|(t)dt < +∞. 

r 

Therefore, up to replacing w(r) with w(r) − D 2 −1 f , we have shown 

r 2 +1 

Lemma 2.1. Let f ∈ C 1 ([0, +∞)) such that ∫ +∞ 

0 

t| log t||f|(t)dt < +∞. 

There exists a C 2 radial solution w(r) of equation 

∆w + 

8 

(1 + |y| 2 w = f(|y|) in R2 

) 2

such that as r → +∞ 

( ∫ ) 

+∞ 

w(r) = t t2 − 1 

t 2 + 1 f(t)dt log r + O( 

and 

0 

∂ r w(r) = 

( ∫ +∞ 

0 

∫ +∞ 

r 

s| log s||f|(s)ds + 

| log r| 

r 2 ) 

) ∫ 

t t2 − 1 1 +∞ 

t 2 + 1 f(t)dt r + O(1 | log r| 

s|f|(s)ds + 

r r 

r 3 ). 

By means of Lemma 2.1, since f 0 has at most logarithmic growth at infinity 

(see (2.6)), we can define w 0 (r) as a radial function satisfying 

w 0 (y) = C 0 log |y| + O( 1 ) as |y| → +∞, (2.7) 

|y| 

where C 0 = 4 ∫ +∞ 

0 

t t2 −1 

(t 2 +1) 3 log 2 ( 

8 

(1+t 2 ) 2 ) = 12 − 4 log 8. 

More precisely, since we will need the exact expression of w 0 , we have that 

w 0 (y) = 1 2 v2 ∞(y) + 6 log(|y| 2 2 log 8 − 10 

+ 1) + 

|y| 2 + 1 

( 

+ |y|2 − 1 

|y| 2 2 log 2 (|y| 2 + 1) − 1 + 1 

2 log2 8 + 4 

) 

−8 log |y| log(|y| 2 + 1) , 

∫ +∞ 

|y| 2 

9 

(2.8) 

ds 

s + 1 log s + 1 

s 

as we can see by direct inspection. From (2.7) we get that also f 1 growths at 

most logarithmically at infinity and w 1 can be defined as a radial function 

satisfying 

for a suitable constant C 1 . 

w 1 (y) = C 1 log |y| + O( 1 ) as |y| → +∞, (2.9) 

|y| 

1 

We will see now that the profile p 

p p−1 

a better approximation for a solution to the equation 

∆v + v p = 0 

( 

) 

p + v ∞ (y) + 1 p w 0(y) + 1 w 

p 2 1 (y) 

in the region |y| ≤ Ce p 8 . Indeed, by Taylor expansions of exponential and 

logarithmic function, we have that, for |y| ≤ Ce p 8 , 

( 

1 + a p + b 

p 2 + c ) [ 

p 

p 3 = e a 1 + 1 a2 

(b − 

p 2 ) + 1 ( 

p 2 c − ab + a3 

(2.10) 

3 

+ b2 2 + a4 

8 − a2 b 

2 

) 

+ O 

( 

log 6 )] 

(|y| + 2) 

p 3 

provided −4 log(|y| + 2) ≤ a(y) ≤ C and |b(y)| + |c(y)| ≤ C log(|y| + 2). 

Observe that in our case 2C ≥ a + b p + c 

p 2 ≥ − 3 4 p for |y| ≤ Ce p 8 . 

is


Hence, by the choice of v ∞ , w 0 and w 1 and expansion (2.10), we obtain that 

∆v + v p = O( 1 p 4 log 6 (|y| + 2) 

(1 + |y| 2 ) 2 ) in |y| ≤ Ce p 8 . 

As before, we will see now that a proper choice of the parameter µ will automatically 

imply that this approximation for v is also good for the boundary 

condition to be satisfied. Indeed, observe that by (2.7), (2.9) and Lemma 

2.1 we get: for i = 1, 2 

( 

P w i ( x − ξ ) 

) = w i ( x − ξ ) − 2πC i H(x, ξ) + C i log δ + O(δ) in C 1 (¯Ω)(2.11) 

δ 

δ 

( 

P w i ( x − ξ ) 

) = −2πC i G(x, ξ) + O(δ) in C 1 

δ 

loc (¯Ω \ {ξ}), 

provided ξ is bounded away from ∂Ω. If we take µ as a solution of 

( 

log(8µ 4 ) = 8πH(ξ, ξ) 1 − C 0 

4p − C ) 

1 

4p 2 + log δ ( 

C 0 + C ) 

1 

, 

p p 

we get that 

u(x) = e 2(p−1) 

p 

p p 

p−1 µ 2 

p−1 

[P U δ,ξ (x) + 1 p P ( 

w 0 ( x − ξ 

δ 

) 

) + 1 ( 

p 2 P 

w 1 ( x − ξ 

δ 

)] 

) 

is a good first approximation in order to construct a solution for (1.1) with 

just one concentration point. 

Let us remark that µ bifurcates, as p gets large, by ¯µ = e − 3 4 e 2πH(ξ,ξ) , solution 

of equation 

More precisely, 

log(8µ 4 ) = 8πH(ξ, ξ) − C 0 

4 

= 8πH(ξ, ξ) − 3 + log 8. 

µ = e − 3 4 e 2πH(ξ,ξ) (1 + O( 1 p )). 

Let us see now how things generalize if we want to construct a solution to 

problem (1.1) which exhibits m points of concentration. Let ε > 0 fixed and 

take an m−uple ξ = (ξ 1 , . . . , ξ m ) ∈ O ε , where 

O ε = {ξ = (ξ 1 , . . . , ξ m ) ∈ Ω m : dist (ξ i , ∂Ω) ≥ 2ε, |ξ i − ξ j | ≥ 2ε, i ≠ j} . 

Define 

U ξ (x) = 

where 

m∑ 

j=1 

γµ 

1 

γ = p p 

2 

p−1 

j 

[P U δj ,ξ j 

(x) + 1 ( 

p P w 0 ( x − ξ ) 

j 

) + 1 ( 

δ j p 2 P w 1 ( x − ξ ) ] 

j 

) , 

δ j 

p−1 e − p 

2(p−1) 

and δ j = µ j e − p 1 

4 , 

C ≤ µ j ≤ C. (2.12)

The parameters µ j will be chosen later. Observe that for any j ≠ i and 

x = δ i y + ξ i 

1 


(x) + 1 ( 

p P w 0 ( x − ξ ) 

j 

) + 1 ( 

δ j p 2 P w 1 ( x − ξ )] 

j 

) 

δ j 

γµ 

2 

p−1 

j 

= 8π 

γµ 

2 

p−1 

j 

[ 

G(ξ i , ξ j ) 1 − C 0 

4p − C ] 

1 

4p 2 

p + O( e− 4 + |x − ξ i | 

). 

γ 

Hence, we get that the function U ξ (x) is a good approximation for a solution 

to problem (1.1) exhibiting m points of concentration provided 

log(8µ 4 i ) = 

( 

8πH(ξ i , ξ i ) 1 − C 0 

4p − C ) 

1 

4p 2 + log δ ( 

i 

C 0 + C ) 

1 

p p 

+8π ∑ 2 

p−1 [ 

µ i 

2 

G(ξ j , ξ i ) 1 − C 0 

p−1 

4p − C ] 

1 

4p 

j≠i µ 

. 

A direct computation shows that, for p large, µ satisfies 

j=1 

γµ 

2 

p−1 

j 

j 

∑ 

µ i = e − 3 2πH(ξ 

4 e i ,ξ i )+2π 

j≠i G(ξ j,ξ i ) 1 (1 + O( )). (2.13) 

p 

Indeed, with this choice of the parameters µ i , we have that 

m∑ 1 


(x) + 1 ( 

p P w 0 ( x − ξ ) 

j 

) + 1 ( 

δ j p 2 P 

= 1 

γµ 

2 

p−1 

i 

for x = δ i y + ξ i . 

w 1 ( x − ξ j 

δ j 

) 

(p + v ∞ (y) + 1 p w 0(y) + 1 p 2 w 1(y) + O(e − p 4 |y| + e 

− p 4 ) 

) 

(2.14) 

Remark 2.1. Let us remark that U ξ is a positive function. Since |v ∞ + 

1 

p w 0 + 1 w 

p 2 1 | ≤ C in |y| ≤ ɛ δ i 

, by (2.14) we get that U ξ is positive in B(ξ i , ɛ) 

for any i = 1, . . . , m. Moreover, by (2.11) we get that 

( 

P w i ( x − ξ ) 

j 

) → −2πC i G(·, ξ j ) 

δ j 

in C 1 -norm on |x − ξ j | ≥ ɛ, i = 0, 1, and then 

P U δj ,ξ j 

+ 1 ( 

p P w 0 ( x − ξ ) 

j 

) + 1 ( 

δ j p 2 P w 1 ( x − ξ ) 

j 

) 

δ j 

→ 8πG(·, ξ j ) 

in C 1 -norm on |x−ξ j | ≥ ɛ. Hence, since ∂G 

∂ n (·, ξ j) < 0 on ∂Ω, U ξ is a positive 

function in Ω. 

)] 

11


We will look for solutions u of problem (1.1) in the form u = U ξ + φ, where 

φ will represent an higher order term in the expansion of u. Let us set 

W ξ (x) = pU p−1 

ξ 

(x). 

In terms of φ, problem (1.1) becomes 

⎧ 

⎨ L(φ) = −[R + N(φ)] in Ω, 

where 

and 

⎩ 

φ = 0 

on ∂Ω, 

(2.15) 

L(φ) := ∆φ + W ξ φ (2.16) 

R ξ := ∆U ξ + U p ξ , N(φ) = [(U ξ + φ) p − U p ξ 

p−1 

− pUξ 

φ]. 

(2.17) 

The main step in solving problem (2.15) for small φ, under a suitable choice 

of the points ξ i , is that of a solvability theory for the linear operator L. 

