The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities, in two space dimensions

The role of Onofri type inequalities in the symmetry properties of extremals for Caffarelli-Kohn-Nirenberg inequalities,

in two space dimensions

JEANDOLBEAULT, MARIAJ. ESTEBAN ANDGABRIELLATARANTELLO

Abstract. We first discuss a class of inequalities of Onofri type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than 1. Without symmetry assumption, it holds if and only if the parameter is in the interval( 1,0].

The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Caffarelli-Kohn-Nirenberg inequality, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method and a careful analysis of the convergence to a solution of a Liouville equation. In this way, the Onofri inequality appears as a limit case of the Caffarelli-Kohn- Nirenberg inequality.

Mathematics Subject Classification (2000):26D10 (primary); 46E35, 58E35 (secondary).

1. Introduction

The Onofri inequality on the sphere S², see for instance [1, 14, 15], states that Z

S²

e^{2u 2RS2u d} d  e^kruk

2

L2(S2,d ), (1.1) for allu2E= {u2L¹(S²,d ): |ru|2L²(S²,d )}, whered denotes the mea- sure induced by Lebesgue’s measure inR^{3 S2}, normalized so thatR

S²d = 1.

Using the stereographic projection fromS²ontoR², we see that (1.1) is equivalent to the following Onofri type inequality inR^2:

Z

R²e^{v RR2vdµ} dµ  e

1

16⇡krvk²_L2(R2,dx), Received October 2, 2007; accepted February 19, 2008.

for all v 2 D = {v 2 L¹(R^2,dµ) : |rv| 2 L²(R^2,dx)}wheredµdenotes the probability measure

dµ= dx

⇡(1+ |x|²)².

In this paper, we first note how the above inequality can be generalized to the family of probability measures

dµ↵ = ↵+1

⇡

|x|^2↵dx (1+ |x|^2(↵+1))², for↵> 1, and investigate when the inequality

Z

R²e^{v RR2vdµ↵} dµ↵  e

1

16⇡(↵+1)krvk²_L2(R2,dx), (1.2) holds in the space

E^↵=n

v2L¹(R^2,dµ↵) : |rv|2L²(R^2,dx)o.

In Section 2, we prove that (1.2) always holds for functions inE↵which are radially symmetric about the origin. Meanwhile, without symmetry assumption, inequal- ity (1.2) holds inE↵if and only if↵2( 1,0].

The Moser-Trudinger inequality was initially proved by N. Trudinger in [18], and then established in a sharp form by J. Moser in [14] using symmetrization techniques. In dimension two it involves a term exp(4⇡|u|²). However, in [14], J. Moser also establishes (1.1), up to a non sharp constant, and this is why (1.1) is sometimes called a Moser or Moser-Trudinger inequality in the literature. Onofri’s proof relies on conformal invariance and provides the sharp constant on S². In- equalities of the type (1.1) are known to hold over any compact surface (see [10]), but onS², the advantage of Onofri’s inequality is that it holds with the best possible constant.

We use the above information to investigate possible symmetry breaking phe- nomena for extremal functions of the Caffarelli-Kohn-Nirenberg inequality (see [3]), in two space dimensions. Namely,

✓Z

R²

|u|^p

|x|^bp dx

◆2/p

 C_a,b Z

R²

|ru|²

|x|^2a dx 8u2Da,b, witha<ba+1, p= 2

b a,

Da,b= {|x| ^bu 2L^p(R^2,dx): |x| ^a|ru|2L²(R^2,dx)},

(1.3)

and an optimal constant Ca,b. Typically (1.3) is stated with a < 0 (see [3]) so that the space Da,b is obtained as the completion of C_c¹(R²), the space of smooth functions in R² with compact support, with respect to the norm kuk² =

k|x| ^buk²p+k|x| ^aruk²₂. Actually (1.3) holds also fora > 0 (see Section 2), but in this caseDa,bis obtained as the completion with respect tok·kof the space {u2C_c¹(R²⁾:supp(u)⇢R²\ {0}}. We know that forb=a+1, the best constant in (1.3) is given byCa,b=a+1 =a²and it is never achieved (see [4, Theorem 1.1, (ii)]). On the contrary, fora < b < a +1, the best constant in (1.3) is always achieved, say at some functionua,b 2Da,b that we will call anextremal function.

