Subcritical approximation of the Sobolev quotient and a related concentration result

Subcritical Approximation of the Sobolev Quotient and a Related Concentration Result

GIAMPIEROPALATUCCI

ABSTRACT- LetVbe a general, possibly non-smooth, bounded domain ofR^N,N3.

Let2ˆ2N=(N 2) be the critical Sobolev exponent. We study the following variational problem

S_e ˆsup Z

V

juj^{2 e}dx: Z

V

jruj²dx1;uˆ0on@V 8<

:

9=

;;

investigating its asymptotic behavior asegoes to zero, by means ofG^‡-con- vergence techniques. We also show that sequences of maximizersueconcentrate energy at one pointx02V.

1. Introduction.

LetVbe ageneral bounded, possible non-smooth, bounded open set in R^N;N3. Consider the elliptic Dirichlet problem with nearly critical nonlinearity

Dueˆlu²_e ^{1 e} inV;

u_e>0 inV;

u_eˆ0 on@V;

8>

<

>: (1:1)

where 2ˆ2N=(N 2);lˆN(N 2). We know that whene>0 the above problem has a solutionue, but wheneˆ0 it becomes delicate. In particular, the existence and the properties of the solutions strongly depend on the domainV (see [14, 9, 2, 5] and others).

Indirizzo dell'A.: LATP, Universite Aix-Marseille III, Faculte des Sciences et Techniques Saint-JeÂroÃme, Avenue Escadrille Normandie-NieÂmen, 13397 Marseille cedex 20, France

E-mail: giampiero.palatucci@univ-cezanne.fr

In view of this qualitative change wheneˆ0, it is interesting to analyze the asymptotic behavior of the subcritical solutionsueof (1.1) asegoes to zero.

In the case of smooth domains V, this analysis was carefully in- vestigated by Han [8] and Rey [15]. They showed that the solutionsu_e of (1.1), that are maximizing for the following variational problem

S_e:ˆsup Fe(u): Z

V

jruj²dx1;uˆ0 on@V 8<

:

9= (1:2) ;

with F_e(u)ˆ Z

V

juj^{2 e}dx;

(1:3)

concentrate at exactly one pointx₀ inV; i.e., the measuresjru_ej²dxcon- verge (up to subsequences) in the sense of measures to a Dirac mass at x₀2V. They also showed thatx₀is acritical point of the Robin function ofV (the diagonal of the regular part of the Green function), answering to a conjecture by Brezis and Peletier in [3], in which was analyzed the same problem in the case ofVbeing aspherical domain.

In order to obtain this concentration result, even without localizing the blowing-up, the cited authors also utilize some standard elliptic regularity techniques that require to work in smooth domains.

We also stress that regularity assumptions on the domain V have a strong impact on concentration results. In fact, in the case of smooth do- mains, the Robin function has no critical points in a neighborhood of the boundary and thus in [3, 8, 15] concentration may occur only inV. In [7], Flucher, Garroni and MuÈller were able to construct an example of a non- smooth domain V, with its Robin function achieving its infimum on the~ boundary; and then Pistoia and Rey showed that concentration can occur on the boundary, analyzing the asymptotic behavior of the maximizing solutions of the elliptic Dirichlet problem (1.2) in suchV~ (see [13]).

Here we can obtain the same concentration result in general bounded domain, not depending from any strong hypotheses of regularity, as a consequence of Lions' Concentration-compactness principle ([10]) together with the convergence of maxF_e to supF₀ (with F₀(u)ˆR

Vjuj²dx; see Section 2).

We also investigate the asymptotic behavior ase!0 of the subcritical Sobolev quotient S_e in (1.2) by means of De Giorgi's G^‡-convergence techniques. We can find this kind of approach, that is using G^‡-con- vergence techniques to study concentration phenomena linked to critical

Sobolev exponent, in [1], where Amar and Garroni, following Flucher and MuÈller in [6], studied the variational problem

S^g_e ˆsupG_e(u):u2H¹₀(V);kruk_L2(V)1 (1:4)

with Ge(u)ˆe ² Z

V

g(eu)dx;

(1:5)

where gis anon-negative upper semi-continuous function bounded from above bycjtj²:They obtained aG^‡-convergence result, that implies con- centration phenomena arising in critical growth problems.

Although it is not possible to reduce our problem (1.2) to (1.4), theG^‡- convergence of the functionalF_eis related to theG^‡-convergence ofF₀to its upper semi-continuous envelope sc^‡F₀, obtained by Amar and Garroni in [1] (see Section 3).