In developing this theory, we will take into account the invariance, under 

translations and dilations, of the problem ∆v + e v = 0 in R 2 . We will 

perform the solvability theory for the linear operator L in weighted L ∞ 

spaces, following [12]. For any h ∈ L ∞ (Ω), define 

⎛ 

⎞ 

‖h‖ ∗ = sup ∣ 

x∈Ω 

−1 

∣ m∑ δ 

⎝ 

j ⎠ 

j=1 (δj 2 + |x − ξ h(x) ∣∣ . (2.18) 

j| 2 ) 3 2 

We conclude this section by proving an estimate of R in ‖ · ‖ ∗ . 

Proposition 2.1. For fixed ε > 0, there exist C > 0 and p 0 > 0 such that 

for any ξ ∈ O ε and p ≥ p 0 

‖∆U ξ + U p ξ ‖ ∗ ≤ C p 4 . (2.19) 

Proof Observe that 

m∑ 1 

∆U ξ = 

j=1 γµ j 

m∑ 1 

= 

j=1 γµ 

2 

p−1 

2 

p−1 

j 

( 

( 

−e U j 

+ 1 

pδj 

2 ∆w 0 ( x − ξ j 

) + 1 

δ j p 2 δj 

2 ∆w 1 ( x − ξ ) 

j 

) 

δ j 

−e U j 

+ 1 

pδ 2 j 

˜f 0 ( x − ξ j 

) + 1 

δ j p 2 δj 

2 

) 

− 1 p eU j 

w 0 ( x − ξ j 

) − 1 δ j p 2 eU j 

w 1 ( x − ξ j 

) 

δ j 

˜f 1 ( x − ξ j 

δ j 

) 

, (2.20) 

where, for j = 1, . . . , m, U j = U δj ,ξ j 

and, for i = 0, 1, ˜fi (y) = 

1 

(1+|y| 2 ) 2 f i (y) 

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, 

⎛ 

⎞−1 m∑ 

∣ 

δ ( 

⎝ 

j ⎠ 

(δj 2 + |x − ξ ∆U 

j| 2 ) 3 ξ + U p ξ 

2 

j=1 

) 

(x) ∣ ( 

≤ Ce p 4 ( C ) 

p )p + pe − p 2 

13 

= O(pe − p 4 ), (2.21) 

and, if |x − ξ i | ≤ ε for some i = 1, . . . , m, 

|∆U ξ + U p ξ | = ∣ ( 

1 

8 

γδi 2 2 

− 

µ 

p−1 (1 + |y| 2 ) 2 + 1 p ˜f 0 (y) + 1 p ˜f 2 1 (y) (2.22) 

i 

− 1 8 

p (1 + |y| 2 ) 2 w 0(y) − 1 ) 

8 

p 2 (1 + |y| 2 ) 2 w 1(y) 

+U p ξ (δ iy + ξ i ) + O(pe − p 2 ) 

∣ ∣ , 

where we denote y = x−ξ i 

δ i 

. By (2.14) we deduce that, for x = δ i y + ξ i 

( 

U p ξ (x) = ( p 

2 

) p 1 + 1 

p−1 p v ∞(y) + 1 p 

γµ 

2 w 0(y) + 1 p p ) p 

p 3 w 1(y) + O( e− 4 

p |y| + e− 4 

p ) . 

Since ( 

i 

p 

2 

p−1 

γµ i 

U p ξ (x) = 1 

) p = 

1 

2 , by (2.10) we get for |x − ξ i | ≤ ε √ δ i 

γδi 2µ p−1 

i 

( 

γδ 2 i µ 2 

p−1 

i 

8 

(1 + |y| 2 ) 2 [1 + 1 p 

w 0 (y) − 1 ) 

8 

2 log2 ( 

(1 + |y| 2 ) 2 ) 

+ 1 8 

(w 

p 2 1 − log( 

(1 + |y| 2 ) 2 )w 0 + 1 8 

3 log3 ( 

(1 + |y| 2 ) 2 ) 

+ w2 0 

2 + 1 8 

8 log4 ( 

(1 + |y| 2 ) 2 ) − w ) 

0 8 

2 log2 ( 

(1 + |y| 2 ) 2 ) 

( 

log 6 )] 

(|y| + 2) 

+O 

p 3 + p 2 e − p 4 y + p 2 e − p 4 , y = x − ξ i 

. 

δ i 

Hence, in this region we obtain that 

⎛ 

⎞−1 m∑ 

∣ 

δ ( 

⎝ 

j ⎠ 

(δj 2 + |x − ξ ∆U 

j| 2 ) 3 ξ + U p ξ 

2 

j=1 

≤ ∣ (δ2 i + |x − ξ i| 2 ) 3 2 

(∆U ξ + U p ) 

ξ 

(x) ∣ δ i 

) 

(x) ∣ ∣ (2.23) 

≤ C ( 

γ (1 + |y|2 ) 3 1 log 6 ) 

(|y| + 2) 

2 O 

p 3 (1 + |y| 2 ) 2 ≤ C p 4 , y = x − ξ i 

. 

δ i 

On the other hand, if ε √ δ i ≤ |x − ξ i | ≤ ε we have that 

( p 

) 

U p e 

ξ (x) = O 2 1 

γ (1 + |y| 2 ) 2 , y = x − ξ i 

, 

δ i


since (1 + s p )p ≤ e s . Thus, in this region 

⎛ 

⎞−1 m∑ 

∣ 

δ ( ) 

⎝ 

j ⎠ 

j=1 (δj 2 + |x − ξ ∆U 

j| 2 ) 3 ξ + U p ξ 

(x) ∣ (2.24) 

2 

( 

) 

p 

= O 

≤ Cpe − p x − ξ i 

(1 + |y| 2 ) 1 8 , y = . 

2 

δ i 

By (2.21), (2.23) and (2.24) we obtain the desired result. 

3. Analysis of the linearized operator 

In this Section, we prove bounded invertibility of the operator L, uniformly 

on ξ ∈ O ε , by using L ∞ -norms introduced in (2.18). Let us recall that 

L(φ) = ∆φ + W ξ φ, where W ξ (x) = pU p−1 

ξ 

(x). For simplicity of notation, we 

will omit the dependence of W ξ on ξ. 

As in Proposition 2.1, we have for the potential W (x) the following expansions. 

If |x − ξ i | ≤ ε for some i = 1, . . . , m 

p 

W (x) = p( 

γµ 

2 

p−1 

i 

+O( e− p 4 

p |y| + e− p 4 

p ) ) p−1 

= δ −2 

i 

( 

) p−1 (1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) (3.1) 

1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4 

p |y| + e− p 4 

p ) ) p−1 

where again we use the notation y = x−ξ i 

δ i 

. In this region, we have that 

W (x) ≤ C δi 

2 e v∞(y) e − 1 p v∞(y) ( ) 

= O e U i(x) 

, 

since v ∞ (y) ≥ −2p. Indeed, by Taylor expansions of exponential and logarithmic 

functions as in (2.10), we obtain that, if |x − ξ i | ≤ ε √ δ i (and 

|y| ≤ √ ε 

δi 

), 

W (x) = δ −2 

i 

= 

( 

1 + 1 p v ∞(y) + 1 p 2 w 0(y) + 1 p 3 w 1(y) + O( e− p 4 

p |y| + e− p 4 

p ) ) p−1 

( 

8 

δi 2(1 + |y|2 ) 2 1 + 1 p (w 0 − v ∞ − 1 2 v2 ∞) + O( log4 (|y| + 2) 

p 2 ) 

If |x − ξ i | ≥ ε for any i = 1, . . . , m: 

Summing up, we have that 

W (x) = O(p( C p )p−1 ). 

) 

. 

□

Lemma 3.1. Let ε > 0 be fixed. There exist D 0 > 0 and p 0 > 0 such that 

W (x) ≤ D 0 m ∑ 

e U j(x) 

j=1 

for any ξ ∈ O ε and p ≥ p 0 . Furthermore, 

( 

8 

W (x) = 

δi 2(1 + |y|2 ) 2 1 + 1 p (w 0 − v ∞ − 1 ) 

2 v2 ∞) + O( log4 (|y| + 2) 

p 2 ) 

for any |x − ξ i | ≤ ε √ δ i , where y = x−ξ i 

δ i 

. 

Remark 3.1. As for W , let us point out that, if |x − ξ i | ≤ ε for some 

i = 1, . . . , m, there holds 

p(U ξ + O( 1 p 

p 3 ))p−2 ≤ Cp( 

2 

) p−2 e p−2 

p v ∞( x−ξ i ) δ i = O 

(e ) U i(x) 

. 

p−1 

γµ 

i 

Since this estimate is true if |x − ξ i | ≥ ε for any i = 1, . . . , m, we have that 

p(U ξ + O( 1 m∑ 

p 3 ))p−2 ≤ C e Uj(x) . (3.2) 

j=1 

In an heuristic way, the operator L is close to ˜L defined by 

( m 

) 

∑ 

˜L(φ) = ∆φ + φ. 

e U i 

i=1 

The operator ˜L is ”essentially” a superposition of linear operators which, 

after a dilation and translation, approach, as p → ∞, the linear operator in 

R 2 , 

8 

φ → ∆φ + 

(1 + |y| 2 ) 2 φ, 

namely, equation ∆v+e v 8 

= 0 linearized around the radial solution log . 

(1+|y| 2 ) 2 

Set 

z 0 (y) = |y|2 − 1 

|y| 2 + 1 , z y i 

i(y) = 4 

1 + |y| 2 i = 1, 2. 

The first ingredient to develop the desired solvability theory for L is the well 

known fact that any bounded solution of L(φ) = 0 in R 2 is precisely a linear 

combination of the z i , i = 0, 1, 2, see [3] for a proof. 

The second ingredient is a detailed analysis of L − ˜L. It has been proved in 

[12] and [14] that the operator ˜L is invertible in the set of functions which, 

roughly speaking, are orthogonal to the functions z i for i = 1, 2, and the 

operatorial norm of ˜L −1 behaves like p as p → +∞. Since L is close to ˜L up 

to terms of order at least 1 p 

(see Lemma 3.1), the invertibility of L becomes 

delicate and non trivial. 

In [12] and [14] there were established a priori estimates respectively in 

weighted L ∞ −norms and in H0 1 (Ω)−norms. We will follow the approach in 

[12] since the estimates there are stronger and in this context very helpful. 

