Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions

Semiclassical Limit for a Quasilinear Elliptic Field

Equation: One-Peak and Multi-Peak Solutions∗

Marino Badiale† Vieri Benci‡ Teresa D’Aprile§

October 15, 1999

Abstract

This paper deals with the existence of one-bump and multi-bump solutions for thefollowing nonlinear field equation

−∆u+ V (hx)u−∆pu+W ′(u) = 0

where u : RN → R

N+1, N ≥ 2, p > N, h > 0, the potential V is positive and Wis an appropriate singular function. Existence results are established provided that his sufficiently small and we find solutions exhibiting a concentration behaviour in thesemiclassical limit (i.e. as h→ 0+) at any prescribed finite set of local minima, possiblydegenerate, of the potential. Such solutions are obtained as local minima for the asso-ciated energy functional. No restriction on the global behaviour of V is required exceptthat it is bounded below away from zero. In the proofs of these results we use a varia-tional approach and the method relies on the study of the behaviour of sequences withbounded energy, in the spirit of the concentration-compactness principle.

1 Introduction

This paper can be considered as a continuation of the study already begun in [2]; indeed, asin [2], we are concerned with the existence and the concentration behaviour of bound states(i.e. solutions with finite energy) for the following nonlinear elliptic system:

− h2∆v + V (x)v − hp∆pv +W ′(v) = 0, (1.1)

where

• h > 0,• v : R

N −→ RN+1,

• N ≥ 2, p > N,• V : R

N −→ R,• W : Ω −→ R with Ω ⊂ R

N+1 an open set, denoting W ′ the gradient of W.∗The first author was supported by M.U.R.S.T., “Variational and Nonlinear Differential Equations”. The

second and the third authors were supported by M.U.R.S.T., “Equazioni Differenziali e Calcolo delle Vari-azioni”.

†Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa (Italy), e-mail: [email protected]‡Dipartimento di Matematica Applicata “U. Dini”, via Bonanno 25/b, 56126 PISA (Italy), e-mail:

[email protected]§Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 PISA (Italy), e-mail: [email protected]

1

Here ∆v = (∆v1, ...,∆vN+1), being ∆ the classical Laplacian operator, while ∆pv denotesthe (N+1)-vector whose j-th component is given by

div (|∇v|p−2∇vj).

By making the change of variables x→ hx, (1.1) can be rewritten as

−∆u+ Vh(x)u−∆pu+W ′(u) = 0 (1.2)

where Vh(x) = V (hx) and u(x) = v(hx).Equations like (1.1) or (1.2) have been introduced in a set of recent papers (see [3]-[10]). Insuch works the authors look for soliton-like solutions; the solitons are interpreted as particlesand their dynamics is studied in order to provide some examples of classical models whichexhibit a quantistic behaviour. We refer to [4], [8] and [10] for a more precise description ofsuch developments.Equations of the form (1.1) arise also in the study of standing waves for nonlinear Schrodingerequation where the presence of a small diffusion parameter h becomes natural: an interest-ing question, already raised in [2], which motivates the present work, is wheter one canfind solutions which concentrate around local minima, possibly degenerate, of the poten-tial V. In [2] we studied the semiclassical limit for equation (1.1) under the assumptionlim inf |x|→+∞ V (x) > infx∈RN V (x), and we proved the existence of at least a solution forsmall h > 0; furthermore this solution concentrates around an absolute minimum of V ash → 0+, in the sense that its shape is a sharp peak near that point, while it vanishes ev-erywhere else. The object of this paper is to go further in this recearch in order to removeany global assumption on V except for infx∈RN V (x) > 0 and to construct solutions withmultiple peaks which concentrate at any prescribed finite set of local minimum points of V.In other words any local minimum point (or any finite set of local minimum points) is able toattract a suitable family of concentrating bound states as h goes to zero, without requiringany assumption at infinity for V.

The phenomenon described above is connected with other concentration phenomenaknown in literature for related elliptic equations. For instance, a large number of works hasbeen devoted in studying single and multiple spike solutions for the Schrodinger equation:

ih∂ψ

∂t= − h2

2m∆ψ + V (x)ψ − γ|ψ|p−1ψ, (1.3)

where γ > 0, p > 1 and ψ : RN → C. Looking for standing waves of (1.3), i.e. solutions of

the form ψ(x, t) = exp(−iEt/h)v(x), the equation for v becomes

− h2∆v + V (x)v − |v|p−1v = 0 (1.4)

where we have assumed γ = 2m = 1 and the parameter E has been absorbed by V. The firstresult in this line, at our knowledge, is due to Floer and Weinstein ([14]). These authorsconsidered the one-dimentional case and constructed for small h > 0 such a concentratingfamily via a Lyapunov-Schmidt reduction around any nondegenerate critical point of thepotential V , under the condition that V is bounded and p = 3. In [18] and [19] Oh generalizedthis result to higher dimensions when 1 < p < N+2

N−2 (N ≥ 3) and V exhibits “mild oscillations”at infinity. Variational methods based on variants of Mountain-Pass Lemma are used in [20]to get existence results for (1.4) where V lies in some class of highly oscillatory V ’s whichare not allowed in [18]-[19]. Under the condition lim inf |x|→+∞ V (x) > infx∈RN V (x) in [21]Wang established that these mountain-pass solutions concentrate at global minimum points

2

of V as h → 0+; moreover a point at which a sequence of solutions concentrate must becritical for V. This line of research has been extensively pursued in a set of papers by DelPino and Felmer ([11]-[13]). We also recall the nonlinear finite dimensional reduction usedin [1] and a recent paper by Grossi ([15]). The most complete and general results for thiskind of problems seem due to Del Pino and Felmer ([12]) and Li ([16]).

In this work we use variational methods relying on topological tools and permitting toobtain good localization results under relatively minimal assumptions. Throughout this paperwe always assume the following hypotheses

a) V ∈ C1(RN ,R) and infx∈RN V (x) > 0;

b) W ∈ C2(Ω, R) where Ω = RN+1 \ ξ for some ξ with |ξ| = 1;

c) W (ξ) ≥W (0) = 0 for all ξ ∈ Ω;

d) there exist c, r > 0 such that

|ξ| < r ⇒W (ξ + ξ) > c|ξ|−q

where1q

=1N− 1p, N ≥ 2, p > N ;

e) there exists ε ∈ (0, 1) such that for every ξ ∈ Ω with |ξ| ≤ ε :

W (ξ) ≤ infW (η) | η ∈ Ω, |η| > ε;

f) for every ξ, η ∈ Ω :|ξ| ≤ |η| ≤ ε⇒W (ξ) ≤W (η).

For N = 3 and p = 6 (hence q = 6) a simple function W which satisfies assumptions b),c) and d) is the following

W (ξ) =|ξ|2

|ξ − ξ|6.

We observe that no restriction on the global behaviour of V is required other than a). In thispaper we look for bound states, namely solutions with bounded energy; under the regularityassumption on V and W it is standard to check that the weak solutions of (1.2) correspondto the nontrivial critical points for the associated energy functional

Eh(u) =∫

RN

(12

(|∇u|2 + Vh(x)|u|2

)+

1p|∇u|p +W (u)

)dx. (1.5)

The basic idea of our arguments is simple: to get concentrated solutions we consider theopen set of the functions that are small away from a local minimum (or a finite set oflocal minima) of V and prove the existence of a minimum of the energy functional Eh inthis open set for small h > 0. Similar arguments work for both isolated and non isolatedminima with just only a change of technical devices (see theorem 5.2 and 6.2). Howeverthe process is technically delicate since it needs, to be pursued, a detailed analysis of thebehaviour of minimizing sequences with exact estimates of the value of the energy functional,developing methods based on a version of the concentration-compactness principle ([17]). Wethink to have achieved in [2] and in this work satisfactory results as regards concentrationnear local and global minimum points of V. The problem of finding solutions concentrated

3

around critical points that are not minima remains open. We expect that the minimizationtechniques we used in [2] and in this paper do not work. However we guess that also themethods developed for nonlinear Schrodinger equation like (1.3), such as the finite dimentionalreduction of [1], [14], [15] and [16], or the penalization of the functional as in [11], [12] and [13],cannot be immediately applied. Probably the approach has to be changed.

