Multiple solutions for a class of elliptic equations with jumping nonlinearities

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www.elsevier.com/locate/anihpc

Multiple solutions for a class of elliptic equations with jumping nonlinearities

^✩

Riccardo Molle

^a,^∗

, Donato Passaseo

^b

aDipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy bDipartimento di Matematica “E. De Giorgi”, Università di Lecce, PO Box 193, 73100 Lecce, Italy

Received 15 May 2009; received in revised form 17 September 2009; accepted 17 September 2009 Available online 30 September 2009

Dedicated to Luce Spennato, in memory of her kindness, goodness and generosity

Abstract

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integerkwe prove that, when a parameter is large enough, there exists a solution which presentskinterior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.

Résumé

Nous considérons un problème de Dirichlet semi-linéaire avec le terme non linéaire qui interfère avec les valeurs propres de l’opérateur linéaire. Avec des méthodes variationnelles, nous montrons que le nombre de solutions est arbitrairement grand pourvu que le nombre de valeurs propres qui interfèrent avec le terme non linéaire soit suffisamment grand. Pour la démonstration nous prouvons que pour toutk∈Nle problème a une solution qui présentekpics quand un paramètre est suffisamment grand. Nous décrivons aussi le comportement asymptotique et la forme de cette solution quand ce paramètre tend à l’infini.

MSC:35J20; 35J60; 35J65

Keywords:Jumping nonlinearities; Multiplicity of solutions; Variational methods

1. Introduction

Several papers have been devoted to study existence and multiplicity of solutions for semilinear elliptic problems of the following type

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

* Corresponding author.

E-mail address:[email protected] (R. Molle).

doi:10.1016/j.anihpc.2009.09.005

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u+g(u)=ξ inΩ,

u=0 on∂Ω, (1.1)

whereΩis a bounded connected domain ofRⁿ,ξ∈L²(Ω)andg:R→Ris a continuous function such that

t→−∞lim g(t )

t =α and lim

t→+∞

g(t )

t =β (1.2)

withαandβ inR. We assume, for example,αβ(the caseαβis similar).

We denote byλ_i(or also byλ_i(Ω)) the eigenvalues of the Laplace operator−inH₀¹(Ω). SinceΩis a connected domain, we haveλ₁< λ₂λ₃. . ..

If there exists some eigenvalueλ_i such thatλ_i ∈ ]α, β[, we say thatg is a jumping nonlinearity and thatλ_i is a jumped eigenvalue. It is well known that, ifg∈C¹(R)andg(t )=λ_i ∀t∈R,∀i∈N, then there exists a unique solution u∈H₀¹(Ω)for every ξ ∈L²(Ω). In fact, in this case one can apply for example a well-known result of Caccioppoli (see [9]).

The situation is very different ifg(t )meets some eigenvalue λ_i, what happens, for example, ifgis a jumping nonlinearity. The first paper concerning this case is due to Ambrosetti and Prodi (see [4]). They consider in [4] the problem

u+g(u)=ξ₀+t e₁ inΩ,

u=0 on∂Ω, (1.3)

whereg∈C²(R),ξ₀∈L²(Ω),t∈Rande₁is a positive eigenfunction corresponding to the first eigenvalueλ₁. Under the assumptions thatg(t ) >0 for allt∈Rand

0< lim

t→−∞g(t ) < λ₁< lim

t→+∞g(t ) < λ₂, (1.4)

they prove that there exists a functiont¯:L²(Ω)→Rsuch that problem (1.3) has exactly two solutions ift >t (ξ¯ ₀), exactly one solution ift= ¯t (ξ₀)and no solution ift <t (ξ¯ ₀).

After the result of Ambrosetti and Prodi, several authors have studied semilinear problems where the nonlinear terms interfere with the spectrum of the linear operator and in particular (especially in the early 1980s) elliptic equations with jumping nonlinearities (see [1–3], [5–8], [10–18,20,21,23–32], [35–38], etc.). They apply in these papers analytical, topological and variational methods and exploit several tools (as topological degree, Morse index, Ry- bakowski index, etc.) in order to describe the right-hand side membersξ for which the problem has solution and to estimate the number of solutions. The literature on this subject is really very extensive and in recent years there has been a new growing interest in these problems (see [8,18]). Here we recall only the following results.

