A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics ∗ Marino Badiale† and Gabriella Tarantello‡
Abstract In this paper we analyze the existence and non existence of cylindrical solutions for a nonlinear elliptic equation in IR3 (see problem (1) below), which has been proposed as a model for the dynamics of galaxies. We prove a general integral inequality of Sobolev-Hardy type that allows us to use variational methods when in (1) the power p belongs to the interval [4, 6]. We find solutions in the range 4 < p ≤ 6. The value p = 4 seems to have, for problem (1), characteristics similar to those of the critical Sobolev exponent p = 6.
1
Introduction
In this paper we are concerned with solutions of cylindrical symmetry for the following problem −∆u(x) = φ(r)|u|p−2 u in IR3 , (1) u(x) > 0 in IR3 R p−1 φ(r)u dx < +∞ IR3
with p > 1. We adopt standard notations to denote cylindrical coordinates in IR3 , and for p 3 2 x1 + x22 , z = x3 and u = u(r, z) for a cylindrically x = (x1 , x2 , x3 ) ∈ IR , we write r = symmetric function. The weight function φ is a non negative continuous function, depending on r only, vanishing both at zero and at infinity (see (3) below for the precise hypotheses we adopt on φ). This equation has been proposed to us by two astrophysicists (G.Bertin, Scuola Normale Superiore, Pisa, Italy, and L. Ciotti, Osservatorio Astronomico, Bologna, Italy) as a model describing the dynamics of galaxies. It is derived from the usual Poisson equation 4ϕ = 4πGρ relating the gravitational potential ϕ with the density of matter ρ, in case one assumes the dependance of ρ with respect to the variables ϕ and r, namely ρ = ρ(r, ϕ). By ∗ This
research was supported by MURST “Variational Methods and Nonlinear Differential Equations” di Matematica, Universit` a di Torino. Via Carlo Alberto 10, 10123 Torino, Italy. email:
[email protected] ‡ Dipartimento di Matematica, Universit` a di Roma II, Via della Ricerca Scientifica, 00133 Roma. email:
[email protected] † Dipartimento
1
phisical reasonings Bertin and Ciotti proposed a special dependance of ρ on ϕ, which leads to equation (1) with r2α φ(r) = (α ≥ 0). (2) 1 (1 + r2 ) 2 +α The cylindrical symmetry of the problem is Rderived from the assumption that the galaxies are axial symmetric. Moreover, the condition IR3 φ(r)up−1 dx < +∞ guarantees that the given solution carries a finite total mass. We refer to the book [2] for a general treatment of these problems, and the forthcoming lecture notes [5] for a derivation of (1). Various equations similar to (1) have been proposed to model several phenomena of interest in astrophyisics. We recall here Eddington’s and Matukuma’s equations, which have attracted much interest in recent years (see [12, 13, 14, 15, 20, 22, 23, 24, 25, 26, 27, 28]). In [1] various astrophysical models are introduced and discussed, including some generalizations of Matukuma’s equation. However, these equations have been proposed to model globular clusters of stars, and so radial symmetry is the natural symmetry assumption in this context. As far as we know, much less is known in case one treats other types of symmetry such as the cylindrical symmetry considered here. The only result we are aware of, which is somewhat related to our problem, has been obtained by W.M.Ni in [21] (theorem 1.4), and concerns the equation −4u = φ(|x0 |)up
in IRN ,
with x = (x0 , z) ∈ IRk × IRN −k and k ≥ 3. However, Ni’s result does not appy to equation (1), where we need to take k = 2. Beside, Ni in [21] establishes positive solutions bounded away from zero, and therefore they cannot satisfy the finite mass assumption in (1). Motivated by the Bertin and Ciotti choice (2) for φ, we are going to work with the following set of assumptions: φ ∈ C(IR+ ),
φ ≥ 0,
φ(r) = 0
if and only if r = 0,
rφ(r) ∈ L∞ (IR+ ),
(3)
where as usual we set IR+ = [0, +∞). The functions in (2) satisfy (3) for α > 0. We mention however that our approach can be used to handle more general situations. Notice that a delicate aspect in the analysis of problem (1) is represented by the requirement on the solution to carry a finite total mass. Under assumption (2) (where φ(r) = O(1/r) as r → +∞), for a given solution u ≥ 0, this amounts to guarantee the integrability of up−1 /r away from the x3 − axis (i.e. r = 0). Interestingly enough, we have found that, when p ∈ [4, 6], then problem (1) admits a variational formulation in the Sobolev space D1,2 (IR3 ), which also guarantees the desired finite total mass condition. This fact rests upon a new Sobolev-Hardy inequality, which we have derived in a rather general framework and for any dimension N ≥ 3. It extends an inequality established by Caffarelli, Kohn and Nirenberg ([4]) by interpolation between the well known Sobolev and Hardy inequalities. In fact, our inequality may be viewed as the “cylindrical” version of the Caffarelli-Kohn-Nirenberg’s inequality, in the sense it takes its “most” natural formulation among function with cylindrical symmetry in the same way that the Caffarelli-Kohn-Nirenberg’s inequality does for functions with radial symmetry. We believe that our version of the Sobolev-Hardy inequality (given in (8) below) has independent interest and it is worthwile investigating. For instance we have proved that, except for the case where it reduces to the mere Hardy’s inequality, there always exist extremal functions for such inequality. Thus, the best constant for our inequality (8) is always attained, with the exception of the Hardy’s inequality, where it is well known that this is never the case. The analogous result for the Sobolev-Hardy inequality derived by Caffarelli, Kohn and Nirenberg was established by 2
Ghoussoub and Yuan in [11]. In [11] the authors took advantage of the radial symmetry of the problem to use Schwarz symmetrization and reduce to work with radially symmetric functions. In this way they were able to derive explicitly the expression for the (radial) extremal functions, see [11]. On the contrary, for our “cylindrical” Sobolev-Hardy inequality, we show existence of the corresponding extremal functions by suitable concentration-compactness principles, in the same spirit of the pioneering works of P.L.Lions ([16, 17, 18, 19]) for Sobolev’s inequality. Unfortunately, for our extremals we are not able to claim any symmetry property of cylindrical type, but leave this as an interesting open question to investigate. For dimension N = 3 and 4 ≤ p ≤ 6 our general inequality, (see (8) below) implies that Z
2
IR3
|∇u| dx ≥ Cp
Z
p
φ(r)|u| dx
IR3
2/p
,
(4)
for a suitable constant Cp > 0 and φ satsfying (3). By means of (4) and a general asymptotic result of Egnell ([7])we can also conclude that, when p ∈ [4, 6] and u satisfies 3 −∆u(x) = φ(r)|u|p−2 u in IR , 3 (5) u(x) > 0 in IR u ∈ D1,2 (IR3 )
then
lim sup |x| u(x) < +∞, |x|→+∞
R (see lemma 5.1). Consequently, the finite total mass assumption IR3 φ(r)up−1 dx < +∞ is automatically satisfied in this case by solutions of (5). Thus, to obtain the desired solutions to (1) we need only to solve (5). To this purpose we take advantage of (4) and (weakly) formulate problem (5) in terms of a variational principle in D1,2 (IR3 ). For instance, we see that any extremal function for the minimization problem Z Z inf |∇u|2 dx u ∈ D1,2 (IR3 ), φ(r)|u|p dx = 1 , (6) IR3
IR3
yields a solution for (1) provided that p ∈ [4, 6] and φ satisfies (3). It is not difficult to see that, in case 4 < p < 6, the extremal problem (6) always attains its infimum. We do not expect this to be the case when p = 4 or p = 6 which appear as critical cases. This is not too surprising in the case p = 6, the critical Sobolev exponent in dimension N = 3. Indeed for φ satisfying (3) the infimum in (6) cannot be attained when p = 6, since the corresponding minimizing sequence must concentrate at a point of the cylinder x21 + x22 = r02 with φ(r0 ) = maxIR+ φ (see lemma 4.6). Concerning the power p = 4, it occurs as a “critical” exponent for our general Sobolev-Hardy inequality ((8) below), which in the special case suitable for problem (1) takes the form: Z
IR3
2 Z |∇u|2 dx ≥ C
IR3
|u|4 dx , r
u ∈ D1,2 (IR3 )
(7)
with C > 0 a suitable constant. As a consequence of our general result (see theorem 2.5) we know that the best constant in (7) is attained by some extremal functions. In other words (6) attains its infimum when p = 4 and rφ(r) = 1. On the other hand, if we assume that the function rφ(r) is not a constant but it is monotone increasing (as in (2)), then we show explicitly (see theorem 3.2 (iii)) that (6) cannot be attained. Thus, the existence or non existence of extremal functions for (6) might both occur according to the nature of the function φ(r) (or rather rφ(r)).