A fundamental key point of theG^‡-convergence is specifying the set- ting for the limit functional, with the choice of the topology. Here, in particular, the topology has to be sufficiently weak to assure the con- vergence of maximizing sequences and sufficiently strong to allow us to find the concentration.

Due to the presence of the constraint on the Dirichlet energy, the variational problem (1.2) suggests to study every sequence ue weakly converging to some function u in the Sobolev space H¹₀(V). This con- vergence is not sufficient to describe the problem, so we also have to study every sequence jru_ej²dx converging to some measure m in the sense of measures.

We will show that the asymptotic behavior of the functional (1.3), in term of G^‡-convergence, is described by the following functional F that depends by the two variablesu, belonging toH¹₀(V), andm, being inM(V), the set of non-negative bounded total variation Borel measures, with its atomic coefficientsm_i:

F(u;m)ˆ Z

V

juj²dx‡SX¹

iˆ0

m²_i²; (1:6)

where S is the best Sobolev constant of the embeddingH¹₀(V),!L²(V), that is

Sˆsup Z

V

juj²dx: Z

V

jruj²dx1;uˆ0 on@V 8<

:

9=

;: (1:7)

Now, a natural question arises: can we localize the blowing up? Our hope is that it will possible to prove thatx0is a``harmonic center'' ofV, that is the minimum of the Robin function ofV. We could work by means of a reasonable second order development of the G^‡-limit, in order to obtain more informations about the asymptotic behavior of the maximizing se- quences. We surely need to look at some techniques introduced in [6] and [7], where Flucher, Garroni and MuÈller showed that the concentration at the critical points of the Robin function is a very general phenomenon.

Second, again using a similar variational approach, in [11] the author generalizes the asymptotic analysis of subcritical solutions of the analo- gous problem to (1.1) for thep-Laplacian operator.

Finally, in a work in progress with A. Pisante ([12]) we study non-local concentration phenomena in the same spirit of this paper.

The remaining of the paper is divided in two sections. Section 2 is de- voted to the concentration result. In Section 3 we state and prove theG^‡- convergence result.

2. The concentration result.

In this section, we will analyze the asymptotic behavior of the se- quencesuethat are maximizers for the nearly critical Sobolev quotientS_e, as stated in the following theorem.

THEOREM2.1. Let u_ebe a maximizer sequence for the Sobolev quotient S_e defined in(1.2).Then u_econcentrates at some x₀2V, i.e.,

jruej²dx* dx0 inM(V):

This result is a consequence of Lions's Concentration-compactness principle (see forthcoming Lemma2.2) together with the convergence of S_e toSasegoes to zero (see Proposition 2.4).

LEMMA2.2 ([10, Lemma I.1]). LetV R^Nbe an open subset and let u_n be a sequence in H¹₀(V)weakly converging to u as n! 1and such that

jrunj²dx* m and junj²dx* n inM(V):

Then, either u_n !u in L²(V)or there exists a finite set of distinct points x₀;. . .;x_kinVand positive numbersn₀;. . .;n_ksuch that we have

nˆ juj²dx‡X^k

iˆ0

n_id_xi; n²_i(S)²(S)²:

Moreover, there exist a positive measure ~m2 M(V)with spt~mV and positive numbersm₁;. . .;m_k such that

mˆ jruj²dx‡~m‡X^k

iˆ0

m_id_xi; n_iSm²_i²:

Armed with the concentration-compactness alternative, we can de- scribe the asymptotic behavior of the optimal sequencesu_n of the critical Sobolev quotient S. The key of the proof is the well-known convexity argument by Lions (see also [6, Lemma 10]).

COROLLARY 2.3. Let u_n be a maximizing sequence for the critical Sobolev quotient S defined in (1.7). Thenu_n concentrates at one point x02V.

Now, we will prove that the subcritical Sobolev quotientsS_econverge to S asegoes to zero.

PROPOSITION2.4. For everye>0, letS_e be defined by (1.2)and letS be defined by(1.7).Then

lime!0S_eˆS: (2:1)

PROOF. First, using the fact thatV is bounded together with HoÈlder inequality, we can show that

lim sup

e!0 S_e S: (2:2)

Takeue2H¹₀(V) maximizers forS_e; we have S_e ˆ F_e(u_e)ˆ

Z

V

ju_ej^{2 e}dx

Z

V

ju_ej² 0

@

1 A

2 e 2

jVj^2e

(S)^22ejVj^2e:

Thus, passing to the limit asegoes to zero, it follows inequality (2.2).