If no eigenvalue of − in H₀¹(Ω) belongs to the interval [α, β], then a well-known theorem of Rabinowitz (see [34]) applies and guarantees that problem (1.1) has at least one solution for everyξ∈L²(Ω).

Ifα < λ1< β, there exists a functiont¯:L²(Ω)→Rsuch that problem (1.3) has at least two solutions ift >t (ξ¯ 0), at least one solution ift= ¯t (ξ0)and no solution ift <t (ξ¯ 0)(see [4,7,3], etc.). Ifα < λ1< λ2< β, there exist at least four solutions of problem (1.3) fort >0 large enough (see [27,38], etc.).

Ifn=1 (i.e.Ω is an interval) andα < λ₁< λ_i < β, then (see [29,12,36]) problem (1.3) has at least 2i distinct solutions fort >0 large enough (indeed, exactly 2isolutions if suitable additional conditions are satisfied). Notice that this result does not hold in the casen >1. In fact, in [14] Dancer considered problem (1.3) withg(u)= −αu⁻+βu⁺ (whereu⁺=max{u,0}andu⁻=u⁺−u) and showed that for everyi2 there exists a smooth bounded domainΩ_i inRⁿ, withn >1, and a functionξ₀∈L²(Ω_i)such that problem (1.3), withΩ=Ω_i, has only four solutions fort >0 large enough even ifα < λ₁(Ω_i) < λ_i(Ω_i) < β.

In the present paper our aim is to show that the number of solutions of a problem with jumping nonlinearity may be arbitrarily large, for any fixed domainΩ, provided the number of jumped eigenvalues is large enough. Therefore, we consider the following problem

u−αu⁻+βu⁺=e₁ inΩ,

u=0 on∂Ω, (1.5)

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where we fixα < λ₁ and letβ→ +∞ (notice that, if in problem (1.3) we replaceu byt uand lett → +∞, we obtain (1.5) as limit problem). We show that, for any fixed domain Ω, the number of distinct solutions tends to infinity asβ→ +∞. In fact, for every positive integer kwe construct, for β >0 large enough, a solutionu_k,β of problem (1.5), which presentskpeaks and converges asβ→ +∞to the solution_α₋^e¹_λ

1 (see also Remark 3.9).

The main result of this paper is presented in the following theorem.

Theorem 1.1.LetΩ be a bounded connected domain ofRⁿ, withn3. Let us fixα < λ1. Then, for every positive integer k there exists β¯k >0 such that for all β >β¯k problem (1.5) has a solutionuk,β satisfying the following properties:

(I) there exist a positive constantr¯andkpointsx₁^β, . . . , x_k^βinΩ, with dist

x^β_i, ∂Ω

> r¯

√β fori=1, . . . , k, and x_i^β−x_j^β 2r¯

√β fori=j, (1.6)

such that, for everyβ >β¯_k,u_k,β(x)0∀x∈Ω\_k

i=1B(x^β_i,√^r^¯

β)whileu⁺_k,β≡0inB(x_i^β,√^r^¯

β)fori=1, . . . , k;

(II) the pointsx₁^β, . . . , x_k^β satisfy, in addition,

β→+∞lim e₁ x_i^β

=max

Ω e₁ fori=1, . . . , k, and lim

β→+∞

βx_i^β−x^β_j= ∞ fori=j; (III) for everyβ >β¯_k, we haveu_k,β(x) >_α^e¹₋^(x)_λ

1 ∀x∈Ω and

β→+∞lim sup

u_k,β(x)− e₁(x) α−λ1

: x∈Ω\ k i=1

B x_i^β, ρ

=0 ∀ρ >0; (1.7)

(IV) for everyi=1, . . . , k, the functionsU

k,β,x_i^β, defined in√

β(Ω−x_i^β)by

Uk,β,x_i^β(x)=uk,β

x

√β +x^β_i

∀x∈ β

Ω−x_i^β ,

converge asβ→ +∞to the nonconstant radial solutionUof the problem

⎧⎨

⎩

U+U⁺=0 inRⁿ,

|xlim|→∞U (x)= 1 α−λ₁max

Ω e1. (1.8)

Moreover, the convergence is uniform on the compact subsets ofRⁿ.