3
Now recall that, in view of our physical motivations, we seek solutions to (1) with cylindrical symmetry. Notice that, in general, we cannot expect extremals to (6) to be automatically cylindrical symmetric. To see this, let us consider the Sobolev critical case, where we take p = 6 into (1). We have already argued that, under assumption (3), the infimum is not attained. However, if we restrict to consider in (6) only functions u = u(r, z) with cylindrical symmetry, then we can rule out the possibility that a “concentration” phenomenon occurs along a minimizing sequence. In fact, among cylindrically symmetric functions, the infimum in (6) is attained when p = 6 (and clearly also when 4 < p < 6, see lemmas 5.7 and 5.9). However, the corresponding best constant (over cylindrically symmetric functions) must be strictly larger than the best constant relative to the whole space D1,2 (IR3 ). Such a gap must persist also when the power p is close to the critical power p = 6, yielding to a non cylindrically symmetric minimizer for (6) in this case. Thus, to make sure to obtain cylindrically symmetric solutions for (1), we shall investigate the extremal problem (6) over the subspace Dc1,2 (IR3 ) ⊂ D1,2 (IR3 ) of cylindrically symmetric functions and derive: Theorem 1.1 Let 4 < p ≤ 6 and φ satisfy (3), then problem (1) admits a cylindrically symmetric solution. We refer to theorem 3.2 below, for the complete statement of our results concerning (1). As already mentioned, the power p = 4 appears as a “critical exponent” case for (1), and at this point we can claim a solution to (1) only when φ(r) = 1r . It is clear from our discussion that a particularly delicate situation occurs when p = 4 and φ satisfies (2), where solutions to (1) (with or without cylindrical symmetry) no longer can be obtained by a mere minimization procedure, as illustrated above. We leave this very interesting problem for further investigation. The paper is organized a follows: after the introduction (section 1), in section 2 we prove a general inequality of Sobolev-Hardy type and provide an existence result for the corresponding extremal functions which attain the best constant in our inequality. In section 3 we use a special case of this Sobolev-Hardy inequality to derive a variational formulation for (1). We analyze the critical case p = 4 and establish a nonexistence result. We also state our main existence result in theorem 3.2, whose proof is given in section 4 relative to the case 4 < p ≤ 6. In section 5 we study the asymptotic behavior of the solutions found in the previous section, proving in particular that they have finite total mass. We wish to thank Giuseppe Bertin and Luca Ciotti, for suggesting the problem and for fruitful discussions. Notations We collect below a list of the main notations used throughout the paper. • For q > 1, D1,q (IRN ) is the closure of C0∞ (IRN ) with respect to the norm ||u|| =
R
IRN
|∇u|q dx
• Lq (IRN ), Lqloc (IRN ) are the usual Lebesgue spaces. • For 1 < q < N , q ∗ = NqN −q defines the critical exponent for the Sobolev embedding and Sq denotes the corresponding best constant, namely the greatest constant C satisfying C
Z
IRN
∗
|u|q dx ≤
for all u ∈ C0∞ (IRN ).
4
Z
IRN
q∗ /q |∇u|q dx
1/q
.
• If f : B → IR is a C 1 functional on the Banach space B, df (u) : B → IR defines its Fr´echet differential computed at u ∈ B. • For any P ∈ IRN and r > 0 we denote by B(P, r) or Br (P ) the open ball in IRN with center P and radius r, while Br = B(0, r). • We will write σ to denote a quantity vanishing when → 0 and σn for a quantity vanishing when n → +∞. • We will use C to indicate any fixed positive constant. • x · y is the usual scalar product in IRN . • N is the set of natural numbers. • We set IR+ = [0, +∞[. • S N −1 = {x ∈ IRN |x| = 1}.
2
A Sobolev-Hardy type inequality
Set IRN = IRk × IRN −k , with 2 ≤ k ≤ N , and write x = (x0 , z) ∈ IRk × IRN −k . For given real −s) numbers q, s such that 1 < q < N , 0 ≤ s ≤ q, and s < k, set q∗ (s, N, q) = q(N N −q . For sake of simplicity we will write q∗ (s) for q∗ (s, N, q). Theorem 2.1 There exists a positive constant C = C(s, q, N, k) such that for all u ∈ D1,q (IRN ) we have Z
IRN
|u|q∗ (s) dx ≤ C |x0 |s
Z
IRN
N −s N −q |∇u| dx .
