**SINGULAR LIMITS FOR** 2-DIMENSIONAL **ELLIPTIC PROBLEM WITH**

**EXPONENTIALLY DOMINATED** **NONLINEARITY AND A QUADRATIC**

**CONVECTION TERM**

**Sami Baraket, Imen Bazarbacha, Saber Kharrati and Taieb Ouni**

**Abstract**

We study existence of solutions with singular limits for a two-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a quadratic convection non linear gradient term, imposing Dirichlet boundary condition. This paper extends previous results obtained in [1], [3], [4] and some references therein for related issues.

**1** **Introduction and statement of the results**

In this work we are concerned with the following types of stationary singular
problems: { *−*∆u = Φ(x, u,*∇u)* in Ω

*u* = 0 on *∂Ω,*

(1)
where Φ is a smooth nonlinear function and Ω is an open smooth bounded
subset of R^{2}. We focus on Problem (1) and we establish several recent
contributions in the study of this equation. The most natural case where

Key Words: singular limits, Green’s function, gradient term with quadratic growth, nonlinear domain decomposition method.

2010 Mathematics Subject Classification: 35J60, 53C21, 58J05.

Received: January, 2012.

Accepted: February, 2012.

19

Φ(x, u,*∇u) =λ|∇u|*^{2}+*ε*^{2}*V*(x)
(

*e** ^{u}*+

*e*

*)*

^{γu}where*ε*and *λ*are small parame-
ters, *γ* *∈*(0,1) and *V* : Ω *−→*[0,+*∞*) is a positive function. We deal with
very general growth for*V*(x) like, for example,

*V*(x) =

∏*N*
*i=1*

*|x−p**i**|*^{2β}^{i}*f*(x),

where *f* is a smooth positive function such that*f*(p*i*)*>*0, for *i*= 1,*· · ·* *, N*
and (β* _{i}*)

_{1}

_{≤}

_{i}

_{≤}*are positive numbers, so that problem (1) becomes*

_{N}

*−*∆u*−λ|∇u|*^{2} = *ε*^{2}

∏*N*
*i=1*

*|x−p*_{i}*|*^{2β}^{i}*f*(x)
(

*e** ^{u}*+

*e*

*)*

^{γu}in Ω

*u* = 0 on *∂Ω.*

(2)

Denote by Λ :=*{p*1*,· · ·* *, p**N**} ⊂*Ω the set of singular sources. We are interested
in solutions which “concentrate” as the parameters *ε* and *λ* tend to 0 (in a
sense that will be determined later).

The aim of this paper is to study the inﬂuence of the non-linear gradient
term *|∇u|*^{2}. It turns out that the presence of this term can have signiﬁcant
inﬂuence on the existence of a solution, as well as on its asymptotic behavior.

The general question is: Does there exist a family of solutions (u*ε,λ,β** _{i}*)

*ε>0,λ>0*

of some singularly perturbed problem like (2) which converges to a non trivial
singular function (on some set Λ) as the parameters*ε*and*λ*tend to 0?

Given ¯*β* = min

1*≤**i**≤**N**β*_{i}*>* 0, for (β* _{i}*)

_{1}

_{≤}

_{i}

_{≤}

_{N}*∈*R+

*\N*, we will suppose in the following that

*λ*satisﬁes

(H* _{λ}*) : lim

*ε−→0**λ−→0*

*λ ε*^{−}^{δ/( ¯}* ^{β+1)}*= 0, for any

*δ∈*(0,1).

For*β*= 0, we will suppose in the following that *λ*satisﬁes
(C*λ*) : If 0*< ε < λ,* then lim

*λ**−→*0*λ*^{1+δ/2}*ε*^{−}* ^{δ}* = 0, for any

*δ∈*(0,1).

For *λ≡*0, Esposito in [13] gave a positive answer to the above question for
the following problem:

*−*∆u = *ε*^{2}*e*^{u}*−*4π

∑*N*
*i=1*

*β*_{i}*δ*_{p}* _{i}* in Ω

*u* = 0 on *∂Ω,*

(3)

where again Ω *⊂* R^{2} is an open smooth bounded subset of R^{2}, *δ**p** _{i}* are the
Dirac mass at

*p*

*i*and Λ :=

*{p*1

*,· · ·*

*, p*

*N*

*} ⊂*Ω is the set of singular sources.

Notice that using suitable transformation, problem (3), is equivalent to the following

*−*∆u = *ε*^{2}

∏*N*
*i=1*

*|x−p**i**|*^{2β}^{i}*f(x)e** ^{u}* in Ω

*u* = 0 on *∂Ω,*

(4)

where *f* : Ω *−→* R is a smooth function such that *f*(p* _{i}*)

*>*0 for any

*i*= 1,

*· · ·*

*, N*and

*β*

*are positives numbers.*

_{i}This type of equation has been studied by Bartolucci et al. in [8] and by Chen
and Lin in [10]. They obtained the existence, sharp estimates and construction
of multiple bubbles to (4). That the construction of nontrivial branches of
solutions of such semilinear elliptic equations with exponential nonlinearities is
equivalent to prove the existence of a conformal change of metric for which the
corresponding mean curvature surfaces in the Euclidean space is*non constant*
*function.*

Baraket et al. in [3] extended these results for special nonlinearities of type
*e** ^{u}*+

*e*

*for*

^{γu}*γ*

*∈*(0,1) instated of

*e*

*. Problem (4)*

^{u}

_{|}

_{β}

_{i}_{=0}(a.e: Λ is empty) is

given by {

*−*∆u = *ε*^{2}*e** ^{u}* in Ω

*u* = 0 on *∂Ω.* (5)

The above equation was ﬁrst studied by Liouville in 1853, see [15], in which he
derived a representation formula for all solutions of (5) which are deﬁned in
the hole spaceR^{2}. When the parameter*ε*tends to 0, the asymptotic behavior
of nontrivial branches of solutions of (5) is well understood thanks to the
pioneer work of Suzuki [20] who characterizes the possible limit of nontrivial
branches of solutions of (5). See also [18]. The existence of nontrivial branches
of solutions was ﬁrst proved by Weston [24] and then a general result has been
obtained by Baraket and Pacard [6] and Baraket and Ye [7] for *e** ^{u}* +

