Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps

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Ann. I. H. Poincaré – AN 32 (2015) 23–40

www.elsevier.com/locate/anihpc

Nonlinear scalar field equations: Existence of a positive solution with infinitely many bumps

Giovanna Cerami

, Donato Passaseo

, Sergio Solimini

aPolitecnico di Bari, Dipartimento di Meccanica, Matematica e Management, Via Orabona 4, 70125 Bari, Italy

bUniversità del Salento, Dipartimento di Matematica e Fisica, Ex Collegio Fiorini, Via per Arnesano, 73047 Monteroni di Lecce (LE), Italy Received 7 March 2013; accepted 23 August 2013

Available online 15 October 2013

Abstract

In this paper we consider the equation

(E) −u+a(x)u= |u|^p−1u inR^N,

whereN2,p >1,p <2^∗−1= ^N_N⁺₋²₂, ifN3. During last thirty years the question of the existence and multiplicity of solutions to(E)has been widely investigated mostly under symmetry assumptions ona. The aim of this paper is to show that, differently from those found under symmetry assumption, the solutions found in[6]admit a limit configuration and so(E)also admits a positive solution of infinite energy having infinitely many ‘bumps’.

©2013 Elsevier Masson SAS. All rights reserved.

Résumé

Dans ce papier nous considérons l’équation

(E) −u+a(x)u= |u|^p−1u enR^N,

oùN2,p >1,p <2^∗−1=^N_N⁺₋²₂, siN3. Pendant les trente dernières années la question de l’existence et de la multiplicité de solutions d’(E)a été largement examinée surtout conformément aux suppositions de symétrie sura. Le but de ce papier est de montrer que, différemment de ceux trouvés conformément à la supposition de symétrie, les solutions trouvées dans[6]admettent une configuration de limite et donc(E)admet aussi une solution positive d’énergie infinie ayant une infinité de ‘bumps’.

©2013 Elsevier Masson SAS. All rights reserved.

MSC:35J20; 35J60; 35Q55

Keywords:Variational methods; Solutions with infinitely many bumps; Schrödinger equation

✩ Work supported by the Italian national research project “Metodi variazionali e topologici nello studio di fenomeni non lineari”.

* Corresponding author.

E-mail addresses:[email protected](G. Cerami),[email protected](D. Passaseo),[email protected](S. Solimini).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.08.008

1. Introduction

In this paper we consider the equation (E) −u+a(x)u= |u|^p−1u inR^N,

whereN2,p >1,p <2^∗−1=^N_N⁺₋²₂, ifN3, and the potentiala(x)is a positive function that is not required to enjoy symmetry.

During the past years there has been a considerable amount of research on this kind of questions; the interest comes, essentially, from two reasons: their specific mathematical difficulties, that make them challenging to the researchers, and, moreover, the fact that equations as (E) arise naturally in several branches of mathematical physics. Indeed the solutions of(E)can be seen as stationary states (corresponding to solitary waves) in nonlinear equations of the Klein–Gordon type

∂²ϕ

∂t² −ϕ+

a(x)+ω²

ϕ− |ϕ|^p−2ϕ=0 (1)

and of Schrödinger type i∂ϕ

∂t −ϕ+

a(x)+ω²

ϕ− |ϕ|^p−2ϕ=0 (2)

(whereϕ=ϕ(t, x)) is a complex function defined onR×R^N.

Let us consider, for instance, Eq.(1): it corresponds to the Lagrangian density L(ϕ)= −1

2|ϕt|²+1

2|∇ϕ|²+1 2

a(x)+ω²

|ϕ|²− 1 p|ϕ|^p.

Thus, looking for a solitary wave, of the standing wave form, means searching solutionsϕ(x, t )=e^iωtu(x), with u:R^N→R, hence one is led exactly to the equation considered in(E). Analogously searching for stationary states of(2)leads again to(E).

Furthermore, we recall that, besides the above mentioned problems, equations like (E), which are also called Euclidean scalar field equations, appear in several other context of physics: nonlinear optics, laser propagations, constructive field theory, etc.