However ua,b is not explicitly known unless we have the additional information thatua,bis radially symmetric about the origin. In the class of radially symmetric functions, the extremals of (1.3) are given (see [4,6]) up to scalar multiplication and dilation, by

u^rad_a,b(x)=⇣

1+ |x| ^{2a(1b a+a b)}⌘ ₁₊^{b a}_{a b}

. (1.4)

See [4] for more details and for a “modified inversion symmetry” property of ex- tremal functions, based on a generalized Kelvin transformation. Also we refer to [12, 13, 16] for further partial symmetry results. On the other hand, extremals are known to be non-radially symmetric for a certain range of parameters (a,b) identified first in [4] and subsequently improved in [9]. Those results provide a rather satisfactory information about the symmetry breaking phenomenon forua,b

when |a|is sufficiently large. Also they apply to any dimension N 3, where inequality (1.3) reads as follows:

✓Z

R^N

|u|^p

|x|^bp dx

◆2/p

 C_a,b^N Z

R^N

|ru|²

|x|^2a dx, 8u2Da,b^N , (1.5) with p=_{(N 2)}²₊^N_{2(b a)},D_a,b^N = {|x| ^bu 2L^p(R^N,dx): |x| ^a|ru|2L²(R^N,dx)}, an optimal constantC_a,b^N , anda,b2R^{such thata< (N 2)/2,a} ba+1.

Again we observe that inequality (1.5) makes sense also if a > (N 2)/2 and a ba+1, where now the spaceD_a,b^N is given by the completion with respect tok·kof the set{u2C_c¹(R²⁾:supp(u)⇢R²\ {0}}.

Inequality (1.5) is sometimes called the Sobolev-Hardy inequality see [6], as for N >2 it interpolates between the usual Sobolev inequality (a=0, b=0) and the Hardy inequality (a =0, b=1); or the weighted Hardy inequalities (see [4]), since for b = a+ 1 it furnishes a family of Hardy-type inequalities involving weights.

For N 3 and 0a < (N 2)/2, the extremalu_a,bof (1.5) (which again exists for everya < b < a+1) is always radially symmetric (see [6], and for a survey on previous results, see [4]). On the other hand, whena<0, this is ensured only in some special cases described in [12, 13]. Also see [16, Theorem 4.8] for an earlier but slightly less general result.

In this paper, we focus on the less investigated bidimensional case N = 2, and besides symmetry breaking phenomena, we explore the possibility of ensuring radial symmetry for the extremal ua,b, a property which cannot be handled as in [12, 13, 16], (see in particular [16, Remark 4.9]).

To this purpose, first we check in Section2.2 that (1.3) (or more generally (1.5)) holds for all a 6= 0 (ora 6= (N 2)/2 if N 3) and not only fora < 0 (or a < (N 2)/2) as it is usually found in literature. In this way we can analyze radial symmetry ofu_a,bin the rangea6=0 and for allb2(a,a+1).

Theorem 1.1. Leta 6= 0andN = 2. Ifa <b < h(a)= a+ p^|a|

1+a², then(1.3) admits onlynon radially symmetricextremals.

As in [4, 9], Theorem 1.1 follows by analyzing the linearized operator around the radial extremal u^rad_a,b and show that it yields to a saddle (and not a minimum) type solution.

Since as |a| ! +1, 0 < a+1 h(a) ! 0, it is reasonable to look for radially symmetric extremals when|a|is small. But so far, forN =2, there was no result identifying a set of parameters(a,b)for whichua,b is shown to be radially symmetric. Here we provide a contribution in this direction which asserts that the above curveh(a)is asymptotically optimal as a!0. More precisely, we show that ifa!0₊, thenh⁰₊(0)=2 (or ifa!0 , thenh⁰ (0)=0) gives the optimal value of the ratio b/a that signs the transition between radial symmetry and symmetry breaking.

Theorem 1.2. Leta6=0andN =2. For every">0, there exists >0such that for|a|2(0, ),b2(a,a+1), if one of the following conditions holds:

(i) a>0andb/a>2+", (ii) a<0andb/a< ",

then the extremals of (1.3)areradially symmetric,and given, up to scalar multipli- cation and dilation, byu^rad_a,bdefined in(1.4).

Note that, as a consequence of Theorem 1.1, we can also state , for small |a|, the following counterpart of Theorem 1.2 in case of symmetry breaking.

Corollary 1.3. Let N = 2. For every" > 0, there exists > 0 such that if

|a|2(0, ),b2(a,a+1), if one of the following conditions holds: (i) a>0andb/a<2 ",

(ii) a<0andb/a>",

then(1.3)cannot admit a radially symmetric extremal.