15


Given h ∈ C(¯Ω), we consider the linear problem of finding a function 

φ ∈ W 2,2 (Ω) such that 

2∑ m∑ 

L(φ) = h + c ij e U j 

Z ij , in Ω, (3.3) 

i=1 j=1 

∫ 

φ = 0, on ∂Ω, (3.4) 

e U j 

Z ij φ = 0, for all i = 1, 2, j = 1, . . . , m, (3.5) 

Ω 

for some coefficients c ij , i = 1, 2 and j = 1, . . . , m. Here and in the sequel, 

for any i = 0, 1, 2 and j = 1, . . . , m we denote 

⎧ 

⎪⎨ 

Z ij (x) := z i ( x − ξ j 

δ j 

) = 

The main result of this Section is the following: 

⎪⎩ 

|x−ξ j | 2 −δ 2 j 

δ 2 j +|x−ξ j| 2 if i = 0 

4δ j (x−ξ j ) i 

δ 2 j +|x−ξ j| 2 if i = 1, 2. 

Proposition 3.1. Let ε > 0 be fixed. There exist p 0 > 0 and C > 0 such 

that, for h ∈ C(¯Ω) there is a unique solution to problem (3.3)-(3.5), for any 

p > p 0 and ξ ∈ O ε , which satisfies 

‖φ‖ ∞ ≤ Cp‖h‖ ∗ . (3.6) 

Proof. The proof of this result consists of six steps. 

1 st Step. The operator L satisfies the maximum principle in ˜Ω := Ω \ 

∪ m j=1 B(ξ j, Rδ j ) for R large, independent on p. Namely, 

if L(ψ) ≤ 0 in ˜Ω and ψ ≥ 0 on ∂ ˜Ω, then ψ ≥ 0 in ˜Ω. 

In order to prove this fact, we show the existence of a positive function Z 

in ˜Ω satisfying L(Z) < 0. We define Z to be 

( ) 

m∑ a(x − ξj ) 

Z(x) = z 0 , a > 0. 

δ j 

j=1 

First, observe that, if |x−ξ j | ≥ Rδ j for R > 1 a 

, then Z(x) > 0. On the other 

hand, we have: 

⎛ 

m∑ 

W (x)Z(x) ≤ D 0 ⎝ 

e U j(x) 

j=1 

⎞ 

⎠ Z(x) ≤ D 0 Z(x) 

m∑ 

j=1 

8δ 2 j 

|x − ξ j | 4 , 

where D 0 is the constant in Lemma 3.1. Further,by definition of z 0 , 

m∑ 

−∆Z(x) = a 2 8δ2 j (a2 |x − ξ j | 2 − δj 2) 

(a 2 |x − ξ j | 2 + δj 2)3 

≥ 1 3 

j=1 

m∑ 

j=1 

8a 2 δ 2 j 

(a 2 |x − ξ j | 2 + δ 2 j )2 ≥ 4 27 

m∑ 

j=1 

8δ 2 j 

a 2 |x − ξ j | 4

provided R > 

√ 

2 

a . Hence 

LZ(x) ≤ 

( 

− 4 

27 

) 

1 

a 2 + D ∑ m 

0 

j=1 

8δ 2 j 

|x − ξ j | 4 < 0 

provided that a is chosen sufficiently small, but independent of p. 

function Z(x) is what we are looking for. 

17 

The 

2 nd Step. Let R be as before. Let us define the “inner norm” of φ in the 

following way 

‖φ‖ i = sup 

x∈∪ m j=1 B(ξ j,Rδ j ) 

|φ|(x). 

We claim that there is a constant C > 0 such that, if L(φ) = h in Ω, 

h ∈ C 0,α (¯Ω), then 

‖φ‖ ∞ ≤ C[‖φ‖ i + ‖h‖ ∗ ], 

for any h ∈ C 0,α (¯Ω). We will establish this estimate with the use of suitable 

barriers. Let M = 2 diam Ω. Consider the solution ψ j (x) of the problem: 

{ 

−∆ψj = 2δ j 

|x−ξ j 

in Rδ 

| 3 j < |x − ξ j | < M 

ψ j (x) = 0 on |x − ξ j | = Rδ j and |x − ξ j | = M. 

Namely, the function ψ j (x) is the positive function defined by: 

where 

and 

ψ j (x) = − 

2δ j 

|x − ξ j | + A + B log |x − ξ j|, 

B = 2( δ j 

M − 1 R ) 1 

log( M Rδ j 

) < 0 

A = 2δ j 

− B log M. 

M 

Hence, the function ψ j (x) is uniformly bounded from above by a constant 

independent of p, since we have that, for Rδ j ≤ |x − ξ j | ≤ M, 

Define now the function 

ψ j (x) ≤ A + B log(Rδ j ) = 2δ j 

M − B log M Rδ j 

= 2 R . 

∑ 

˜φ(x) m = 2‖φ‖ i Z(x) + ‖h‖ ∗ ψ j (x), 

where Z was defined in the previous Step. First of all, observe that by the 

definition of Z, choosing R larger if necessary, 

˜φ(x) ≥ 2‖φ‖ i Z(x) ≥ ‖φ‖ i ≥ |φ|(x) for |x − ξ j | = Rδ j , j = 1, . . . , m, 

and, by the positivity of Z(x) and ψ j (x), 

j=1 

˜φ(x) ≥ 0 = |φ|(x) for x ∈ ∂Ω.


Since by definition of ‖ · ‖ ∗ we have that 

⎛ 

⎞ 

m∑ δ 

⎝ 

j ⎠ 

(δj 2 + |x − ξ ‖h‖ 

j| 2 ) 3 ∗ ≥ |h(x)|, (3.7) 

2 

j=1 

finally we obtain that 

L ˜φ 

∑ m m ( 

∑ 

≤ ‖h‖ ∗ Lψ j (x) = ‖h‖ ∗ − 

2δ ) 

j 

|x − ξ 

j=1 

j=1 

j | 3 + W (x)ψ j(x) 

m ( 

∑ 

≤ ‖h‖ ∗ − 

2δ j 

|x − ξ 

j=1 

j | 3 + 2mD ) 

0 

R 

eU j(x) 

⎛ 

⎞ 

m∑ δ 

≤ −‖h‖ ∗ ⎝ 

j ⎠ 

(δj 2 + |x − ξ ≤ −|h(x)| ≤ |Lφ|(x) 

j| 2 ) 3 2 

j=1 

provided R > 16mD 0 and p large enough. Hence, by the maximum principle 

in Step 1 we obtain that 

|φ|(x) ≤ ˜φ(x) for x ∈ ˜Ω, 

and therefore, since Z(x) ≤ 1 and ψ j (x) ≤ 2 R , 

‖φ‖ ∞ ≤ C[‖φ‖ i + ‖h‖ ∗ ]. 

3 rd Step. We prove uniform a priori estimates for solutions φ of problem 

Lφ = h in Ω, φ = 0 on ∂Ω, when h ∈ C 0,α (¯Ω) and φ satisfies (3.5) and in 

addition the orthogonality conditions: 

∫ 

e U j 

Z 0j φ = 0, for j = 1, . . . , m. (3.8) 

Ω 

Namely, we prove that there exists a positive constant C such that for any 

ξ ∈ O ε and h ∈ C 0,α (¯Ω) 

‖φ‖ ∞ ≤ C‖h‖ ∗ , 

for p sufficiently large. By contradiction, assume the existence of sequences 

p n → ∞, points ξ n ∈ O ε , functions h n and associated solutions φ n such that 

‖h n ‖ ∗ → 0 and ‖φ n ‖ ∞ = 1. 

Since ‖φ n ‖ ∞ = 1, Step 2 shows that lim inf n→+∞ ‖φ n ‖ i > 0. Let us set 

ˆφ n j (y) = φ n(δj ny + ξn j ) for j = 1, . . . , m. By Lemma 3.1 and (3.7), elliptic 

estimates readily imply that ˆφ n j converges uniformly over compact sets to a 

bounded solution ˆφ ∞ j of the equation in R 2 : 

∆φ + 

8 

(1 + |y| 2 ) 2 φ = 0. 

This implies that ˆφ ∞ j is a linear combination of the functions z i , i = 0, 1, 2. 

Since ‖ ˆφ n j ‖ ∞ ≤ 1, by Lebesgue theorem the orthogonality conditions (3.5) 

and (3.8) on φ n pass to the limit and give ∫ R 2 8 

(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 > 

0. 

4 th Step. We prove that there exists a positive constant C > 0 such that 

any solution φ of equation Lφ = h in Ω, φ = 0 on ∂Ω, satisfies 

‖φ‖ ∞ ≤ Cp‖h‖ ∗ , 

when h ∈ C 0,α (¯Ω) and we assume on φ only the orthogonality conditions 

(3.5). Proceeding by contradiction as in Step 3, we can suppose further that 

p n ‖h n ‖ ∗ → 0 as n → +∞ (3.9) 

but we loss in the limit the condition ∫ 8 

R 2 z 

(1+|y| 2 ) 2 0 (y) ˆφ ∞ j 

have that 

19 

= 0. Hence, we 

ˆφ n j → C j 

|y| 2 − 1 

|y| 2 + 1 in C0 loc (R2 ) (3.10) 

for some constants C j . To reach a contradiction, we have to show that 

C j = 0 for any j = 1, . . . , m. We will obtain it from the stronger condition 

(3.9) on h n . 

To this end, we perform the following construction. By Lemma 2.1, we 

8 

find radial solutions w and t respectively of equations ∆w + w = 

(1+|y| 2 ) 2 

8 

8 

8 

z 

(1+|y| 2 ) 2 0 (y) and ∆t + t = in R 

(1+|y| 2 ) 2 (1+|y| 2 ) 2 , such that as |y| → +∞ 

2 

w(y) = 4 3 

1 

1 

log |y| + O( ) , t(y) = O( 

|y| |y| ), 

since 8 ∫ +∞ 

0 

t (t2 −1) 2 

dt = 4 (t 2 +1) 4 3 and 8 ∫ +∞ 

0 

t t2 −1 

dt = 0. 