During last thirty years the question of the existence and multiplicity of solutions to(E)has been widely investi- gated and most results have been obtained under symmetry assumptions ona. However, some considerable progress has been performed also in the case in whicha(x)is not required to fulfill symmetry properties. We just mention that, formerly, the existence of a positive solution has been shown in[7,1,2], while the existence of infinitely many changing sign solutions has been proven in[5].

We refer the interested reader either to [4] or to[6] for a more detailed description of the development of the researches as well as a quite exhaustive list of references.

Very recently, an answer to the question of the existence of infinitely many positive solutions to(E)has been given in[6]where the following result has been proved:

Theorem 1.Let assumptions

(h₁) a(x)→a_∞>0as|x| → ∞, (h₂) a(x)a₀>0,∀x∈R^N, (h₃) a∈L^N/2_loc (R^N),

(h4) ∃ ¯η∈(0,√

a_∞): lim_|x|→+∞(a(x)−a_∞)e^η¯|x|= +∞

be satisfied.

Then there exists a positive constant,A=A(N,η, a¯ ₀, a_∞)∈R, such that, when a(x)−a_∞

N/2,loc:= sup

y∈R^N

a(x)−a_∞

L^N/2(B₁(y))<A,

equation(E)has infinitely many positive solutions belonging toH¹(R^N).

The solutions, whose existence is asserted in Theorem 1, are multi-bump functions. The claim is proved just showing, by purely variational methods, that, for allk∈N\ {0}, there exists a solution of (E)which belongs to a special class consisting of functions having exactlykbumps.

The aim of this paper is to show that(E)admits also a positive solution of infinite energy having infinitely many

‘bumps’.

In order to be more precise about our achievement we need to introduce some notation.

We denote byw(x)∈H¹(R^N)the ground state solution of the limit equation (E_∞) −u+a_∞u= |u|^p−1u inR^N.

For all functionu∈H¹(R^N)and for fixedδ, we set u^δ(x):=(u−δ)⁺(x)

and

u_δ(x):=u(x)−u^δ(x)

and we callu^δ(x)theemerging partofuaboveδ,u_δ(x)thesubmerged partofuunderδ.

Fixingδ andρ, we say thatu∈H¹(R^N)isemerging around thekpointsx₁, x₂, . . . , x_k (inkballs of radiusρ) if

u^δ(x)= k

i=1

u^δ_i(x)

where for alli∈ {1,2, . . . , k},u^δ_i 0,u^δ_i ≡0,u^δ_i ∈H₀¹(Bρ(xi)),Bρ(xi)∩Bρ(xj)= ∅ifi=j. Now our result can be stated as follows:

Theorem 2.Let assumptions ofTheorem1be satisfied.

Then there exists a solution of(E),u¯∈H_loc¹ (R^N), which is emerging around an unbounded sequence of points (x¯n)n,x¯n∈R^N(x¯n= ¯xmform=n).

Moreoveru¯and(x¯n)nhave the following properties:

nlim→∞min

| ¯x_n− ¯x_m|: m∈N, m=n

= +∞, (3)

nlim→∞u(x¯ + ¯xn)=w(x) uniformly on all compact subsets ofR^N. (4) We remark that, while the multi-bump solutions obtained inTheorem 1have a variational characterization, solu- tionu¯ is obtained as limit, ask→ +∞, of a sequence(u_k)_k of multi-bump solutions to(E)given byTheorem 1.

The reason of the success of our procedure is strongly related to the nature of the solutionsu_k. Actually, ak-bump solution is obtained first minimizing the action functional on sets of functions emerging around a fixedk-tuple of points ofR^N, then considering the maxima of the minima when thek-tuples vary in a suitable subset ofR^N. In this argument the role played by the ‘slow decay’ assumption ona(x)is basic, because the attractive effect ofa(x), due to assumption (h4), is dominating on the repulsive disposition, due to the maximization procedure, of positive masses with respect to each other and makes the bumps not to escape at infinity.

So, relation(3)is just a consequence of the above described situation and indicates the bumps ofu¯ rarefy, as the distance from the origin increases, giving rise to a quite new phenomenon. Indeed, for instance, multi-bump positive solutions, obtained under assumptions of radial symmetry ona(x)in[8], clearly do not converge as the number of the bumps increases, on the contrary the solutions are built in such a way that the bumps, as their number increases, spread out, going far away from each other and far away from the origin in a uniform way.