We will directly prove the weaker statement in Corollary 1.3 as a consequence of the Onofri type inequality (1.2). We emphasize that such an approach makes no use of the linearized problem around the radial solution (1.4) and could be helpful in other contexts. To prove the more complete result stated in Theorem 1.1, we use the Emden-Fowler transformation in order to formulate (1.3) (or more generally (1.5)) as the Gagliardo-Nirenberg inequality on the cylinder R⇥S¹ (or more generally

R⇥ S^{N 1}). In this way we can analyze the linearized elliptic problem around the solution corresponding to (1.4) and see in which case it does not yield to a local minimizer. We shall obtain a precise description of the linearized problem in Section 3. This information will lead us directly to the proof of Theorem 1.1. It will be useful also to handle the more interesting part of our contribution given by Theorem 1.2, that will be derived from an argument by contradiction using a blow- up method and a careful analysis of the convergence to a solution of a Liouville equation

To emphasize the relevance of Theorem 1.2 and the advantage of our approach, we notice that it provides an alternative and direct proof of Onofri’ s inequality (see Section 5) without any use of the conformal invariance, but rather by identifying it as a limiting case of the Caffarelli-Kohn-Nirenberg inequalities.

In concluding we wish to bring the reader’s attention to an Onofri type inequal- ity in the cylinderR⇥S¹(see Proposition 5.1 in Section 5). We believe it helps to illustrate the nature of the symmetry breaking phenomenon.

ACKNOWLEDGEMENTS. This work has been partially supported by European Programs HPRN-CT # 2002-00277 & 00282, by the projects ACCQUAREL and IFO of the French National Research Agency (ANR) and by M.U.R.S.T. project:

Variational Methods and Non Linear Differential Equation, Italy. The third author wishes also to express her gratitude to Ceremade for the warm and kind hospitality during her visits.

2. The Onofri inequality in connection to the Caffarelli-Kohn-Nirenberg inequality

Consider the measureµ_↵and the Banach spaceE↵,↵ > 1, defined in Section 1.

Here and from now on, we letkvk2denotekvkL²(R^2,dx).

2.1. Onofri inequalities inR²

Proposition 2.1. Let↵> 1. For allv2E↵, there holds Z

R²e^{v RR2vdµ↵} dµ↵  e^{16⇡1(↵+1)}

⇣

krvk²₂+↵(↵+2)k_r¹@✓vk²₂⌘

. (2.1)

Proof. We use polar coordinates in R² ⇡ C^{. For x} 2 R^{2, we let x} = r e^i✓, r 0,✓ 2 [0,2⇡). We also consider cylindrical coordinates inR³, so that for (y,z)2R²⇥R^{, we let y}=⇢e^i✓,⇢ 0,✓ 2[0,2⇡)andz 2R. In this way, we can writeR^{3 S2}= {(⇢e^i✓,z):⇢²+z²=1 and✓ 2[0,2⇡)}. We recall that the

inverse6₀of the usual stereographic projection fromS²ontoR²is defined by 6₀ r e^i✓ =(⇢e^i✓,z)= 2r e^i✓

1+r², r² 1 1+r²

! .

Ifuis defined onS², thenv=u 6₀is defined inR²and for any continuous real function f inR^{, we have}

⇡ Z

S²

f(u)d = Z

R²

f(v)

(1+ |x|²)² dx and 4⇡

Z

S²|ru|²d = Z

R²|rv|²dx whenever f(u)and|ru|²belong toL¹(S²).

In order to prove the proposition, we are going to use the inverse of adilated stereographic projection given for all↵ > 1 by the function6_↵ :R² !S²such that

6_↵ r e^i✓ = 2r^↵+1e^i✓

1+r^2(↵+1), r^2(↵+1) 1 1+r^2(↵+1)

! .

Note that for anyr 0,✓ 2[0,2⇡),6_↵(r e^i✓)=6₀(r^1+↵e^i✓)and, for any⇢ 0,

✓ 2[0,2⇡)andz2[ 1,1],

6_↵¹ (⇢e^i✓,z) =

✓ ⇢

1 z

◆1/(↵+1)

e^i✓.

Now, if f is a continuous real function in R^{, f(u),} |ru|² 2 L¹(S²) andv = u 6_↵, then an elementary computation (see the Appendix) shows that

Z

S²

f(u)d = Z

R² f(v)dµ↵, 4⇡

Z

S²|ru|²d = 1

↵+1 Z

R² |rv|²+↵(↵+2) 1 r @_✓v

2! dx. The result follows from Onofri’s inequality (1.1).