*e*

*,*

^{γu}*γ∈*(0,1) instated of

*e*

*.These results were extended, applying to the Chern- Simons vortex theory in mind, by Esposito et al. in [14] and Del Pino et al.*

^{u}in [12] to handle equations of the form

*−*∆u=*ε*^{2}*V e*^{u}

where*V* is a non constant (positive) function and*ε*is a small parameter. Let us
mention that the construction of nontrivial branches of solutions of semilinear
elliptic equations with exponential nonlinearities allowed Wente to provide
counter examples to a conjecture by Hopf [23] concerning the existence of
compact (immersed)*constant*mean curvature surfaces in the Euclidean space,
see also [22] and [25] and some references therein.

To describe our result, we ﬁrst introduce the Green’s function*G(x, x** ^{′}*) deﬁned

on Ω*×*Ω, to be solution of

{ *−*∆G(x, x* ^{′}*) = 8πδ

*x=x*

*′*in Ω

*G(x, x** ^{′}*) = 0 on

*∂Ω*

and let

*H*(x, x* ^{′}*) =

*G(x, x*

*) + 4 log*

^{′}*|x−x*

^{′}*|,*its regular part.

Given a ﬁnite subset *S* =*{q*1*,· · ·* *, q**K**}* of Ω,*m∈*Nand *s∈ {*1,*· · ·, N}*, we
set

F(x_{1}*,· · ·* *, x** _{m}*) =

∑*m*
*j=1*

*H*(x_{j}*, x** _{j}*) +∑

*i**̸*=j

*G(x*_{i}*, x** _{j}*) (6)

+ 4

∑*s*
*i=1*

∑*m*
*j=1*

*β** _{i}*log(

*|x*

_{j}*−p*

_{i}*|*) + 2

∑*m*
*j=1*

log(f(x* _{j}*))
which is well deﬁned in (Ω

*\*Λ)

*for any*

^{m}*x*

*i*

*̸*=

*x*

*j*with

*i̸*=

*j*and let

G(x_{1}*,· · ·* *, x*_{m}*, w*_{1}*,· · ·* *, w** _{s}*) =

∑*m*
*j=1*

∑*s*
*i=1*

(1 +*β** _{i}*)G(x

_{j}*, w*

*). (7) Gis well deﬁned for*

_{i}*x*

*j*

*̸*=

*w*

*i*with

*x*

*j*

*∈*Ω, w

*i*

*∈*Ω.

The main result of this paper is the following.

**Theorem 1.** *Given* *α∈*(0,1). LetΩ *be an open smooth bounded domain in*
R^{2}*,f* *a smooth positive function and*(β* _{i}*)

_{1}

_{≤}

_{i}

_{≤}

_{n}*∈*R+

*\N*

^{∗}*be real numbers.*

*(a) LetS* =*{p*_{j}_{1}*,· · ·* *, p*_{j}_{s}*} ⊂*Λ, given*λ* *satisfying the condition*(H* _{λ}*), then

*there exist*

*ε*

_{0}

*>*0,

*λ*

_{0}

*>*0, and {

*u*_{ε,λ,β}* _{ji}*}

0<ε<ε0 0<λ<λ0

*of solutions of problem*
(2) *such that*

*ε−→0*lim

*λ−→0*

*u**ε,λ,β** _{ji}* =

∑*s*
*i=1*

(1 +*β**j** _{i}*)G(

*·, p*

*j*

*)*

_{i}*in*C

^{2,α}

*loc*(Ω

*\S).*

*(b) LetS* =*{q*_{1}*,· · ·, q*_{m}*} ⊂*Ω*\*Λ *and*(q_{1}*,· · ·, q** _{m}*)

*be a nondegenerate criti-*

*cal point of the functional*F

*, given*

*λsatisfying the condition*(C

*), then*

_{λ}*there exist*

*ε*

_{0}

*>*0,

*λ*

_{0}

*>*0, and

*{u*

*:= (u*

_{ε,λ}*)*

_{ε,λ,0}*}*0<ε<ε0

0<λ<λ0

*of solutions of*
*problem*(2) *such that*

*ε−→0*lim

*λ−→0*

*u**ε,λ*=

∑*m*
*i=1*

*G(·, q**i*) *in* C^{2,α}*loc*(Ω*\S).*

*(c) Let* *S* *be such that* *S* *∩*Λ = *{p**j*_{1}*,· · ·, p**j*_{s}*}, S\*Λ = *{q*1*,· · ·* *, q**m**}* *and*
(q1*,· · ·* *, q**m*) *be a nondegenerate critical point of the function* F(*·*) +
G(*·, p**j*_{1}*,· · ·, p**j** _{s}*), given

*λsatisfying the conditions*(H

*λ*)

*and*(C

*λ*), then

*there exist*

*ε*0

*>*0,

*λ*0

*>*0, and {

*u**ε,λ,β*_{ji}

}

0<ε<ε0 0<λ<λ0

*of solutions of*(2)*such*
*that*

*ε−→0*lim

*λ−→0*

*u**ε,λ,β** _{ji}* =

∑*s*
*k=1*

(1 +*β**j** _{k}*)G(

*·, p*

*j*

*) +*

_{k}∑*m*
*i=1*

*G(·, q**i*) *in* C^{2,α}* _{loc}*(Ω

*\S).*

In the case where Λ is empty and taking *f* a positive constant funcion,
part (b) of Theorem 1 extends the result in [1].

Observe that the nondegeneracy assumption on critical points, if it exists, is a rather mild assumption since it is certainly fulﬁlled for a generic choice of the domain Ω.

**2** **Construction of the approximate solution**

We ﬁrst describe the rotationally symmetric approximate solutions of

∆v+*λ|∇v|*^{2}+*ε*^{2}*|x|*^{2β}(

*e** ^{v}*+

*e*

*)*

^{γv}= 0 (8)

inR^{2} where*β* *∈*R+*\N*and*γ∈*(0,1).

