Sharp constant in a Sobolev inequality

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Printed

Analysis, Theory, in Great Britain.

Methods

& Appltcafions,

Vol. 20, No. 3, pp. 261.268,

SHARP CONSTANT

0362-546X/93 $6.00+ .@_I @ 1993 Pergamon Press Lfd

1993.

IN A SOBOLEV

INEQUALITY

XU-JIA WANG Department of Mathematics, Zhejiang University, Hangzhou, People’s Republic of China (Received 1 August 1991; received in revised form 6 January 1992; received for publication 28 April 1992) Key words and phrases: Sobolev inequality, elliptic equation, maximum principle.

1. INTRODUCTION LET Q BE A domain

Sobolev

inequality

in R”. In this paper we are concerned Il~l]~~*~c, I S IIAuJI,~(,), i.e.

with the best constant

S in the

where 1 < p < n/2, p* = np/(n - 2p), A denotes the Laplacian operator, and Wi8p(CJ) is the completion of C,“(Q) in W2’p(Cl). It is easy to see that S is independent of the domain Q and depends only on n. Whenp = 2, integrating by parts gives IlAz.& (c) = I(D*u]]~z~~). For general p 2 1, by the elliptic LP estimates it follows that, for u E W~~P(Q),~]A~l],P(,, is a norm equivalent to lI~*4ILq,, . The main result of this paper is the following theorem. THEOREM 1.1. When

Q = R”, the infimum

S is achieved

by the unique

positive

radial solution

of the problem A(IAulP-’

AU) = Iu]~*-*u

i u(0) = 1, When p = 2, the solution

24-+Oas

in R”,

Ix] + 43.

(I -2)

is given by u(x) = (1 + c],#4-n)‘2,

where c = [(n + 2)n(n

- 2)(n - 4)1-l’*, s = 7?(n + 2)n(n

The best constant

3 in the simplest

and the best constant - 2)(n - 4) I- n l-(n) [ (2>/ Yn*

Sobolev

(1.3)

inequality

IIUIIf”P/‘“-P’(p) 5 SIIDz4IIp(~n,

(1.4)

was discussed by Talenti [l]. Some other sharp constants related to the Sobolev embedding W’~P(Q) L, L@(“-p)(Q) h as also been discussed by Escobar [2, 31, Lions [4], and Lions et al. [5]. It should be mentioned that Lions [4] studied sharp constants in various inequalities such as the Sobolev inequalities (1 .l), (1.4), and the Sobolev trace inequalities. He proved that the sharp constants are achieved and the extremal functions exist. But the extremal functions corresponding to the Sobolev inequalities were not given there, hence the values of the best constants were not known, except for the simplest Sobolev inequality (1.4). 261

262

XU-JIA WANG

To establish the best constant s” in (1.4), a usual method is to employ the technique of the Schwarz symmetric rearrangement by which one can show that the best constant S, is reached by a spherically symmetric function, and then derive this function. This procedure can also be used to establish inequalities of type (1.1). In [6] Talenti derived the sharp constant C in the inequality

Ilull U*(P) 5 CI~AUIIL~,~), where L(p,p) stands for Lorentz space (see theorem 2 of [6]). In this paper we present an approach different from those used in the articles mentioned above. We do not use the device of rearrangement, but the elliptic maximum principle instead, which we describe briefly as follows. We first consider the following approximating problem

(1.5) where p W2*p(Q) positive converges constant

-c q < p*, B is the unit ball in V, and WtPP(sZ) which vanish on X2. We will prove in Section 2 that radial function u,. Let q -+ p* we then, in Section to a function u which is the unique radial solution S in (1.1) is achieved. 2. THE

INFIMUM

denotes the set of functions in the infimum S, is achieved by a 3, show that, after resealing, uq of (1.2), and by which the best

S,

For p 5 q 5 p*, let s, =

inf u E wp(B)

B

IAt@‘~;

It

is not hard to show that S, is continuous for q E [p, p*]. By llD2u/l Lo 5 CllA&(,, 7 hence S, > 0. Next observe that the ;;;mprt for q E [p,p*). If (~4~) is a minimizing sequence of uk &) -+ S,, then uk (or a subsequence) converges to a and strongly in Lq(B). Hence by the weak lower semicontinuity (2. l)q. We have the following lemma. LEMMA

2.1. For p 5 q < p*, the infimum

ProoJ: From

(2.11,

l]ull~~c~, = 1 . 1

S, is achieved

the elliptic Lp estimates we have embedding W2*p(B) - Lq(B) is (2.1),, i.e. if /Iu~JI~~(~) = 1 and function u weakly in W29p(B) u reaches the infimum S, in

by a positive

function

uq.