(t 2 +1) 3 

For simplicity, from now on we will omit the dependence on n. For j = 

1, . . . , m, define now 

u j (x) = w( x − ξ j 

) + 4 δ j 3 (log δ j)Z 0j (x) + 8π 3 H(ξ j, ξ j )t( x − ξ j 

) 

δ j 

and denote by P u j the projection of u j onto H0 1(Ω). Since u j − P u j + 

4 

3 log 1 

|·−ξ j | 

= O(δ j ) on ∂Ω (together with boundary derivatives), by harmonicity 

we get 

The function P u j solves 

P u j = u j − 8π 3 H(·, ξ j) + O(e − p 4 ) in C 1 (¯Ω), 

P u j = − 8π 3 G(·, ξ j) + O(e − p 4 ) in C 1 loc (¯Ω \ {ξ j }). 

∆P u j + W (x)P u j = e U j 

Z 0j + 

(3.11) 

( 

) 

W (x) − e U j 

P u j + R j , (3.12) 

where 

R j (x) = 

( 

P u j − u j + 8π ) 

3 H(ξ j, ξ j ) e U j 

.


Multiply (3.12) by φ and integrate by parts to obtain: 

∫ 

∫ ( 

) ∫ ∫ 

e U j 

Z 0j φ + W (x) − e U j 

P u j φ = P u j h − 

Ω 

Ω 

First of all, by Lebesgue theorem and (3.10) we get that 

∫ 

∫ 

e U j 

8(|y| 2 − 1) 2 

Z 0j φ → C j 

Ω 

R 2 

The more delicate term is ∫ Ω 

we have that 

∫ 

Ω 

− 8π 3 

( 

W (x) − e U j 

( 

) ∫ 

W (x) − e U j 

P u j φ = 

∑ 

k≠j 

= 4 log δ j 

3 p 

− 8π 3 

∑ 

k≠j 

= − C j 

3 

Ω 

Ω 

R j φ. (3.13) 

(1 + |y| 2 ) 4 = 8π 3 C j. (3.14) 

) 

P u j φ. By Lemma 3.1 and (3.11) 

B(ξ j ,ε √ δ j ) 

∫ 

G(ξ k , ξ j ) 

B(ξ k ,ε √ W (x)φ + O(e − p 8 ) 

δ k ) 

∫ 

B(0, √ ε ) 

δ j 

∫ 

G(ξ k , ξ j ) 

( 

) 

W (x) − e U j 

P u j φ 

( 

8 

(1 + |y| 2 ) 2 w 0 − v ∞ − 1 2 v2 ∞ 

ε 

B(0, √ ) δk 

8 

(1 + |y| 2 ) 2 ˆφ k + O( 1 p ) 

) 

z 0 (y) ˆφ j 

8(|y| 

∫R 2 − 1) 2 ( 

2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 ) 

2 v2 ∞ (y) + o(1) 

since Lebesgue theorem and (3.10) imply: 

∫ 

and 

∫ 

B(0, √ ε ) 

δ j 

ε 

B(0, √ ) δk 

( 

8 

(1 + |y| 2 ) 2 w 0 − v ∞ − 1 2 v2 ∞ 

) 

z 0 (y) ˆφ j → 

∫ 

8(|y| 2 − 1) 2 ( 

C j 

R 2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 ) 

2 v2 ∞ 

8 

(1 + |y| 2 ) ˆφ 

∫ 

2 k → C k 

R 2 8 

(1 + |y| 2 ) 2 |y| 2 − 1 

|y| 2 + 1 = 0. 

In a straightforward but tedious way, by (2.8) we can compute: 

8(|y| 

∫R 2 − 1) 2 ( 

2 (1 + |y| 2 ) 4 w 0 − v ∞ − 1 ) 

2 v2 ∞ (y) = −8π, 

so that we obtain ∫ ( 

) 

W (x) − e U j 

P u j φ = 8π 

Ω 

3 C j + o(1). (3.15) 

As far as the R.H.S. in (3.13), we have that by (3.11) 

∫ 

∫ 

) 

m∑ δ k 

| P u j h| = O 

(‖h‖ ∗ ( 

Ω 

Ω (δk 2 + |x − ξ )|P u 

k| 2 ) 3 j | = O(p‖h‖ ∗ ) (3.16) 

2 

k=1

21 

since ∫ Ω |P u j| = O(| log δ j |) = O(p) and 

∫ 

∫ 

δ j 

1 

B(ξ j ,ε) (δj 2 + |x − ξ |u 

j| 2 ) 3 j | ≤ 

|u 

2 R 2 (1 + |y| 2 ) 3 j |(δ j y + ξ j ) = O(p). 

2 

Finally, by (3.11) 

∫ 

R j φ = O 

Ω 

(∫ 

Ω 

e U j 

(|x − ξ j | + e − p 4 ) 

) 

= O(e − p 4 ). (3.17) 

Hence, inserting (3.14)-(3.17) in (3.13) we obtain that 

16π 

3 C j = o(1) 

for any j = 1, . . . , m. Necessarily, C j = 0 and the claim is proved. 

5 th Step. We establish the validity of the a priori estimate: 

‖φ‖ ∞ ≤ Cp‖h‖ ∗ (3.18) 

for solutions of problem (3.3)-(3.5) and h ∈ C 0,α (¯Ω). 

gives 

⎛ 

⎞ 

2∑ m∑ 

‖φ‖ ∞ ≤ Cp ⎝‖h‖ ∗ + |c ij | ⎠ 

since 

i=1 j=1 

‖e U j 

Z ij ‖ ∗ ≤ 2‖e U j 

‖ ∗ ≤ 16. 

The previous Step 

Hence, arguing by contradiction of (3.18), we can proceed as in Step 3 and 

suppose further that 

2∑ m∑ 

p n ‖h n ‖ ∗ → 0 , p n |c n ij| ≥ δ > 0 as n → +∞. 

i=1 j=1 

We omit the dependence on n. It suffices to estimate the values of the 

constants c ij . For i = 1, 2 and j = 1, . . . , m, multiply (3.3) by P Z ij and, 

integrating by parts, get: 

2∑ 

m∑ 

l=1 h=1 

∫ 

( ) 

c lh P Zlh , P Z ij + hP Z 

H 1 ij = 

0 Ω 

− 

∫ 

Ω 

∫ 

Ω 

W (x)φP Z ij (3.19) 

e U j 

Z ij φ, 

since ∆P Z ij = ∆Z ij = −e U j 

Z ij . For i = 1, 2 and j = 1, . . . , m we have the 

following expansions: 

( ) 

∂H 

P Z ij = Z ij − 8πδ j ∂(ξ j ) i 

(·, ξ j ) + O δj 

3 ( ) (3.20) 

P Z 0j = Z 0j − 1 + O 

δ 2 j


in C 1 (¯Ω) and 

( ) 

∂G 

P Z ij = −8πδ j ∂(ξ j ) i 

(·, ξ j ) + O δj 

3 ( ) 

P Z 0j = O δj 

2 (3.21) 

in C 1 loc (¯Ω \ {ξ j }). By (3.20)-(3.21) we deduce the following “orthogonality” 

relations: for i, l = 1, 2 and j, h = 1, . . . , m with j ≠ h, 

( 

) 

(P Z ij , P Z lj = 64 ∫ ) 

|y| 2 

δ 

H 1 0 (Ω) (1+|y| 2 ) 4 il + O(δj 2) 

) 

R (P 2 Z ij , P Z lh = O(δ jδ h ) (3.22) 

H 1 0 (Ω) 

and 

(P Z 0j , P Z lj 

) 

H 1 0 (Ω) = O(δ2 j ) 

(P Z 0j , P Z lh 

) 

H 1 0 (Ω) = O(δ jδ h ) 

(3.23) 

uniformly on ξ ∈ O ε , where δ il denotes the Kronecker’s symbol. In fact, we 

have that 

∫ 

(P Z ij , P Z lj 

)H = e U j 

Z ij P Z lj 

1 

0 (Ω) Ω 

∫ 

( 

) 

= e U j 

∂H 

Z ij Z lj − 8πδ j (ξ j , ξ j ) + O(δ j |x − ξ j | + δj 3 ) + O(δj 4 ) 

∂(ξ j ) l 

B(ξ j ,ε) 

∫ 

= 128 

R 2 

B(ξ j ,ε) 

y i y l 

(1 + |y| 2 ) 4 + O(δ2 j ) = 

⎛ 

∫ 

|y| 

⎝64 

2 

R 2 

(1 + |y| 2 ) 4 ⎞ 

⎠ δ il + O(δ 2 j ), 

∫ 

(P Z ij , P Z lh 

)H = e U j 

Z ij P Z lh 

1 

0 (Ω) Ω 

∫ 

( 

) 

= e U j 

∂G 

Z ij −8πδ h (ξ j , ξ h ) + O(δ h |x − ξ j | + δ 3 

∂(ξ h ) 

h) + O(δj 3 ) 

l 

= O(δ j δ h ), 

∫ 

(P Z 0j , P Z lj 

)H = e U j 

Z 0j P Z lj 

1 

0 (Ω) Ω 

∫ 

( 

) 

= e U j 

∂H 

Z 0j Z lj − 8πδ j (ξ j , ξ j ) + O(δ j |x − ξ j | + δj 3 ) + O(δj 3 ) 

∂(ξ j ) l 

B(ξ j ,ε) 

= O(δ 2 j )

23 

and 

∫ 

(P Z 0j , P Z lh 

)H = e U j 

Z 0j P Z lh 

1 

0 (Ω) Ω 

∫ 

( 

) 

= e U j 

∂G 

Z 0j −8πδ h (ξ j , ξ h ) + O(δ h |x − ξ j | + δ 3 

∂(ξ h ) 

h) + O(δj 2 ) 

l 

B(ξ j ,ε) 

= O(δ j δ h ). 