This paper is organized as follows: section 2 is devoted to the description of the abstractsetting, i.e. the functional set in which the energy functional Eh is defined. In section 3 and 4we provide the topological devices we need for our arguments. Section 5 and 6 are concernedwith the proof of the existence of one-peak and multi-peak solutions in the semiclassical limit.The main results are given in theorem 5.1 and theorem 6.1. Notice that, in order to avoidlengthy technicalities, the proof of the multi-peak result (theorem 6.1) is given just only inthe case of two peaks (i.e. in the case of two local minima of the potential V ) but the samearguments hold for any finite set of peaks.

Notations

We fix the following notations we will use from now on.

• |x| is the Euclidean norm of x ∈ RN . Analogously |A| is the Euclidean norm of a m × n

real matrix A.

• For u : RN → R

N+1, ∇u is the (N + 1) × N real matrix whose rows are given by thegradient of each component function uj .

• For any U ⊂ RN , int(U) is its internal part and U is its closure.

• If x ∈ RN and r > 0, then Br(x) or B(x, r) is the open ball with center in x and radius r.

• For a Banach space H we denote its dual by H ′.

• If H is a Banach space , by F [u] we indicate the duality between F ∈ H ′ and u ∈ H. Inthe same way for a continuous bilinear map a : H ×H → R, we put a[u]2 ≡ a(u, u).

2 Functional Setting

In order to minimize the functional Eh we choose a suitable Banach space: let Hh denote thesubspace of W 1,2(RN ,RN+1) consisting of functions u such that

‖u‖Hh≡(∫

RN

(|∇u|2 + Vh(x)|u|2

)dx

)1/2

+(∫

RN|∇u|p dx

)1/p

< +∞. (2.6)

The space Hh can also be defined as the closure of C∞0 (RN ,RN+1) with respect to the

norm (2.6). The main properties of Hh are summarized in the following lemma.

Lemma 2.1 The following statements hold:i) Hh is continuously imbedded in W 1,2(RN ,RN+1) and W 1,p(RN ,RN+1).ii) There exist two constants C0, C1 > 0 such that, for every u ∈ Hh,

‖u‖L∞ ≤ C0‖u‖Hh

and|u(x)− u(y)| ≤ C1 |x− y|(p−N)/p‖∇u‖Lp ∀x, y ∈ R

N . (2.7)

iii) For every u ∈ Hh

lim|x|→+∞

u(x) = 0 (2.8)

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iv) If un converges weakly in Hh to some function u, then it converges uniformly on everycompact set contained in R

N .

The proof is a direct consequence of the Sobolev embedding theorems.

Notice that from (2.7) we derive the following property we are going to use several timesin the proofs of our results: given uα ⊂ Hh a family of functions verifying ‖∇uα‖Lp ≤ Mfor some M ≥ 0, then there results: for every ε > 0 there exists δ > 0 such that

∀α, ∀x, y ∈ RN : |x− y| ≤ δ ⇒ |uα(x)− uα(y)| ≤ ε ∀α.

We refer to the above property as to the “equi-uniform continuity” of the family uα.

Since the functions in Hh are continuous, we can consider the set

Λh =u ∈ Hh

∣∣∣ ∀x ∈ RN : u(x) 6= ξ

.

By ii) and iii) of Lemma 2.1, it is easy to obtain that Λh is open in Hh. The boundary ofΛh is given by

∂Λh =u ∈ Hh

∣∣∣ ∃x ∈ RN : u(x) = ξ

.

Now we want to give a topological classification of the maps u ∈ Λh. More precisely weintroduce a topological invariant with suitable “localization” properties in the sense that,roughly speaking, it depends on the compact region where u is concentrated. This invariantconsists of an integer number called “topological charge” and it will be defined by means ofthe topological degree.

3 Topological Charge

In this section we take from [9] some crucial definitions and results. In the open set Ω =R

N+1 \ ξ we consider the N-sphere centered at ξ

Σ =ξ ∈ R

N+1∣∣∣ |ξ − ξ| = 1

.

On Σ we take the north and the south pole, denoted by ξN and ξS , with respect to the axisjoining the origin with ξ, i.e., since |ξ| = 1,

ξN = 2ξ, ξS = 0.

Then we consider the projection P : Ω → Σ defined by

∀ ξ ∈ Ω : P (ξ) = ξ +ξ − ξ

|ξ − ξ|.

Notice that, by definition, it follows:

P (ξ) = 2ξ ⇔ ξ =(1 + |ξ − ξ|

)ξ,

which leads toP (ξ) = 2ξ ⇒ |ξ| > 1 (3.9)

Using the above-mentioned notation we can give the following definition.

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Definition 3.1 Given u ∈ Λh, we define the (topological) charge of u as the following integernumber

ch(u) = deg(P u, int (K(u)) , 2ξ

),

where K(u) is the following compact set

K(u) =x ∈ RN

∣∣∣ |u(x)| > 1

(3.10)

Notice that the choice of the value 1 in definition 3.1 depends on the norm of the singularityξ.

In other words, the topological charge is the Brouwer topological degree of P u in theset K(u) with respect to the north pole of Σ, and it is well defined thanks to (2.8) and (3.9).The following lemma shows how the topological charge is a “global” invariant.

Lemma 3.1 For every u ∈ Λh and for every R > 0 such that K(u) ⊂ BR(0) :

ch(u) = deg(P u, BR(0), 2ξ

).

The proof is the same as in Proposition 3.3 of [9] and is based on the excision property ofthe topological degree.

From well known properties of the topological degree we get other useful properties of thetopological charge. For example notice that if K(u) consists of m connected componentsK1, ...,Km, i.e. u has the energy concentrated in different regions of the space, and if we put

ch (u, Kj) = deg(P u, int(Kj), 2ξ

),

then by the additivity property of the degree we get

ch(u) =m∑

j=1

ch (u, Kj) . (3.11)

Another consequence is that the topological charge is stable under uniform convergence, asthe following lemma states.

Lemma 3.2 For every u ∈ Λh there exists r = r(u) > 0 such that, for every v ∈ Λh

‖u− v‖L∞ ≤ r ⇒ ch(u) = ch(v).

The proof can be found in [2], lemma 3.2.

Finally we setΛ∗

h =u ∈ Λh

∣∣∣ ch(u) 6= 0.

By lemma 3.2 Λ∗h is open in Hh and by (3.9) we deduce

∀u ∈ Λ∗h : ‖u‖L∞ > 1.

Our aim is to minimize the functional Eh defined by (1.5) in some subset of the class Λ∗h in

order to obtain the existence of solutions for (1.2) in the set of the fields u with nontrivialcharge.

6

4 The Energy Functional

Now we are going to study the properties of the functional Eh; first of all we want to showthat it is well defined in the space Λh, i.e. for every u ∈ Λh we have

Eh(u) < +∞. (4.12)

Indeed, by hypothesis b), the Taylor expansion of the function W near 0 gives

W (ξ) = W ′′(0)[ξ]2 + o(|ξ|2)

with lim|ξ|→0o(|ξ|2)|ξ|2 = 0. Then there exist %, c, ε > 0 such that, for all ξ ∈ R

N+1,

|W ′′(0)[ξ]2| ≤ c|ξ|2, |ξ| ≤ %⇒ |o(|ξ|2)| ≤ ε|ξ|2

which implies ∫RN

W (u) dx ≤ c

∫RN

|u|2 dx+ ε

∫|u|≤%

|u|2 dx+∫|u|>%

o(|u|2) dx.

By (2.8) the set |u| > % has compact closure, while o(|u|2) is continuous, then we ob-tain (4.12). Obviously Eh is bounded from below and is coercive in the Hh-norm:

lim‖u‖Hh

→+∞Eh(u) = +∞. (4.13)

Moreover in [2] we have proved that the energy functional Eh belongs to the class C1(Λh,R)under the assumptions a)-d). An immediate corollary is that the critical points (particularlythe minima) u ∈ Λh for the functional Eh are weak solutions of equation (1.2).