Corollary 2.2. If↵2( 1,0], then(1.2)holds true for anyv2E^↵.

Proof. It is an immediate consequence of Proposition 2.1 since for↵2( 1,0], we have↵(↵+2)0.

This result is optimal. While (1.2) remains valid for all↵> 1 among radially symmetric functions (about the origin), in general it fails to hold inE↵for↵>0. In view of the proof of Corollary 2.2, this is a consequence of the conformal invariance and of the positivity of↵(↵+2)for↵>0, but this can also be seen from a more analytical point of view as follows.

Proposition 2.3. If↵>0, then inequality(1.2)fails to hold inE↵.

Proof. Let us exhibit a counter-example to (1.2), which is valid for all↵ >0. For any"2(0,1), let us consider the functionv_":R²!R^{defined by}

2v"= 8>

>>

<

>>

>: log

✓ "

("+⇡|x x¯|²)²

◆

if |x x¯|1 log

✓ "

("+⇡)²

◆

if |x x¯|>1

where x¯ denotes the point(1,0). For this function we can calculate the various terms of (1.2).

First we compute the left hand side, and see that

µ_↵(e^2v")=

Z

R²e^2v" dµ↵ =I↵,"+A↵

"

("+⇡)² where

I↵," = 1

"

Z

|x x¯|<1

1 1+⇡ x x¯

p"

2!2 dµ↵

and A↵ =R

|x x¯|>1dµ↵ is finite for all↵ > 1. Now, by the change of variables x = ¯x+p

"yand dominated convergence, we find

"lim!0

Z

|y|<1

| ¯x+p

"y|^2↵

1+ | ¯x+p

"y|^{2(↵+1) 2} 1+⇡|y|^{2 2} dy = 1 4

Z

R²

dy (1+⇡|y|²)². So, for the functionv_", the left hand side of (1.2) satisfies

"lim!0µ_↵(e^2v")= lim

"!0I_↵," = ↵+1

4⇡ .

Next we compute the r.h.s. of (1.2), that is _4⇡(↵¹₊₁₎krv_"k²₂+2µ_↵(v_")and see that

krv_"k²2 =4⇡ log

✓"+⇡

"

◆ 4⇡² ("+⇡) and

2µ_↵(v_")= J↵,"+A↵ log "

("+⇡)², where

J↵," = Z

|x x¯|<1

log

✓ "

("+⇡|x x¯|²)²

◆ dµ↵.

UsingA↵ =1 R

|x x¯|<1dµ↵, we get

2µ_↵(v_")=log "

("+⇡)² +B↵,", B↵," = Z

|x x¯|<1

log

✓ "+⇡

"+⇡|x x¯|²

◆2

dµ↵,

"lim!0B↵," = Z

|x x¯|<1

log

✓ 1

|x x¯|⁴

◆ dµ↵. Hence

1

4⇡(↵+1)krv_"k²2+2µ_↵(v_")= ↵

1+↵log"+O(1) as "!0, and comparing with the estimate above, we violate (1.2) for">0 small enough.

2.2. The extended Caffarelli-Kohn-Nirenberg inequality

The range in which inequalities (1.3) and (1.5) are usually considered can be ex- tended as follows.

Lemma 2.4. IfN = 2, then inequality(1.3)holds for anya 6= 0andbsuch that a<ba+1. IfN 3, then inequality(1.5)holds for anya6=(N 2)/2andb such thataba+1.

Proof. We use Kelvin’s transformation and deal with the caseN=2. Ifu2Da,b, then v(x) = u x/|x|² is such that|x|^a|rv| 2 L²(R^2,dx). Hence, fora > 0, b2(a,a+1], definea⁰= a,b⁰ =b 2a2( a, a+1]and apply (1.3) to the pair(a⁰,b⁰)withp=2/(b⁰ a⁰)to obtain

Z

R²

✓ |v|^p

|x|^b0pdx

◆2/p

 C_a0,b⁰

Z

R²

|rv|²

|x|^2a0 dx in Da⁰,b⁰. Now, we make the change of variablesy=x/|x|²and get

Z

R²

✓ |u|^p

|y|^{4 b0p} dy

◆2/p

 Ca⁰,b⁰

Z

R²

|ru|²

|y| ^2a0 dy in Da,b. Thus we arrive at the desired conclusion withCa,b =Ca⁰,b⁰, since

4 b⁰p=bp, 2a⁰ =2a and p=2/(b⁰ a⁰)=2/(b a).