Remark 1.2. It is clear that it follows from property (I)that the number of distinct solutions tends to infinity as β→ +∞. From property(II)we infer that the peaks concentrate near the maximum points ofe₁and, if the distance between two peaks tends to zero asβ→ +∞, the approaching rate is less than the concentration rate. Property(III) shows that the peaks “are based” on the solution _α₋^e¹_λ

1, which is the minimal solution of the problem (as one can easily verify). Finally, property(IV)describes the asymptotic profile of the rescaled peaks. Let us point out that only ifn3 the problem (1.8) has a nontrivial solution (since any bounded super-harmonic function inRⁿwithn <3 is a constant function). In the casesn=1 andn=2 new, more refined, arguments have to be used in order to construct k-peak solutions and describe their asymptotic behaviour (see [33]).

The method we use for the proof of Theorem 1.1 is completely variational. The solutions are obtained as critical points of the functionalf :H₀¹(Ω)→Rdefined by

f (u)=1 2

Ω

|Du|²−α u⁻2

−β u⁺2

dx+

Ω

e₁u dx ∀u∈H₀¹(Ω). (1.9)

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Notice that similar phenomena occur also in other fields of Nonlinear Analysis. For example,k-peak solutions with similar properties as in Theorem 1.1 appear in many superlinear problems such as singular perturbation problems, nonlinear Schrödinger equations, nonlinear scalar field equations, elliptic equations involving critical or supercrit- ical Sobolev exponents, etc. Solutions of this type are usually obtained by using a Lyapunov–Schmidt type finite- dimensional reduction method. In particular, we recall the paper [18] of Dancer and Yan, where they consider the problem

u+ |u|^p=t e₁ inΩ,

u=0 on∂Ω, (1.10)

withp∈ ]1,ⁿ_n⁺₋²₂[(here the nonlinear term satisfies (1.2) withα= −∞andβ= +∞). For every positive integerk, they prove the existence of a k-peak solution of (1.10) fort >0 large enough (thus proving a well-known Lazer–

McKenna conjecture). Concerning superlinear problems of this type, several results have been obtained in the last few years (see for example [19,22,40] and the references therein).

2. Notation and preliminary results

We look for solutionsu∈H₀¹(Ω)of the following type. For everyβ >0, setr_β=^√³^r^¯_β¹ wherer¯₁is the radius of the ball inRⁿfor which the first eigenvalue of the Laplace operator is equal to 1, i.e.

inf

B(0,r¯₁)

|Du|²dx: u∈H₀¹

B(0,r¯₁) ,

B(0,r¯₁)

u²dx=1

=1. (2.1)

For every positive integerk, consider the set Ω_k,β=

(x₁, . . . , x_k)∈Ω^k: |x_i−x_j|2rβ ifi=j, dist(xi, ∂Ω)r_βfori=1, . . . , k

. (2.2)

It is clear that, for every fixedk∈N,Ω_k,β= ∅forβlarge enough and the ballsB(x₁, r_β), . . . , B(x_k, r_β)are pairwise disjoint and included inΩ if(x₁, . . . , x_k)∈Ω_k,β.

We say that a function u∈H₀¹(Ω) is ak-peak function, with respect to the balls B(x₁, r_β), . . . , B(x_k, r_β), if u⁺=_k

i=1u⁺_i where, for everyi=1, . . . , k,u⁺_i is a nonnegative function inH₀¹(Ω)such thatu⁺_i ≡0 andu⁺_i (x)=0

∀x∈Ω\B(x_i, r_β).

One can easily verify that, if a k-peak functionuof this form is a solution of problem (1.5), then for everyi= 1, . . . , kthe functiont→f (u+t u⁺_i )has fort=0 the unique maximum point in the set[−1,+∞[andf(u)[u⁺_i ] =0, that is

Ω

Du⁺_i ²dx−β

Ω

u⁺_i 2

dx+

Ω

e1u⁺_i dx=0. (2.3)

Therefore, it is natural to consider the subsetsViofH₀¹(Ω)consisting of all thek-peak functionsu, with respect to the ballsB(x₁, r_β), . . . , B(x_k, r_β), such thatf(u)[u⁺_i ] =0, and to look for critical points of the functionalf constrained on the subsetsV_i (even if they are not smooth manifolds and this fact gives some more problems when we have to prove that the constrained critical points actually give solutions of (1.5)).

For thek-peak functionsuof the form described above we use also constraints of the following type (a barycenter type constraint)

Ω

u⁺_i (x)2

(x−xi) dx=0. (2.4)

For everyi=1, . . . , k, we denote byBithe subset ofH₀¹(Ω)consisting of all thek-peak functionsu, with respect to the ballsB(x1, rβ), . . . , B(xk, rβ), such that (2.4) holds.