The next three propositions deal with some other properties of the functional Eh; we omitthe proofs because they are the same as in [9], provided that we substitute “Eh” for “E” and“Λh” for “Λ”. The first deals with the behaviour of Eh when u approaches the boundary ofΛh.

Proposition 4.1 Let un ⊂ Λh be bounded in the Hh-norm and weakly convergent to u ∈∂Λh, then ∫

RNW (un) dx→ +∞ as n→ +∞.

(See [9], Lemma 3.7, pg. 326)

The second proposition follows from the previous and assures a useful compactness prop-erty for the set Λh.

Proposition 4.2 Let un ⊂ Λh be weakly converging to u and such that Eh(un) is bounded,then u ∈ Λh.

(See [9], Proposition 3.8, pg. 327)

Finally the last result states the lower semicontinuity of the energy functional Eh.

Proposition 4.3 For every u ∈ Λh and for every sequence un ⊂ Λh, if un weaklyconverges to u, then

lim infn→+∞

Eh(un) ≥ Eh(u).

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(See [9], proposition 3.10, pg. 328)

Before going on with the statements of our resuts we fix some notations. In what followswe’ll consider two points x0, x1 ∈ R

N with properties specified in each situation; then weintroduce the sets

H = W 1, 2(RN , RN+1) ∩W 1, p(RN , R

N+1)

andΛ =

u ∈ H

∣∣∣ ∀x ∈ RN : u(x) 6= ξ

and the following two functionals defined in Λ :

E0(u) =∫

RN

(12

(|∇u|2 + V0|u|2

)+

1p|∇u|p +W (u)

)dx,

E1(u) =∫

RN

(12

(|∇u|2 + V1|u|2

)+

1p|∇u|p +W (u)

)dx,

where we have putV0 ≡ V (x0) and V1 ≡ V (x1).

E0 and E1 are the functionals associated with equation (1.2) replacing Vh respectively by V0

and V1. Since these equations are translation invariant, if we define the set

Λ∗ = u ∈ Λ | ch(u) 6= 0

the splitting lemma (lemma 4.1 of [9]) is verified for both functionals. Let us now define:

E∗0 = inf

u∈Λ∗E0(u), E∗

1 = infu∈Λ∗

E1(u).

By theorem 4.2 in [9] we deduce that both the minima E∗0 and E∗

1 are attained. We call theminimizing functions respectively u0 and u1, namely

u0, u1 ∈ Λ∗, E0(u0) = E∗0 , E1(u1) = E∗

1 . (4.14)

Now we are able to provide in next section the first existence and concentration result of thiswork.

5 Concentration Near One Local Minimum: One-Peak Solu-tions

In this section we solve the problem of finding a family of solutions vh for equation (1.1)exhibiting a concentration behaviour around a local minimum point x0 of V. First assume x0

be an isolated minimum; therefore let r0 > 0 such that

∀x ∈ B(x0, r0) \ x0 : V (x) > V (x0) ≡ V0. (5.15)

In [2] under hypotheses a)−f) and moreover lim inf |x|→+∞ V (x) > infx∈RN V (x) we haveproved the existence of solutions vh of equation (1.1) for small h > 0 which exhibit a concen-tration behaviour around an absolute minimum of V ; these solutions have been obtained asminima of the energy functional Eh in the open set Λ∗

h. This result suggests that solutionssatisfying our current requirements must be search as local minima for the functional Eh in

8

some open subset of Λh smaller than Λ∗h. The idea is to define the set in which it is convenient

to minimize Eh by localizing in some sense the region in which the functions are concentrated.More precisely for every h > 0 we set

Λ∗h =

u ∈ Λ∗

h

∣∣∣∣∣ ∀x ∈ RN \B

(x0

h,r0h

): |u(x)| < 1

.

The continuous immersion Hh ⊂ L∞ assures that Λ∗h is open in Λh. Then we define

E∗h = inf

u∈Λ∗h

Eh(u).

The object is to show that Λ∗h is the desired open set. To this aim we need the estimate

provided by the following lemma.

Lemma 5.1 lim suph→0+ E∗h ≤ E∗

0

Proof. Fix R > 0 with K(u0) ⊂ BR(0), where u0 has been defined in (4.14) and K(u)in (3.10). Now take a ϕR ∈ C∞

0 (RN ,R) such that:

ϕR ≡ 1 in BR(0), ϕR ≡ 0 in RN \BR+1(0), 0 ≤ ϕ ≤ 1, |∇ϕR| ≤ c

where c is independent of R. Let uR = ϕRu0 and w(x) = uR

(x− x0

h

). It is obvious that,

provided h is small enough as to verify r0h > R+ 1, there results: w ∈ Λ∗

h. Now observe:

E∗h ≤ Eh(w) = E0(w) +

12

∫RN

(Vh(x)− V0) |w|2 dx =

= E0(uR) +12

∫RN

(V (hx+ x0)− V0) |uR|2 dx. (5.16)

Since Vh converges to V (0) uniformly on compact sets as h→ 0+, then, if we take h sufficientlysmall, the integral in (5.16) is close to zero. This implies

lim suph→0+

E∗h ≤ E0(uR).

Obviously E0(uR) → E0(u0) ≡ E∗0 as R→ +∞. Thus

lim suph→0+

E∗h ≤ E∗

0

and the conclusion follows. 2

Now we are able to prove the main result of this section.

Theorem 5.1 Assume that a)−f) hold. Then there exists h0 > 0 such that the minimumE∗

h is attained in the open set Λ∗h for all h ∈ (0, h0). Furthermore if for every h ∈ (0, h0) we

setvh(x) = uh

(x

h

)(5.17)

where uh ∈ Λ∗h is the minimizing function for E∗

h, then vh is a solution of (1.1) and the familyvh concentrates at the local minimum x0 in the following sense: vh has at least one localmaximum point xh with |vh(xh)| > 1 and xh → x0 as h→ 0+; also, for every δ > 0, it holds:

vh → 0 as h→ 0+ uniformly in the set |x− x0| ≥ δ.

9

Proof. Fix h > 0 arbitrarily and consider uhk a minimizing sequence in Λ∗

h; it has obviouslybounded energy because of (4.13). Then up to subsequence we have

uhk uh weakly in Hh as k → +∞.

Proposition 4.2 implies uh ∈ Λh. Observe that by the definition of Λ∗h it follows:

ch(uhk) = ch

(uh

k , B

(x0

h,r0h

)).

Since uhk → uh uniformly on compact sets, we deduce

|uh(x)| ≤ 1 ∀x ∈ RN \B

(x0

h,r0h

)(5.18)

which implies

ch(uh) = ch

(uh, B

(x0

h,r0h

)).

Now, by using the continuity of the topological charge with respect to the uniform conver-gence, we easily infer that the sequence ch(uh

k) is definitively constant and moreover, for klarge enough,

ch(uh) = ch(uhk) 6= 0,

namely uh ∈ Λ∗h. It remains to prove that, at least for small h > 0, uh is the desired

family of minimizing functions and that the family vh related to uh by (5.17) satisfiesthe concentration property announced in the theorem.From now on we focus our attention on a generic sequence hn → 0+.For sake of clarity we divide the remaining argument into 7 Steps.

Step 1. There exist l ∈ N, R1, ...., Rl > 0, l sequences of points x1hn, ...., xl

hn ⊂ R

N andu1, ...., ul ∈ Λ such that, up to subsequence,

|uhn(xihn

)| > 1 ∀n ∈ N, ∀ i = 1, ..., l; (5.19)

xihn

is a local maximum point for uhn ; (5.20)

|xihn− xj

hn| → +∞ as n→ +∞ for i 6= j; (5.21)

uhn(·+ xihn

) ui weakly in H as n→ +∞; (5.22)

∀x ∈ RN \

(l⋃

i=1

BRi(xihn

)

): |uhn(x)| ≤ 1; (5.23)

ch(uhn) =l∑

i=i

ch(ui). (5.24)

The proof is an easy consequence of lemma 4.1 of [9]. Indeed we repeat the same argumentfor the sequence uhn and the functional

Einf (u) =∫

RN

(12

(|∇u|2 +

(inf

x∈RNV (x)

)|u|2

)+

1p|∇u|p +W (u)

)dx.