We deﬁne the function

*v**ε,β*(x) := log 8(β+ 1)^{2}*−*2 log(ε^{2}+*|x|*^{2(1+β)}) (9)

solution of the equation

*−*∆v=*ε*^{2}*|x|*^{2β}*e** ^{v}* inR

^{2}

*.*(10) Notice that equation (10) is invariant under dilation in the following sense : If

*v*is a solution of (10) and if

*τ >*0, then

*v(τ·*) + 2(β+ 1) log

*τ*is also a solution of (10). With this observation in mind, we deﬁne for all

*τ >*0

*v** _{ε,τ,β}*(x) := 2(β+ 1) log

*τ*+ log 8(β+ 1)

^{2}

*−*2 log(ε

^{2}+

*|τ x|*

^{2(1+β)}) (11) and

*v** _{ε,τ}*(x) :=

*v*

*(x). (12)*

_{ε,τ,0}**2.1** **A linearized operator on** *B*1

We set*B*_{1}* ^{∗}*=

*B*

_{1}

*− {*0

*}*. Then we introduce the following space:

**Definition 1.** *Given* *k∈* N*,α∈* (0,1) *and* *µ∈* R*, we introduce the H¨older*
*weighted spaces* C^{k,α}*µ* ( ¯*B*_{1}* ^{∗}*)

*as the space of functions in*C

^{k,α}*( ¯*

_{loc}*B*

_{1}

*)*

^{∗}*for which the*

*following norm*

*∥u∥*_{C}^{k,α}_{µ}_{( ¯}_{B}*∗*

1)= sup

*r**≤*1/2

(*r*^{−}^{µ}*∥u(r·*)*∥*_{C}* ^{k,α}*( ¯

*B*

_{2}

*−*

*B*

_{1})

)*,*

*is ﬁnite.*

We then deﬁne the subspace of radial functions in C^{k,α}*µ* ( ¯*B*^{∗}_{1}) by
C^{k,α}*rad,µ*( ¯*B*_{1}* ^{∗}*) =

*{f*

*∈*C

^{k,α}*µ*( ¯

*B*

_{1}

*);*

^{∗}*such that f(x) =f*(

*|x|*),

*∀*

*x∈B*¯

_{1}

^{∗}*}.*We deﬁne the linear second order elliptic operatorL

*β*by

L*β*:= ∆ +*ε*^{2}*|x|*^{2β}*e*^{v}^{ε,τ,β}

which corresponds to the linearization of ∆u+*ε*^{2}*|x|*^{2β}*e** ^{u}*= 0 about the radial
symmetric solutions

*v*

*ε,τ,β*for both cases

*β*= 0 and

*β̸∈*N.

When *k* *≥* 2, we let [C^{k,α}*µ* ( ¯Ω)]0 be the subspace of functions *w* *∈* C^{k,α}*µ* ( ¯Ω)
satisfying*w*= 0 on*∂*Ω. We recall the result in [13] which states as:¯

**Proposition 1.** *[6],[13]*

*(i) Let* *µ >* 0, *µ* *̸∈* N *and* *β* *̸∈* N*, then* L*β* : [C^{2,α}*µ* (B1)]0 *−→* C^{0,α}*µ**−*2(B1) *is*
*surjective.*

*(ii) Letµ >*0,*µ̸∈*N*andβ*= 0, thenL0: [C^{2,α}*rad,µ*(B_{1})]_{0}*−→*C^{0,α}*rad,µ**−*2(B_{1})*is*
*surjective.*

*(iii) Let* *µ >* 1, *µ* *̸∈* N *and* *β* = 0, then L0 : [C^{2,α}*µ* (B1)]0 *−→* C^{0,α}*µ**−*2(B1) *is*
*surjective.*

In the following, we denote byG*µ,β* a right inverse ofL*β*. Let
*δ∈*(0,1),

we deﬁne

*r**ε,β*:= max(ε^{2(1}^{−}^{γ)}^{−}^{δ/2(β+1)}*, λ*^{1/2}*ε*^{−}^{δ/2(β+1)}*, λ*^{1/2}*, ε*^{2/(2β+3)})
and

*r**ε*:= max(ε^{2(1}^{−}^{γ)}^{−}^{δ/2}*, λ*^{1/2}*ε*^{−}^{δ/2}*, λ*^{1/2}*, ε*^{1/2}).

Remark that by assumptions (H*λ*) and (C*λ*),*r**ε,β*and*r**ε*tend to 0 respectively
as*ε*and*λ*tend to 0.

We would like to ﬁnd a solution*v* of

∆v+*λ|∇v|*^{2}+*ε*^{2}*|x|*^{2β}(

*e** ^{v}*+

*e*

*)*

^{γv}= 0 (13)

in*B**r** _{ε,β}* of the form

*v(x) =v**ε,τ,β*(x) +*h(x).*

Equation (13) yields

L*β**h* = *−*8(β+ 1)^{2}*τ*^{2(β+1)}*ε*^{2}*|x|*^{2β}

(ε^{2}+*|τ x|*^{2(β+1)})^{2} (e^{h}*−h−*1)

*−* ^{8}^{γ}^{(β+1)}_{(ε}2+^{2γ}*|**τ x*^{τ}^{2γ(β+1)}*|*^{2(β+1)})^{ε}^{2γ}^{2}^{|}^{x}^{|}^{2β}*e*^{γh}*−λ|∇*(v*ε,τ,β*+*h)|*^{2}

(14)

in*B**r** _{ε,β}*.