(2.1), we have S, = inf{J(u);

(2.2)

u E Weep],

where

Let u be a minimizer

and let u E Wi*p(B) -Au

By the elliptic renormalization

= /Au1

maximum principle of u gives a positive

be the solution in B,

we have minimizer

of the following

problem

v = 0 on dB.

0 I Iu( 5 v in B and n of (2. l)q.

J(v) I J(u).

Hence

a

Sobolev

Remark

2.1. Clearly

uq is a weak solution A(IAulp-*

263

inequality

of the problem in B,

AU) = S&l”-”

u=Au=O Set u, = -IAulp-*

(2.3)

on aB.

Au, then (uq, vq) is a weak solution

of

-AU

= IV]“@-‘)

in B,

-Au

= S,luj”-“U

in B,

i u=u=o

(2.4)

on dB.

By the maximum principle and by the elliptic regularity theory we conclude positive in B, and uq, vq E C 4+a(B) fl C*+“(8) for some CYE (0, 1). LEMMA

2.2. U, is spherically

symmetric

and is radially

that U, and u, are

decreasing.

Proof. We follow the idea of Gidas et al. [7], and Troy [8], i.e. using the maximum principle and the technique of moving parallel planes to a critical position and then showing that the solution uq is symmetric about the limiting plane. For any given direction y, we have to prove that uq is symmetric about the plane y *x = 0. Without loss of generality we may suppose that y = (1, 0, . .., 0). Let TX denote the plane y - x = A and Cx denote the piece lx E B; x1 > A). Forx E Z, let xi denote the reflection of xin the plane T,, and CL denote the reflection of Xx in T, . Then for 0 I A < 1, Cl is nonempty and CL is contained in B. by the Hopf lemma we have that (a/ax&,(e) < 0, (a/ax,)v,(e) < 0, Since uq , uq E C*‘“(B), where e = (1, 0, . . . , 0). Hence for I < 1 and sufficiently close to 1 we also have (a/ax&, < 0, (a/ax,)u, < 0 in C,. Set A0 = inf

1; $

uq I 0, $1

uq I 0, u,(xJ

2 u,(x),

and u,(xJ

2 uq(x) in Ex

.

1

Then A, < 1 and we only need to prove & = 0. We prove it by contradiction. Indeed, if I, > 0, let p,(x) = uq(xx) - U,(X), ul,(x) = v,(xJ - V,(X), then pq and I,U, are nonnegative in C,, and (q,, v,) satisfies Ap4 = Iu~(x~)]~‘@-‘) - ]v~(x)I”~-‘) i Aw, = S,(]U,(X~)~~-~ -

Iu,(x)(~-‘)

2 0 2 0

in Xx, in &

with o)~ = u/, = 0 on T, n B, pq > 0, ‘y, > 0 on aX,\T,. Hence by the maximum principle it follows v)~ > 0, wq > 0, in Xx, and (a/ax&, > 0, (a/ax&, > 0 on TXf~ B. Next observe by the Hopf lemma that (a/ax&, < 0, (a/ax&, < 0 on T, n dB. By the definition of &, we therefore reach a contradiction. Hence uq is spherically symmetric. By the maximum principle uq is obviously radially decreasing. n We remark that the symmetric by Troy [8], where the functions continuous.

result for systems of elliptic equations has been demonstrated on the right-hand side of the equations are assumed to be C’

264

Xu-JIA 3.

WANG

THE INFIMUM

S

In this section we show that, after resealing, the solution uq obtained in the previous section converges to a positive radial solution u of (1.2) which reaches the infimum S, and then show that (1.2) has a unique radial solution. We first prove the following lemma. LEMMA 3.1.

Proof.

uq converges

to zero weakly

in WzsP(B).

We first verify that lim (Iz4QIIL~*(~j = 1. 4-p*

This is because

on the one hand we have 1 =

and so lim

(3.1)

JJuq 1)Lp*(Bjr

I14JIL~~B~ 5 II~qllL~*~~~I~~~~~~~I@*~q”P*

1. On the other hand

observing

that

4+p* sq

by the continuity If t = wq Au,

=

of S, we obtain

IbqIbp(Bj

2

~p*~bq~b~B~,

@* (]u~I(~~*(~)II 1. Hence

(3.1) holds.

there exists a subsequence of L, Tstill denoted by uq) so that U, + U* f 0 in W2*p(B), set ](u*]ILp*(,), then 0 < r I 1. We claim that r = 1. Indeed, if r < 1, let uq = u* + wq, then monotone, it follows 0 weakly in W2sp(B) and in LP*(B). Since Au, is radially -+ Au* a.e. in B. From [9] we have

s

Ju,Ip*dx=

B

Iz4*J%x+

IwqJp*dx+

jAwqjpdx + o(l).

jAuqlpdx = B

s B

B

o(l),

.i’B

i B

Hence IAuqjpdx 2 S,,

S, =

[

(1.