Now, since 

∫ 

| 

∫ 

hP Z ij | ≤ C ′ |h| ≤ C‖h‖ ∗ , 

Ω 

Ω 

by (3.22) the L.H.S. of (3.19) is estimated as follows: 

L.H.S. = Dc ij + O ( e − p 2 

2∑ m∑ 

|c lh | ) + O(‖h‖ ∗ ), (3.24) 

l=1 h=1 

where D = 64 ∫ R 2 |y| 2 

(1+|y| 2 ) 4 . Moreover, by Lemma 3.1 the R.H.S. of (3.19) 

takes the form: 

R.H.S. = 

= 

∫ 

√ 

B(ξ j ,ε δ j ) 

∫ 

= 1 p 

B(0, √ ε ) 

δ j 

∫ 

B(ξ j ,ε √ δ j ) 

( 

W (x) − e U j 

∫ 

√ 

W (x)φP Z ij − e U j 

φZ ij + O( δ j ‖φ‖ ∞ ) (3.25) 

Ω 

) ∫ 

√ 

φP Z ij + e U j 

φ (P Z ij − Z ij ) + O( δ j ‖φ‖ ∞ ) 

Ω 

( 

) 

32y i 

(1 + |y| 2 ) 3 w 0 − v ∞ − v2 ∞ ˆφ j + O( 1 2 p 2 ‖φ‖ ∞) 

in view of (3.20), where ˆφ j (y) = φ(δ j y + ξ j ). Inserting the estimates (3.24) 

and (3.25) into (3.19), we deduce that 

Dc ij + O ( e − p 2 

Hence, we obtain that 

2∑ m∑ 

|c lh | ) = O(‖h‖ ∗ + 1 p ‖φ‖ ∞). 

l=1 h=1 

2∑ 

l=1 h=1 

m∑ 

|c lh | = O(‖h‖ ∗ + 1 p ‖φ‖ ∞). (3.26) 

Since ∑ 2 

l=1 

∑ mh=1 

|c lh | = o(1), as in Step 4 we have that 

ˆφ j → C j 

|y| 2 − 1 

|y| 2 + 1 in C0 loc (R2 )


for some constant C j , j = 1, . . . , m. Hence, in (3.25) we have a better 

estimate since by Lebesgue theorem the term 

∫ 

( 

32y i 

(1 + |y| 2 ) 3 w 0 − v ∞ − 1 ) 

2 v2 ∞ (y) ˆφ j (y) 

converges to 

B(0, √ ε ) 

δ j 

∫ 

32y i (|y| 2 ( 

− 1) 

C j 

(1 + |y| 2 ) 4 w 0 − v ∞ − 1 ) 

2 v2 ∞ (y) = 0. 

R 2 

Therefore, we get that the R.H.S. in (3.19) satisfies: R.H.S. = o( 1 p 

), and in 

turn, ∑ 2 ∑ mh=1 

l=1 |c lh | = O(‖h‖ ∗ ) + o( 1 p 

). This contradicts 

and the claim is established. 

2∑ m∑ 

p |c ij | ≥ δ > 0, 

i=1 j=1 

6 th Step. We prove the solvability of (3.3)–(3.5). To this purpose, we 

consider the spaces: 

and 

2∑ m∑ 

K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m} 

i=1 j=1 

∫ 

Kξ ⊥ = {φ ∈ L 2 (Ω) : 

Let Π ξ : L 2 (Ω) → K ξ defined as: 

Ω 

e U j 

Z ij φ = 0 for i = 1, 2, j = 1, . . . , m}. 

2∑ m∑ 

Π ξ φ = c ij P Z ij , 

i=1 j=1 

where c ij are uniquely determined (as it follows by (3.22)-(3.23)) by the 

system: 

∫ 

Ω 

e U h 

Z lh 

⎛ 

⎝φ − 

2∑ 

m∑ 

i=1 j=1 

c ij P Z ij 

⎞ 

⎠ = 0 for any l = 1, 2 , h = 1, . . . , m. 

Let Π ⊥ ξ = Id − Π ξ : L 2 (Ω) → Kξ ⊥ . Problem (3.3)–(3.5), expressed in a weak 

form, is equivalent to find φ ∈ Kξ 

⊥ ∩ H1 0 (Ω) such that 

∫ 

(φ, ψ) H 1 

0 (Ω) = (W φ − h) ψ dx, for all ψ ∈ Kξ ⊥ ∩ H0 1 (Ω). 

Ω 

With the aid of Riesz’s representation theorem, this equation gets rewritten 

in Kξ 

⊥ ∩ H1 0 (Ω) in the operatorial form 

(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 

in Kξ ⊥ ∩ H1 0 (Ω). The homogeneous equation φ = K(φ) in K⊥ ξ ∩ H1 0 (Ω), 

which is equivalent to (3.3)-(3.5) with h ≡ 0, has only the trivial solution in 

view of the a priori estimate (3.18). Now, Fredholm’s alternative guarantees 

unique solvability of (3.27) for any ˜h ∈ Kξ ⊥ . Moreover, by elliptic regularity 

theory this solution is in W 2,2 (Ω). 

At p > p 0 fixed, by density of C 0,α (¯Ω) in (C(¯Ω), ‖ · ‖ ∞ ), we can approximate 

h ∈ C(¯Ω) by smooth functions and, by (3.18) and elliptic regularity theory, 

we can show that (3.6) holds for any h ∈ C(¯Ω). The proof is complete. □ 

Remark 3.2. Given h ∈ C(¯Ω), let φ be the solution of (3.3)-(3.5) given by 

Proposition 3.1. Multiplying (3.3) by φ and integrating by parts, we get 

∫ ∫ 

‖φ‖ 2 H 1 0 (Ω) = W φ 2 − hφ. 

By Lemma 3.1 we get 

Ω 

‖φ‖ H 1 

0 (Ω) ≤ C (‖φ‖ ∞ + ‖h‖ ∗ ) . 

4. The nonlinear problem 

We want to solve the nonlinear auxiliary problem 

∆(U ξ + φ) + (U ξ + φ) p = 

2∑ m∑ 

c ij e U j 

Z ij , in Ω, (4.1) 

i=1 j=1 

U ξ + φ > 0, in Ω (4.2) 

∫ 

Ω 

Ω 

φ = 0, on ∂Ω, (4.3) 

e U j 

Z ij φ = 0, for all i = 1, 2, j = 1, . . . , m, (4.4) 

for some coefficients c ij , i = 1, 2 and j = 1, . . . , m, which depend on ξ. 

Recalling that 

N(φ) = |U ξ + φ| p − U p ξ 

we can rewrite (4.1) in the form 

− pU 

p−1 

ξ 

φ , R = ∆U ξ + U p ξ , 

2∑ m∑ 

L(φ) = − (R + N(φ)) + c ij e U j 

Z ij . 

i=1 j=1 

Using the theory developed in the previous Section for the linear operator 

L, we prove the following result: 

Lemma 4.1. Let ε > 0 be fixed. There exist C > 0 and p 0 > 0 such that, 

for any p > p 0 and ξ ∈ O ε , problem (4.1)-(4.4) has a unique solution φ ξ 

which satisfies 

25 

‖φ ξ ‖ ∞ ≤ C p 3 . (4.5)


Further, there holds: 

2∑ m∑ 

|c ij (ξ)| ≤ C p 

i=1 j=1 

4 , ‖φ ξ‖ H 1 

0 (Ω) ≤ C p 3 . (4.6) 

Proof. Let us denote by C ∗ the function space C(¯Ω) endowed with the 

norm ‖ · ‖ ∗ . Proposition 3.1 implies that the unique solution φ = T (h) of 

(3.3)-(3.5) defines a continuous linear map from the Banach space C ∗ into 

C 0 (¯Ω), with norm bounded by a multiple of p. Problem (4.1)-(4.4) becomes 

φ = A(φ) := −T (R + N(φ)). 

For a given number γ > 0, let us consider the region 

We have the following estimates: 

F γ := {φ ∈ C 0 (¯Ω) : ||φ|| ∞ ≤ γ p 3 }. 

‖N(φ)‖ ∗ ≤ Cp‖φ‖ 2 ∞ 

‖N(φ 1 ) − N(φ 2 )‖ ∗ ≤ Cp (max i=1,2 ‖φ i ‖ ∞ ) ‖φ 1 − φ 2 ‖ ∞ , 

for any φ, φ 1 , φ 2 ∈ F γ . In fact, by Lagrange theorem we have that: 

( 

|N(φ)| ≤ p(p − 1) U ξ + O( 1 ) p−2 

p 3 ) φ 2 

|N(φ 1 ) − N(φ 2 )| ≤ p(p − 1) 

(U ξ + O( 1 ) p−2 ( ) 

p 3 ) max |φ i| |φ 1 − φ 2 | 

i=1,2 

(4.7) 

for any x ∈ Ω, and hence, by (3.2) we get (4.7) since ‖ ∑ m 

j=1 e U j 

‖ ∗ = O(1). 

By (4.7), Propositions 2.1 and 3.1 imply that 

and 

‖A(φ)‖ ∞ ≤ D ′ p (‖N(φ)‖ ∗ + ‖R‖ ∗ ) ≤ O(p 2 ‖φ‖ 2 ∞) + D p 3 

) 

‖A(φ 1 )−A(φ 2 )‖ ∞ ≤ C ′ p ‖N(φ 1 )−N(φ 2 )‖ ∗ ≤ Cp 

(max 

2 ‖φ i‖ ∞ ‖φ 1 −φ 2 ‖ ∞ 

i=1,2 

for any φ, φ 1 , φ 2 ∈ F γ , where D is independent of γ. Hence, if ‖φ‖ ∞ ≤ 2D 

p 3 , 

we have that 

‖A(φ)‖ ∞ = O( 1 p ‖φ‖ ∞) + D p 3 ≤ 2D 

p 3 . 

Choose γ = 2D. Then, A is a contraction mapping of F γ since 

‖A(φ 1 ) − A(φ 2 )‖ ∞ ≤ 1 2 ‖φ 1 − φ 2 ‖ ∞ 

for any φ 1 , φ 2 ∈ F γ . Therefore, a unique fixed point φ ξ of A exists in F γ . 