By Proposition 1, solving the equation (14) is equivalent to ﬁnd a ﬁxed point
*h, in a small ball of*C^{2,α}*δ* (B*r** _{ε,β}*), solution of

*h*=G*δ,β* *◦*R(h) (15)

where

R(h) = *−*8(β+ 1)^{2}*τ*^{2(β+1)}*ε*^{2}*|x|*^{2β}

(ε^{2}+*|τ x|*^{2(β+1)})^{2} (e^{h}*−h−*1)

*−* 8* ^{γ}*(β+ 1)

^{2γ}

*τ*

^{2γ(β+1)}

*ε*

^{2}

*|x|*

^{2β}

(ε^{2}+*|τ x|*^{2(β+1)})^{2γ} *e*^{γh}*−λ|∇*(v*ε,τ,β*+*h)|*^{2}*.*

(16)

We have

*|*R(0)*| ≤c**κ**ε*^{2}*|x|*^{2β}(ε^{2}+*|τ x|*^{2(β+1)})^{−}^{2γ}+*c**κ**λ|∇v**ε,τ,β**|*^{2}*,*
this implies that for*γ∈*(0,1) and*|x|*=*r*we have

sup

*r**≤**r*_{ε,β}

*r*^{2}^{−}^{δ}*|*R(0)*| ≤c**κ**ε*^{2} sup

*r**≤**r*_{ε,β}*|x|*^{2β+2}^{−}* ^{δ}*(ε

^{2}+

*|τ x|*

^{2(β+1)})

^{−}^{2γ}+c

*κ*sup

*r**≤**r*_{ε,β}

*λ|x|*^{4β+4}^{−}* ^{δ}*(ε

^{2}+

*|τ x|*

^{2(β+1)})

^{−}^{2}

*≤c**κ**ε*^{4(1}^{−}^{γ)}^{−}* ^{δ/(β+1)}*+

*c*

*κ*

*λε*

^{−}

^{δ/(β+1)}*≤c*

*κ*

*r*

_{ε,β}^{2}

*.*

(17)

For*h*1*, h*2in*B(0,*2c*κ**r*_{ε,β}^{2} ) ofC^{2,α}*rad,δ*(B*r** _{ε,β}*),we have for

*γ∈*(0,1), that sup

*r**≤**r**ε,β*

*r*^{2}^{−}^{δ}*|*R(h2)*−*R(h1)*|*

*≤c**κ**ε*^{2} sup

*r**≤**r**ε,β*

*r*^{2β+2}^{−}* ^{δ}*(ε

^{2}+

*|τ x|*

^{2(β+1)})

^{−}^{2}e

^{h}^{2}

*−e*

^{h}^{1}+

*h*1

*−h*2 +c

_{κ}*ε*

^{2}sup

*r**≤**r*_{ε,β}

*r*^{2β+2}^{−}* ^{δ}*(ε

^{2}+

*|τ x|*

^{2(β+1)})

^{−}^{2γ}

*|e*

^{γh}^{1}

*−e*

^{γh}^{2}

*|*+c

*κ*sup

*r**≤**r*_{ε,β}

*r*^{2}^{−}^{δ}*λ(|∇*(v*ε,τ,β*+*h*2)*|*^{2}*− |∇*(v*ε,τ,α*+*h*1)*|*^{2})

*≤c**κ**ε*^{δ/(β+1)}*r*^{2}_{ε,β}*∥h*2*−h*1*∥*_{C}^{2,α}

*δ* (B* _{rε,β}*)

+c*κ**ε*^{4(1}^{−}^{γ)}^{−}^{δ/(β+1)}*∥h*2*−h*1*∥*_{C}^{2,β}

*δ* (B* _{rε,β}*)+

*c*

*κ*

*λ∥h*2

*−h*1

*∥*

_{C}

^{2,β}

*δ* (B* _{rε,β}*)

*≤c**κ**r*_{ε,β}^{2} *∥h*2*−h*1*∥*_{C}^{2,α}

*δ* (B* _{rε,β}*)

*.*

Thus, applying a classical ﬁxed point argument, when*ε* is small enough, we
prove the existence and uniqueness of*h, solution of (15). We summarize this*
in the following result.

**Proposition 2.** *i)Given* *δ∈*(0,1*−γ],β* *̸∈*N*,λsatisfying* (H*λ*)*andκ >*1,
*then there exist* *c*_{κ}*>* 0 *independent of* *ε* *and a solution* *h* *∈*C^{2,α}*rad,δ*(B_{r}* _{ε,β}*)

*of*(15)

*satisfying*

*∥h∥*_{C}^{2,α}

*δ* (B* _{rε,β}*)

*≤*2c

_{κ}*r*

^{2}

_{ε,β}*.*

*Moreover* *u**ε,τ,β*(x) =*v**ε,τ,β*(x) +*h(x)is a solution of (13) inB**r*_{ε,β}*.*

*ii)* *Given* *δ* *∈* (0,1), *β* = 0, *λ* *satisfying* (C*λ*) *and* *κ >* 1, then there exist
*c*_{κ}*>*0 *independent ofε* *andh∈*C^{2,α}*rad,δ*(B_{r}* _{ε}*)

*solution of*(15)

_{|}

_{β=0}*satisfying*

*∥h∥*_{C}^{2,α}

*δ* (B* _{rε}*)

*≤*2c

*κ*

*r*

^{2}

_{ε}*.*

*Moreover* *u**ε,τ*(x) =*v**ε,τ*(x) +*h(x)is a solution of* (13)_{|}_{β=0}*in* *B**r*_{ε}*.*
**2.2** **Analysis of the Laplace operator in weighted spaces [4]**

In this section, we study the mapping properties of the Laplace operator in
weighted H¨older spaces. Given *q*1*, . . . , q**K* *∈*Ω, we set**q**:= (q1*,· · ·, q**K*) and

Ω¯* ^{∗}*(q) := ¯Ω

*− {q*1

*, . . . q*

*K*

*},*

and we choose*r*_{0}*>*0 so that the balls*B*_{r}_{0}(q* _{j}*) of center

*q*

*and radius*

_{j}*r*

_{0}are mutually disjoint and included in Ω. For all

*r∈*(0, r

_{0}), we deﬁne

Ω¯*r*(q) := ¯Ω*−*

∪*K*
*j=1*

*B**r*(q*j*).