Similarly in dimensionN 3, argue as above witha= N 2 a⁰,b p=2N b⁰p and p=2N/(N 2 2(b⁰ a⁰))=2N/(N 2 2(b a)).

Surprisingly, the case a > 0 if N = 2, ora > (N 2)/2 if N 3, has apparently never been considered. According to our argument, it requires to de- fine with care the space Da,b. Indeed if a function u 2 C_c¹(R^N)\Da,b for a > (N 2)/2, N 2, then umust satisfyu(0) = 0. Although optimal func- tions for inequality (1.5), a > (N 2)/2, N 2, have not been studied, it has been noted in [4, Theorem 1.4] that whenever u > 0 satisfies the corresponding Euler-Lagrange equations, then, up to a scaling, it satisfies the “modified inversion symmetry” property, that is, there exists⌧ >0 such that

u(x)= x

⌧

(N 2 2a)

u

✓

⌧² x

|x|²

◆

8x2R^N.

The transformationu 7!|x| ^{(N 2 2a)} u(x/|x|²)is sometimes called the general- ized Kelvin transformation, seee.g.[6]. The modified inversion symmetry formula can be shown for an optimal function u using the fact thatvgiven in terms of u as in the proof of Lemma 2.4 is also an optimal function for inequality (1.5), with parametersa⁰,b⁰.

2.3. The Onofri inequality as a limit case of the Caffarelli-Kohn-Nirenberg inequality inR²

We now relate inequalities (1.2) and (1.3). In this section, we will only consider the casea<0. The casea>0 follows by Lemma 2.4.

For N = 2,↵ > 1," 2(0,1), let us make the following special choice of parameters:

a= "

1 "(↵+1), b=a+" and p= 2

". (2.2)

Letu" =u^rad_a,bbe given in (1.4), that is

u"(x)=⇣

1+ |x|^2(↵+1)⌘ ₁^"_"

. We consider the functions

f" =

 u"

|x|^a+"

2/"

, g" =

|ru"|

|x|^a

2

, and the integrals

_" = Z

R² f" dx and "=

Z

R²g" dx.

Straightforward computations show that

_" = Z

R²

|x|^2↵

1+ |x|^{2(1+↵) 2} u²_"

|x|^2a dx = ⇡

↵+1 Z ₁

0

s^1""

(1+s)^12" ds,

" =4a²

Z

R²

|x|^{2(2↵+1 a)} 1+ |x|^2(1+↵)

2 1 "

dx.

Notice that we can use Euler’s Gamma function 0(x) = R₁

0 s^{x 1}e ^sds, and on the basis of the well known identity:

2 Z ₁

0

s^{2a 1}(1+s²) ^bds= 0(a)0(b a)

0(b) ,

deduce for "the following expression:

" =4⇡|a|

0

✓2 "

1 "

◆ 0

✓ 1 1 "

◆

0

✓ 2 1 "

◆ .

Lemma 2.5. Let ↵₀ > 1, v 2 C_c¹(R^{2), w"} = (1+"v)u". With the above notations, we have

1

_"

Z

R²

|w_"|^p

|x|^bp dx = Z

R²|1+"v|^2" f" dx Z

R² f" dx and, as"!0, uniformly with respect to↵ ↵₀,

Z

R²

|rw_"|²

|x|^2a dx = ^"+"²

"

8(1+↵)² (1 ")²

Z

R²

u^2/"" v

|x|^{2(a ↵)} dx+ Z

R|r² v|² u²_"

|x|^2a dx+O(a²")

# . Proof. By definition ofg_", we can write

Z

R²

|rw_"|²

|x|^2a dx = "+2"

Z

R²ru"·r(u"v) dx

|x|^2a

| {z }

(I)

+"² Z

R²|r(u"v)|² dx

|x|^2a

| {z }

(II)

.

A simple algebraic computation shows that r·

✓ru"

|x|^2a

◆

= 4a²

" u

2

" 1

" |x|^{2(↵ a)}. (2.3)

Using (2.3) and an integration by parts, we obtain (I)= 4a²

"

Z

R²|x|^{2(↵ a)}u^2/"_" vdx. As for(II), we expand|r(u"v)|²and write

(II)= Z

R²

h

v²|ru"|²+u"r(v²)·ru"+u²_"|rv|² i dx

|x|^2a

where the first two terms can be evaluated as above using (2.3) and an integration by parts. Hence,

Z

R²

⇣

v²|ru"|²+u"r(v²)·ru"

⌘ dx

|x|^2a = 4a²

"

Z

R²|x|^{2(↵ a)}u^2/"_" v²dx. To complete the proof we just remark that the function|x|^{2(↵ a)}u^2/"" is uniformly bounded for↵ ↵₀> 1.