q
(8)
Proof. Note that for s = 0, then q∗ (0) = q ∗ is the Sobolev embedding exponent, and inequality (8) reduces to the usual Sobolev embedding. So we assume s > 0 and to simplify notations we write q∗ in place of q∗ (s). We begin by introducing cylindrical coordinates in IRN as follows x0 = ρy,
ρ ∈ [0, +∞[,
y ∈ S k−1 ,
dx = dx0 dz = ρk−1 dρ dσ dz,
where dσ is the surface element of S k−1 . By an approximation procedure it will suffices to establish (8) for u ∈ C0∞ (IRN ). We can also assume u ≥ 0. Thus, Z +∞ d q∗ q∗ q∗ |u(x)| = |u(ρy, z)| = − u (λρy, z) dλ = dλ 1
5
−q∗
+∞
Z
uq∗ −1 (λρy, z)∇x0 u(λρy, z) · y ρ dλ,
1
where ∇x0 is the gradient with respect to the x0 −variables. By suitable change of variables we get Z uq∗ (x0 , z) 0 dx dz = |x0 |s IRN Z +∞ Z Z Z +∞ q∗ −1 u (λρy, z) −q∗ ∇x0 u(λρy, z) · yρk−1 dρ = dλ dz dσ s−1 N −k ρ k−1 1 IR S 0 Z Z +∞ Z Z +∞ q∗ −1 1 u (ry, z) k−1 0 −q∗ dλ ∇ u(ry, z) · yr drdσdz = x λk−s+1 rs−1 1 IRN −k S k−1 0 Z Z q∗ uq∗ −1 (x0 , z) x0 0 0 u(x , z) · − ∇ dx0 dz. (9) x k − s IRN −k IRk |x0 |s−1 |x0 | We distinguish the following two cases: Case 1. 1 ≤ s < k. Let us first assume s < q. In view of (9) by H¨older inequality we find Z Z uq∗ (x0 , z) 0 q∗ uq∗ −1 (x0 , z) dx dz ≤ |∇x0 u(x0 , z)|dx0 dz = |x0 |s k − s IRN |x0 |s−1 IRN Z s−1 q∗ uq∗ s (x0 , z) q∗ −1 0 u s (x , z)|∇x0 u(x0 , z)|dx0 dz ≤ k − s IRN |x0 |s−1 Z s−1 Z q−s Z 1/q s sq q∗ uq∗ (x0 , z) 0 q∗ 0 q dx dz u |∇x0 u(x , z)| . k−s |x0 |s IRN IRN IRN Therefore, by Sobolev inequality we obtain Z uq∗ (x0 , z) 0 dx dz ≤ |x0 |s IRN Z s/q s Z q−s q q∗ q∗ |∇x0 u(x0 , z)|q dx ≤ u dx k−s IRN IRN
q∗ k−s
s
1 q−s
Sq q
− q−s Sq q
Z
IRN
q∗ k−s
q∗ s q−s q q +q |∇u(x)| dx =
q
s Z
N −s N −q |∇u(x)| dx ,
q
IRN
so the desired inequality is established with 1
C(s, q, N, k) =
q−s
Sq q
q∗ k−s
Similar, and simpler, arguments give the result when s = q.
6
s
.
(10)
Case 2. 0 < s < 1. We treat this case by interpolation between the case s = 0 (Sobolev’s inequality) and the case s = 1 (established above). For this purpose notice that q∗ (s) = (1 − s)q ∗ + sq∗ (1), where we recall q∗ (1) = Z
IRN
1 Sq1−s
q(N −1) N −q .
Consequently,
q∗ (1) s Z u (x) uq∗ (x) 0 (1−s)q ∗ dx dz = u (x) dx0 dz ≤ |x0 |s |x0 | IRN Z 1−s Z s uq∗ (1) (x) q∗ dx ≤ u dx |x0 | IRN IRN
1 Sq
s q−1 q 1 Sq
q−s q
N −1 s Z qq∗ (1−s)+ N −q s q∗ (1) q |∇u| dx = k−1 IRN N −s s Z N −q q∗ (1) q |∇u| dx . k−1 IRN
The desired inequality in this case follows as well by taking s q∗ (1) 1 . C(s, q, N, k) = q−s k−1 Sq s
(11)
Remark 2.2 Inequality (8) generalizes the analogous inequality obtained by Caffarelli, Kohn and Nirenberg ([4]) for the case k = N , as an interpolation between the Hardy’s inequality, where s = q, and the Sobolev’s inequality corresponding to s = 0.
Remark 2.3 Notice in particular that for 1 < q < k and s = q we have the following extension of Hardy’s inequality: Z Z uq (x) dx ≤ C |∇u|q dx. (12) 0 |q N N |x IR IR For k = N it is well known (e.g. [9]) that the best constant in (12) is given by q q C = CN,q = N −q and it is never attained. So the estimate for C as given in (10) is sharp in this case. This suggests that, in case 2 ≤ k ≤ N and 1 ≤ q < k, the best constant in our generalized version of Hardy’s inequality (12) is given by q q C = Ck,q = . k−q
7
Remark 2.4 Notice that (10) and (11) give an estimate for the best constant in (8) in terms of the best Sobolev’constant Sq respectively when s ≥ 1 and 0 < s < 1.
In analogy to the Sobolev’s best constant we introduce the best constant in (8) by setting Z Z |u|q∗ N q 1,q S = inf |∇u| dx u ∈ D (IR ), dx = 1 . (13) 0 s IRN IRN |x | Clearly, S depends on k, q, s and N and reduces to the best Sobolev constant for s = 0 and to the best Hardy constant when s = q and k = N . At this point it is natural to ask whether or not the best constant S is attained by some extremal functions. The answer to this question is positive except for the case where (8) reduces to the Hardy inequality, i.e. s = q, where we know that this is never the case (at least when k = N ).
Theorem 2.5 Assume 1 ≤ k ≤ N , 1 < q < N , x = (x0 , z) ∈ IRk × IRN −k , 0 ≤ s < q and s < k. Then the extremal problem (13) attains its infimum at a function u ∈ D1,q (IRN ) which satisfies Z Z |u|q∗ |∇u|q dx = S, dx = 1. 0 s IRN IRN |x |
Remark 2.6 As an immediate consequence of Theorem 2.5 we get that the problem 1 q−2 q −1 −div(|∇v| ∇v) = |x0 |s v ∗ 3 v(x) > 0 in IR v ∈ D1,q (IRN )
(14)
1
admits the non trivial solution v = S − q∗ −q u where u is an extremal function for (13) as given by Theorem 2.5.
Remark 2.7 Notice that, for every λ > 0 and ζ ∈ IRk , the extremal problem (13), as well as (14), are invariant under the trasformation u → uλ,ζ with uλ,ζ (x0 , z) = λ
N −q q
u(λx0 , λ(z − ζ)).
(15)
In particular, if u minimizes (13) so does uλ,ζ for every λ > 0 and ζ ∈ IRk .
Remark 2.8 In the particular case where k = N , the analogous of Theorem 2.5 was establisehd by Ghoussoub and Yuan in [11], where an explicit expression for all (radial) extremal functions is derived which includes the Sobolev case, corresponding to s = 0, as a particular case.
8
We will obtain the proof of Theorem 2.5 through several lemmata. In particular we will use a suitable formulation of the two Concentration-Compactness principles by P.L.Lions. We recall that these principles are general tools that must be worked out in each particular situation. However, the proof of the first Concentration-Compactness principle we use (lemma 2.9 below) coincides exactly with that one finds in the quoted papers ([16, 17]), so we just state the result without proof, in this case. On the other hand we give a complete proof for the particular version of the second Concentration-Compactness we need in lemma 2.11 below. Throughout this section by {un }n we denote a minimizing sequence for(13), and we will use the same notations for any of its subsequences. So we assume that Z Z |un |q∗ |∇un |q dx → S, dx = 1. (16) 0 s IRN IRN |x | For R > 0 and ζ ∈ IRk define ΩR (ζ) = {(x0 , z) ∈ IRk × IRN −k |x0 | + |z − ζ| < R}.