**Definition 2.** *Given* *k* *∈*R*,* *α∈*(0,1) *and* *ν* *∈*R*, we introduce the H¨older*
*weighted space*C^{k,α}*ν* ( ¯Ω* ^{∗}*(q))

*as the space of functionsw∈*C

^{k,α}*loc*( ¯Ω

*(q))*

^{∗}*for with*

*the following norm*

*∥w∥*_{C}^{k,α}_{ν}_{( ¯}_{Ω}*∗*(q)): =*∥w∥*_{C}* ^{k,α}*( ¯Ω

*−∪*

^{K}*j=1*

*B*

_{r}_{0}

*(q*

_{/2}*))*

_{j}+

∑*K*
*j=1*

sup

0<r*≤**r*_{0}*/2*

*r*^{−}^{ν}*∥w(q**j*+*r·*)*∥*_{C}* ^{k,α}*( ¯

*B*2

*−*

*B*1)

(18)

*is ﬁnite.*

When *k≥*2, we denote by [C^{k,α}*ν* ( ¯Ω* ^{∗}*(q))]

**the subspace of functions**

_{0}*w∈*C

^{k,α}*ν*( ¯Ω

*(q)) satisfying*

^{∗}*w*= 0 on

*∂Ω.*

**Proposition 3.** *Assume thatν <*0 *andν* *̸∈*Z*, then*
L*ν* : [C^{2,α}*ν* ( ¯Ω* ^{∗}*(q))]

**0**

*−→*C

^{0,α}

*ν*

*−*2( ¯Ω

*(q))*

^{∗}*w* *7−→* ∆*w*

*is surjective. .*

In the following, we denote by ˜G*ν* a right inverse of L*ν*.

**Remark 1.** *Observe that, when* *ν <*0, *ν /∈*Z*, a right inverse is not unique*
*and depends smoothly on the points* *q*1*, . . . , q**m**, at least locally. Once a right*
*inverse is ﬁxed for one choice of the points* *q*1*, . . . , q**K**, a right inverse for*
*another choice of pointsq*˜1*, . . . ,q*˜*K* *close toq*1*, . . . , q**K* *can be obtained by using*
*a simple perturbation argument.*

**2.3** **Harmonic extensions [4]**

We study the properties of interior and exterior harmonic extensions. Given
*φ∈*C^{2,α}(S^{1}), we deﬁne*H** ^{i}*(=

*H*

*(φ;*

^{i}*·*)) to be the solutions of

{ ∆*H** ^{i}* = 0 in

*B*1

*H** ^{i}* =

*φ*on

*∂B*1

*.*

(19)
We denote by*e*1*, e*2 the coordinate functions on*S*^{1}.

**Lemma 1.** *i) If we assume that*

∫

*S*^{1}

*φ dv**S*^{1}= 0 (20)

*then there existsc >*0 *such that*

*∥H** ^{i}*(φ;

*·*)

*∥*

_{C}

^{2,α}

1 ( ¯*B*_{1}* ^{∗}*)

*≤c∥φ∥*

_{C}

^{2,α}(S

^{1})

*.*

*ii) If we assume that*

∫

*S*^{1}

*φ dv** _{S}*1 = 0

*and*

∫

*S*^{1}

*φ e*_{ℓ}*dv** _{S}*1= 0

*for*

*ℓ*= 1,2 (21)

*then there existsc >*0

*such that*

*∥H** ^{i}*(φ;

*·*)

*∥*

_{C}

^{2,α}

2 ( ¯*B*_{1}* ^{∗}*)

*≤c∥φ∥*C

^{2,α}(S

^{1})

*.*

Given*φ∈*C^{2,α}(S^{1}) we deﬁne*H** ^{e}*(=

*H*

*(φ;*

^{e}*·*)) to be the solution of { ∆

*H*

*= 0 in R*

^{e}^{2}

*−B*1

*H** ^{e}* =

*φ*on

*∂B*1

(22) which decays at inﬁnity.

**Definition 3.** *Given* *k* *∈* N*,* *α* *∈* (0,1) *and* *ν* *∈* R*, we deﬁne the space*
C^{k,α}*ν* (R^{2} *−B*_{1}) *as the space of functions* *w* *∈* C^{k,α}*loc*(R^{2} *−B*_{1}) *for which the*
*following norm*

*∥w∥*_{C}^{k,α}_{ν}_{(}_{R}2*−**B*_{1})= sup

*r**≥*1

(

*r*^{−}^{ν}*∥w(r·*)*∥*_{C}^{k,α}_{ν}_{( ¯}_{B}_{2}_{−}_{B}_{1}_{)})
*,*

*is ﬁnite.*

**Lemma 2.** *If we assume that*

∫

*S*^{1}

*φ dv** _{S}*1= 0. (23)

*Then there existsc >*0*such that*

*∥H** ^{e}*(φ,;

*·*)

*∥*

_{C}

^{2,α}

*−1*(R^{2}*−**B*_{1})*≤c∥φ∥*_{C}^{2,α}(S^{1})*.*

If *F* *⊂L*^{2}(S^{1}) is a space of functions deﬁned on *S*^{1}, we deﬁne the space
*F** ^{⊥}* to be the subspace of functions of

*F*which are

*L*

^{2}(S

^{1})-orthogonal to the functions

*e*

_{1}

*, e*

_{2}.

**Lemma 3.** *The mapping*

P: C^{2,α}(S^{1})^{⊥}*−→* C^{1,α}(S^{1})^{⊥}*φ* *7−→* *∂**r**H*^{i}*−∂**r**H*^{e}*whereH** ^{i}*=

*H*

*(φ;*

^{i}*·*)

*andH*

*=*

^{e}*H*

*(φ;*

^{e}*·*), is an isomorphism.

**3** **The nonlinear interior problem**

We are interested in equations of type

∆*w*+*λ|∇w|*^{2}+*ε*^{2}*V*(x)
(

*e** ^{w}*+

*e*

*)*

^{γw}= 0. (24)

First, we will treat the case where 0 is a zero of the potential*V*. Then, we can
write*V*(x) =*|x|*^{2β}*K(x) whereK*is some smooth function such that*K(0)>*0.