The paper is organized as follows: in Section2 the variational framework of the paper is introduced and some useful facts as well as some results contained in [6]are collected; Section 3contains some preliminary estimates;

Section4is devoted to the proof, subdivided in several steps, ofTheorem 2.

2. Notation, variational framework and useful facts

Throughout the paper we make use of the following notation:

• Given a measurable subsetDofR^N,|D|denotes its Lebesgue measure.

• WhenDis open,H¹(D)denotes the closure ofC^∞(D)with respect to the norm uD:=

D

|∇u|²+u² dx

1/2

; the norm inH¹(R^N)is denoted by · .

• L^p(D), 1p+∞, denotes the Lebesgue space with norm| · |p,D

|u|p,D:=

D

|u|^pdx 1/p

; the norm inL^p(R^N)is denoted by| · |p.

• B_r(x), whenxbelongs to a metric spaceX, denotes the open ball ofXhaving radiusrand centered atx.

• L^p_loc(R^N), 1p <+∞, andH_loc¹ (R^N)denote the sets of functionsusuch that∀x∈R^Nthere exists an open set DR^N, so thatx∈Dandu_|D∈L^p(D), respectively,u_|D∈H¹(D).

• Given any functionu:R^N→R, suppudenotes its support (defined as in[3, Sec. 4]); ifuandv are real valued functions defined inD⊆R^N, we denote, as usual, byu∨v andu∧v the functions defined by(u∨v)(x):=

max(u(x), v(x))and(u∧v)(x):=min(u(x), v(x))for allx∈D.

• C,c,c,˜ c,ˆ c_i denote various positive constants.

The variational functionals, related to problems (P )and (P_∞), are denoted respectively byI and I^∞ and are defined inH¹(R^N)as

I (u)=1 2

R^N

|∇u|²+a(x)u²

dx− 1 p+1

R^N

|u|^p+1dx,

and

I^∞(u)=1 2

R^N

|∇u|²+a_∞u²

dx− 1 p+1

R^N

|u|^p+1dx.

The following lemma contains some known facts about(P_∞), see f.i.[4].

Lemma 3.(P_∞)has a positive, ground state, solutionw, unique up to translation, radially symmetric, decreasing when the radial coordinate increases and such that

|x|→+∞lim D^jw(x)|x|^N2−1e^√a∞|x|=dj>0, dj∈R, j=0,1. (5) In what follows we use the notation

m_∞:=I^∞(w) and wy(x):=w(x−y); (6)

moreover,we set

α(x):=a(x)−a_∞.

In the rest of the paper, according to[6], the real numbersδ >0 andρ >0 are fixed in such a way that δ <min

w(0)/3,1, (a0/p)^1/(p−1), a₀^1/(p−1)/2,

a_∞− ¯η²1/(p−1)

(7)

and

w(x) < δ, ∀x∈R^N\Bρ/2(0).

Considering the setsKk,k∈N\ {0}defined as Kk=

R^N whenk=1;

{(x₁, . . . , x_k)∈(R^N)^k: |x_i−x_j|3ρ, i, j=1,2, . . . , k, i=j} whenk >1, (8) we set, for all(x₁, . . . , x_k)∈Kk,

Sx₁,...,x_k = u∈H¹

R^N

: uemerging aroundx1, x2, . . . , xk andI(u) u^δ_i

=0, βi(u)=0,∀i=1,2, . . . , k , where

β_i(u)= 1

|u^δ_i|²₂

R^N

(x−x_i) u^δ_i(x)2

dx

is a barycenter type map.

The arguments developed in[6]show thatTheorem 1can be completed in the following way:

Theorem 4. Let the assumptions of Theorem1 be satisfied. Then, for all k∈N\ {0}, there exists (at least) one solutionu¯_k of(E)which is emerging aroundkpoints(x¯^k₁, . . . ,x¯_k^k)∈Kk.

Moreover,u¯_k is found as a critical point ofI and is characterized as I (u¯_k)= min

Sxk¯1,...,xk¯k

I (u)=max

Kk

Sxmin1,...,xk

I (u).