Finally, let us denote by S_x^β₁_,...,x_k the set of all the k-peak functions u∈ H₀¹(Ω), with respect to the balls B(x₁, r_β), . . . , B(x_k, r_β), such thatu∈V_i∩B_i for everyi=1, . . . , k.

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Proposition 2.1. Let k be a positive integer, β >0 large enough so thatΩ_k,β= ∅, α < λ₁ and consider a point (x₁, . . . , x_k)∈Ω_k,β.

ThenSx^β₁,...,x_k= ∅and the minimum of the functionalf on the setSx^β₁,...,x_k is achieved.

Proof. First notice that S_x^β₁_,...,x_k = ∅ because of the choice of the radiusr_β. In fact, since√

βr_β >r¯₁, there exist nonnegative functionsv_i∈H₀¹(Ω)such thatv_i=0 inΩ\B(x_i, r_β),

Ω|Dv_i|²dx < β

Ωv_i²dxand

Ω(v_i(x))²(x− x_i) dx=0 (for example, we can choose asv_i a positive eigenfunction related to the first eigenvalue of the Laplace operator inH₀¹(B(x_i,r˜_β)), with √^r^¯¹

β <r˜_β < r_β).

Therefore,u=_k

i=1t_iv_i∈S_x^β₁_,...,x_k for suitablet_i>0.

For every functionu∈S_x^β₁_,...,x_k, we have f (u)=f

−u⁻ +

k i=1

f u⁺_i

(2.5) where f (−u⁻)f (_α₋^e¹_λ

1) (since α < λ1) and f (u⁺_i ) >0 for i =1, . . . , k (because u∈V_i implies f (u⁺_i )= max{f (t u⁺_i ): t0}). It follows that inf_Sβ

x1,...,xkf >−∞.

Let us consider a minimizing sequence(uj)j forf onSx^β₁,...,x_k. Taking into account thatuj∈Vi, namely

Ω

Du⁺_j,i²dx−β

Ω

u⁺_j,i2

dx+

Ω

e1u⁺_j,idx=0, (2.6)

and thate1>0 inΩ, we have

Ω

Du⁺

j,i²dx < β

Ω

u⁺_j,i2

dx. (2.7)

Therefore, if we setvj,i=_u+^u⁺^j,i j,i_L2(Ω)

the sequence(vj,i)j is bounded inH₀¹(Ω). As a consequence, up to a subse- quence, it converges asj→ ∞to a functionv_i∈H₀¹(Ω)inL²(Ω), weakly inH₀¹(Ω)and almost everywhere inΩ (thusv_i0 inΩandv_iL²(Ω)=1). Notice thatu_j∈V_i ∀j∈Nimplies

lim inf

j→∞u⁺

j,i

L²(Ω)>0 fori=1, . . . , k. (2.8)

In fact, arguing by contradiction, assume that for some i∈ {1, . . . , k}, up to a subsequence,u⁺_j,iL²(Ω)→0 as j→ ∞. Then, from (2.6) we obtain

u⁺

j,i

L²(Ω)

Ω

|Dv_j,i|²dx−βu⁺

j,i

L²(Ω)+

Ω

e₁v_j,idx=0 (2.9)

and, asj → ∞,

Ωe₁v_idx=0 which gives a contradiction becausee₁>0,v_i0 inΩandv_i≡0.

Now, let us prove that lim sup

j→∞

u⁺

j,i

L²(Ω)<+∞ fori=1, . . . , k. (2.10)

Arguing again by contradiction, assume that for some i∈ {1, . . . , k}, up to a subsequence, u⁺_j,iL²(Ω)→ ∞ as j→ ∞. Thus, from (2.6) we have

Ω

|Dvj,i|²dx−β+u⁺_j,i⁻¹

L²(Ω)

Ω

e1vj,idx=0 (2.11)

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which, asj→ ∞, implies

jlim→∞

Ω

|Dv_j,i|²dx=β. (2.12)