By (3.26), we get that 

2∑ m∑ 

|c ij (ξ)| = O 

(‖N(φ ξ )‖ ∗ + ‖R‖ ∗ + 1 ) 

p ‖φ ξ‖ ∞ ≤ C p 4 

i=1 j=1

27 

and by Remark 3.2 we deduce that 

‖φ ξ ‖ H 1 

0 (Ω) = O (‖φ ξ‖ ∞ + ‖N(φ ξ )‖ ∗ + ‖R‖ ∗ ) ≤ C p 3 . 

The proof is now complete since φ ξ solves (4.1)-(4.4): in order to show 

the validity of (4.2), let us remark that p|φ ξ | → 0 in C(¯Ω) and by elliptic 

regularity theory p|φ ξ | → 0 in C 1 (¯Ω \ ∪ m j=1 B(ξ j, ɛ)) and so, we can proceed 

as in Remark 2.1 to show that U ξ + φ ξ > 0 in Ω. 

□ 

Let ξ 1 , ξ 2 ∈ O ε . Since 

∆(φ ξ1 − φ ξ2 ) + pU p−1 

ξ 1 

(φ ξ1 − φ ξ2 ) = ((U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p ) 

( 

+ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1 

) 

ξ 1 

(φ ξ2 − φ ξ1 ) + ∆(U ξ2 − U ξ1 ) 

2∑ m∑ 

+ (c ij (ξ 1 ) − c ij (ξ 2 )) e U j 

(ξ 1 )Z ij (ξ 1 ) 

+ 

i=1 j=1 

2∑ m∑ 

i=1 j=1 

and by (3.2) 

( 

) 

c ij (ξ 2 ) e U j 

(ξ 1 )Z ij (ξ 1 ) − e U j 

(ξ 2 )Z ij (ξ 2 ) 

( 

) 

‖ (U ξ1 + φ ξ2 ) p − (U ξ1 + φ ξ1 ) p − pU p−1 

ξ 1 

(φ ξ2 − φ ξ1 ) ‖ ∗ 

≤ C p 2 ‖φ ξ 1 

− φ ξ2 ‖ ∞ ‖p(U ξ1 + O( 1 p 3 ))p−2 ‖ ∗ 

= o( 1 p ‖φ ξ 1 

− φ ξ2 ‖ ∞ ) 

uniformly in O ε , by Proposition 3.1 and (4.6) we get 

‖φ ξ1 − φ ξ2 ‖ ∞ ≤ Cp‖(U ξ2 + φ ξ2 ) p − (U ξ1 + φ ξ2 ) p ‖ ∗ 

+ C 2∑ m∑ 

p 3 ‖e U j 

(ξ 1 )Z ij (ξ 1 ) − e U j 

(ξ 2 )Z ij (ξ 2 )‖ ∗ 

i=1 j=1 

+ Cp‖∆(U ξ2 − U ξ1 )‖ ∗ , 

for any p ≥ p 0 and ξ 1 , ξ 2 ∈ O ε (here, ‖ · ‖ ∗ is considered with respect to ξ 1 ). 

Hence, for fixed p ≥ p 0 , the map ξ → φ ξ is continuous in C 0 (¯Ω) and, in view 

of Remark 3.2, in H0 1(Ω). Further, this map is a C1 -function in C 0 (¯Ω) as it 

follows by the Implicit Function Theorem applied to the equation: 

[ 

( ) p ] 

G(ξ, φ) := Π ⊥ ξ U ξ + Π ⊥ ξ φ + ∆ −1 U ξ + Π ⊥ ξ φ + Π ξ φ = 0, φ ∈ C 0 (¯Ω), 

where Π ξ , Π ⊥ ξ 

are the maps from L2 (Ω) respectively onto 

2∑ m∑ 

K ξ = { c ij P Z ij : c ij ∈ R for i = 1, 2, j = 1, . . . , m} 

i=1 j=1


and 

∫ 

Kξ ⊥ = {φ ∈ L 2 (Ω) : 

Ω 

e U j 

Z ij φ = 0 for i = 1, 2, j = 1, . . . , m} 

(see the notations in Step 6 in the proof of Proposition 3.1). Let us remark 

that Π ξ φ ∈ C0 k(¯Ω), for any k ≥ 0. Indeed, G(ξ, φ ξ ) = 0 and the linearized 

operator: 

∂G 

∂φ (ξ, φ ξ) = Π ⊥ ξ 

[ 

( )] 

Id + p∆ −1 (U ξ + φ ξ ) p−1 Π ⊥ ξ + Π ξ 

is invertible for p large. In fact, easily we reduce the invertibility property to 

uniquely solve the equation ∂G 

∂φ (ξ, φ ξ)[φ] = h in Kξ 

⊥ for any h ∈ K⊥ ξ ∩C 0(¯Ω). 

By Fredholm’s alternative, we need to show that in Kξ ⊥ ∩C 0(¯Ω) there is only 

the trivial solution for the equation ∂G 

∂φ (ξ, φ ξ)[φ] = 0, or equivalently for 

( 

Lφ = p U p−1 

ξ 

− (U ξ + φ ξ ) p−1) 2∑ m∑ 

φ + c ij e U j 

Z ij , 

i=1 j=1 

for any choice of the coefficients c ij , since by elliptic regularity theory φ ∈ 

C0 2(¯Ω). By Proposition 3.1 and (3.2), we derive that 

( 

‖φ‖ ∞ ≤ C ′ p‖p U p−1 

ξ 

− (U ξ + φ ξ ) p−1) φ‖ ∗ 

≤ C ′ p 2 ‖φ‖ ∞ ‖φ ξ ‖ ∞ ‖p(U ξ + O( 1 p 3 ))p−2 ‖ ∗ < ‖φ‖ ∞ 

and hence, φ = 0. Similarly, we have also that ξ → φ ξ is a C 1 -function in 

H 1 0 (Ω). 

5. Variational reduction 

After problem (4.1)-(4.4) has been solved, we find a solution of (2.15) (and 

hence for (1.1)) if ξ is such that 

c ij (ξ) = 0 for all i = 1, 2 , j = 1, . . . , m, (5.1) 

where c ij (ξ) are the coefficients in (4.1). Problem (5.1) has a variational 

structure. Associated to (1.1), let us consider the energy functional J p given 

by 

J p (u) = 1 ∫ 

|∇u| 2 dx − 1 ∫ 

|u| p+1 dx , u ∈ H0 1 (Ω), 

2 Ω 

p + 1 Ω 

and the finite dimensional restriction 

F (ξ) := J p (U ξ + φ ξ ) (5.2) 

where φ ξ is the unique solution to problem (4.1)-(4.4) given by Lemma 4.1. 

Critical points of F correspond to solutions of (5.1) for large p, as the following 

result states: 

Lemma 5.1. The functional F (ξ) is of class C 1 . Moreover, for all p sufficiently 

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 

H 1 0 (Ω) and then, F (ξ) is a C1 -function of ξ. 

Since D ξ F (ξ) = 0, we have that 

∫ 

0 = − 

= − 

= − 

Ω 

2∑ m∑ 

i=1 j=1 

2∑ m∑ 

i=1 j=1 

(∆(U ξ + φ ξ ) + (U ξ + φ ξ ) p ) (D ξ U ξ + D ξ φ ξ ) 

∫ 

c ij (ξ) e U j 

Z ij (D ξ U ξ + D ξ φ ξ ) 

Ω 

∫ 

c ij (ξ) e U j 

Z ij D ξ U ξ + 

Ω 

2∑ 

m∑ 

i=1 j=1 

∫ 

c ij (ξ) D ξ (e U j 

Z ij )φ ξ 

Ω 

since ∫ Ω eU j 

Z ij φ ξ = 0. By the expression of U ξ , we have that: 

m∑ 

[ 

1 

∂ (ξj ) i 

U ξ = − 

2 

P 

p−1 

s=1 γµ s 

2 

+ 

p 2 (p − 1) w 1 + 1 p ∇w 0 · y + 1 p 2 ∇w 1 · y 

( 

1 

+ 

2 

P 

p−1 

γδ j µ j 

( 

1 

= 

2 

P 

p−1 

γδ j µ 

j 

2 

p − 1 U δ s ,ξ s 

− 2Z 0s + 

) ∣∣y= 

x−ξs 

δs 

Z ij − 1 p ∂ iw 0 ( x − ξ j 

) − 1 δ j p 2 ∂ iw 1 ( x − ξ j 

) 

δ j 

Z ij − 1 p ∂ iw 0 ( x − ξ j 

) − 1 δ j p 2 ∂ iw 1 ( x − ξ j 

) 

δ j 

29 

( 

2 

p(p − 1) w 0 (5.3) 

] 

∂ (ξj ) i 

log µ s 

) 

) 

+ O( 1 γ ), 

since P : L ∞ (Ω) → L ∞ 1 

(Ω) is a continuous operator (apply to 

p−1 U δ s ,ξ s 

, 

Z 0s , 1 p w j( x−ξs 

δ s 

), ∇w j ( x−ξs 

δ s 

) · ( x−ξs 

δ s 

) for j = 0, 1 which are bounded in Ω in 

view of Lemma 2.1), and 

∂ (ξj ) i 

( 

e U h 

Z lh 

) 

( 

= −4δ h e U δ 

h 

il 

δh 2 + |x − ξ h| 2 − 6(x − ξ ) 

h) l (x − ξ j ) i 

(δh 2 + |x − ξ h| 2 ) 2 δ hj 

+3e U h 

Z 0h Z lh ∂ (ξj ) i 

log µ h , (5.4) 

where δ hj denotes the Kronecker’s symbol. Hence, by (5.3)-(5.4) for i = 1, 2 

and j = 1, . . . , m we get that 

1 

2∑ m∑ 

) 

0 = ∂ (ξj ) i 

F (ξ) = − 

2 

c lh (ξ) 

(P Z ij , P Z lh 

p−1 

H 

γδ j µ l=1 h=1 

1 0 (Ω) 

j 

( ) 

1 

+O + ‖φ ξ ‖ ∞ |∂ 

pγδ (ξj ) i 

(e 

j 

∫Ω 

U ∑ 2 m∑ 

h 

Z lh )| |c lh (ξ)| 

l=1 h=1


since ∂ i w j ( x−ξ s 

δ s 

) is bounded in Ω, j = 0, 1, in view of Lemma 2.1. Taking 

into account (3.22)-(3.23), (4.5) and (5.4) we get 

0 = 

64 

γδ j µ 

2 

p−1 

j 

( ∫ 

R 2 |y| 2 

(1 + |y| 2 ) 4 dy ) 

c ij (ξ) + O 

⎛ 

⎝ 1 

pγδ j 

2∑ 

m∑ 

l=1 h,s=1 

⎞ 

|c lh (ξ)| ⎠ 

which implies for p large (independent of ξ ∈ O ε ) that c ij (ξ) = 0 for any 

i = 1, 2 and j = 1, . . . , m. 