For a given ↵ > 1, we now investigate the limit as " ! 0. We prove that inequality (1.2) is a limiting case of inequality (1.3), whenever (1.3) admits a radially symmetric extremal for any"small enough. In such a case, we can write (1.3) as follows:

1

_"

Z

R²

|w|^p

|x|^bp dx  1

"

Z

R²

|rw|²

|x|^2a dx

!1/"

. (2.4)

Thus, if we takew=w_" =(1+"v)u", then we have:

1

_"

Z

R²

|w_"|^p

|x|^bp dx  1+"²

"

"

8(1+↵)² (1 ")²

Z

R²

u^2/"" v

|x|^{2(a ↵)} dx+ Z

R²

|rv|²u²_"

|x|^2a dx

#!1/"

+O(a²"²).

In particular, observe that

|x| ^bp f"dx

RR² f" dx ⇠ ↵+1

⇡ |x|^2↵u^2/"_" dx ⇠dµ↵(x) as "!0₊.

Proposition 2.6. Let us fix ↵ > 1 and suppose that there exists a sequence ("_n)_n2Nconverging to0such that the radial extremal functionu"n is also extremal for(1.3)with(a,b,p)=(an,bn,pn)specified a follows,

pn = 2

"_n, an= "_n

1 "_n (↵+1), bn =an+"_n. Then, inequality(1.2)holds true inE↵.

Proof. Asn! 1, we have

"n=4⇡|an| +o("n), _"n = ⇡

↵+1+o(1).

Using Lebesgue’s theorem of dominated convergence repeatedly and Lemma 2.5, for anyv2C_c¹(R^2)andw"n =(1+"_nv)u"n, we have

1

_"n Z

R²

|w_"n|^pn

|x|^bnpn dx = Z

R²|1+"_nv|^"2n f"n dx Z

R² f"n dx

! Z

R²e^2vdµ↵, 1

"n

Z

R²

|rw_"n|²

|x|^2an dx =1+"_n

✓Z

R²2vdµ↵+ 1

4(1+↵)⇡ krvk²2

◆

+O("_n²) asn!+1. The proposition follows by applying inequality (1.3) with(a,b,p)= (a_n,b_n,p_n). By density we can finally choosevin the larger spaceE↵.

Remark 2.7. Incidentally let us note that if we temporarily admit the result in The- orem 1.2, then we find a sequence of optimal functions as required by Proposi- tion 2.6. In particular, for ↵ = 0, this gives an alternative proof of the Onofri inequality in R² as a consequence of Caffarelli-Kohn-Nirenberg inequality (1.3).

Using the inverse6₀of the stereographic projection, this also proves Onofri’s in- equality (1.1) onS².

Let us now consider another asymptotic regime in which ↵ ! 1. Proposi- tions 2.6 and 2.8 will be useful for the proof of Corollary 1.3 (symmetry breaking).

Proposition 2.8. If("_n)_n2Nand(↵_n)_n2Nare two sequences of positive real num- bers such that asn!+1,

n!+1lim "_n =0, lim

n!+1↵_n = +1 and an = "_n

1 "_n(1+↵_n) !

n!+10 , then for nlarge enough, the radially symmetric extremal u"n cannot be a global extremal for inequality(1.3).

Proof. We argue by contradiction and assume that (2.4) holds with respect to the given choice of parameters. By definition of "n,_"n, and Lebesgue’s theorem of dominated convergence, we know that

n!+1lim

"n

|an|=4⇡ and lim

n!+1(↵_n+1)_"n =⇡.

Ifv2C_c¹(R²), then by a direct computation, we find:

(↵_n+1) Z

R²

|u"n(1+"_nv)|^pn

|x|^bnpn dx

=(↵_n+1) Z 2⇡

0

Z ₊₁

0

r^{2↵1n+""nn+1} 1+"_nv(rcos✓,rsin✓) ^2/"n

1+r^{2(↵n+1) 12"n}

dr d✓

= Z 2⇡

0

Z ₊₁

0

t^11+""nn

(1+t²)^12"n

1+"_nv(t^1+1↵n cos✓, t^1+1↵n sin✓) ^2/"n dt d✓.