We state, without proof, the first form of P.L.Lions’ Concentration-Compactness Principle. Lemma 2.9 Let {un }n satisfy (16). Along a subsequence (still denoted by {un }n ), one of the following three alternatives hold: i) There exist a sequence {ζn }n ⊂ IRN −k such that, for all > 0 there exist n ∈ N and R > 0 such that Z |un |q∗ dx > 1 − , ∀ n ≥ n , ∀R ≥ R . 0 s ΩR (ζn ) |x | h R ii) For all R > 0 limn supζ∈IRN −k ΩR (ζ)
|un |q∗ |x0 |s
i dx = 0 .
iii) There exists α ∈]0, 1[ such that for all > 0 there exist R > 0, a sequence of positive numbers Rn → +∞ and a sequence {ζn }n ⊂ IRN −k such that Z Z |un |q∗ |un |q∗ α) (17) α < , dx − (1 − dx − < , IRN \ΩRn (ζn ) |x0 |s ΩR (ζn ) |x0 |s Z |un |q∗ dx < . (18) 0 s ΩRn (ζn )\ΩR (ζn ) |x | Our first task is to prove that only alternative i) can actually occur. Indeed, we immediately rule out case ii) in view of the invariance of (13) under the trasformation (15). In fact, by replacing un N −q
with λn q u(λn x0 , λn (z − ζn )) (still a minimizing sequence satisfying (16)), for a suitable choice of λn > 0 and ζn ∈ IRk , we can always assume that Z Z Z Z |un |q∗ 0 |un |q∗ 0 1 sup dx dz = dx dz = . (19) 0 s 0 |s |x | |x 2 N −k 0 0 |z−ζ| 1, we derive Z Z Z q q q |∇un | dx = ϕn |∇un | dx + Kn \Ωn
Kn \Ωn
Z
(1 − ϕqn )|∇un |q dx ≥
Kn \Ωn
ϕqn |∇un |q dx +
Kn \Ωn
Z
(1 − ϕn )q |∇un |q dx.
Kn \Ωn
Consequently Z
q
IRN
|∇un | dx ≥
Z
ϕqn |∇un |q dx
+
Z
IRN \Ωn
Kn
Z
ϕqn |∇un |q dx +
IRN
Z
IRN
(1 − ϕn )q |∇un |q dx =
(1 − ϕn )q |∇un |q dx,
and (23) is establisehd. At this point we can use the Calculus lemma to obtain Z
IRN
ϕqn |∇un |q dx Z
IRN
+
Z
IRN
C
q
IRN
q
(1 − ϕn ) |∇un | dx =
IRN
IRN
|∇(ϕn un ) − un ∇ϕn |q dx+
|∇(1 − ϕn )un + un ∇ϕn | dx ≥
Z
Z
|un |q |∇ϕn |q dx−
|∇(1 − ϕn )un |q dx + 2
|∇(1 − ϕn )un |q−1 |un | |∇ϕn |dx +
Z
Z
q
IRN
C
Z
q−1
|∇(ϕn un )|
Z
IRN
Z
IRN
|un | |∇ϕn |dx +
Z
IRN
IRN
|∇(ϕn un )|q dx+
|∇(1 − ϕn )un ||un |q−1 |∇ϕn |q−1 dx − q−1
|∇(ϕn un )||un |
q−1
|∇ϕn |
dx .
(24)
We prove that the last terms in the previous inequality vanish as → 0, uniformly in n. For this purpose first notice that Z
q−1
IRN
|∇(ϕn un )|
q−1
|un | |∇ϕn |dx ≤ 2
Z
q
IRN
q
|un | |∇ϕn | dx +
Z
q−1
IRN
|∇un |
and Z
IRN
q−1
|∇(ϕn un )||un |
q−1
|∇ϕn |
dx ≤
Z
IRN
q
q
|un | |∇ϕn | dx + 11
Z
IRN
|un | ϕq−1 n |∇ϕn |dx
|∇un ||un |q−1 ϕn |∇ϕn |q−1 dx.
We estimate each term above separately. Clearly, q Z Z 1 |∇ϕn |q |un |q dx ≤ C |un |q dx, N R − R n IR An, where An, = ΩRn (ζn )\ΩR (ζn ). By an application of H¨older inequality, and recalling (18), we obtain Z Z 1 |un |q dx = |un |q 0 sq/q |x0 |sq/q∗ dx ≤ ∗ |x | An, An, !q/q∗ 1 |un | dx |x0 |s An,
Z
Z
q∗
0
|x |
sq q∗ −q
! q∗q−q ∗
dx
q/q∗
≤
Z
An,
0
|x |
sq q∗ −q
! q∗q−q ∗
dx
.
(25)
An,
Notice that in Lemma 2.9, case iii) we can always choose Rn ≥ 2R . Setting γ = Z Z Z 0 γdx |x | ≤ dz |x0 |γ dx0 ≤ CRnN −k
Z
we have
|x0 | 0, we may extract a subsequence and assume that Z |un |q∗ dx → α , 0 s ΩR (ζn ) |x | as n → ∞. Notice that such a sequence {un }n also depends on . By the (contradiction) assumption R (17) obviously |α − α| ≤ , so that α → α as → 0. Set σn = IRN |∇un |q dx − S. Clearly σn also depends on and, by (16), σn → 0 as n → ∞. By Lemma 2.10 and (18) we have Z Z Z S + σn = |∇un |q dx ≥ |∇ϕun |q dx + |∇(1 − ϕn )un |q dx − σ ≥ IRN
"Z
S
IRN
S
Z ΩR (ζn )
|ϕn un |q∗ |x0 |s
IRN
IRN
q/q∗ # |(1 − ϕn )un |q∗ dx − σ ≥ |x0 |s IRN !q/q∗ Z |ϕn un |q∗ − σ . 1− dx + σ |x0 |s ΩR (ζn )
q/q∗ Z dx +
!q/q∗ |ϕn un |q∗ dx + |x0 |s
Consequently, for a fixed > 0, if we pass to the limit for n → ∞, we find S ≥ S αq/q∗ + (1 − α + σ )q/q∗ − σ . At this point we can let → 0 to obtain S ≥ S αq/q∗ + (1 − α)q/q∗ , which gives the desired contradiction, as α ∈]0, 1[. In conclusion, we see that we can always arrange a minimizing sequence for (13) to satisfy alternative i) of Lemma 2.9. More precisely, replacing un (x0 , z) with un (x0 , z + ζn ) (still a minimizing sequence for (13)), we can always assume the following holds:
13
Z
∀ > 0 ∃n ∈ N, ∃R > 0 such that ∀n ≥ n , ∀R ≥ R :
ΩR (0)
|un |q∗ > 1 − . |x0 |s
(27)
To go on with the proof of our main result we have now to apply the second form of the Concentration-Compactness principle. As we mentioned above, we state and prove a suitable form of such principle relative to the case s > 0 and refer to [18, 19] for the original version which applies directly for s = 0. Lemma 2.11 Let {un }n ⊂ D1,q (IRN ) be a sequence with the property that there exist two Radon measures µ, ν and a function u ∈ D1,q (IRN ) such that, for some s > 0, i) un * u weakly in D1,q (IRN ). ii) νn = |un |q∗ |x10 |s dx * ν, weakly in the sense of measures. iii) µn = |∇un |q dx * µ, weakly in the sense of measures. Then, there exist an at most countable set of indices J, a corresponding set of points {zj ∈ IRN −k | j ∈ J}, and two sets {µj }j , {νj }j of non negative numbers such that ν = |u|q∗
X 1 dx + νj δj , |x0 |s
µ ≥ |∇u|q dx +
j∈J
X
µj δj
and
q/q∗
µj ≥ Sνj
∀j ∈ J,
j∈J
where δj = δ(0,zj ) is the Dirac measure with pole at (0, zj ) ∈ IRN . Proof. Nq N −s ∗ ∗ Notice first that, for s > 0, q∗ = q∗ (s) = q N −q < q , where q = N −q is the critical Sobolev exponent. Hence we can further assume, by compactness of Sobolev’embedding, that ∗ un → u in Lqloc (IRN ) and pointwise a.e.