We would like to ﬁnd*w*solution of

∆*w*+*λ|∇w|*^{2}+*ε*^{2}*|x|*^{2β}*K(x)*
(

*e** ^{w}*+

*e*

*)*

^{γw}= 0 (25)

in*B**r** _{ε,β}*. Given

*φ∈*C

^{2,α}(S

^{1}) satisfying (20) and (21), we deﬁne

**v**:=

*u*

_{ε,τ,β}*−*log(K(0)) +

*H*

*(φ,*

^{i}*·/r*

*)*

_{ε,β}then we look for a solution of (25) of the form *w*=**v**+*v* and using the fact
that*H*_{β}* ^{i}* is harmonic, this amounts to solve

*−L**β**v*=*S(v)*
where

*S(v)* = 8(β+ 1)^{2}*τ*^{2(β+1)}*ε*^{2}*|x|*^{2β}

(ε^{2}+*|τ x|*^{2(1+β)})^{2} *e*^{H}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)+h+v}
(*K(x)*

*K(0)* *−*1
)

+ 8(β+ 1)^{2}*τ*^{2(β+1)}*ε*^{2}*|x|*^{2β}
(ε^{2}+*|τ x|*^{2(1+β)})^{2} *e*^{h}

(

*e*^{H}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)+v}*−v−*1
)

+ 8(β+ 1)^{2}*τ*^{2(β+1)}*ε*^{2}*|x|*^{2β}
(ε^{2}+*|τ x|*^{2(1+β)})^{2}

(*e*^{h}*−*1)
*v*

+ 8* ^{γ}*(β+ 1)

^{2γ}

*τ*

^{2γ(β+1)}

*ε*

^{2}

*|x|*

^{2β}(ε

^{2}+

*|τ x|*

^{2(β+1)})

^{2γ}

*e*

^{γh}(*K(x)*

*K(0)*^{γ}*e*^{γH}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)+γv}*−*1
)

+ *λ∇*[

*v**ε,τ,β*(x) +*H** ^{i}*(φ, x/r

*ε,β*) +

*h(x) +v(x)]*

^{2}

*−* *λ|∇*(v*ε,τ,β*(x) +*h(x))|*^{2}*.*

By Proposition 1, it is suﬃcient to ﬁnd*v∈*C^{2,α}*µ* (B_{r}* _{ε,β}*) solution of

*v*=*−G**µ,β**◦S(v)* (26)

in*B**r** _{ε,β}*. We denote byN(=N

*ε,τ,β,φ*) the nonlinear operator appearing on the right hand side of the above equation. Given

*κ >*0 (whose value will be ﬁxed later on), we further assume that the functions

*φ*satisfy

*||φ||*_{C}^{2,α} *≤κ r**ε,β* for*β̸∈*N (27)
and

*||φ||*_{C}^{2,α} *≤κ r*_{ε}^{2} for*β*= 0. (28)
Then, we have the following result.

**Lemma 4.** *i)Letβ* *̸∈*N*, under the above assumptions, there exists a constant*
*c**κ**>*0 *such that forµ∈*(0,1*−γ]*

*∥N*(0)*∥*_{C}^{2,α}_{µ}_{(B}

*rε,β*)*≤c*_{κ}*r*_{ε,β}*and*

*∥N*(v2)*−*N(v1)*∥*_{C}^{2,α}_{µ}_{(B}_{rε,β}_{)}*≤c**κ**r**ε,β**∥v*2*−v*1*∥*_{C}^{2,α}_{µ}_{(B}_{rε,β}_{)}
*providedv*1*, v*2*∈*C^{2,α}*µ* (B*r** _{ε,β}*)

*satisfy*

*∥v*

*i*

*∥*

_{C}

^{2,α}

_{µ}_{(B}

_{rε,β}_{)}

*≤*2

*c*

*κ*

*r*

*ε,β*

*.*

*ii)* *Let* *β* = 0, under the above assumptions, there exists a constant *c**κ* *>*0
*such that forµ∈*(1,2)

*∥N*(0)*∥*_{C}^{2,α}_{µ}_{(B}

*rε*)*≤c**κ**r*^{2}_{ε}*.*
*and*

*∥N*(v_{2})*−*N(v_{1})*∥*_{C}^{2,α}* _{µ}* (B

*)*

_{rε}*≤c*

_{κ}*r*

^{2}

_{ε}*∥v*

_{2}

*−v*

_{1}

*∥*

_{C}

^{2,α}

*(B*

_{µ}*)*

_{rε}*providedv*1*, v*2*∈*C^{2,α}*µ* (B*r** _{ε}*)

*satisfying*

*∥v*

*i*

*∥*

_{C}

^{2,α}

*(B*

_{µ}*)*

_{rε}*≤*2

*c*

*κ*

*r*

_{ε}^{2}

*.*

**Proof .** The proof of the ﬁrst estimate of part*i) follows from the result of*
Lemma 1 part*i) together with the assumption on the norms ofφ. We brieﬂy*
comment on how these are used: it follows from Lemma 1, part *i) that for*
*β̸∈*N, we have

*∥H** ^{i}*(φ,

*·/r*

*)*

_{ε,β}*∥*

_{C}

^{2,α}( ¯

*B*

*)*

_{rε,β}*≤c*

_{κ}*r r*

_{ε,β}

^{−}^{1}

*∥φ∥*

_{C}

^{2,α}(S

^{1})

for all*r≤r**ε,β**/2. Then using (27), we get*

*∥H** ^{i}*(φ,

*·/r*

*ε,β*)

*∥*

_{C}

^{2,α}

1 ( ¯*B** _{rε,β}*)

*≤c*

*κ*

*r*

*ε,β*

for all*r≤r**ε,β**/2.*

Using the fact that*∥h∥ ≤*2c*κ**r*_{ε,β}^{2} which tends to 0 as*ε*tends to 0, then for
*µ∈*(0,1*−γ], we get*