Obviously Einf is defined in Λ and is translation-invariant. Notice that uhn ⊂ Λ∗ and fromproposition 4.3 we have

Einf (uhn) ≤ Ehn(uhn) ≤ lim infk→+∞

Ehn(uhnk ) = E∗

hn, (5.25)

10

hence lemma 5.1 assures that Einf (uhn) is bounded. So all the hypotheses of the quotedlemma are satisfied.

Before going on with Step 2 observe that, since ch(uhn) 6= 0, from (5.24) we deduce thatthere is i = 1, ..., l, for semplicity we mean i = 1, such that ch(u1) 6= 0.

Now for every n ∈ N let

whn(x) = uhn(x+ x1hn

) = vhn(hn x+ hn x1hn

).

In order to simplify the notation we put w0 ≡ u1 and, according to (5.22), we obtain

whn w0 weakly in H as n→ +∞,

w0 ∈ Λ∗.

In what follows we’ll assume, without loss of generality,

whn → w0 a.e. in RN as n→ +∞.

Step 2. Up to subsequence, there results: hn x1hn→ x0 as n→ +∞.

Combining (5.19) with the definition of Λ∗hn

it immediately follows:

x1hn∈ B

(x0

hn,r0hn

)∀n ∈ N

by which the sequence hn x1hn is obviously bounded in R

N . Then, setting xhn ≡ hn x1hn,

there exists a point x ∈ RN such that, up to subsequence,

xhn → x as n→ +∞.

We claim that x ≡ x0. First of all observe that Fatou’s lemma implies

lim infn→+∞

∫RN

V (hnx+ xhn)|whn |2 dx ≥∫

RNV (x)|w0|2 dx,

andlim infn→+∞

∫RN

W (whn) dx ≥∫

RNW (w0) dx.

Now lemma 5.1, proposition 4.3 and the second inequality in (5.25) yield

E∗0 ≥ lim sup

n→+∞E∗

hn≥ lim sup

n→+∞Ehn(uhn) =

= lim supn→+∞

∫RN

(12

(|∇whn |2 + V (hnx+ xhn)|whn |2

)+

1p|∇whn |p +W (whn)

)dx ≥

≥∫

RN

(12

(|∇w0|2 + V (x)|w0|2

)+

1p|∇w0|p +W (w0)

)dx. (5.26)

On the other hand, since x1hn ⊂ B(x0, r0), it holds x ∈ B(x0, r0). Suppose by contradiction

that x 6= x0. Then the choice of r0 in (5.15) would imply: V (x) > V0 ≡ V (x0). So by (5.26)we would obtain:

E∗0 >

∫RN

(12

(|∇w0|2 + V0|w0|2

)+

1p|∇w0|p +W (w0)

)dx.

11

Since k is arbitrary, this would mean that whn is not bounded in L2 which is a contradiction:Step 3 is proved.

Step 4. For every η > 0 there exists Rη > R1 such that, eventually passing to a subsequence,

|whn(x)| ≤ η ∀x ∈ RN with |x| ≥ Rη.

Fix 0 < η < ε (ε is given in hypothesis e)) arbitrarily and consider Rη > R1 as in theclaim of Step 3. For sake of semplicity we continue to denote by whn(x) the subsequenceverifying the thesis of last step. Then we have to prove that the following stronger version ofStep 3 holds: for large n

|whn(x)| ≤ η ∀x ∈ RN \BRη(0). (5.29)

By (5.23) straightforward calculation shows

|whn(x)| ≤ 1 ∀x ∈ RN \(BR1(0) ∪

l⋃i=2

BRi

(xi

hn− x1

hn

))(5.30)

and, by (5.21), taking into account that whn → w0 a.e. in RN ,

|w0(x)| ≤ 1 ∀x ∈ RN \BR1(0).

Since w0 ∈ Λ∗ and since the topological degree is continuous with respect to the uniformconvergence, we can assume

ch (whn , BR1(0)) 6= 0 ∀n ∈ N . (5.31)

In order to simplify the notation, for all ε > 0 we consider the function

Tε : RN+1 → RN+1 (5.32)

defined by

Tε(z) =

sign z1 min |z1| , ε

....

....sign zN+1 min |zN+1| , ε

∀ z =

z1......

zN+1

∈ RN+1.

Now suppose that (5.29) is not verified for some n ∈ N large enough as to satisfy, accordingto (5.21),

BRη(x1hn

) ∩BRi(xihn

) = ∅ ∀ i = 2, ..., l,

so that, because of (5.23) and (5.31),

ch(whn

, BRη(0))

= ch(whn

, BR1(0))6= 0. (5.33)

As (5.29) is not verified, there is a point zhnwith |zhn

| > Rη and |whn(zhn

)| > η. By continuity,we can find two positive numbers ρ, ν such that

|whn(z)|2 > η2 + 3 ν ∀ z ∈ B(zhn

, ρ), ρ < |zhn| −Rη,

√η2 + ν ≤ ε.

13

Remember that whn= uhn

(· + x1hn

) where uhnis the weak limit in Hhn

of a minimizing

sequence in Λ∗hn. Denote such minimizing sequence by u

hnk . Since uhn

k → uhnuniformly on

compact sets we have that for k sufficiently large

|uhnk (z + x1

hn)|2 > η2 + 2 ν ∀ z ∈ B(zhn

, ρ)

and, by Step 3,

|uhnk (x+ x1

hn)|2 ≤ η2 + ν ∀x ∈ RN with |x| = Rη.

Now consider the function sequence

unk =

u

hnk in BRη(x1

hn),

T√η2+ν

(u

hnk

)in RN \BRη(x1

hn).

By (5.33) and by construction it’s immediate to prove that for large k there results unk ∈ Λ∗

hn,

precisely ch(unk) = ch

(un

k , BRη(x1hn

))

= ch(whn

, BRη(0))6= 0. Furthermore

|unk(x)| ≤ |uhn

k (x)| < 1 ∀x ∈ RN \B(x0

hn,r0hn

)and this implies un

k ∈ Λ∗hn

definitively. The object is to show that for k sufficiently large

Ehn(un

k) ≤ Ehn(uhn

k )− γ (5.34)

for some γ > 0, in contradiction with the minimizing property of uhnk . This contradiction

will give the thesis.So it remains to prove (5.34). From now up to the end of Step 4 we’ll tacitly consider

indexes k large. Notice that, since B(zhn, ρ) ⊂ RN \B(0, Rη) we have∫

B(zhn,ρ)V(hn

(x+ x1

hn

))|un

k(·+ x1hn

)|2 dx ≤(η2 + ν

) ∫B(zhn

,ρ)V(hn

(x+ x1

hn

))dx ≤

≤∫

B(zhn,ρ)V(hn

(x+ x1

hn

))|uhn

k (·+ x1hn

)|2 dx− ν

∫B(zhn

,ρ)V(hn

(x+ x1

hn

)). (5.35)

Now put γ = ν2

∫B(zhn

,ρ) V(hn

(x+ x1

hn

))dx > 0. Because of hypotheses e)−f) and by

construction∫RN

12|∇un

k |2 +1p|∇un

k |p +W (unk) dx ≤

∫RN

12|∇uhn

k |2 +1p|∇uhn

k |p +W (uhnk ) dx. (5.36)

On the other hand from (5.35) it follows

12

∫RN

Vhn(x)|un

k |2 dx ≤12

∫RN

Vhn(x)|uhn

k |2 dx− γ. (5.37)

Finally combining (5.36) with (5.37) we obtain (5.34).

Step 5. Up to subsequence, whn verifies: for every η > 0 there exists Rη > 0 such that, forlarge n,

|whn(x)| < η ∀x ∈ RN \BRη(0),

14

i.e. whn(x) is arbitrary small provided |x| is sufficiently big and n is large enough.

Remark. We notice that, contrary to last Step, in the claim of Step 5 the subsequence whnis independent of η. This fact will play a very important role in Step 7 when we’ll prove theconcentration result.