In particular |un |q∗
1 1 → |u|q∗ 0 s pointwise a.e. 0 s |x | |x |
Let ϕ ∈ C0∞ (IRN ), we have Z
q∗
IRN
|ϕ|
1 1 |un | dx ≤ q /q |x0 |s S ∗ q∗
By the Calculus lemma we estimate Z Z |∇ϕun |q dx =
IRN
Z
IRN
q
q
|ϕ| |∇un | dx +
Z
Z
IRN
IRN
(28)
Z
|ϕ|q |∇un |q dx+
|ϕ|q−1 |∇ϕ| |∇un |q−1 |un |dx + C
Z
|ϕ||∇ϕ|q−1 |∇un ||un |q−1 dx ≤
IRN
|∇ϕ|q |un |q dx + C
IRN
q∗ /q |∇ϕun | dx . q
|ϕ∇un + un ∇ϕ|q dx ≤
IRN
Z
Z
q
q
|∇ϕ| |un | dx + C
Z
IRN
14
q
IRN
IRN
q
|ϕ| |∇un | dx
q−1 Z q
IRN
q1 |∇ϕ| |un | dx + q
q
C
Z
IRN
C
Z
IRN
q
q
|ϕ| |∇un | dx
Z q−1 q |ϕ|q dµ
q1 Z
IRN
q−1 Z q |∇ϕ| |un | dx ≤ q
IRN
q
q
IRN
|∇ϕ|q |u|q dx
q1
+C
Z
IRN
|ϕ| dµ +
q1 Z |ϕ|q dµ
IRN
Z
IRN
|∇ϕ|q |u|q dx+
|∇ϕ|q |u|q dx
q−1 q
+ o(1) (29)
as n → ∞, since Z
IRN
|ϕ|q |∇un |q dx →
Z
IRN
|ϕ|q dµ and un → u in Lqloc (IRN )
as 1 < q ≤ q∗ . We start by analizing the case where u = 0. From (28) and (29) we get Z
IRN
|ϕ| dν ≤
Z
1
q∗
S q∗ /q
IRN
q
|ϕ| dµ
q∗ /q
and by an approximation procedure we derive 1 µ(Ω)q∗ /q , S q∗ /q for all bounded borelian sets Ω. Let us now write X ν = ν0 + νj δxj , ν(Ω) ≤
(30)
j∈J
where νj = ν({xj }), {xj | j ∈ J} is the set of atoms of ν and ν0 is free of atoms. As ν(IRN ) < +∞, we derive that necessarily J is countable. Also ν0 is absolutely continuous with respect to µ, hence ν0 = f dµ with f ∈ L1 (IRN , dµ) and f (x) = lim
r→0
ν0 (Br (x)) µ − a.e. µ(Br (x))
(31)
As ν0 ≤ ν, from (30) and (31) we obtain that, if x is not an atom of µ, then f (x) = 0. From this we easily derive that ν0 = 0. Indeed, since µ(IRN ) < +∞, as above we see that the set E of atoms for µ is at most countable. Since ν0 is free of atoms, ν0 (E) = 0. Thus, we conclude that, for any borelian set B, Z ν0 (B) = ν0 (B ∩ E) + ν0 (B\E) = ν0 (B\E) = f (x)dµ = 0, B\E
since f (x) = 0 in B\E. Consequently ν=
X
νj δxj ,
j∈J
and, by setting µj = µ({xj }), from (30) we derive νj ≤
1 q /q µ∗ . S q∗ /q j
Clearly, µ≥
X
µj δxj .
j
15
Hence, we obtain that the lemma is proved in the case u = 0 (excepted for the fact that xj = (0, zj ), which we will prove later). Now we turn to the general case. From the well known result of Brezis and Lieb ([3]) we obtain |un |q∗
1 1 1 dx − |un − u|q∗ 0 s dx − |u|q∗ 0 s dx * 0 0 s |x | |x | |x |
weakly in the sense of measures. On the other hand we can apply the previous results to the sequence {un − u}n and obtain |un − u|q∗
X 1 dx * ν˜ = νj δxj , 0 s |x | j∈J
so that ν = |u|q∗
X 1 1 dx + ν˜ = |u|q∗ 0 s dx + νj δxj . 0 s |x | |x | j∈J
To obtain the rest of our lemma, we are going to exploit (28) and (29) with an appropriate choice of test function. To this purpose let ϕ ∈ C0∞ (B2 ), 0 ≤ ϕ ≤ 1 and ϕ = 1 in B1 . For > 0 define x − xj ϕ (x) = ϕ , for a given j ∈ J. We apply (28) and (29) and obtain "Z Z Z 1 q∗ q ϕ dν ≤ q /q ϕ dµ + |u|q |∇ϕ |q dx+ S ∗ IRN IRN IRN Z Z q−1 #q∗ /q 1/q
C
IRN
|u|q |∇ϕ |q dx
+C
IRN
|u|q |∇ϕ |q dx
q
.
(32)
Notice that
Z
IRN
q
q
|u| |∇ϕ | dx =
Z
q
q
|u| |∇ϕ | dx ≤
Z
B2 (xj )
On the other hand we get
R
B2 (xj ) q∗
B2 (xj )
|u|
!q/q∗
B2 (xj )
An easy change of variables shows that Z Z |∇ϕ |N dx = B2 (xj )
q∗
Z
N
|∇ϕ | dx
!q/N
.
B2 (xj )
Z 1 x − xj N dx = |∇ϕ|N dy. (∇ϕ) N B2
|u| dx → 0 as → 0. Hence, passing to the limit in (32), as → 0,
1 µ({xj })q∗ /q , S q∗ /q that is to see that xj is also an atom for the measure µ and ν({xj }) ≤
q/q∗
µj ≥ S νj
.