Furthermore, settingd(x)=dist(x,[supp(u¯k)^δ∪suppα⁻]), the relation

0< (u¯_k)_δ(x) < Cδe^{− ¯ηd(x)} (9)

holds, withC >0depending only onη,¯ a_∞,N.

For what follows it is useful to remark that, for allu∈H¹(R^N), emerging aroundkpointsx1, . . . , xk, the func- tionalI can be written as

I (u)=I (u_δ)+J_δ u^δ

=I (u_δ)+ k

i=1

J_δ u^δ_i

where the functionalJδis defined, for allvbelonging toH¹(R^N)and having compact support, as J_δ(v)=1

2R^N

|∇v|²+a(x)v² dx+

R^N

a(x)δv dx

− 1 p+1

suppv

δ+ |v|p+1

dx+ 1

p+1δ^p+1|suppv|.

Next lemmas describe some important features of the functionalsI andJ_δ, as well as the nature of the ‘natural nonsmooth constraint’I(u)[u^δ_i] =0,i=1, . . . , k, imposed to the functions belonging to the setsS_x1,...,xk.

Lemma 5.Let assumptions(h₁),(h₂)and(h₃)be satisfied. The functionalI is coercive, convex and, hence, weakly lower semicontinuous on the set

H:=

u∈H¹ R^N

: u(x)δ, ∀x∈R^N .

Proof. Because of the choice(7)ofδ, for any functionu∈Hwe have I (u)=1

2

R^N

|∇u|²+a(x)u²

dx− 1 p+1

R^N

|u|^p+1dx

1

2

R^N

|∇u|²dx+a0

1

2− 1

p+1

R^N

u²dx >0, (10)

thus the assertion follows. 2

Lemma 6. Let assumptions(h₁), (h₂) and (h₃) be satisfied. For all u∈H¹(R^N) such that u^δ=0, the function Iu: [0,+∞)→Rdefined as

Iu(t)=I

u_δ+t u^δ

has a unique maximum pointt_u∈(0,+∞).

Lemma 7.Let assumptions(h₁),(h₂)and(h₃)be satisfied. Letk∈N\ {0}and(x₁, . . . , x_k)belong to(R^N)^k. Let u∈H¹(R^N)be a function emerging around the pointsx₁, . . . , x_k, then for allj∈ {1, . . . , k}the function

t→I

u_δ+

i=j

u^δ_i +t u^δ_j

has a unique maximum pointt_u^j∈(0,+∞).

Lemma 8.Let assumptions(h1),(h2)and(h3)be satisfied. Then, for allk∈N\ {0}, for all(x1, . . . , xk)∈Kk, for all i∈ {1, . . . , k},

u∈S_x1,...,xk ⇒

I (u) >0, J_δ

u^δ_i

>0. (11)

For the proofs of the above lemmas we refer the reader to[6].

We also remark that the proof ofLemma 6showsu_δ+t_uu^δis the only point in the half line{u_δ+t u^δ:t∈(0,+∞)}, for which

I

uδ+tuu^δ u^δ

=0.

Analogously,Lemma 7gives, foruemerging aroundx₁, . . . , x_k, that, for anyj∈ {1,2, . . . , k}, in the half line{u_δ+

i=ju^δ_i +t u^δ_j: t∈(0,+∞)}there is just one pointuδ+

i=ju^δ_i +t_u^ju^δ_j, for which I

u_δ+

i=j

u^δ_i +t_u^ju^δ_j u^δ_j

=0.

Given anyu∈H¹(R^N)such thatu^δ=0, we callu_δ+t_uu^δthe projection ofuon the set{u∈H¹(R^N): I(u)[u^δ] = 0}and we setϑ (u):=t_u.

Analogously, ifu∈H¹(R^N)is emerging aroundkpoints we calluδ+

i=ju^δ_i +tu^ju^δ_j the projection ofuon the set{u∈H¹(R^N): I(u)[u^δ_j] =0}and we setϑj(u):=t_u^j.