Taking into account that f

u⁺_j,i

=max f

t u⁺_j,i : t0

=max t²

2

Ω

|Dv_j,i|²dx−β

+t

Ω

e₁v_j,idx: t0

=1 2

β−

Ω

|Dv_j,i|²dx ₋1

Ω

e₁v_j,idx 2

, (2.13)

from (2.12) we infer that limj→∞f (u⁺_j,i)= +∞. As a consequence, we have also limj→∞f (u_j)= +∞, in contradiction with the fact that(u_j)_jis a minimizing sequence. Therefore, the sequence(u⁺_j)_jis bounded inL²(Ω). Notice that also the sequence(u⁻_j)_j is bounded inL²(Ω)as one can easily verify taking into account thatα < λ₁. Thus, the sequence(uj)jis bounded inL²(Ω)and, as a consequence, also inH₀¹(Ω)(because sup_j_∈Nf (uj) <+∞). It follows that, up to a subsequence,uj converges to a functionu∈H₀¹(Ω)inL²(Ω), weakly inH₀¹(Ω)and a.e. inΩ. The convergence inL²(Ω)implies thatuis ak-peak function with respect to the ballsB(x₁, r_β), . . . , B(x_k, r_β)(notice thatu⁺_i ≡0 because of (2.8)) and thatu∈B_i for everyi=1, . . . , k. Now we prove that, asj→ ∞,

Ω

Du⁻_j²dx→

Ω

Du⁻²dx and

Ω

Du⁺_j,i²dx→

Ω

Du⁺_i ²dx fori=1, . . . , k (2.14) (namely, that u_j →u also in H₀¹(Ω)) which allows us to conclude that u∈V_i for i=1, . . . , k and that u is a minimizing function forf onS_x^β₁_,...,x_k.

For the proof we argue by contradiction and assume that (up to a subsequence)

jlim→∞

Ω

Du⁻_j²dx >

Ω

Du⁻²dx or

jlim→∞

Ω

Du⁺_j,i²dx >

Ω

Du⁺_i ²dx for somei∈ {1, . . . , k}. (2.15)

Notice that, since f (uj)=f

−u⁻_j +

k i=1

f u⁺_j,i

, (2.16)

we have (up to a subsequence) inf

S_x^β_1,...,xk

f = lim

j→∞f (u_j)= lim

j→∞f

−u⁻_j +

k i=1

jlim→∞f u⁺_j,i

(2.17) where

jlim→∞f

−u⁻_j

=f

−u⁻

if lim

j→∞

Ω

Du⁻_j²dx=

Ω

Du⁻²dx (2.18)

while

jlim→∞f

−u⁻_j

> f

−u⁻

if lim

j→∞

Ω

Du⁻_j²dx >

Ω

Du⁻²dx. (2.19)

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Fori=1, . . . , kwe have

jlim→∞f u⁺_j,i

=f u⁺_i

andf(u) u⁺_i

=0 if lim

j→∞

Ω

Du⁺

j,i²dx=

Ω

Du⁺

i ²dx (2.20)

(becausef(u)[u⁺_i ] =limj→∞f(u_j)[u⁺_j,i], in this case, andf(u_j)[u⁺_j,i] =0∀j ∈Nsinceu_j∈V_i).

On the contrary, if limj→∞

Ω|Du⁺_j,i|²dx >

Ω|Du⁺_i |²dx, we have f(u)[u⁺_i ]<0. As a consequence, there existst¯_i∈ ]0,1[such thatf(¯t_iu⁺_i )[¯t_iu⁺_i ] =0. It follows that

ft¯_iu⁺_i

=1 2t¯_i

Ω

e₁u⁺_i dx <1 2

Ω

e₁u⁺_i dx= lim

j→∞

1 2

Ω

e₁u⁺_j,idx= lim

j→∞f u⁺_j,i

(2.21) (where the first and the last equality hold because, ifwis a nonnegative function inH₀¹(Ω),f(w)[w] =0 implies f (w)=¹₂

Ωe1w dx).

Therefore we infer that, if (2.15) occurs, there exists a functionu¯∈S_x^β₁_,...,x_k (of the formu¯= −u⁻+k i=1t¯iu⁺_i witht¯i∈ ]0,1]fori=1, . . . , k) such that

f (u) <¯ lim

j→∞f (u_j)= inf

S_x^β₁_,...,xk

f (2.22)

which is a contradiction.