To conclude notice that, by weak convergence, µ ≥ |∇u|q dx. Clearly, µ ≥ µj δxj , and since the measures |∇u|q and the δxj ’s are orthogonal to each other, we derive 16
µ ≥ |∇u|q dx +
X
µj δxj .
j
At this point all it remains to prove is that xj = (0, zj ) for suitable zj ∈ IRN −k . For this purpose ∗ we notice that, since un → u in Lqloc (IRN ), then for all > 0 we have Z Z Z Z 1 1 q∗ |un − u| dx ≤ s |un − u|q∗ dx → 0. (33) 0 |s |x 0 0 |z| 0 such that tu u ∈ M (respectively tu u ∈ Mc ). The value tu is determined by the equation Z t||u||2 − tp−1 φ |u|p dx = 0 IR3
which gives tu =
R
||u||2 φ |u|p IR3
1 p−2
.
Then, for u 6= 0 we get I(tu u) =
1 1 − 2 p
t2u ||u||2 =
1 1 − 2 p
19
||u||2
R
||u||2 φ |u|p IR3
2 p−2
=
1 1 − 2 p
||u||2
R
IR3
φ |u|p
2/p
p ! p−2
.
Therefore, if we define the best constant in (34) as given by Z Z N 2 1,2 p L = inf φ(r)|u| dx = 1 ∇u| dx | u ∈ D (IR ) , IR3
respectively
(39)
IR3
Lc = inf
Z
IR3
Z N 1,2 |∇u| dx u ∈ Dc (IR ) , 2
IR3
φ(r)|u| dx = 1 p
(40)
then
inf I(u) =
u∈M
1 1 − 2 p
L
p p−2
respectively
inf I(u) =
u∈Mc
1 1 − 2 p
p p−2
Lc
,
and I attains its infimum on M (respectively Ic attains its infimum on Mc ) if and only if the best constant L (respectively Lc ) is attained. We show that this is always the case when 4 < p < 6 (for both minimization problems (37) and (38)) leading to a solution for (1). On the other hand, extremals to (37) or equivalently (39) are not to be expected when p = 6 or p = 4, which appear as “critical” values. This is not surprisingly when p = 6, the critical Sobolev exponent in dimension N = 3. In fact in this case, if we normalize φ so that maxIR+ φ = 1, then the best constant coincides exactly with the best Sobolev constant S2 . So it will be attained if and only if φ ≡ 1, which is clearly not the case when φ satisfies (3). However, if we limit our attention to function with cylindrical symmetry, then the relative extremal problem (38) (or, equivalently, (40)) does attain its extremal yielding to a cylindrically symmetric solution to (1) with p = 6. The situation is much more delicate for the value of the power p = 4. In fact, neither one of the above minimization problems can attain their infimum in this case, when rφ(r) is assumed a non-constant, monotone increasing function, as it is the case for the original choice of Bertin and Ciotti given in (2). We summarize the situation in the following statement. Theorem 3.2 Let φ satisfy (3). i) If 4 < p < 6 then I (respectively Ic ) attains its infimum over M (respectively Mc ) at a solution u (respectively a cylindrically symmetric solution uc ) for (1). These solutions are also extremals for the best constant in (34) over D1,2 (IRN ) (respectively Dc1,2 (IRN )). Furthermore, there exists p0 ∈ (4, 6) such that if p0 < p < 6 then L < Lc ; therefore u and uc define two different solution for (1) in this case, and the best constant in (34) is attained at a function which is not cylindrically symmetric. ii) If p = 6 then I cannot attain its infimum over M . On the contrary, Ic attains its infimum over Mc at a cylindrically symmetric solution for (1) which corresponds to an extremal for the best constant in (34) over Dc1,2 (IRN ). iii) If p = 4 and the function r → rφ(r) is assumed increasing and not constant then neither I nor Ic can attain their infimum over M or Mc respectively. Before turning to the proof of Theorem 3.2 we wish to draw the attention once more to the particularly delicate case p = 4. In this situation, at this point we can claim a solution for (1) 20
only when φ(r) = 1r , as given by the extremal functions of our Sobolev-Hardy inequality (see Theorem 2.1 with N = 3, k = 2 and s = 1). While, the case where φ(r) =
r2α 1
(1 + r2 ) 2 +α
α ≥ 0,
as proposed by Bertin and Ciotti, remains completely open for further investigations. We treat case i), ii), iii) of theorem 3.2 separately. We start with the proof of theorem 3.2 iii). Proof of iii). We argue by contradiction and assume that there exist w ∈ M such that I(u) = inf M I. This means that Z 2 ||w|| = φ(r)|w|4 dx IR3
and ||w||2 ≤ ||u||2 , ∀u ∈ M. We can also assume w ≥ 0 in IR3 . For λ > 0, let wλ (x) = λ1/2 w(λx) (notice that ||w|| = ||wλ || and w = w1 ) and set 12 ||wλ ||2 R tλ = . φ |wλ |4 IR3 Thus, tλ wλ ∈ M , and therefore ||w||2 ≤ ||tλ wλ ||2 = t2λ ||wλ ||2 = t2λ ||w||2 , hence t2λ ≥ 1, so that 2
||wλ || ≥
Z
IR3
φ |wλ |4 dx.
On the other hand, introducing cylindrical coordinates over IR3 and using the monotonicity of rφ(r), we have Z 2π Z Z +∞ r 1 r r 4 4 φ |wλ | dx = φ |w| dx = φ w (rcosθ, rsinθ, z) drdzdθ > 3 3 λ λ λ λ IR IR 0 IR 0
Z
Z
4
Z 0
2π
Z Z IR
+∞
φ(r)r w4 (rcosθ, rsinθ, z)drdzdθ =
Z
IR3
0
a contradiction. The same argument works for Ic and Mc .
21
φ(r) w4 dx = ||w||2 = ||wλ ||2 ,
4
The case 4 < p ≤ 6
We devote this section to derive part i) and ii) of theorem 3.2. For this purpose we recall a well known application of Ekeland’s principle to select the “best” minimizing sequence for a functional over the corresponding Nehari’s manifold. We refer to [10] (corollary 1.10 p. 7) for the proof. Lemma 4.1 Let f be a C 2 functional on an Hilbert space H that is coercive and bounded from below on the set Mf = {u ∈ H | u 6= 0, ψ(u) := df (u)u = 0}. Suppose dψ(u)u 6= 0 for any u ∈ Mf and that for every sequence {un }n that is minimizing for f on Mf , we have that {dψ(un )}n is bounded in the dual space H 0 and lim supn |dψ(un )un | > 0. Then, for every minimizing sequence {vn }n for f in Mf , there exists a sequence {un }n in Mf such that f (un ) ≤ f (vn ), limn ||un − vn || = 0 and limn ||df (un )|| = 0 (this last norm is that of H 0 ). It is easy to check that the hypotheses of lemma 4.1 are verified in our problem, when we set H = D1,2 (IR3 ) (respectively H = Dc1,2 (IR3 )), f = I (respectively f = Ic ), Mf = M (respectively Mf = Mc ). Clearly I (respectively Ic ) is C 2 , bounded R from below and also coercive on M (respectively Mc ). Furthermore dψ(u)u = 2||u||2 − p IR3 φ|u|p dx = (2 − p)||u||2 6= 0 on M (respectively on Mc ). Assume {un }n is a minimizing sequence for I in M (respectively for Ic in Mc ). Then for suitable constants 0 < γ ≤ C we have Z Z 2 0 0, and all the hypotheses of lemma 4.1 are verified, so we obtain the following lemma: Lemma 4.2 There is a minimizing sequence {un }n for I on M (respectively for Ic on Mc ) satisfying limn ||dI(un )|| = 0 (respectively limn ||dIc (un )|| = 0). We investigate further properties of such minimizing sequences that yield to the strong convergence (of a subsequence) to the desired minimizer. Lemma 4.3 Let {un }n be a minimizing sequence asp given by lemma 4.2. There exist R0 > r0 > 0 such that, if Ω0 = {x = (x1 , x2 , x3 ) ∈ IR3 | r0 < r = x21 + x22 < R0 }, then Z φ|un |p dx ≥ γ/2. Ω0
22
Proof. Write Z
IR3
φ|un |p dx =
Z
φ|un |p dx +
Z
Ω0
IR3 \Ω0
φ|un |p dx
and set Ω1 = IR3 \Ω0 . Consequently, Z
p
φ|un | dx ≤
Z
C
4
IR3
φ|un | dx
Z 6−p 2
φ|un | dx
6−p 2
6
φ|un |
p−4 2
≤
Ω1
Ω1
Ω1
Z
4
||un ||3(p−4) (max{φ(r) | r ∈ [0, r0 ] ∪ [R0 , +∞)})
C||un ||p (max{φ(r) | r ∈ [0, r0 ] ∪ [R0 , +∞)})
p−4 2
p−4 2
≤
.