Remark 1.We point out that, whena(x)=a_∞, the definition ofS_x1,...,xk still makes sense andLemmas 6, 7 and 8 hold. In this case we use the notationS_x^∞

1,...,x_k. Moreover, for allu∈H¹(R^N),u^δ=0, we denote byuδ+ϑ^∞(u)u^δ, the projection of u on the set{u∈H¹(R^N): (I^∞)(u)[u^δ] =0}, and, when u is emerging aroundk points, uδ+

i=ju^δ_i +ϑ_j^∞(u)u^δ_j denotes the projection ofuon the set{u∈H¹(R^N): (I^∞)(u)[u^δ_j] =0}.

We close this section by a lemma, whose proof can be found in[6], that gives useful information about the set of the projections (on the constraintI(u)[u^δ] =0) of the family of ground state solutions of(P_∞).

Lemma 9.Let assumptions(h₁),(h₂)and(h₃)be satisfied. Then{ϑ (w_y): y∈R^N}is a bounded set.

3. Preliminary estimates

For allk∈N\ {0}and for all(x₁, . . . , x_k)∈Kk, we put μ(x₁, . . . , x_k):=min

I (u): u∈S_x1,...,xk

; M_x1,...,xk=

u∈S_x1,...,xk: I (u)=μ(x₁, . . . , x_k)

; μk=max

Kk

μ(x1, . . . , xk)=max

Kk

S_xmin_1,...,xkI (u).

Lemma 10.Let assumptions(h₁),(h₂),(h₃)and(h₄)be satisfied. Let(x₁, . . . , x_k)∈Kk and u∈M_x1,...,xk. Then, settingd(x)=dist(x,[suppu^δ∪suppα⁻]), relation

0< u_δ(x) < Cδe^{− ¯ηd(x)} (12)

holds, withC >0depending only onη,¯ a_∞,N.

The proof of the above lemma can be found in[6, Lemma 3.4].

Lemma 11. Let assumptions (h₁), (h₂), and (h₃) be satisfied. Let (x_n)_n, x_n ∈R^N be a sequence such that limn→+∞|x_n| = +∞and letv_n∈M_xn, then

I (v_n)m_∞+o(1). (13)

Proof. Let us considerw˜_n=(w_xn)_δ+ϑ (w_xn)(w_xn)^δ (wxn as defined in(6)), then we havew˜_n∈S_xn and, by using Lemma 6andRemark 1, we get

μ(xn)=I (vn)I (w˜n)=I^∞(w˜n)+1 2

R^N

α(x)

˜ wn(x)2

dx

I^∞(w)+1 2

R^N

α(x)

˜ wn(x)2

dx

m_∞+1

2 max

1, ϑ (wx_n)2

R^N

α(x)

wx_n(x)2

dx

. (14) Now, for allε >0 anr=r(ε) >0 can be found so thatα(x) < εas|x|> r; hence, for largen, the relation

R^N

α(x)

wx_n(x)2

dx sup

Br(0)

wx_n(x)2

Br(0)

α(x) dx +ε

R^N

w(x)2

dx

Cε,¯ (15)

holds, because sup_Br(0)wx_n(x)−−−−−→_n→+∞ 0. Then, since by Lemma 9, supϑ (wx_n)∈R,(13)follows combining(14) and(15). 2

Lemma 12.Let assumptions(h1),(h2), and(h3)be satisfied. Let((x₁ⁿ, . . . , xⁿ_k))nbe a sequence ofk-tuples belonging toKksuch that

n→+∞lim

xⁿ₁, . . . , x_kⁿ

=(x1, . . . , xk).

Then

n→+∞lim μ

x₁ⁿ, . . . , x_kⁿ

=μ(x₁, . . . , x_k). (16)

Proof. Let us consideru_n∈M_xⁿ

1,...,x_kⁿandu∈M_x1,...,xk. Set, for alln, ˆ

u_n(x):=

ˆ u_n(x)

δ+ k

i=1

ϑ_i(uˆ_n)(uˆ_n)^δ_i(x) where, for alli∈ {1, . . . , k},

(uˆ_n)^δ_i(x)=u^δ_i

x+x_i−x_iⁿ

and(uˆ_n)_δis the unique positive minimizer for the minimization problem min

I (u): u∈H¹ R^N

, |u|δ, u=δon k

i=1

supp(uˆ_n)^δ_i

.