Let us fix 0 < ε < ε. Combining Step 3 and Step 4 we deduce that there exists R1ε > R1

sufficiently large so that, eventually passing to a subsequence w1hn,

|w1hn

(x)| ≤ ε ∀x ∈ RN with |x| ≥ R1ε, ∀n ∈ N .

Repeating the same argument for ε2 > 0, there exists R2ε > R1

ε and a subsequencew2

hn

ofw1

hn

verifying

|w2hn

(x)| ≤ ε2 ∀x ∈ RN \BR2ε(0), ∀n ∈ N

and so on for ε3, ε4 etc. Now we apply a diagonal method and consider the sequence

whn = wnhn

∀n ∈ N .

Obviously whn is a subsequence of the original whn. It remains to prove that whnsatisfies the desired property of Step 5. Let us fix η > 0; there is α ∈ N such that εα < η.By definition

|whβ(x)| < η ∀β > α, ∀x ∈ RN \BRα

ε(0),

which is the thesis of Step 5.

Step 6. Up to a subsequence there results: uhn ∈ Λ∗hn.

We already know that for n sufficiently large uhn ∈ Λ∗hn

and

|uhn(x)| ≤ 1 ∀x ∈ RN \B(x0

hn,r0hn

)(5.38)

In order to conclude we want to show that, up to subsequence, inequality (5.38) is strict.Applying Step 4 we deduce the existence of R > 0 such that for a subsequence it holds

|whn(x)| < 1 ∀x ∈ RN \BR(0).

Now fix δ < r0. Since hnx1hn→ x0 as n→ +∞ we can choose n sufficiently large so that for

every n > n the following two inequalities are verified:∣∣∣∣x1hn− x0

hn

∣∣∣∣ < δ

hnand

r0 − δ

hn> R. (5.39)

We claim that for every n > n it is: uhn ∈ Λ∗hn. Indeed take n > n and x ∈ RN with∣∣∣x− x0

hn

∣∣∣ ≥ r0hn

; from inequalities (5.39) it follows

|x− x1hn| ≥

∣∣∣∣x− x0

hn

∣∣∣∣− ∣∣∣∣x0

hn− x1

hn

∣∣∣∣ > r0hn

− δ

hn> R

by which we conclude

∀n > n, ∀x ∈ RN \B(x0

hn,r0hn

): |uhn(x)| = |whn(x− x1

hn)| < 1

15

which is the thesis.

Step 7. End of the proof.

Up to now we have proved that, for a generic sequence hn → 0+, up to subsequence wehave uhn ∈ Λ∗

hn. But uhn is the weak limit of a minimizing sequence in Λ∗

hn. Then the weakly

lower semicontinuity of the energy functional Ehn implies

Ehn(uhn) = E∗hn,

i.e. the minimum E∗hn

is attained in Λ∗hn

by uhn . Since the sequence hn is arbitrary, the firstpart of the theorem follows. The second part is a direct consequence of Step 5. Indeed letwhn still denote the subsequence verifying the claim of Step 5; if we put

vhn(x) = uhn

(x

hn

)= whn

(x− x1

hn

hn

)then, taking into account of Step 2, it is easy to prove that vhn decays uniformly for xoutside every fixed neighborhood of x0 as n → +∞. Moreover combining (5.20) togetherwith Step 2 we deduce the existence of at least one local maximum point x1

hnof vhn with

x1hn→ x0 as n→ +∞. Finally the fact that hn is arbitrary allows us to conclude. 2

It is possible to achieve an analogous existence and concentration result for local minimumpoints of V not necessarily isolated. More precisely, given a bounded open A such that

infx∈A

V (x) < infx∈∂A

V (x),

consider the new set

˜Λ∗h =

u ∈ Λ∗

h

∣∣∣∣∣ ∀x ∈ RN \ 1hA : |u(x)| < 1

,

where, with obvious notation,1hA ≡

1hx∣∣∣ x ∈ A .

For every h > 0, ˜Λ∗h is open in Λh; now we define˜

E∗h = inf

u∈˜Λ∗

h

Eh(u).

Theorem 5.1 can be reformulated in the following way.

Theorem 5.2 Under hypotheses a)−f) there exists h0 > 0 such that the minimum ˜E∗h is

attained by a function uh ∈˜Λ∗

h for every h ∈ (0, h0); then each vh related to uh by (5.17) isa solution of (1.1). Furthermore for every sequence hn → 0+ there exists a subsequence, stilldenoted by hn, such that the family vhn concentrates around a minimum of V in A in thefollowing sense: vhn has at least one local maximum point xhn with |vhn(xhn)| > 1 and

xhn → x0 as n→ +∞, x0 ∈ A with V (x0) = infx∈A

V (x).

Also, for every δ > 0, it holds:

vhn → 0 as n→ +∞ uniformly in the set |x− x0| ≥ δ.

The proof is essentially the same of theorem 5.1; the only difference lies in the fact thatdifferent sequences hn → 0+ can bring to different concentration minimum points in A.

16

6 Concentration Near a Finite Set of Local Minima: Multi-Bump Solutions

Throughout this section, if not stated otherwise, we’ll assume x0 and x1 be two local isolatedminima of V. Then we can choose r0, r1 > 0 verifying

∀x ∈ B(x0, r0) \ x0 : V (x) > V (x0) ≡ V0 (6.40)

and∀x ∈ B(x1, r1) \ x1 : V (x) > V (x1) ≡ V1. (6.41)

Obviously it is not restrictive to suppose

|x0 − x1| > r0 + r1

so that the two balls are disjoint. Under these hypotheses it is natural to wonder if there existsolutions vh of equation (1.1) for small h > 0 with a “two-bump” shape, i.e. which concentratenear these two local minima. In order to construct these solutions it is reasonable to look forthem among local minima of the energy functional Eh and so the first step is, as usual, todefine a suitable subset of Λh in which we are going to minimize Eh. As in the case of onelocal minimum, the idea is to consider the space of functions localized in some sense aroundthe minima x0 and x1 : for every h > 0 we define

Λ∗h =

u ∈ Λh

∣∣∣∣∣ ∀x ∈ RN \(B

(x0

h,r0h

)∪B

(x1

h,r1h

)): |u(x)| < 1,

ch

(u, B

(x0

h,r0h

))6= 0, ch

(u, B

(x1

h,r1h

))6= 0

.

Then the functions in Λ∗h not necessarily belong to Λ∗

h (as in the case for Λ∗h) but they have

nontrivial charge localized in B(x0

h ,r0h

)and B

(x1h ,

r1h

)which does not allow them to vanish

in none of these balls. Obviously Λ∗h is open in Λh because of the continuous immersion

Hh ⊂ L∞ and since the topological degree is continuous respect to the uniform convergence.Now for every h > 0 we put

E∗h = inf

u∈Λ∗h

Eh(u).

Before going on we provide the analogous of lemma 5.1.

Lemma 6.1 lim suph→0+ E∗h ≤ E∗

0 + E∗1 .

Proof. As in the proof of lemma 5.1, fix R > 0 with K(u0), K(u1) ⊂ BR(0), where u0 andu1 have been defined in (4.14) and K(u) in (3.10). Now take a ϕR ∈ C∞

0 (RN ,R) such that:

ϕR ≡ 1 in BR(0), ϕR ≡ 0 in RN \BR+1(0), 0 ≤ ϕ ≤ 1, |∇ϕR| ≤ c

where c is independent of R. Let u0R = ϕRu0 and u1

R = ϕRu1. Then put

w0(x) = u0R

(x− x0

h

), w1(x) = u1

R

(x− x1

h

).