By (41) we obtain Z
p
φ|un | dx ≤ C
Ω1
max
r≤r0 ,R0 ≤r
p−4 2 φ(r)
for suitable C > 0. Since φ(r) → 0 as r → 0 or r → +∞ and p > 4, we may choose r0 and R0 such that Z φ|un |p dx ≤ γ/2, Ω1
and immediately derive the conclusion.
Let us define now Q = {x ∈ IR3 r0 < r < R0 , 0 < x3 < 1},
Qj = ζj + Q,
where ζj = (0, 0, j), j an integer. We also define Z sk = sup φ |uk |p dx. j
Qj
Notice that ∪j Qj = Ω0 and Qj ∩ Qk = ∅ for j 6= k. Lemma 4.4 lim inf k sk > 0. Proof. By hypothesis (3) there exist constants 0 < c0 ≤ c1 such that 0 < c0 ≤ φ(r) ≤ c1
∀r ∈ [r0 , R0 ].
(42)
For all j, k ∈ N set vk,j (x) = uk (x + ζj ). By (42) we have Z
p
!4/p
φ|uk | dx
≤
Z
4/p c1
Qj
p
|uk | dx
!4/p
Qj
C
"Z
4/p
= c1
Z
4/p |vk,j |p dx ≤
Q
1/2 Z 1/2 #4 2 |∇vk,j | dx + |vk,j | dx ≤ 2
Q
Q
23
C
"Z
|∇vk,j |2 dx
1/2
+
Z
Q
Q
C
Z
!2
2
|∇uk | dx
+
1/4 #4 Z 4 |vk,j | dx ≤C
!1/2
|∇uk |2 dx
Z
+
Qj
Z
Qj
4
Qj
C
"
2
|uk | dx ≤ C ||uk ||
"Z
2
|∇uk | dx +
Z
Qj
Qj
Z
2
|∇uk | dx +
Z
Qj
!1/4 4 ≤ |uk |4 dx 4
#
!4/p
≤
φ|uk | dx ≤
Qj
#
4
φ|uk | dx .
Qj
As a consequence we obtain Z
φ|uk |p dx =
Ω0
" X Z 1−4/p
Csk
φ|uk | dx
#1−4/p "Z
XZ j
|∇uk |2 dx +
Qj
Ω0
Qj
Z
IR3
≤
1−4/p sk
X Z
1−4/p
φ|uk |4 dx = Csk
|∇uk |2 dx +
Z
IR3
Z
φ|uk |4 dx
p
φ|uk | dx
Qj
j
XZ j
1−4/p
R
φ|uk | dx
#4/p
Qj
Csk Since
p
φ|uk |p dx =
Qj
j
Qj
j
p
XZ
|∇uk |2 dx +
Ω0
Z
φ|uk |4 dx ≤
Ω0 1−4/p
≤ Csk
.
φ|un |4 dx ≥ γ/2 by lemma 4.3 and p > 4, we immediately get the desired conclusion.
By lemma 4.4 there exist k0 ∈ N and γ1 > 0 such that sk > γ1 , for all k ≥ k0 . Notice that for any k we find jk ∈ N such that Z sk = φ|uk |p dx. Qjk
Define ζjk = (0, 0, jk ) and let vk (x) = uk (x + ζjk ). Since our problem is invariant under translations along the x3 −axis, we have that also vk is Ra minimizing sequence and dI(vk ) → 0 (respecR tively dIc (vk ) → 0). Furthermore Q φ|vk |p dx = Qj φ|uk |p dx = sk > γ1 for suitable γ1 > 0 and k k ≥ k0 . Lemma 4.5 Let 4 < p < 6. Then, up to a subsequence, the sequence {vn }n converges strongly to v0 which defines a minimizer for I over M (respectively for Ic over Mc ). Proof. Since also the sequence {vn }n satisfies the estimate (41), along a subsequence we may assume that vn * v0 weakly in D1,2 (IR3 ) (respectively Dc1,2 (IR3 )), vn (x) → v0 (x) pointwise a.e. and in Lploc (IR3 ). Recall that by assumption p < 6 and therefore Z Z γ1 ≤ lim φ|vk |p dx = φ|v0 |p dx. k
Q
Q
So v0 6= 0, and since dI(vk ) → 0 (respectively dIc (vk ) → 0), we have that v0 satisfies the equation
24
−∆v0 = φ|v0 |p−2 v0 ,
in D1,2 (IR3 )
respectively in Dc1,2 (IR3 ) ,
which implies v0 ∈ M (respectively v0 ∈ Mc ). Consequently,
1 1 − 2 p
2
||v0 || = I(v0 ) ≥ inf I = lim I(vk ) = M
k
1 1 − 2 p
2
||vk || ≥
1 1 − 2 p
||v0 ||2 .