We remark that such a minimizer exists and is unique; indeed the same argument ofLemma 5shows that the func- tionalI is coercive and convex on the convex set{u∈H¹(R^N): |u|δ, u=δon k

i=1supp(uˆn)^δ_i}.

Now, we have thatuˆn∈S_xⁿ

1,...,x_kⁿand lim sup

n→+∞μ

x₁ⁿ, . . . , x_kⁿ

=lim sup

n→+∞I (u_n) lim

n→+∞I (uˆ_n)=I (u)=μ(x₁, . . . , x_k). (17) On the other hand, we set

˜

un(x)=(u˜n)δ(x)+ k

i=1

ϑi(u˜n)(u˜n)^δ_i(x), where, for alli∈ {1, . . . , k},

(u˜n)^δ_i(x)=(un)^δ_i

x+x_iⁿ−xi

and(u˜n)δis the unique positive minimizer for the minimization problem min

I (u): u∈H¹ R^N

, |u|δ, u=δon k

i=1

supp(u˜_n)^δ_i

. Then we obtainu˜n∈Sx₁,...,x_k and

I (u)I (u˜n), lim

n→+∞

I (u˜n)−I (un)

=0.

Thus

μ(x₁, . . . , x_k)lim inf

n→+∞μ

x₁ⁿ, . . . , x_kⁿ holds and, together with(17), gives(16). 2

Lemma 13.Let assumptions(h1),(h2),(h3)and(h4)be satisfied, let(x1, . . . , xk)∈Kk;then a pointy∈R^N exists so that(x1, . . . , x_k, y)∈Kk+1and

μ(x₁, . . . , x_k)+m_∞< μ(x₁, . . . , x_k, y). (18)

Proof. The argument of this proof is similar to that used in Proposition 4.4 of[6], however we repeat it here in order to make the paper more self-contained.

Let us considery_n=σ_nτ, withσ_n∈R,σ_n−−−−−→_n→+∞ +∞,τ ∈R^N,|τ| =1 and set Σ_n=

x∈R^N: σ_n

2 −1< (x·τ ) <σ_n 2 +1

.

For largen∈N,(x₁, . . . , x_k, y_n)∈Kk+1; then we takeu_n∈M_x1,...,xk,yn;u_ncan be written as

u_n(x)=(u_n)_δ(x)+ k

i=1

(u_n)^δ_i(x)+(u_n)^δ_k+1(x),

where(u_n)^δ_i and(u_n)^δ_k+1are the emerging parts ofu_naroundx_i andy_nrespectively. We remark that supp(un)^δ_i ⊂Bρ(xi); βi(un)=0, ∀i∈ {1, . . . , k}

and

supp(un)^δ_k+1⊂B_ρ(y_n); β_k+1(u_n)=0;

moreover, for largen, we can assume that k

i=1

B_ρ(x_i)⊂

x∈R^N: (x·τ ) <σn

2 −1

,

B_ρ(y_n)⊂

x∈R^N: (x·τ ) >σ_n 2 +1

.

Let us definev_n(x)=χ_n(x)u_n(x), whereχ_n∈C^∞(R^N,[0,1]),χ_n(x)=χ (|(x·τ )−^σ₂ⁿ|)andχ∈C^∞(R⁺,[0,1])is a function such thatχ (t )=0 ift1/2,χ (t )=1 ift1. Let us evaluateI (v_n):

I (v_n)=I (χ_nu_n)

=1 2

R^N

|χ_n∇u_n+u_n∇χ_n|²+a(x)(χ_nu_n)²

dx− 1 p+1

R^N

|χ_nu_n|^p+1dx

=1 2R^N

χ_n²

|∇u_n|²+a(x)u²_n dx+1

2R^N

|∇χ_n|²u²_ndx

+1 4

R^N

∇χ_n²∇u²_ndx− 1 p+1

R^N

(u_n)^p+1dx+ 1 p+1

R^N

1−χ_n^p+1

(u_n)^p+1dx

μ(x₁, . . . , x_k, y_n)+ ¯c₁

Σn

(u_n)_δ2

dx+ ¯c₂

Σn

(u_n)_δp+1

dx

μ(x₁, . . . , x_k, y_n)+O e^{− ¯ησn}

, (19)

where last inequality is obtained using the exponential decay ofu_n(seeLemma 10) and observing that, by(h₄), for largen, suppα⁻∩Σ_n= ∅.