It is obvious that, provided h is small enough as to verify

r0h> R+ 1,

r1h> R+ 1,

|x0 − x1|h

> 2(R+ 1),

17

then there results:

ch

(w0, B

(x0

h,r0h

))= ch

(u0, B

(0,r0h

))= ch(u0) 6= 0,

ch

(w1, B

(x1

h,r1h

))= ch

(u1, B

(0,r1h

))= ch(u1) 6= 0

andx ∈ RN |w0(x) 6= 0 ∩ x ∈ RN |w1(x) 6= 0 = ∅

which implies w0 + w1 ∈ Λ∗h. Notice now:

E∗h ≤ Eh(w0 + w1) = E0(w0) + E1(w1)+

+12

∫RN

(Vh(x)− V0) |w0|2 dx+12

∫RN

(Vh(x)− V1) |w1|2 dx = E0(u0R) + E1(u1

R)+

+12

∫RN

(V (hx+ x0)− V0) |u0R|2 dx+

12

∫RN

(V (hx+ x1)− V1) |u1R|2 dx. (6.42)

Since Vh converges to V (0) uniformly on compact sets as h→ 0+ then, if we take h sufficientlysmall, the integrals in (6.42) are close to zero. This implies

lim suph→0+

E∗h ≤ E0(u0

R) + E1(u1R).

Obviously E0(u0R) → E0(u0) ≡ E∗

0 and, analogously, E1(u1R) → E1(u1) ≡ E∗

1 as R → +∞.Thus

lim suph→0+

E∗h ≤ E∗

0 + E∗1

and the conclusion follows. 2

Next theorem gives the desired existence and concentration result. The proof is quitetechnical and lengthy. However the basic ideas are similar to those of theorem 5.1 eventhough the estimates are more delicate. For sake of clarity we’ll follow the same scheme ofthe quoted theorem referring to it for the analogous parts.

Theorem 6.1 Assume that a)−f) hold. Then there exists h0 > 0 such that the minimumE∗

h is attained in the open set Λ∗h for all h ∈ (0, h0). Furthermore if for every h ∈ (0, h0) we

setvh(x) = uh

(x

h

)(6.43)

where uh ∈ Λ∗h is the minimizing function for E∗

h, then vh is a solution of (1.1) and the familyvh concentrates at the local minima x0 and x1 in the following sense: vh has at least twolocal maximum points x0

h and x1h with |vh(x0

h)| > 1, |vh(x1h)| > 1 and x0

h → x0, x1h → x1 as

h→ 0+; also, for every δ > 0, it holds:

vh → 0 as h→ 0+ uniformly in the set |x− x0| ≥ δ ∩ |x− x1| ≥ δ.

Proof. Let us fix h > 0 and consider uhk a minimizing sequence in Λ∗

h; it has obviously boundedenergy. Then up to subsequence we have

uhk uh weakly in Hh as k → +∞.

18

Proposition 4.2 implies uh ∈ Λh. Since uhk → uh uniformly on compact sets, we deduce

|uh(x)| ≤ 1 ∀x ∈ RN \(B

(x0

h,r0h

)∪B

(x1

h,r1h

)). (6.44)

Now, using the continuity of the topological charge with respect to the uniform convergence,we easily infer that the sequences

ch(uh

k , B(x0

h ,r0h

))and

ch(uh

k , B(x1

h ,r1h

))are con-

stant for large k′s and moreover

ch

(uh, B

(x0

h,r0h

))= ch

(uh

k , B

(x0

h,r0h

))6= 0, (6.45)

andch

(uh, B

(x1

h,r1h

))= ch

(uh

k , B

(x1

h,r1h

))6= 0. (6.46)

Furthermore the weakly lower semicontinuity of the energy functional Eh implies

Eh(uh) ≤ lim infk→+∞

Eh(uhk) = E∗

h. (6.47)

It remains to prove that, at least for small h > 0, uh belongs to the class Λ∗h and that the

family vh related to uh by (6.43) satisfies the concentration property announced in thetheorem.

From now on we focus our attention on a generic sequence hn → 0+.For sake of clarity we divide the remaining argument into 8 Steps.

Step 1. There exist l ∈ N , R1, ...., Rl > 0, l sequences of points x1hn, ...., xl

hn ⊂ RN and

u1, ...., ul ∈ Λ such that, up to subsequence,

|uhn(xihn

)| > 1 ∀n ∈ N , ∀ i = 1, ..., l; (6.48)

xihn

is a local maximum point for uhn ; (6.49)

|xihn− xj

hn| → +∞ as n→ +∞ for i 6= j; (6.50)

uhn(·+ xihn

) ui weakly in H as n→ +∞; (6.51)

∀x ∈ RN \(

l⋃i=1

BRi(xihn

)

): |uhn(x)| ≤ 1. (6.52)

The proof is analogous to that of Step 1 in theorem 5.1. The only difference lies in thefact that the sequence uhn in general is not contained in Λ∗, but it is immediate to provethat the splitting lemma in [9] also works for sequences in Λ with L∞-norm bigger than one,which is exactly our case. Notice that from (6.47) we have

Einf(uhn) ≤ Ehn(uhn) ≤ lim infk→+∞

Ehn(uhnk ) = E∗

hn.

Hence lemma 6.1 assures that Einf (uhn) is bounded, which is enough to conclude.

From (6.44) and (6.48) we deduce that for each n ∈ N there results

xihn∈ B

(x0

hn,r0hn

)∪B

(x1

hn,r1hn

)∀i ∈ 1, ..., l.

19

Furthermore the definition of Λ∗h implies that each ball contains at least one of the points

xihn. Then, up to a subsequence, we assume that the following partition holds

xihn∈ B

(x0

hn,r0hn

)∀i ∈ 1, ..., l0,

xihn∈ B

(x1

hn,r1hn

)∀i ∈ l0 + 1, ..., l,

where 1 < l0 < l. By (6.51) and (6.52) for every i ∈ 1, ..., l the sequencech(uhn , BRi

(xi

hn

))is constant for large n′s. From (3.11), we obtain

l0∑i=1

ch(uhn , BRi

(xi

hn

))= ch

(uhn , B

(x0

hn,r0hn

)).

In a similar way

l∑i=l0+1

ch(uhn , BRi

(xi

hn

))= ch

(uhn , B

(x1

hn,r1hn

)).

Because of (6.45) and (6.46) we obtain the existence of two indexes i0 ∈ 1, ..., l0, i1 ∈l0 + 1, ..., l so that

ch(uhn , BRi0

(xi0

hn

))6= 0, ch

(uhn , BRi1

(xi1

hn

))6= 0 for large n.

Step 2. There resultsui0 , ui1 ∈ Λ∗,

∀x ∈ RN \BRi0(0) : |ui0(x)| ≤ 1,

∀x ∈ RN \BRi1(0) : |ui1(x)| ≤ 1.

By (6.51) we have uhn(· + xi0hn

) → ui0 uniformly on compact sets as n → +∞; thenfrom (6.50) and (6.52) it follows that

|ui0(x)| ≤ 1 ∀x ∈ RN \BRi0(0)

and, taking into account that the topological degree is continuous with respect to the uniformconvergence, this leads to the following inequality which holds for n large enough:

ch(ui0) = ch(uhn , BRi0

(xi0

hn

))6= 0

i.e. ui0 ∈ Λ∗. The same argument works for ui1 too.

Step 3. Up to subsequence, it holds:

hn xi0hn→ x0, hn x

i1hn→ x1 as n→ +∞

20

We already know that

xi0hn∈ B

(x0

hn,r0hn

), xi1

hn∈ B

(x1

hn,r1hn

)∀n ∈ N ,

by which the sequences hn xi0hn and hn x

i1hn are obviously bounded in RN . Then we deduce

that there exist two points x0, x1 ∈ RN such that, up to subsequence,

hn xi0hn→ x0, hn x

i1hn→ x1 as n→ +∞.

We claim that x0 ≡ x0 and x1 ≡ x1. Hereafter, for sake of semplicity, for every A ⊂ RN wedenote by E

0|A, E1|A and Eh|A, the functionals defined in the same way as E0, E1 and Eh

respectively but with their integrals restricted to the set A. For example if u ∈ Λ we have:

E0|A(u) =

∫A

(12

(|∇u|2 + V0|u|2

)+

1p|∇u|p +W (u)

)dx.

Let us fix η > 0, there exists ρ > maxRi0 , Ri1 such that

E0|RN\Bρ(0)

(ui0) <η

2and E

1|RN\Bρ(0)(ui1) <

η

2.