We pass to the limit asn!+1and obtain:

n!+1lim 1

_"n Z

R²

|u"n(1+"_nv)|^pn

|x|^bnpn dx = 1

⇡ Z 2⇡

0

e^{2v(cos✓,sin✓)}d✓

Z ₊₁

0

t dt (1+t²)²

= 1 2⇡

Z 2⇡

0

e^{2v(cos✓,sin✓)}d✓.

Analogously, (↵_n+1)

Z

R²

u^2/""n ⁿ

|x|^{2(an ↵n)}vdx

=(↵_n+1) Z

R²|x|²

↵n+"n

1 "n v(x)

(1+ |x|^2(1+↵n))^12"n

dx

= Z 2⇡

0

Z ₊₁

0

t^11+""nn

(1+t²)^12"n

v(t^↵n+11cos✓, t^↵n+11sin✓)dt d✓.

By Lemma 2.5, we see that 1

"n

Z

R²

|r[u"n(1+"_nv)] |²

|x|^2an dx =1+ "_n²

"n

8(↵_n+1)² (1 "_n)²

Z

R²

u^2/""n ⁿ

|x|^{2(an ↵n)}vdx + O

✓ "_n 1+↵_n

◆

+O "_na²_n

"n

! , and so

n!+1lim 1

"n

Z

R²

|r[u"n(1+"_nv)] |²

|x|^2an dx

!1/"n

=e

2

⇡ R2⇡

0 v(cos✓,sin✓)d✓R₊₁ 0 t dt

(1+t2)2

=e^{⇡1 R02⇡v(cos✓,sin✓)d✓}.

Hence the validity of (2.4) would imply that for allv2C¹_c (R²), there holds:

1 2⇡

Z 2⇡

0

e^{2v(cos✓,sin✓)}d✓ e^{⇡1 R02⇡v(cos✓,sin✓)d✓}.

But this is clearly impossible, since such an inequality is violated for instance by the function v(x) = v(x1,x2)= x₁²⌘(x), with⌘a standard cut-off function such that⌘(x)=1 if|x|1,⌘(x)=0 if|x| 2.

3. Symmetry breaking

This section is devoted to the proof of Theorem 1.1. We start by establishing Corol- lary 1.3, which is weaker but follows as an easy consequence of the results of Sec- tion 2.

3.1. Proof of Corollary 1.3

By Lemma 2.4 and Kelvin’s transformation, we can reduce the proof to the case a < 0. Let us argue by contradiction and assume that there exists "₀ 2 (0,1), an ! 0 andbn such that"₀ < ^b_aⁿ

n < 1 anduan,bn is radially symmetric. Set

"_n =b_n a_n >0 and define↵_nsuch that↵_n+1= a_n(1 "_n)/"_n. Notice that

"_n!0₊while↵_n+1=a_n a_n/(b_n a_n)=a_n (b_n/a_n 1) ¹>a_n+(1 "₀) ¹. Hence, lim inf_n!+1↵_n ↵₀ = "₀/(1 "₀) > 0. But this is impossible since it contradicts Proposition 2.8 in case lim infn!+1↵_n = +1, or Propositions 2.3 and

2.6 if lim sup_n!+1↵_n <+1. ⇤

3.2. Proof of Theorem 1.1

It is well known (see [4]) that by means of the following Emden-Fowler transfor- mations:

t =log|x|, ✓ = x

|x|2 S^{N 1}, w(t,✓)= |x|^{N 2 2a2} v(x), (3.1) inequality (1.5) foruis equivalent to the Sobolev inequality forwonR⇥S^{N 1}. Namely,

kwk²_L^p_(R⇥S^{N 1}₎  C_a,b^N



krwk²_L²_(R⇥S^{N 1}₎+1

4(N 2 2a)²kwk²_L²_(R⇥S^{N 1}₎ , for w 2 H¹(R⇥ S^{N 1}), with p = 2N/[(N 2)+2(b a)] and the same optimal constantC_a,b^N as in (1.5). This inequality is consistent with the statement of Lemma 2.4, as it makes sense for anya6=(N 2)/2, independently of the sign of

N 2 2a.