Thusu_j→uinH₀¹(Ω),u∈S^β_x₁_,...,x_k andf (u)=min_S^β

x1,...,xkf. 2

Proposition 2.1 allows us to introduce the functionϕ_β:Ω_k,β→Rdefined by ϕ_β(x₁, . . . , x_k)= min

S^β_x

1,...,xk

f ∀(x₁, . . . , x_k)∈Ω_k,β. (2.23)

Proposition 2.2.For every positive integerkand forα < λ₁, fixβ >0 large enough so thatΩ_k,β= ∅. Then there exists(x₁^β, . . . , x_k^β)∈Ωk,βsuch thatϕβ(x₁^β, . . . , x_k^β)=maxΩ_k,βϕβ (see(2.23)).

Proof. Let us consider a sequence(x1,j, . . . , x_k,j)_j inΩ_k,βsuch that

jlim→∞ϕ_β(x_1,j, . . . , x_k,j)=sup

Ω_k,β

ϕ_β. (2.24)

Since Ω_k,β is a compact set, there exists (x₁^β, . . . , x_k^β)∈Ω_k,β such that, up to a subsequence, (x_1,j, . . . , x_k,j)→ (x₁^β, . . . , x_k^β)asj→ ∞.

By Proposition 2.1, there exists uβ ∈S^β

x₁^β,...,x^β_k such that f (uβ)=min_S^β

xβ 1,...,xβ

k

f. For everyj ∈N, consider the function u¯_j ∈S^β_x_1,j_,...,x_k,j such that u¯_j = − ¯u⁻_j +_k

i=1u¯⁺_j,i, whereu¯⁺_j,i(x)=u⁺_β,i(x+x_i^β −x_i,j) and− ¯u⁻_j is the minimizing function for the minimum

min

f (v): v∈H₀¹(Ω), v0 inΩ,

Ω

vu¯⁺_j,idx=0 fori=1, . . . , k

(2.25) (sinceα < λ₁, there exists a unique minimizing function which depends continuously on the point(x_1,j, . . . , x_k,j)∈ Ω_k,β).

Standard arguments show that u¯_j → u_β and f (u¯_j)→f (u_β) as j → ∞. Thus, taking into account that min_Sβ

x1,j ,...,xk,j

f f (u¯_j)∀j∈Nbecauseu¯_j∈S_x^β_1,j_,...,x_k,j, we obtain sup

Ωk,β

ϕ_β= lim

j→∞ϕ_β(x_1,j, . . . , x_k,j) lim

j→∞f (u¯_j)=f (u_β)=ϕ_β

x₁^β, . . . , x_k^β

(2.26) which impliesϕ_β(x^β₁, . . . , x_k^β)=maxΩk,βϕ_β. 2

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In order to describe the behaviour of the problem as β→ +∞, we need also some preliminary results on the capacity.

For every smooth bounded domainAinRⁿwithn3, the capacity ofAis defined by cap(A)=min

Rⁿ

|Du|²dx: u∈D^1,2 Rⁿ

, u1 a.e. inA

. (2.27)

It is well known that there exists a unique minimizing functionuA. Moreover, we haveuA=1 inA, 0< uA1 in Rⁿ\A,uA=0 inRⁿ\ ¯A.

Notice that we have also cap(A)= −

∂A

(Du_A·ν) dσ (2.28)

whereνdenotes the outward normal on∂A.

Lemma 2.3.LetA₁, A₂, . . . , A_s, withs >1, bespairwise disjoint smooth bounded domains inRⁿwithn3. Then, we have

cap _s

i=1

A_i

<

s i=1

cap(Ai). (2.29)

Proof. For everyi=1, . . . , s, consider the minimizing functionuA_i for cap(Ai); then introduce the functionμ∈ D^1,2(Rⁿ)defined by

μ(x)=max

u_A_i(x): i=1, . . . , s

∀x∈Rⁿ. (2.30)

Notice thatμis subharmonic (but not harmonic) inRⁿ\_s

i=1A¯_i. Therefore, if we setA=_s

i=1A_i, we haveμu_A, μ≡u_Aand

−

∂A

(Du_A·ν) dσ <− s i=1

∂A_i

(Du_A_i·ν) dσ (2.31)

becauseuA(the minimizing function for cap(A)) is harmonic inRⁿ\ ¯A. Thus we obtain cap(A)=

Rⁿ

|Du_A|²dx= −

∂A

(Du_A·ν) dσ <− s i=1

∂A

(Du_A_i·ν) dσ= s i=1

Rn

|Du_A_i|²dx= s

i=1

cap(Ai), (2.32) which completes the proof. 2

3. Asymptotic behaviour and proof of the main results

Our next aim is to prove that, if (x₁^β, . . . , x_k^β)∈Ω_k,β and u_β ∈ S^β

x₁^β,...,x_k^β is a function such that f (u_β)= ϕ_β(x₁^β, . . . , x_k^β)=maxΩk,βϕ_β (see Proposition 2.2), thenu_β is a solution of problem (1.5) forβlarge enough. There- fore, we need to study the behaviour of u_β as β→ +∞and to describe the asymptotic profile of the function u_β (suitably rescaled).