Thus, limk ||vk ||2 = ||v0 ||2 , that is, {vk }k converges strongly to v0 in D1,2 (IR3 ) (respectively Dc1,2 (IR3 ) and I(v0 ) = inf M I (respectively Ic (v0 ) = inf Mc Ic ). The first part of theorem 3.2 i) is therefore established. The second part of the statement i) will be postponed at the end of the section, after we have established part ii) of theorem 3.2. For this purpose notice that lemma 4.5 relies in a crucial way upon the assumption that p < 6 and it fails to hold for p = 6. Lemma 4.6 Let p = 6 and φ satisfy (3), then the functional I cannot attain its infimum on M . Proof. It is equivalent to prove that the extremal problem (39) cannot attain its infimum. For this purpose, note that by assumption (3) φ attains its maximum value in IR+ , and, without loss of generality, we may take maxIR+ φ = 1. Comparing the best constant for (34), as defined in (39), with the best Sobolev constant S2 we immediately derive L ≥ S2 . On the other hand, if we take the extremal functions for the Sobolev inequality as test functions in (39), we can assure also the reverse inequality as follows. Let λ > 0 and x0 = (x0,1 , x0,2 , x0,3 ) ∈ IR3 , it is well known that the function (3λ)1/4 uλ (x) = p λ + |x − x0 |2
satisfies
R |∇uλ |2 dx |∇u1 |2 dx IR3 = 1/3 = S2 . R R 1/3 u6 dx u6 dx IR3 λ IR3 1
R
IR3
On the other hand, by a simple change of variables we find Z q Z 6 2 2 x0,1 + x0,2 u61 dx as λ → +∞. φ(r)uλ dx → φ IR3
IR3
q Hence, by choosing x0 ∈ IR such that r0 = x20,1 + x20,2 satisfies φ(r0 ) = maxIR+ φ = 1, we conclude R R |∇u1 |2 dx |∇uλ |2 dx IR3 IR3 = → S2 as λ → +∞. L≤ R R 6 dx 1/3 6 dx + o(1) 1/3 φu u 3 3 1 λ IR IR 3
That is, L = S2 , and so L is attained if and only if φ ≡ 1, which is certainly not the case when φ satisfy (3). It is clear from the proof of lemma 4.6 that the reason which prevents the infimum in (39) to be attained is given by a “concentration” phenomena occurring for a minimizing sequence. It is also clear that, if we restrict to the class of cylindrically symmetric function such a behavior is no longer allowed for the corresponding minimizing sequence and we have the following:
25
Lemma 4.7 If p = 6 and φ satisfies (3) then the functional Ic attains its infimum on Mc . Proof. By virtue of lemmata 4.3 and 4.4, Ic admits a minimizing sequence {vk }k ⊂ Dc1,2 (IR3 ) with the following properties. R R i) 0 < γ ≤ IR3 |∇vk |2 dx = IR3 φ(r)|vk |6 dx ≤ C for all k ∈ N. R ii) dIc (vk ) → 0 and there exist γ1 > 0 and k0 ∈ N such that Q φ(r)|vk |6 dx ≥ γ1 > 0, for all k ≥ k0 . Hence, passing, if necessary, to a subsequence, we can furter assume that vk → v
strongly in L2loc (IR3 ),
weakly in L2loc (IR3 ),
Di vk * Di v
and there exist two Radon measures µ, ν such that |vk |6 dx * ν,
|∇vk |2 * µ,
weakly in the sense of measures. Furthemore we can apply P.L.Lions’ Concentration-Compactness results (see lemma I.1 in [18] or theorem 9 in [8]) to derive a set of indices J, at most countable, a set of points {xj | j ∈ J}, and non negative numbers νj , µj (j ∈ J), such that X X ν = |v|6 dx + νj δj , µ ≥ |∇v|2 dx + µj δj , νj ≤ S2−3 µ3j , j
j
where S2 , as above, is the best constant for the Sobolev embedding. In view of our symmetry assumption we see that the set of concentration points {xj | j ∈ J} must lie along the x3 −axis. Indeed, for every j ∈ J write xj = (x0j , zj ) with x0j ∈ IR2 , zj ∈ IR and suppose that x0i 6= (0, 0) for some i ∈ J. We show that necessarily νj = 0. To this purpose, denote by A the set of all rotations over IR2 and let Ai = {R ∈ A | Rx0j = x0i for some j ∈ J}. Clearly, Ai is a countable set while A has the cardinality of the continuum. So we can pick R0 ∈ A\Ai to satisfy R0 x0j 6= x0i for all j ∈ J. Let R : IR3 → IR3 the rotation over IR3 defined by R(x0 , z) = (R0 x0 , z). Clearly, Rxj 6= xi for all j, and by the cylindrical symmetry of vk we have vk (Rx) = vk (x). For each > 0 let us choose φ ∈ Cc∞ (IR3 ) such that 0 ≤ φ ≤ 1, φ ≡ 1 in B (xi ), φ ≡ 0 in IR3 \B2 (xi ). By the change of variables x = Ry we get Z Z Z φ |vk |6 dx = φ (Ry)|vk (Ry)|6 dy = IR3
IR3
φ (Ry)|vk (y)|6 dy.
IR3
Hence, passing to the limit as k → +∞, we obtain Z Z φ dν = φ (Ry)dν, IR3
IR3
that is Z
IR3
φ (y)|v|6 dy +
X j
νj φ (xj ) =
Z
IR3
φ (Ry)|v|6 dy +
X
νj φ (Rxj ).
j
R R Clearly IR3 φ (y)|v|6 dy → 0 and IR3 φ (Ry)|v|6 dy → 0 as → 0. On the other hand φ (xi ) = 1 while for → 0 φ (xj ) → 0 if xj 6= xi and φ (Rxj ) → 0 for all j, since Rxj 6= xi for all j.
26
Therefore, passing to the limit as → 0, we immediately derive νi = 0. Consequently, for the measure ν we have X ν = |v|6 dx + νj δj j
with Rδj = δ(0,zj ) for zj ∈ IR and j ∈ J. We want to use this information together with the fact that Q φ|vk |6 dx ≥ γ > 0, to conclude v 6= 0. Towards this goal, recall that φ vanishes along the x3 −axis (i.e. φ(0) = 0). Hence, for ψ ∈ Cc (IR3 ) with 0 ≤ ψ ≤ 1 and ψ ≡ 1 on Q, we have
0 0 and necessarily v 6= 0. At this point we can repeat the arguments of lemma 4.5 to conclude our proof. R
An immediate consequence of lemmata 4.6 and 4.7 is that necessarily L < Lc when p = 6. By the continuity of the constants L and Lc in (39) and (40) with respect to the power p, we find that such a gap must persist for the values of p close to the critical value p = 6, and the second part of theorem 3.2 is established.
5
Asymptotic behavior
In this section we study the asymptotic behavior, as |x| → +∞, of the solutions u we have provided above for problem (35). By this analysis we are able to prove that these solutions have finite total mass. We recall that, in the model under study, u is a gravitational potential and R p−1 φu dx gives the corresponding total mass, which must be finite. While, for u ∈ D1,2 (IR3 ), 3 IR this is obvious when p ∈ [5, 6], it is not clear what R happens when p ∈ [4, 5[. We prove in this section that, for |x| large, u(x) ≤ C|x|−1 , so that IR3 φup−1 dx < +∞ for all p ∈ [4, 6]. Lemma 5.1 Assume that p ∈ [4, 6], φ satisfies (3) and let u ∈ D1,2 (IR3 ), u ≥ 0 be a weak solution of the equation −∆u = φup−1 . There exists a positive constant C such that u(x) ≤
C |x|
as |x| → +∞. We apply a result of H.Egnell (theorem 2 p. 38 in [7]), in a form suitable for our purposes.
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Theorem 5.2 [7] Let u ∈ D1,2 (IRN ), u ≥ 0 be a weak solution of the following equation −∆u = f (x, u), where 0 ≤ f (x, u) ≤ b(x)uσ and 1