From (6.50) it follows that the spheres Bρ(xi0hn

) and Bρ(xi1hn

) are disjoint for n large enough,hence we get

lim infn→+∞

Ehn(uhn) ≥ lim infn→+∞

(E

hn|Bρ(xi0hn

)(uhn) + E

hn|Bρ(xi1hn

)(uhn)

)≥

≥ lim infn→+∞

Ehn|Bρ(x

i0hn

)(uhn) + lim inf

n→+∞E

hn|Bρ(xi1hn

)(uhn) =

= lim infn→+∞

(E

0|Bρ(xi0hn

)(uhn) +

12

∫Bρ(x

i0hn

)(Vhn(x)− V0) |uhn |2 dx

)+

+ lim infn→+∞

(E

1|Bρ(xi1hn

)(uhn) +

12

∫Bρ(x

i1hn

)(Vhn(x)− V1) |uhn |2 dx

)=

= lim infn→+∞

(E

0|Bρ(0)(uhn(·+ xi0

hn)) +

12

∫Bρ(0)

(V (hn x+ hn x

i0hn

)− V0

)|uhn(·+ xi0

hn)|2 dx

)+

+ lim infn→+∞

(E

1|Bρ(0)(uhn(·+ xi1

hn)) +

12

∫Bρ(0)

(V (hn x+ hn x

i1hn

)− V1

)|uhn(·+ xi1

hn)|2 dx

)≥

≥ E0|Bρ(0)

(ui0) +12

∫Bρ(0)

(V (x0)− V0) |ui0 |2 dx+

+E1|Bρ(0)

(ui1) +12

∫Bρ(0)

(V (x1)− V1) |ui1 |2 dx ≥

≥ E0(ui0) + E1(ui1)− η +12

∫Bρ(0)

(V (x0)− V0) |ui0 |2 dx+12

∫Bρ(0)

(V (x1)− V1) |ui1 |2 dx ≥

≥ E∗0 + E∗

1 − η +12

∫Bρ(0)

(V (x0)− V0) |ui0 |2 dx+12

∫Bρ(0)

(V (x1)− V1) |ui1 |2 dx. (6.53)

To get the above inequalities we have used the weakly lower semicontinuity of the functionalsE0 and E1 and moreover the result of Step 2. Since x0 ∈ B(x0, r0) and x1 ∈ B(x1, r1), then

21

the two integrals in (6.53) are not negative. Now suppose by contradiction that x0 6= x0 orx1 6= x1; for sake of semplicity assume x0 6= x0. For η → 0 we would get

lim infn→+∞

Ehn(uhn) ≥ E∗0 + E∗

1 +12

∫BRi0

(0)(V (x0)− V0) |ui0 |2 dx.

By Step 2 we deduce that ui0 is not identically zero on BRi0(0). But the choice of r0 in (6.40)

assures that the integral in the second member is positive, and so

lim infn→+∞

Ehn(uhn) > E∗0 + E∗

1 .

On the other hand (6.47) together with lemma 6.1 implies

lim infn→+∞

Ehn(uhn) ≤ lim infn→+∞

E∗hn≤ E∗

0 + E∗1

and a contradiction follows.

Step 4. For every η > 0 there exist Rη > maxi∈1,...,lRi such that, eventually passing to asubsequence,

∀n ∈ N , ∀ i ∈ 1, ..., l ∀x ∈ RN : |x− xihn| = Rη ⇒ |uhn(x)| ≤ η.

We argue by contradiction and assume the existence η > 0 such that, for every R >maxi∈1,...,lRi there exists nR ∈ N verifying

∀n ≥ nR ∃in ∈ 1, ..., l, ∃ zhn ∈ RN with |zhn − xinhn| = R, s.t. |uhn(zhn)| > η.

Let us fix R1 > maxi∈1,...,lRi and take n1 ∈ N such that

∀n ≥ n1 ∃ i1n ∈ 1, ..., l, ∃ z1hn

with |z1hn− x

i1nhn| = R1 : |uhn(z1

hn)| > η.

From (6.50) it follows that, provided n1 is sufficiently large, we can assume

BR1(xi

hn) ∩BR1

(xjhn

) = ∅ ∀n ≥ n1 ∀i, j ∈ 1, ..., l with i 6= j.

Now choose R2 > R1 + 1: there exists n2 ∈ N , n2 > n1, such that

∀n ≥ n2 ∃i2n ∈ 1, ..., l, ∃ z2hn∈ RN with |z2

hn− x

i2nhn| = R2 : |uhn(z2

hn)| > η

and moreover

BR2(xi

hn) ∩BR2

(xjhn

) = ∅ ∀n ≥ n2, ∀ i, j ∈ 1, ..., l with i 6= j.

We easily infer that, for every k ≥ 2, we can choose Rk > Rk−1 + 1 and we obtain theexistence of nk ∈ N , nk > nk−1, such that

∀n ≥ nk ∃ ikn ∈ 1, ..., l, ∃ zkhn∈ RN with |zk

hn− x

iknhn| = Rk : |uhn(zk

hn)| > η.

andBRk

(xihn

) ∩BRk(xj

hn) = ∅ ∀n ≥ nk, ∀ i, j ∈ 1, ..., l with i 6= j.

22

Proceeding as in the proof of (5.27) in Step 3 of theorem 5.1 we deduce

∃ r > 0 such that ∀k ∈ N , ∀n ≥ nk : Br(zkhn

) ⊂x ∈ RN

∣∣∣ |uhn(x)| ≥ η

2

.

Now put r′ = minr, 12. By construction it easily follows that, for all k, k′ ∈ N with k > k′,

it holds

∣∣∣zkhn− zk′

hn

∣∣∣ ≥ ∣∣∣∣xiknhn− x

ik′

nhn

∣∣∣∣− ∣∣∣∣zkhn− x

iknhn

∣∣∣∣− ∣∣∣∣zk′hn− x

ik′

nhn

∣∣∣∣ ≥ 2Rk −Rk −Rk′ ≥ Rk −Rk−1 > 1.

This implies that for all k ∈ N and for all n ≥ nk the balls Br′(zihn

)i=1,...,k are disjoint and

k⋃i=1

Br′(zihn

) ⊂x ∈ RN

∣∣∣ |uhn(x)| ≥ η

2

.

Since k is arbitrary, this would mean that uhn is not bounded in L2 which is a contradiction.

Step 5. For every η > 0 there exists Rη > maxRi0 , Ri1 such that, eventually passing to asubsequence,

|uhn(x)| ≤ η ∀x ∈ RN \(BRη(xi0

hn) ∪BRη(xi1

hn)).

Fix 0 < η < ε and consider Rη > maxi∈1,...,lRi as in the claim of Step 4. For sake ofsemplicity we denote by uhn(x) the subsequence verifying the thesis of last step. Then wehave to prove that the following assertion: for large n

|uhn(x)| ≤ η ∀x ∈ RN \(BRη(xi0

hn) ∪BRη(xi1

hn)). (6.54)

Now suppose that (6.54) is not verified for some n ∈ N large enough as to satisfy the followingassumptions

BRη(xihn

) ∩BRη(xjhn

) = ∅ ∀ i, j ∈ 1, ..., l with i 6= j, (6.55)

B(xi0hn, Rη) ⊂ B

(x0

hn,r0hn

),

⋃i∈I1

B(xi1hn, Rη) ⊂ B

(x1

hn,r1hn

). (6.56)

Assertion (6.55) derives from (6.50). The inclusions of (6.56) are obviously satisfied for nlarge because of Step 3.

We argue by contradiction. If (6.54) does not hold for n, we can take a point zhnwith

|zhn− xi0

hn| > Rη and |zhn

− xi1hn| > Rη such that |uhn

(zhn)| > η. Then, by the continuity of

uhn, there exist two positive numbers ρ, ν such that

|uhn(z)|2 > η2 + 3 ν ∀ z ∈ B(zhn

, ρ),√η2 + ν ≤ ε,

ρ < min(|zhn

− xi0hn| −Rη

),(|zhn

− xi1hn| −Rη

).