□ 

Next Lemma shows that the leading term of the function F (ξ) is given by 

ϕ m (ξ): 

Lemma 5.2. Let ε > 0. The following expansion holds: 

uniformly for ξ ∈ O ε . 

F (ξ) = 4πmp 

γ 2 − 32π2 

γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 4πm 

γ 2 

+ m ∫ 

8 

2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 ) 

R 2 ( 

Proof. First of all, multiply (4.1) by U ξ + φ ξ and integrate by parts to get: 

∫ 

∫ 

(U ξ + φ ξ ) p+1 = 

2∑ m∑ 

∫ 

|∇(U ξ + φ ξ )| 2 + c ij e U j 

Z ij (U ξ + φ ξ ) 

Ω 

Ω 

i=1 j=1 

Ω 

∫ 

= 

2∑ m∑ 

∫ 

|∇(U ξ + φ ξ )| 2 + c ij e U j 

Z ij U ξ 

Ω 

i=1 j=1 

Ω 

in view of (4.4). Since U ξ is a bounded function, by (4.6) we get that 

∫ 

Ω 

∫ 

(U ξ + φ ξ ) p+1 = 

uniformly for ξ ∈ O ε . We can write: 

F (ξ) = ( 1 2 − 1 

p + 1 

∫Ω 

) 

= ( 1 2 − 1 

∫ 

p + 1 

(∫Ω 

) |∇U ξ | 2 + 2 

Ω 

|∇(U ξ + φ ξ )| 2 + O( 1 p 4 ) 

|∇(U ξ + φ ξ )| 2 + O( 1 p 4 ) (5.5) 

Ω 

∫ ) 

∇U ξ ∇φ ξ + |∇φ ξ | 2 + O( 1 

Ω 

p 4 ).

We expand the term ∫ Ω |∇U ξ| 2 : in view of (2.14) and (2.20) we have that 

∫ 

m∑ 

∫ ( 

|∇U ξ | 2 1 

= 

2 

e U j 

− 1 

Ω 

p−1 

j=1 

B(ξ 

γµ 

j ,ε) pδ 2 ∆w 0 ( x − ξ j 

) 

j δ j 

j 

− 1 

p 2 δj 

2 ∆w 1 ( x − ξ ) 

j 

) + O(p 2 e − p 2 ) U ξ + O(e − p 2 ) 

δ j 

m∑ 

∫ ( 

1 

8 

= 

4 

j=1 γ 2 p−1 B(0, 

µ 

ε ) (1 + |y| 2 ) 2 − 1 p ∆w 0 − 1 ) 

p 2 ∆w 1 + O(p 2 e −p ) × 

δ 

j 

j 

× 

(p + v ∞ + 1 p w 0 + 1 ) 

p 2 w 1 + O(e − p 4 |y| + e 

− p 4 ) + O(e − p 2 ) 

= 

m∑ 

j=1 

1 

γ 2 µ 

= 8πmp 

γ 2 

since µ − 4 

p−1 

j 

get that: 

4 

p−1 

j 

− 32π 

γ 2 

( ∫ 

8πp + 

m ∑ 

j=1 

R 2 ( 

8 

(1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 ) 

p ) 

log µ j + m 8 

γ 

∫R 2 ( 

2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 ) 

= 1 − 4 p log µ j + O( 1 p 2 ). Recalling property (2.13) of µ i , then we 

∫ 

Ω 

|∇U ξ | 2 = 8πmp 

γ 2 

31 

− 64π2 

γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 24πm 

γ 2 (5.6) 

+ m 8 

γ 

∫R 2 ( 

2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 ) 

uniformly for ξ ∈ O ε . Hence, using (4.6) and (5.6) we get that 

∫ 

∇U ξ ∇φ ξ + 1 ∫ 

|∇φ ξ | 2 = O( 1 ). (5.7) 

Ω 2 Ω 

Finally, inserting (5.6)-(5.7) in (5.5), we get that 

uniformly for ξ ∈ O ε . 

F (ξ) = 4πmp 

γ 2 − 32π2 

γ 2 ϕ m(ξ 1 , . . . , ξ m ) + 4πm 

γ 2 

+ m ∫ 

8 

2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) + O( 1 p 3 ) 

R 2 ( 

Now, we want to show that the expansion of F (ξ) in terms of ϕ m (ξ) holds 

in a C 1 -sense: 

Lemma 5.3. Let ε > 0. The following expansion holds: 

∇ (ξj ) i 

F (ξ) = − 32π2 

γ 2 ∇ (ξ j ) i 

ϕ m (ξ 1 , . . . , ξ m ) + o( 1 p 2 ) 

uniformly for ξ ∈ O ε , for any j = 1, . . . , m and i = 1, 2. 

p 7 2


Proof. Let j ∈ {1, . . . , m} and i ∈ {1, 2} be fixed. We want to expand the 

derivatives of F (ξ) in ξ: 

∫ ( ) 

∂ (ξj ) i 

F (ξ) = − ∆u ξ + u p ξ 

∂ (ξj ) i 

u ξ , 

Ω 

where u ξ = U ξ + φ ξ . Let us remark that it is very difficult to show directly 

that the expansion of F (ξ) holds in a C 1 −sense since there is a difference 

between the exponential decay of the concentration paremeters δ i = µ i e − p 4 

and the polinomial decay 1 of ‖R 

p 4 ξ ‖ ∗ (see Proposition 2.1). As usual in 

similar contexts (also in higher dimensions), we should be able to show that 

‖∂ ξ φ ξ ‖ ∞ is of order ‖φ ξ‖ ∞ 

δ i 

. Unfortunately, since ‖φ ξ ‖ ∞ is only of order 1 , 

p 3 

∂ ξ φ ξ is not a small function and, at a first glance, there is no hope for a 

C 1 −expansion of F (ξ). To overcome the problem, the idea is the following: 

first, we replace the term ∂ (ξj ) i 

u ξ with ∂ xi u ξ in the expression of ∂ (ξj ) i 

F (ξ) 

in a neighborhood of ξ j , up to higher order terms, and after, we use a 

Pohozaev-type identity based on integration by parts. 

To this purpose, let η be a radial smooth cut-off function such that 0 ≤ η ≤ 1, 

η ≡ 1 for |x| ≤ ɛ and η ≡ 0 for |x| ≥ 2ɛ. In view of (4.1) and (4.4), we can 

write: 

∫ ( ) 

2∑ m∑ 

∫ 

∆u ξ + u p ξ 

∂ (ξj ) i 

φ ξ = c kl e U l 

Z kl ∂ (ξj ) i 

φ ξ (5.8) 

Ω 

= − 

= 

− 

2∑ 

k=1 l=1 

2∑ m∑ 

k=1 l=1 

2∑ m∑ 

m∑ 

∫ 

c kl 

c kl 

∫ 

c kl 

∫ 

k=1 l=1 

2∑ m∑ 

= − 

− 

k=1 l=1 

2∑ m∑ 

k=1 l=1 

c kl 

∫ 

Ω 

Ω 

c kl 

∫ 

Ω 

Ω 

k=1 l=1 

( ) 

∂ (ξj ) i 

e U l 

Z kl φ ξ 

Ω 

∂ xi 

( 

e U l 

Z kl 

) 

η(x − ξ j )φ ξ 

[∂ (ξj ) i 

(e U l 

Z kl 

) 

+ η(x − ξ j )∂ xi 

(e U l 

Z kl 

)] 

φ ξ 

Ω 

e U l 

Z kl ∂ xi (η(x − ξ j )φ ξ ) 

[∂ (ξj ) i 

(e U l 

Z kl 

) 

+ η(x − ξ j )∂ xi 

(e U l 

Z kl 

)] 

φ ξ 

by an integration by parts of the derivative in x i . As for (5.4), we get that 

( ) 

( ) 

∂ (ξj ) i 

e U l 

Z kl + η(x − ξ j )∂ xi e U l 

Z kl = 4δ l e U η(x − ξ l 

j) − δ lj 

δl 2 + |x − ξ l| 2 δ ik 

+24δ l e U l (x − ξ l) k (x − ξ l ) i 

(δ 2 l + |x − ξ l| 2 ) 2 (δ lj − η(x − ξ j )) + 3e U l 

Z 0l Z kl ∂ (ξj ) i 

log µ l 

= 3e U l 

Z 0l Z kl ∂ (ξj ) i 

log µ l + O(e − 3 4 p ),

33 

where δ lj denotes the Kronecker’s symbol, and hence, we get that 

| 

2∑ 

k=1 l=1 

m∑ 

∫ 

c kl 

Ω 

≤ C‖φ ξ ‖ ∞ max |c kl | 

k , l 

[∂ (ξj ) i 

(e U l 

Z kl 

) 

+ η(x − ξ j )∂ xi 

(e U l 

Z kl 

)] 

φ ξ | 

∫ 

Ω 

( 

) 

e U l 

+ O(e − 3 4 p ) = O( 1 p 7 ) 

in view of |Z 0l Z kl | ≤ 2, (4.5)-(4.6). Inserting in (5.8), we get that 

∫ 

Ω 

( 

∆u ξ + u p ) 

ξ 

∂ (ξj ) i 

φ ξ = − 

2∑ 

k=1 l=1 

+O( 1 p 7 ). 

m∑ 

∫ 

c kl 

Ω 

e U l 

Z kl ∂ xi (η(x − ξ j )φ ξ )(5.9) 

Since pφ ξ → 0 in C 1 -norm away from ξ 1 , . . . , ξ m , (5.9) gives that 

∫ 

Ω 

( ) 

∆u ξ + u p ξ 

∂ (ξj ) i 

φ ξ = − 

= − 

2∑ 

k=1 l=1 

∫ 

m∑ 

∫ 

c kl 

B(ξ j ,ɛ) 

always in view of (4.1) and (4.5)-(4.6). 