The remaining inequality easily follows by the pointwise convergence of Fe to F0. This is a standard argument. For every d>0 there exists

ud2H¹₀(V) such thatkrudk²_L2(V)1 a nd F0(ud)>S d:

(2:3)

Clearly, for such functionu_d, we have S_e F_e(u_d):

Then, combining the above inequality with (2.3) and passing to the limit ase goes to zero, we get

lim inf

e!0 S_e lim

e!0Fe(ud)

ˆF0(ud) S d;

that implied the desired inequality thanks to the arbitrariness of d. The

proof is complete. p

COMPLETION OF THE PROOF OFTHEOREM2.1. The concentration result for the sequence ue of maximizers ofS is immediate. Thanks to Propo- sition 2.4 the maximizersueofS_eare a maximizing sequence forF0. Hence, Corollary 2.3 ensures thatjruej²dxconverge to aDirac mass in the sense

of measures. p

3. TheG^‡-convergence result.

The goal of the present section is to describe by means of G^‡-con- vergence the asymptotic behavior as egoes to zero of the functionals F_e defined by

Fe(u)ˆ Z

V

juj^{2 e}dx; 8u2H¹₀(V) such thatkruk²_L2(V) 1:

(3:1)

In order to applyG^‡-convergence, we have to analyze the asymptotic behavior of the sequence Fe(ue) for every sequence ue such that kru_ek²_L2(V)1:The constraint on the Dirichlet energy ofu_e implies that there existsm2 M(V) such thatm(V)1 a ndjru_ej²dx* minM(V) and, by Sobolev embedding, there exists u2H¹₀(V) such that (up to sub- sequences)ue*uinL²(V).

Thanks to the semi-continuity ofL²-norm, we havem jruj²dx, so we can always decomposemin such way:

mˆ jruj²dx‡~m‡X¹

iˆ0

m_id_xi;

wherem_i2[0;1] andxi2Vis such thatxi6ˆxj if i6ˆj;m~can be view as the

``non-atomic part'' ofm jruj²dx.

In analogy with [1], this suggests the setting for the limit functional as the spaceXdefined by

XˆX(V):ˆ

(u;m)2H¹₀(V) M(V):m jruj²dx;m(V)1 ; endowed with the topologytsuch that

(ue;m_e)!^t (u;m) ,^def ue*uinL²(V) m_e* m inM(V):

(

Note that the topologytis compact inX, then theG^‡-convergence of functionals in this space implies the convergence of maxima.

We also note that X is the smallest space where we have to set our problem, as stated in the following proposition by Amar and Garroni.

PROPOSITION3.1 ([1, Proposition 2.3]). Let(u;m)2X, then there exists a sequence u_e in H₀¹(V) such that, for every e>0, R

Vjru_ej²dx1 and (ue;jruej²dx)!^t (u;m)whene!0.

Once we have taken the right space, we need to extend our functionalFe

to the whole spaceX(and we keep the same symbol), in the natural way as follows

Fe(u;m):ˆ

Z

V

juj^{2 e}dx if (u;m)2X:mˆ jruj²dx;

0 otherwise inX:

8>

><

>>

(3:2) :

We prove the existence of the G^‡-limit of the sequenceFe, giving its characterization in the following theorem.

THEOREM 3.2. There exists the G^‡-limit F of the sequence of func- tionals Fedefined by(3.2)and

F(u;m)ˆ Z

V

juj²dx‡SX¹

iˆ0

m²_i²; 8(u;m)2X:

In view of the pointwise convergence ofF_etoF₀and the convergence of S_e toS (see Proposition 2.4), Theorem 3.2 can be proved repeating the strategy in the proof of [1, Theorem 3.1].

First, we state theG^‡-convergence result by Amar and Garroni in [1].

Letg:R![0;1) be an upper semi-continuous function such that (i) 0g(t)cjtj²; g6ˆ0 in theL¹-sense;

(ii) 9aconstantg₀such that

t!0lim^‡ g(t)

t² ˆg₀: (3:3)

LetS^gbe the criticalgeneralized Sobolev constant(see [6, Definition 1]) such that

S^gˆsup Z

V

g(u)dx:u2H¹₀(V);

Z

V

jruj²dx1 8<

:

9=

;: (3:4)

For any positivee, consider the functionalGedefined inH¹₀(V) M(V) by

G_e(u;m)ˆ e ² Z

V

g(eu)dx if (u;m)2X andmˆ jruj²dx;

0 otherwise inX:

8>

><

>>

: (3:5)

TheG^‡-limit of the functionalGewith respect to the topologytis given in the following theorem.