For N = 2, the inequality holds for functionsw = w(t,✓)defined over the two-dimensional cylinderC = R⇥S¹ ⇡ R⇥(R^/2⇡Z^{),i.e.,such thatw(t,}·)is 2⇡-periodic for a.e.t 2R. The inequality then takes the form

kwk²L^p(C)C_a,a+2/p⇣

krwk²_L²_(C)+a²kwk²_L²_(C)

⌘

8w2 H¹(C^{) (3.2)} for alla 6= 0 and p > 2. HereCa,b is the optimal constant in (1.3) which enters in (3.2) withb=a+2/p.

For anya 6= 0 and p >2, inequality (3.2) is attained at an extremal function w_a,p2 H¹(C⁾which satisfies

8<

:

(w_tt+w_{✓ ✓})+a²w=w^{p 1} in R⇥[ ⇡,⇡],

w >0, w(t,·) is 2⇡-periodic 8t 2R^{, (3.3)} and such that

Ca,a+2/p 1

=kw_a,pk^p_L^p₍²_C)= inf

w2H¹(C⁾\{0}F^(w), where the functional

F^(w)= krwk²_L2(C⁾+a²kwk²_L2(C⁾

kwk²_L^p_(C)

is well defined in H¹(C⁾\ {0}. Moreover, according to [4], we can further assume

that 8

>>

>>

><

>>

>>

>:

w_a,p(t,✓)=w_a,p( t,✓) 8t 2R^{, ✓}2[ ⇡,⇡),

@w_a,p

@t (t,✓) <0 8t >0, 8✓ 2[ ⇡,⇡),

R⇥[max⇡,⇡)w_a,p =w_a,p(0,0).

(3.4)

This symmetry result is easy to establish for a minimizer, but the monotonicity re- quires more elaborate tools like the sliding method and we refer to [4] for more de- tails. For a solution of (3.3) which does not depend on✓, the conditions in (3.4) al- low to determine its value at 0 simply by multiplying the ODE byw_t and integrating from 0 to1. In fact, in this way, one deduces the relation:a²w²(0)/2=w^p(0)/p, which uniquely determinesw(0) > 0. In turn this yields to the following unique

✓-independent solution for (3.3) and (3.4):

w_a,p^⇤ (t)= a²p 2

!1/(p 2)  cosh

✓p 2 2 a t

◆ 2/(p 2)

,

as a consequence of the classification result in [4]. Such a solution is an extremal for (3.2) in the set of functions which are independent of the✓-variable, and

kw^⇤_a,pk_L^pp(²R⁾= inf

f2H¹(R⁾\{0}F^{⇤(f) with} F^⇤(f)= kf⁰k²_L2(R⁾+a²kfk²_L2(R⁾

kfk²_L^p_(R) . For simplicity, we will also write F^(f) = (⇡)^{1 2/p}F^⇤(f) for all functions f which are independent of✓. As a useful consequence of the above considerations, we have the following result.

Lemma 3.1. Let p>2. For anya6=0, Ca,a+2/p

p

p 2 =kw_a,pk_L^pp_(C)  kw_a,p^⇤ k^p_L^p_(C)=4⇡(2a)^pp2(a p)^p22cp

wherecpis an increasing function of psuch that cp !0 as p!2₊,

cp ! ¹₂ as p!+1. (3.5)

As a consequence, ifa=a(p)is such thatlimp!1a(p)p=2(↵+1), then

plim!1 p Z

C|w^⇤_a(p),p|^pdx =8(↵+1). (3.6) Proof. Observe that

kw_a,pk^p_L^p(C⁾= C_a,a+2/p ^pp2= F^(wa,p)

p p 2⇣

F^(wa,p^⇤ )⌘_p^p₂

=kw^⇤_a,pk_L^pp(C⁾. On the other hand,

kw^⇤_a,pk_L^pp_(C) =2⇡ a²p 2

!_p^p₂Z ₁

1

 cosh

✓a(p 2)

2 t

◆ _p^2p₂ dt

=4⇡ a²p 2

!_p^p₂Z ₁

0

2^p2p2e ^{a p t} 1+e ^{a(p 2)t}

2p p 2

dt

=4⇡ a²p 2

!_p^p₂ 2^p2p2

a p Z 1

0

ds 1+s^{(p 2)/p p2p2}

. Hence by setting:

c_p = Z 1

0

ds 1+s^{(p 2)/p}

2p p 2

,

we easily check (3.5) and the fact thatcpis monotonically increasing in p. The lim- iting behavior ofcpstated in (3.5) is a direct consequence of Lebesgue’s dominated convergence theorem.

We can now reformulate Theorems 1.1 and 1.2 in the cylinderC, as follows.