Proposition 3.1.For every positive integerk, forα < λ1and forβ >0large enough so thatΩ_k,β= ∅, consider a point(x₁^β, . . . , x_k^β)∈Ω_k,β and a functionu_β ∈S^β

x₁^β,...,x_k^β such thatf (u_β)=ϕ_β(x₁^β, . . . , x_k^β)=maxΩk,βϕ_β. Then we have

(9)

(a) u_β>_α₋^e¹_λ

1 ∀βandu_β→_α₋^e¹_λ

1 a.e. inΩasβ→ +∞;

(b)

β→+∞lim βⁿ⁻²²

f (u_β)− 1 2(α−λ₁)

= k

2(α−λ1)²

maxΩ e₁ 2

cap(r¯₁) (3.1)

where(for short) cap(r¯1)denotes the capacity of the ball of radiusr¯1inRⁿ; (c)

β→+∞lim e₁ x_i^β

=max

Ω e₁ fori=1, . . . , k, (3.2)

β→+∞lim

βx_i^β−x^β_j= ∞ fori=j. (3.3)

Proof. Property (a) follows by standard arguments taking into account that, sincef (u_β)=min_Sβ xβ

1,...,xβ k

f,−u⁻_β is the unique minimizing function for the minimum

min

f (u): u∈H₀¹(Ω), u0 inΩ,

Ω

uu⁺_β dx=0

(3.4)

and thatu⁺_β =0 inΩ\k

i=1B(x_i^β, rβ)withrβ→0 asβ→ +∞.

Notice that, for everyu∈H₀¹(Ω),f (u)=f (−u⁻)+f (u⁺)and f

−u⁻

=1 2

Ω

Du⁻²dx−α 2

Ω

u⁻2

dx−

Ω

e₁u⁻dx.

If we setv= −u⁻−_α₋^e¹_λ₁, taking into account thatf (_α₋^e¹_λ

1)=_2(α₋¹_λ₁₎, a direct computation shows that f

−u⁻

= 1

2(α−λ₁)+1 2

Ω

|Dv|²dx−α 2

Ω

v²dx. (3.5)

For the proof of property (b), we prove first that lim inf

β→+∞βⁿ⁻²²

f (u_β)− 1 2(α−λ1)

k

2(α−λ₁)²

maxΩ e₁ 2

cap(r¯₁). (3.6)

For everyβ >0 (βlarge enough) choose(x¯₁^β, . . . ,x¯_k^β)∈Ωk,βsuch that

β→+∞lim e₁

¯ x_i^β

=max

Ω e₁ fori=1, . . . , k, and lim

β→+∞

βx¯_i^β− ¯x_j^β= ∞ fori=j. (3.7) Consider a functionu¯_β∈S^β

¯

x^β₁,...,x¯_k^βsuch thatf (u¯_β)=ϕ_β(x¯₁^β, . . . ,x¯_k^β). For everyr >0, setΩ_r^β=Ω\_k

i=1B(x¯_i^β,√^r β).

Standard arguments show that

β→+∞lim min

Ω_r^β

|Du|²dx: u∈H₀¹(Ω),

Ω^β_r

u²dx=1

=λ₁. (3.8)

Therefore, sinceα < λ₁, for everyr >3r¯₁there existsβ_r>0 such that f (u¯β)− 1

2(α−λ₁) k i=1

f

¯ u⁺_β,i

+ k i=1

1 2

B(x¯^β_i,r/√ β)

|Dv¯β|²−αv¯²_β

dx ∀β > βr (3.9)

wherev¯_β= − ¯u⁻_β −_α₋^e¹_λ₁. Taking into account thatf (u¯⁺_β,i)0 fori=1, . . . , k, after rescaling we obtain