B(ξ j ,ɛ) 

( 

∆u ξ + u p ξ 

e U l 

Z kl ∂ xi φ ξ + O( 1 p 7 ) 

) 

∂ xi φ ξ + O( 1 p 7 ) (5.10) 

Now, by Lemma 2.1 we get that | 1 δ s 

∂ xi w l ( x−ξs 

δ s 

)| ≤ C for any l = 1, 2 and 

for x away from ξ s . Hence, by the expression of U ξ and (2.2), (2.11) we get 

that for |x − ξ j | ≤ 2ɛ: 

m∑ 

( 

1 

∂ xi U ξ = 

2 

∂ xi P U δs ,ξ s 

+ 1 ( 

p−1 

p ∂ x i 

P w 0 ( x − ξ ) 

s 

) 

(5.11) 

δ 

s=1 γµ 

s 

s 

+ 1 ( 

p 2 ∂ x i 

P w 1 ( x − ξ )) 

s 

) 

δ s 

m∑ 

( 

1 

= 

2 

∂ xi U δs,ξs + 1 ∂ xi w 0 ( x − ξ s 

) + 1 

p−1 

pδ 

s=1 γµ 

s δ s p 2 ∂ xi w 1 ( x − ξ ) 

s 

) + O( 1 δ s δ s γ ) 

s 

( 

1 

= − 

2 

Z ij − 1 

p−1 p ∂ x i 

w 0 ( x − ξ j 

) − 1 δ 

γδ j µ 

j p 2 ∂ x i 

w 1 ( x − ξ ) 

j 

) + O( 1 δ j γ ), 

j 

since ∂ xi U δs ,ξ s 

= − 1 δ s 

Z is . Since, as already observed, ∂ xi w l ( x−ξ j 

δ j 

) = O(δ j ) 

uniformly away from ξ j , l = 1, 2, by (5.11) in particular we get: 

∂ xi U ξ = O( 1 ), (5.12) 

γ


for ɛ ≤ |x − ξ j | ≤ 2ɛ. Moreover, the maximum principle and Lemma 2.1 

imply that 

( 

P ∂ xi w l ( x − ξ ) 

j 

) − ∂ xi w l ( x − ξ j 

) = O(δ j ) 

δ j δ j 

in C(¯Ω) for any l = 1, 2, and hence, by (5.3) and (5.11) we get that: 

1 

∂ (ξj ) i 

U ξ + η(x − ξ j )∂ xi U ξ = 

γδ j µ 

2 

p−1 

j 

(P Z ij − Z ij ) + O( 1 γ ) = O( 1 γ ) (5.13) 

in view of (3.20). Now, by (5.12)-(5.13) we can write that: 

∫ 

+ 

Ω 

( ) 

∆u ξ + u p ξ 

∂ (ξj ) i 

U ξ = − 

2∑ 

m∑ 

∫ 

c kl 

k=1 l=1 

2∑ m∑ 

= − 

= − 

k=1 l=1 

∫ 

B(ξ j ,ɛ) 

Ω 

c kl 

∫ 

2∑ 

k=1 l=1 

m∑ 

∫ 

c kl 

e U l 

Z kl 

( 

∂ (ξj ) i 

U ξ + η(x − ξ j )∂ xi U ξ 

) 

B(ξ j ,ɛ) 

( 

∆u ξ + u p ξ 

Ω 

e U l 

Z kl ∂ xi U ξ + O( 1 p 5 ) 

) 

∂ xi U ξ + O( 1 p 5 ) 

e U l 

Z kl η(x − ξ j )∂ xi U ξ (5.14) 

in view of (4.1) and (4.6). 

Resuming, by (5.10) and (5.14) we have: 

∫ ( ) 

∂ (ξj ) i 

F (ξ) = − ∆u ξ + u p ξ 

(∂ (ξj ) i 

U ξ + ∂ (ξj ) i 

φ ξ ) (5.15) 

Ω 

∫ ( 

= ∆u ξ + u p ) 

ξ 

(∂ xi U ξ + ∂ xi φ ξ ) + O( 1 p 5 ) 

= 

B(ξ j ,ɛ) 

∫ 

B(ξ j ,ɛ) 

( ) 

∆u ξ + u p ξ 

∂ xi u ξ + O( 1 p 5 ). 

Now, the role of ∇ϕ m (ξ) becomes clear by means of the following Pohozaevtype 

identity (we follow some arguments of [13]): for any B ⊂ Ω and for 

any function u 

∫ 

∫ ( 

∆u ∇u = ∂ n u∇u − 1 ) 

2 |∇u|2 n , (5.16) 

B 

∫ 

B 

∂B 

u p ∇u = 1 

p + 1 

∫ 

∂B 

u p+1 n, 

where n(x) is the unit outer normal vector of ∂B at x ∈ ∂B. Let 

ϕ j (x) = H(x, ξ j ) + ∑ l≠j 

G(x, ξ l ) 

for any j = 1, . . . , m. Since, as already observed, pφ ξ → 0 in C 1 -norm away 

from ξ 1 , . . . , ξ m , for our function u ξ by (2.2) and (2.11) we have the following

asymptotic property: 

pu ξ (x) → 8π √ e 

m∑ 

G(x, ξ l ) in C 

loc 1 (¯Ω \ {ξ 1 , . . . , ξ m }). 

l=1 

35 

(5.17) 

Apply now (5.16) on B = B(ξ j , ε), j = 1, . . . , m, and use (5.17) to obtain as 

p → +∞: 

∫ 

∫ 

B 

(∆u ξ + u p ξ )∇u ξ = 

∂B 

= 64π2 e 

p 2 ∫∂B 

− 1 2 | − 1 

2π 

( 

∂ n u ξ ∇u ξ − 1 2 |∇u ξ| 2 n + 1 

[ 

(− 1 

2πɛ + ∂ nϕ j )(− 1 

2π 

] 

x − ξ j 

|x − ξ j | 2 + ∇ϕ j| 2 n 

= − 64π2 e 

p 2 ∇ϕ j (ξ j ) + o( 1 p 2 ), 

p + 1 up+1 ξ 

) 

n 

x − ξ j 

|x − ξ j | 2 + ∇ϕ j) 

+ o( 1 p 2 ) 

since we decompose ∑ m 

l=1 G(x, ξ l ) = − 1 

2π ln |x − ξ j| + ϕ j (x) with ϕ j (x) an 

harmonic function near ξ j . In fact, we have used that 

1 

2πɛ 

∫ 

∂B 

∇ϕ j = ∇ϕ j (ξ j ) 

since ∇ϕ j is harmonic near ξ j , and by (5.16) 

∫ ( 

∂ n ϕ j ∇ϕ j − 1 ) ∫ 

2 |∇ϕ j| 2 n = 

∂B 

Combining with (5.15), finally we get: 

B 

∆ϕ j ∇ϕ j = 0. 

∂ (ξj ) i 

F (ξ) = − 32π2 e 

p 2 ∂ (ξj ) i 

ϕ m (ξ) + o( 1 p 2 ) = −32π2 γ 2 ∂ (ξ j ) i 

ϕ m (ξ) + o( 1 p 2 ) 

since ∇ϕ j (ξ j ) = 1 2 ∇ ξ j 

ϕ m (ξ). The proof is now complete. 

Finally, we carry out the proof of our main result. 

Proof of Theorem 1.2. Let us consider the set D as in the statement of 

the theorem, C the associated critical value and ξ ∈ D. According to Lemma 

5.1, we have a solution of Problem (1.1) if we adjust ξ so that it is a critical 

point of F (ξ) defined by (5.2). This is equivalent to finding a critical point 

of 

[ 

˜F (ξ) = 

γ2 

32π 2 F (ξ) − 4πmp 

γ 2 − 4πm 

γ 2 − m ∫ 

] 

8 

2γ 2 (1 + |y| 2 ) 2 v ∞ − ∆w 0 ) . 

On the other hand, from Lemmata 5.2 and 5.3, we have that for ξ ∈ D ∩ O ε , 

R 2 ( 

˜F (ξ) = ϕ m (ξ) + o(1)Θ p (ξ) 

where Θ p and ∇ ξ Θ p are uniformly bounded in the considered region as 

p → ∞. 

□


Let us observe that if M > C, then assumptions (1.4), (1.5) still hold for 

the function min{M, ϕ m (ξ)} as well as for min{M, ϕ m (ξ) + o(1)Θ p (ξ)}. It 

follows that the function min{M, ˜F (ξ)} satisfies for all p large assumptions 

(1.4),(1.5) in D and therefore has a critical value C p < M which is close to C 

in this region. If ξ p ∈ D is a critical point at this level for ˜F (ξ), then, since 

˜F (ξ p ) ≤ C p < M, 

we have that there exists ε > 0 such that |ξ p,j −ξ p,i | > 2ε, dist(ξ p,j , ∂Ω) > 2ε. 

This implies C 1 -closeness 

( 

of ˜F (ξ) and ϕ m (ξ) 

) 

at this level, hence ∇ϕ m (ξ p ) → 

0. The function u p (x) = U ξp (x) + φ ξp (x) is therefore a solution with the 

qualitative properties predicted by the theorem. 

□ 

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P. Esposito - Dipartimento di Matematica, Università degli Studi “Roma 

Tre”, Largo S. Leonardo Murialdo, 1 – 00146 Roma, Italy. 

M. Musso - Departamento de Matematicas, Pontificia Universidad Catolica 

de Chile, Avenida Vicuna Mackenna 4860, Macul, Santiago, Chile and Dipartimento 

di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 24 

– 10129 Torino, Italy. 

A. Pistoia - Dipartimento di Metodi e Modelli Matematici, Università di 

Roma “La Sapienza”, Via Scarpa, 16 – 00166 Roma, Italy.

CONCENTRATING SOLUTIONS FOR A PLANAR ... - CAPDE

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