Remember that uhnis the weak limit in Hhn

of a minimizing sequence in Λ∗hn. Denote such

minimizing sequence by uhnk . Since uhn

k → uhnuniformly on compact sets we have that for k

large enough|uhn

k (z)|2 > η2 + 2 ν ∀ z ∈ B(zhn, ρ)

23

and, because of Step 4,

|uhnk (x)|2 ≤ η2 + ν ∀x ∈

x ∈ RN

∣∣∣ |x− xi0hn| = Rη

∪x ∈ RN

∣∣∣ |x− xi0hn| = Rη

.

Now consider the function sequence

unk =

u

hnk in BRη(xi0

hn) ∪BRη(xi1

hn),

T√η2+ν

(u

hnk

)in RN \

(BRη(xi0

hn) ∪BRη(xi1

hn)),

where Tε is the function defined in (5.32). The object is to prove that for large k thereresults un

k ∈ Λ∗hn

. Indeed (6.55), (6.56), the construction above and the fact that uhnk → uhn

uniformly on compact sets yield

ch

(un

k , B

(x0

hn,r0hn

))= ch

(un

k , BRη

(xi0

hn

))= ch

(u

hnk , BRη

(xi0

hn

))= ch

(uhn

, BRη

(xi0

hn

))6= 0.

In the same way

ch

(un

k , B

(x1

hn,r1hn

))6= 0.

Furthermore

|unk(x)| ≤ |uhn

k (x)| < 1 ∀x ∈ RN \(B

(x0

hn,r0hn

)∪B

(x1

hn,r1hn

)).

Then for large k it is: unk ∈ Λ∗

hn. Now we want to show that for k sufficiently large

Ehn(un

k) ≤ Ehn(uhn

k )− γ (6.57)

for some suitable γ > 0, in contradiction with the minimizing property of unk. This con-

tradiction will give the thesis. The proof of (6.57) is exactly the same of (5.34) in Step 4 oftheorem 5.1, and the suitable value of γ is given by

γ =ν

2

∫B(zhn

,ρ)V (hn x) dx > 0.

Step 6. Up to subsequence, uhn verifies: for every η > 0 there exists Rη > maxRi0 , Ri1such that, for large n,

|uhn(x)| < η ∀x ∈ RN \(BRη(xi0

hn) ∪BRη(xi1

hn)),

i.e. uhn(x) is arbitrary small provided |x − xi0hn| and |x − xi1

hn| are sufficiently big and n is

large enough.

Remark. We notice that, contrary to last Step, in the claim of Step 6 the subsequence uhnis independent of η.

Let us fix ε > 0. Combining Step 4 and Step 5 we deduce that there exists R1ε >

maxRi0 , Ri1 sufficiently large so that, eventually passing to a subsequence uh1,n,

|uh1,n(x)| ≤ ε ∀x ∈ RN \(BR1

ε(xi0

h1,n) ∪BR1

ε(xi1

h1,n))

∀n ∈ N .

24

Repeating the same argument for ε2 > 0, there exists R2ε > R1

ε and a subsequenceuh2,n

of

uh1,n

verifying

|uh2,n(x)| ≤ ε2 ∀x ∈ RN \(BR2

ε(xi0

h2,n) ∪BR2

ε(xi1

h2,n)), ∀n ∈ N

and so on for ε3, ε4 etc. Now we apply a diagonal method and consider the sequence uhn,n,which is obviously a subsequence of the original uhn. It remains to prove that it satisfiesthe desired property of Step 6. Let us fix η > 0; there is α ∈ N such that εα < η. Bydefinition

|uhβ,β(x)| < η ∀β > α, ∀x ∈ RN \

(BRα

ε(xi0

hβ,β) ∪BRα

ε(xi1

hβ,β)),

which is the thesis of Step 6.

Step 7. Up to a subsequence there results: uhn ∈ Λ∗hn.

We already know that for n sufficiently large (6.44), (6.45) and (6.46) are satisfied. In orderto conclude we want to show that, up to subsequence, inequality (6.44) is strict. ApplyingStep 5 we deduce that there exists R > 0 such that, eventually passing to a subsequence,

|uhn(x)| < 1 ∀x ∈ RN \(BR(xi0

hn) ∪BR(xi1

hn)).

Now fix δ < minr0, r1. Since Step 3 holds, we can choose n sufficiently large so that forevery n > n the following two inequalities are verified:∣∣∣∣xi0

hn− x0

hn

∣∣∣∣ < δ

hn,

r0 − δ

hn> R, (6.58)

∣∣∣∣xi1hn− x1

hn

∣∣∣∣ < δ

hn,

r1 − δ

hn> R.

We claim that for every n > n it is: uhn ∈ Λ∗hn. Indeed take n > n and x ∈ RN with∣∣∣x− x0

hn

∣∣∣ ≥ r0hn

and∣∣∣x− x1

hn

∣∣∣ ≥ r1hn

; from inequalities (6.58) it follows that

|x− xi0hn| ≥

∣∣∣∣x− x0

hn

∣∣∣∣− ∣∣∣∣x0

hn− xi0

hn

∣∣∣∣ > r0hn

− δ

hn> R.

In the same way|x− xi1

hn| > R.

by which we conclude

∀n > n, ∀x ∈ RN \(B

(x0

hn,r0hn

)∪B

(x1

hn,r1hn

)): |uhn(x)| < 1

which exactly corresponds to the thesis.

Step 8. End of the proof.

Up to now we have proved that, considered a generic sequence hn → 0+, up to subsequencewe have uhn ∈ Λ∗

hn. But uhn is the weak limit of a minimizing sequence in Λ∗

hn. Then the

weakly lower semicontinuity of the energy functional Ehn implies

Ehn(uhn) = E∗hn,

25

i.e. the minimum E∗hn

is attained in Λ∗hn

by uhn . Since the sequence hn is arbitrary, the firstpart of the theorem follows.The second part is a direct consequence of Step 6. Indeed let uhn still denote the subse-quence verifying the claim of Step 6; if we put

vhn(x) = uhn

(x

hn

)then, taking into account of Step 3, it is easy to prove that vhn decays uniformly for xoutside every fixed couple of neighborhoods of x0 and x1 as n→ +∞. Furthermore from (6.49)and Step 3 we get the existence of at least two local maximum points for vhn convergingrespectively to x0 and x1 as n → +∞. Once more the fact that hn is arbitrary allows us toconclude. 2

The analogous of theorem 5.2 works in the case of multi-bump solutions too. In factassume that there are two bounded domains A1, A2, such that

infx∈Ai

V (x) < infx∈∂Ai

V (x) for i = 1, 2,

and consider the new set

Λ∗h =

u ∈ Λh

∣∣∣∣∣ ∀x ∈ RN \(

1hA1 ∪

1hA2

): |u(x)| < 1, ch

(u,

1hA1

)6= 0, ch

(u,

1hA2

)6= 0

.

Λ∗h is open in Λh; then we define

E∗h = inf

u∈Λ∗

h

Eh(u).

The new formulation of theorem 6.1 is the following.

Theorem 6.2 Under hypotheses a)−f) there exists h0 > 0 such that the minimum E∗h is

attained by a function uh ∈Λ∗

h for every h ∈ (0, h0); then each vh related to uh by (6.43) isa solution of (1.1). Furthermore for every sequence hn → 0+ there exists a subsequence, stilldenoted by hn, such that the family vhn concentrates around two minima of V in A1 andA2 respectively. In other words vhn has at least two local maximum points x1

hnand x2

hnwith

|vhn(x0hn

)| > 1, |vhn(x0hn

)| > 1 and

x1hn→ x1 as n→ +∞, x1 ∈ A1, V (x1) = inf

x∈A1

V (x),

andx2

hn→ x2 as n→ +∞, x2 ∈ A2, V (x2) = inf

x∈A2

V (x).

Also, for every δ > 0, it holds:

vhn → 0 as n→ +∞ uniformly in the set |x− x1| ≥ δ ∩ |x− x2| ≥ δ.

Remark 6.1 If the search of two-bump solutions of equation (1.1) for small h > 0 differsfrom the one-bump case in some technical aspects, however the passage from the two-bumpto the more general k-bump case is immediate and produce no other difficulties then thosedue to the inevitable heavy notation.

26

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Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions

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Transcript of Semiclassical limit for a quasilinear elliptic field equation: one-peak and multipeak solutions