*ε*^{2}*|x|*^{2β}(ε^{2}+*|τ x|*^{2(β+1)})^{−}^{2} *e** ^{h}*
(

*e*^{H}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)}*−*1)

C^{0,α}* _{µ−2}*( ¯

*B*

*)*

_{rε,β}*≤c**κ**ε*^{(1}^{−}^{µ)/(β+1)}*≤c**κ**r**ε,β**,*

*ε*^{2}*|x|*^{2β}(ε^{2}+*|τ x|*^{2(β+1)})^{−}^{2}*e*^{H}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)+h}
(*K(x)*

*K(0)* *−*1)

C^{0,α}* _{µ−2}*( ¯

*B*

*)*

_{rε,β}*≤c*_{κ}*ε*^{(1}^{−}^{µ)/(β+1)}*≤c*_{κ}*r** _{ε,β}*
and

*ε*^{2}*|x|*^{2β}(ε^{2}+*|τ x|*^{2(β+1)})^{−}^{2γ}*e*^{γh}

(*K(x)*

*K(0)*^{γ}*e*^{γH}^{i}^{(φ,}^{·}^{/r}^{ε,β}^{)}*−*1)

C^{0,β}* _{µ−2}*( ¯

*B*

*)*

_{rε,β}*≤c**κ**ε*^{4(1}^{−}^{γ)}^{−}* ^{µ/(β+1)}*(1 +

*ε*

^{1/(β+1)})

*≤c*

*κ*

*r*

*ε,β*

*.*on the other hand, using the condition (H

*λ*), we get

*λ∇*[

*v** _{ε,τ,β}*(x) +

*H*

*(φ, x/r*

^{i}*) +*

_{ε,β}*h(x)]*

^{2}

*−λ|∇*[v

*(x) +*

_{ε,τ,β}*h(x)]|*

^{2}

C^{0,β}* _{µ−2}*( ¯

*B*

*)*

_{rε,β}*≤c**κ**r*_{ε,β}^{2} *.*

Then the ﬁrst estimate of part *i) follows. On the other hand, we have for*

*∥v**i**∥*_{C}^{2,β}_{µ}_{(B}_{rε,β}_{)}*≤*2*c**κ**r**ε,β*,

*∥N*(v_{2})*−*N(v_{1})*∥*_{C}^{2,α}_{µ}_{(}_{R}2)

*≤c** _{κ}*
(

*ε*^{1/(β+1)}+*ε*^{µ/(β+1)}*r** _{ε,β}*+

*ε*

^{4(1}

^{−}*+*

^{γ)}*ε*

^{4(1}

^{−}*(1 +*

^{µ)}*ε*

^{1/(β+1)})

)*∥v*_{2}*−v*_{1}*∥*_{C}^{2,α}_{µ}_{(}_{R}2)

*≤c*_{κ}*r*_{ε,β}*∥v*_{2}*−v*_{1}*∥*_{C}^{2,α}* _{µ}* (R

^{2})

*.*

This yields the second estimate of part*i).*

Observe that these estimates are uniform in *τ* provided*τ* remains in a ﬁxed
compact subset of (0,*∞*). Applying a contraction mapping argument, we
obtain the following proposition.

**Proposition 4.** *Given* *β* *̸∈*N*,µ∈*(0,1*−γ]* *and* *κ >*0, there exists *ε*_{κ}*>*0,
*λ*_{κ}*>* 0,(depending on *κ) such that for all* *ε* *∈* (0, ε* _{κ}*),

*λ*

*∈*(0, λ

*)*

_{κ}*satisfying*(H

*), for all*

_{λ}*τ*

*in some ﬁxed compact subset of*[τ

_{−}*, τ*

^{+}]

*⊂*(0,

*∞*)

*and for a*

*given*

*φ*

*satisfying (20) and (27), then there exists a unique*

*v*

*β*(= ¯

*v*

*ε,τ,β,φ*)

*solution of (26) satisfying*

*∥v**β**∥*_{C}^{2,α}_{µ}_{(B}_{rε,β}_{)}*≤*2*c**κ**r**ε,β**.*

Notice that it follows from this proposition that the function
**v**+ ¯*v**ε,τ,β,φ*=*u**ε,τ,β*+*h−*log*K(0) +H** ^{i}*(

*·/r*

*ε,β*) + ¯

*v*

*ε,τ,β,φ*

solves (25) in *B**r** _{ε,β}*. Observe that the function ¯

*v*

*ε,τ,β,φ*obtained as a ﬁxed point for contraction mapping, it depends smoothly on the parameter

*τ*and the boundary data

*φ.*

We turn now to the case when 0 is not a zero of *V*, which corresponds to
the case*β* = 0.

**Proposition 5.** *Given* *κ >*0,*µ∈*(1,2) *andβ* = 0, then there exists *ε**κ**>*0,
*λ**κ* *>*0 *(depending on* *κ) such that for all* *ε∈* (0, ε*κ*), *λ* *∈*(0, λ*κ*) *satisfying*
(C*λ*), for all *τ* *in some ﬁxed compact subset of* [τ_{−}*, τ*^{+}] *⊂* (0,*∞*) *and for a*
*givenφsatisfying (21) and (28), then there exists a uniquev(= ¯v**ε,τ,φ*)*solution*
*of* (26)_{|}*β=0* *such that*

*∥v∥*_{C}^{2,α}_{µ}_{(}_{R}2)*≤*2*c*_{κ}*r*^{2}_{ε}*.*

**4** **The nonlinear exterior problem**

Denote by

*r*_{ε,β}* _{i}* = max(ε

^{2(1}

^{−}

^{γ)}

^{−}

^{δ/2(β}

^{i}^{+1)}

*, λ*

^{1/2}

*ε*

^{−}

^{δ/2(β}

^{i}^{+1)}

*, λ*

^{1/2}

*, ε*

^{2/(2β}

^{i}^{+3)}).