THEOREM 3.3 ([1, Theorem 3.1]). There exists the G^‡-limit G of the sequence of functionals Geand

G(u;m)ˆg0

Z

V

juj²dx‡S^gX¹

iˆ0

m²_i²; (3:6)

for every(u;m)2X.

We note that in the particular case ofgbeing the critical power 2; i.e., g(t)ˆ jtj²,8t2R;the definitions in (3.3), (3.4), (3.5) and (3.6) become re- spectively

g₀ˆ1; S^gˆS; G_eˆF₀ andGˆF and then, applying Theorem 3.3, it follows

F₀ !^G‡ sc^‡F₀F;

(3:7)

whereF₀ is extended to the whole spaceX in the same way as in defini- tion (3.2) and we denote by sc^‡F₀the upper semi-continuous envelope ofF₀ with respect to the topologyt.

Using the convexity argument by Lions together with the fact that the critical Sobolev inequality is strict in bounded domains, Amar and Garroni also provide an optimal upper bound inequality for the limit functionalG.

LEMMA3.4 ([1, Lemma 3.6]). For every(u;m)2X, we have G(u;m)S^g

(3:8)

and the equality holds if and only if(u;m)ˆ(0;d_x0)for some x₀2V.

PROOF OFTHEOREM3.2. Let us introduce theG^‡-lim sup and theG^‡- lim inf ofFe, respectively defined for every (u;m)2Xby

F^‡(u;m)ˆsupn lim sup

e!0 Fe(ue;m_e):(ue;m_e)!^t (u;m)o and

F (u;m)ˆsupn lim inf

e!0 F_e(u_e;m_e):(u_e;m_e)!^t (u;m)o :

We can write the notion ofG^‡-convergence in terms of theG^‡-limsup and theG^‡-liminf (see [4] for further details). TheG^‡-limitFexists if and only if F ˆF^‡ and, in this case, FˆF ˆF^‡. Since F F^‡ always holds, in order to prove Theorem 3.2 it is enough to show the following inequalities

FF^‡ (3:9)

FF : (3:10)

The first inequality (3.9) easily follows by the Concentration-compactness principle by Lions.

PROPOSITION3.5. For every(u;m)2X, we have F(u;m)lim sup

e!0 F_e(u_e;m_e);

for every(ue;m_e)X such that(ue;m_e)!^t (u;m).

PROOF. Let (u_e;m_e) be inXsuch that (u_e;m_e)!^t (u;m); i.e., ue*uinL²(V) a nd jruej²dx* mˆ jruj²dx‡m~‡X¹

iˆ0

m_idxi in M(V);

By Lemma2.2, it follows that

juej²dx* nˆ juj²dx‡X^k

iˆ0

n_idxi; with n_iSm²_i²: (3:11)

Using HoÈlder Inequality by arguing as in the proof of Proposition 2.4, we have

lim sup

e!0 Fe(ue;m_e)lim sup

e!0

Z

V

juej²dx ²₂^e

jVj^2e

!

n(V)

Z

V

juj²dx‡SX¹

iˆ0

m²_i² F(u;m);

where we used (3.11). p

The remaining inequality (3.10) can be obtained like in the proof of [1, Theorem 3.1]. For the sake of self-containment, we will sketch here the proof.

According to the idea of Amar and Garroni, the G^‡-liminf inequality will be proved in two separate cases: (u;m)ˆ(u;jruj²dx‡~m) a nd (u;m)ˆ

0;P

i m_id_xi

. This decomposition is the key point in the proof. In fact, we will be able to use strongL² convergence on the pairs of the first type (see the forthcoming Proposition 3.6, in which we useF_e!F₀); while on the pairs with purely atomic measure part, the functionalsF_eare local, so we can compute their limit on each single Dirac mass (see Proposition 3.7, in which we use S_e !S). Finally, Theorem 3.2 will be recovered thanks to aunifying lemma(see Lemma3.8).

PROPOSITION3.6. Let(u;m)2X be such thatmˆ jruj²dx‡m. Then~ F(u;m)lim inf

e!0 F_e(u_e;m_e)

for every sequence(ue;m_e)X such that(ue;m_e)!^t (u;m)ase!0.