On the other hand, I (v_n)=I

v^I_n +I

v_n^II

, (20)

where v_n^I(x)=

0 if(x·τ )^σ₂ⁿ−¹₂, χ_n(x)(u_n)_δ(x)+_k

i=1(u_n)^δ_i if(x·τ ) <^σ₂ⁿ−¹₂ and

v_n^II(x)=

0 if(x·τ )^σ₂ⁿ +¹₂, χn(x)(un)δ(x)+(un)^δ_k+1(x) if(x·τ ) >^σ₂ⁿ +¹₂. By definition,v_n^I∈S_x1,...,xk, thus

I v_n^I

μ(x₁, . . . , x_k). (21)

Analogously, v_n^II∈S_yn; hence, considering the functionv˜_n∈S_y^∞

n defined asv˜_n(x)=(v^II_n)_δ(x)+ϑ^∞(v_n^II)(v_n^II)^δ(x), we get, for largen,

I v^II_n

I (v˜_n)=I^∞(v˜_n)+1 2

R^N

α(x)

˜ v_n(x)2

dx

m_∞+1

2

Bρ(yn)

α(x)

˜ v_n(x)2

dx− sup

Bσn 2 (0)

v˜_n(x)2 suppα−

α(x)dx

.

Now, by assumption(h4), suppα⁻is bounded; moreover, by maximum principle, sup

Bσn 2

(0)

v˜_n(x)2

sup

∂Bσn 2

(0)

v˜_n(x)2

.

Thus, taking into account that, for largen,∂Bσn

2 (0)∩suppα⁻= ∅, and usingLemma 10, we get I

v^II_n

m_∞+1

2

B_ρ(y_n)

α(x)

˜ v_n(x)2

dx−O e^{− ¯ησn}

. (22)

Therefore, combining(19),(20),(21)and(22), we obtain μ(x1, . . . , xk, yn)μ(x1, . . . , xk)+m_∞+1

2

Bρ(yn)

α(x)

˜ vn(x)2

dx−O e^{− ¯ησn}

; (23)

but, for largen, 1 2

Bρ(yn)

α(x)

˜ v_n(x)2

dx−O e^{− ¯ησn}

>0,

because assumption(h₄)impliesα(x)and

Bρ(yn)α(x)(v˜_n(x))²dx decay more slowly thane^{− ¯ησn}. Therefore(18) follows. 2

Lemma 14.Let(u¯_k)_k be a sequence of solutions to(E)obtained as described inTheorem4. For all real number r >0, let us denote byν(u¯_k, r)the number of points around whichu¯_k is emerging and that are contained inB_r(0).

Then, for allh∈Nthere exist a real numberrh>0and a numberkh∈Nsuch thatν(u¯k, rh)h, for allk > kh. Proof. We argue by contradiction and we assume that there existh∈Nand sequences(r_n)_n,r_n∈R⁺\0,(k_n)_n∈N, such thatr_n→ +∞,k_n→ +∞asn→ +∞, andν(u¯_kn, r_n) < hfor alln∈N.

Let us denote by(x¯₁ⁿ, . . . ,x¯_kⁿ

n)the points around which the solutionu¯_kn is emerging. Passing, if necessary, to a subsequence, we can assume that, for somej < h,

¯

x_iⁿ−−−−−→_n→+∞ x¯i, ∀ij, while

x¯_iⁿ−−−−−→_n→+∞ +∞, ∀i > j.

Lety∈R^Nbe a point, whose existence is proved byLemma 13, such that

μ(x¯₁, . . . ,x¯_j)+m_∞< μ(x¯₁, . . . ,x¯_j, y). (24) To get a contradiction we intend to show that the relations

lim sup

n→+∞

μ

¯

x₁ⁿ, . . . ,x¯_kⁿ

n

−μ

¯

xⁿ_j+2, . . . ,x¯_kⁿ

n

μ(x¯₁, . . . ,x¯_j)+m_∞ (25)

and