and*r** _{ε}*= max(ε

^{2(1}

^{−}

^{γ)}

^{−}

^{δ/2}*, λ*

^{1/2}

*ε*

^{−}

^{δ/2}*, λ*

^{1/2}

*, ε*

^{1/2}) and recall that ¯

*β*= min

1*≤**i**≤**N**β** _{i}*.
Let

*G(·,x) be the unique solution of*˜

*−*∆G(*·,x) = 8*˜ *π δ*_{x}_{˜}

in Ω, with*G(·,x) = 0 on*˜ *∂Ω. Recall that the following decomposition holds*
*G(x,x) =*˜ *−*4 log*|x−x*˜*|*+*H*(x,*x)*˜

where *x7−→* *H*(x,*x) is a smooth function. Here we give an estimate of the*˜
gradient of *H*(x,*x) without proof (see [27] and more details in [22], Lemma*˜
2.1), there exists a constant*c >*0, so that

*|∇**x**H(x,x)*˜ *| ≤c*log*|x−x*˜*|.*

Let **˜***η* := (˜*η*^{1}*, . . . ,η*˜* ^{K}*)

*∈*R

*close to 0,*

^{K}*φ*

**˜**:= ( ˜

*φ*

^{1}

*, . . . ,φ*˜

*)*

^{K}*∈*(C

^{2,α}(S

^{1}))

*satisfying (23) and*

^{K}**˜q**:= (p

*j*

_{1}

*, . . . , p*

*j*

_{s}*,q*˜1

*, . . . ,q*˜

*m*) close to

**q**:= (p*j*_{1}*, . . . , p**j*_{s}*, q*1*, . . . , q**m*). Note that the set *{q*1*, . . . , q**m**}*can be empty .

We deﬁne

**˜**
**v**:=

*•*

∑*s*
*i=1*

(1 +*β*_{j}* _{i}*+ ˜

*η*

^{j}*)*

^{i}*G(·, p*

_{j}*) +*

_{i}∑*s*
*i=1*

*χ*_{r}_{0}(*· −p*_{j}* _{i}*)

*H*

*( ˜*

^{e}*φ*

^{j}*; (*

^{i}*· −p*

_{j}*)/r*

_{i}

_{ε,β}*) if*

_{ji}*S∩*Λ =

*S*=

*{p*

*j*

_{1}

*,· · ·*

*, p*

*j*

_{s}*},*

*•*

∑*s*
*i=1*

(1 +*β**j** _{i}*+ ˜

*η*

^{j}*)*

^{i}*G(·, p*

*j*

*) +*

_{i}∑*s*
*i=1*

*χ**r*_{0}(*· −p**j** _{i}*)

*H*

*( ˜*

^{e}*φ*

^{j}*; (*

^{i}*· −p*

*j*

*)/r*

_{i}*ε,β*

*) +*

_{ji}∑*m*
*i=1*

(1 + ˜*η** ^{i}*)G(

*·,q*˜

*i*) +

∑*m*
*i=1*

*χ**r*_{0}(*· −q*˜*i*)*H** ^{e}*( ˜

*φ*

*; (*

^{i}*· −q*˜

*i*)/r

*ε*)

if *S∩*Λ =*{p**j*1*,· · ·* *, p**j**s**}*and*S\*Λ =*{q*1*,· · ·* *, q**m**},*with*s*+*m*=*K,*
(29)
where *χ**r*0 is a cutoﬀ function identically equal to 1 in *B**r*_{0}*/2* and identically
equal to 0 outside*B*_{r}_{0}. We suppose that

{ *q*˜*i*=*q**i*=*p**j** _{i}* if

*q*

*i*

*∈S∩*Λ

*|q*˜*i**−q**i**|< κr**ε* if *q**i**∈S\*Λ (30)

and we let {

*β*_{j}* _{i}*=

*β*

*if*

_{i}*q*

_{i}*∈S∩*Λ

*β** _{i}*= 0 if

*q*

_{i}*∈S\*Λ. (31)

We would like to ﬁnd a solution of

∆*v*+*λ|∇v|*^{2}+*ε*^{2}

∏*N*
*i=1*

*|x−p*_{i}*|*^{2β}^{i}*f*(x)
(

*e** ^{v}*+

*e*

*)*

^{γv}= 0 (32)

in ¯Ω*r*_{ε,}*β*¯(˜**q) := ¯**Ω*− ∪*^{K}*i=1**B**r*_{ε,}*β*¯(˜*q**i*) which is a perturbation of **˜v. Writing** *v* =

**˜**

**v**+ ˜*v, this amounts to solve*

*−*∆ ˜*v*=*ε*^{2}

∏*N*
*i=1*

*|x−p**i**|*^{2β}^{i}*f*(x)
(

*e*^{˜}^{v+˜}* ^{v}*+

*e*

^{γ}

^{v+γ˜}

^{˜}*)*

^{v}+ ∆**˜v**+*λ|∇*(˜**v**+ ˜*v)|*^{2}*.*
We need to deﬁne some auxiliary weighted spaces :

**Definition 4.** *Let* *r*¯*∈*(0, r_{0}*/2),k* *∈*R*,α∈*(0,1) *and* *ν* *∈*R*, we deﬁne the*
*H¨older weighted space* C^{k,α}*ν* ( ¯Ω_{r}_{¯}(q)) *as the set of functions* *w* *∈* C* ^{k,α}*( ¯Ω

_{r}_{¯}(q))

*for which the following norm*

*∥w∥*_{C}^{k,α}_{ν}_{( ¯}_{Ω}_{¯}_{r}_{(q))}:=*∥w∥*_{C}* ^{k,α}*( ¯Ω

_{r}_{0}

*(q))+*

_{/2}∑*K*
*i=1*

sup

*r**∈*[¯*r,r*0*/2)*

(*r*^{−}^{ν}*∥w(q** _{i}*+

*r·*)

*∥*

_{C}

*( ¯*

^{k,α}*B*

_{2}

*−*

*B*

_{1})

)*.*
*is ﬁnite.*