PROOF. We ta ke (u_e;m_e)X such that m_eˆ jru_ej²dx and (u_e;m_e)!^t (u;m) a s e!0. Since the atomic part of m_e is zero, u_e!u in L²(V) a s e!0, so (up to asubsequence) u_e!u a.e. in V.

It follows that for almost everyxinV

ju_e(x)j^{2 e}! ju(x)j² as e!0:

Hence, by Lebesgue Convergence Theorem, we have:

lime!0

Z

V

ju_ej^{2 e}dxˆ Z

V

juj²dxF(u;m);

that gives the desired inequality. p

PROPOSITION3.7. For every(u;m)2Xsuch that(u;m)ˆ 0;P^k

iˆ0m_idxi

, withxi2V, there exists theG^‡-limit of(Fe)and we have

G^‡ lim

e!0Fe

(u;m)ˆF(u;m):

PROOF. First, we prove the following result. For every open setAV, for every x2A and for every (u;m)2X such that (u;m)ˆ(0;d_x), there exists theG^‡-limit of the sequence (F_e) restricted toAand the following equality holds:

G^‡ lim

e!0F_e

(0;d_x;A)ˆS:

Let us fix AV and let eh be such that eh!0 if h! 1. By the compactness property of theG^‡-convergence, it follows that there exists a subsequence (still denoted bye_h) and a functionalF_A:X(A)![0;1) such that

G^‡ lim

h!1Feh

(u;m;A)ˆFA(u;m):

We also have

S_eh ˆsup

X(A)Feh !max

X(A)FA ash! 1 and then, recalling Proposition 2.4,

maxX(A)FAˆS: (3:12)

By Proposition 3.6 and Lemma 3.4, withg(t)ˆ jtj²andVˆA, it follows FA(u;m)5S; 8(u;m)6ˆ(0;dx); 8x2A:

Combining the above inequality with (3.12), it follows the existence of x2Asuch that

F_A(0;dx)ˆS

and the above equality can be proved by density for every (u;m)ˆ(0;d_x) with anyx2A.

Finally, in order to deal with the casemˆP^k

iˆ0m_idxi, it suffices to consider the sequence (ue;m_e) such that

u_eˆX^k

iˆ0



m_i

p uⁱ_e; m_eˆ jru_ej²dx;

where (uⁱ_e;jruⁱ_ej²dx)!^t (0;dxi) are the recovering sequences given by the pre- vious step. A detailed proof is very well written in [1, Proposition 3.7]. p In order to complete the proof of inequality (3.10), we need the following technical lemma by Amar and Garroni, applied to our functionalsFe.

LEMMA 3.8 ([1, Lemma 4.1]).If F(u;m)F (u;m) for every (u;m)2X such that

(1) m(V)51;

(2) mˆ jruj²dx‡m~‡P^k

iˆ0m_id_xi; x_i2V;

(3) dist

supp (juj ‡~m);S^k

iˆ0fx_ig

>0:

Then

F(u;m)F (u;m)for every(u;m)2X:

COMPLETION OF THE PROOF OFTHEOREM3.2. In view of Lemma3.8, it remains to prove theG^‡-liminf inequality for every pair (u;m)2Xsuch that m(V)51,mˆ jruj²‡m~‡P^k

iˆ0m_id_xiand dist

supp (juj ‡~m);S^k

iˆ0fx_ig

>0:

As ameans to do this, it suffices to consider the recovery sequence (u_e;m_e):ˆ(u^A_e ‡u^B_e;m^A_e ‡m^B_e);

where (u^A_e;m^A_e) and (u^B_e;m^B_e) are given by Proposition 3.6 and Proposi- tion 3.7, respectively (see [1, pag. 17]). The proof is complete. p REMARK3.9. We can also deduce the concentration result in Theorem 2.1, by using theG^‡-convergence ofFetoF. In fact, by Theorem 3.2 and

G^‡-convergence properties, we have that every maximizing sequence (ue;jruej²dx) ofFetmust converge to apair (u;m)2Xmaximizer forF:

(u_e;jru_ej²dx)!^t (u;m); with F(u;m)ˆmax

X(V)F:

By Lemma3.4, with g(t)ˆ jtj², we have max

X(V)FˆF(0;dx0), for some x₀2V.

It follows

(ue;jruej²dx)!^t (0;dx0);

that is the desired concentration result. p

Acknowledgements. I would like to thank Prof. Adriana Garroni for suggesting the study of concentration problems related to critical Sobolev exponent.

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Manoscritto pervenuto in redazione il 7 dicembre 2009.