lb*lQJ’p* + (1, Iwql%yp]

+

o(1)

B

>

l.i

spa

Iu*IP*dw+

lBlwql~*djp”*+ C(t)+ O(l)

= sp*(1: luqlp*tiy~p’+ o(1)

= s,*

+ C(r)

+ o(1)

(3.2)

for some positive constant C(r). Passing to the limit we reach a contradiction, hence T = 1. By the weak lower semicontinuity of I~Au]\~~~~, in Wzpp(B) it then follows I~Au*(~&,, = S,, and hence uq -+ U* in W2,p(B). This means that U* is a minimizer of (2.1),*. Thus U* is a weak solution of the problem A(\An*lp-* AU*) = &,*Iu*I~*-~ u* = Au* = 0

in B, on aB.

(3.3)

Sobolev

By remark

2.1 U* E C4+a(B)

IAu*Ipdx = B

Next multiplying

SpjB Iu*lp*dx. by parts and then letting q -+ p* we obtain

(2.3) by C XjDjU* and integrating

s

XiXjDiu*Dj(AU*)P-’

da - T

(Au*(~~

[

aB

Combining

(2.3) by U* and letting q + p*, we obtain

tl C’+Ol(B). Multiplying [

265

inequality

= -9

.i, (U*,J’*&,

B

the above two formulae

.i

yields

5 u* 2 (Au*)p-’ dr ar

aB

But on the other hand by the maximum principle aB, a contradiction. Hence U* = 0. n

da = 0.

we have (a/&-)(Au*)P-’

From lemma 3.1 we obtain immediately that (a/&+@) combined with (3.1), implies S,, 2 S. On the other hand, S,, I S. We thus conclude

< 0, (a/&-)u*

< 0 on

-+ 0 as q + p* for x E aB, which, since Wt,p(B) > WEEP, we have

s,* = s.

(3.4)

where Mg = sup U,(X), R, = Mqo)*-p)‘2p. From (2.5) we have

Now set P,(X) = M;‘u,(R;‘x), A(lA~,lP-’

AeQ

in B(0, R4).

= &~&-~*S,ly7,)~-’

(3.5)

Let I,V~= IAP~~~-‘, from (2.4), (p,,, t,~,) satisfies

Direct computation

-AI+Y~ = M,4-p*S,lq4(q-1

in B(O, R,),

-APQ = ,#@-r)

in B(O, R,),

P)4 = v)q = 0

on dB(0, R4).

(3.6)

gives

s

IAu,lpdx = S,,

IAq~,(~dx =

B(O,R,)

(3.7)

i ,B

(3.8) For any given R > 0, by the elliptic I16&2+u(&)

regularity S

cR

3

it follows h&I~C2+"(&)

that s

CR,

CR independent of q, provided R, is large enough so that the ball of radius R with the centre at the origin. Hence of pq and wq, which we still denote by p4 and wq, so that

for some cy E (0, 1) with the constant

R, > R. Where BR = B(0, R) denotes

there exist subsequences

+

9

in C2(BR),

w4 + cc/ in C2(BR).

(3.9)

XU-JIA WANG

266

Passing

to the limit in (3.6) we infer that in B,, -Aw -Av,

where j3 = limM,4-P*

I

= /3Sq+‘-‘, = ,,,r/@-r),

1. Using the diagonal

method

(3.10)

we deduce that cp and I,Vare well defined

on R” and ia&fy (3.10) on R”. Note by the maximum principle hence v, E C”(lR”). From (3.7) and (3.8), and using the Fatou lemma we deduce

that cp is positive

on R” and

(3.11)

We claim that

IlvllP*(P) =

1.

(3.12)

Indeed, if it is not true, then r = II~JII~~*(~“) < 1. From (3.9) we have q(O) = p,(O) = 1 and so T > 0. Let pq = ~1 + pq, and let v)~, pq, A(pq and Ap, be zero outside B(0, Rq). Then pq - 0 weakly in Lp*(P) and Apq - 0 weakly in Lp(lR”). From [9] we have

s

IPqlp*~+ o(l), m” bqlp*~ = ’IR”IPlp*~ + c i R” I IAvI’ d.x + lR”

.iR”

>

lim S 4-p*

=

[I

lApqIp do + o(l).

R”

n Idp*dx +

iRn

IPqlp* cLqp’“’

lim S( j:” Iv)~I~*dxy”*. 4-p”

From

(3.7) and (3.8), this implies n

lim q+p*

But on the other hand

!

IAuqlpti

> ;;m+S(s.

we have IAuqlp dx = S, .i B

This is a contradiction.

/~q~p*~~‘p*

B

Hence

(3.12) holds.

luqlqdx = s, -+ s.

= S.

Sobolev

inequality

261

of S we thus conclude that ]lAv]lrPcan, = S and so From (3.11), (3.12) and by the definition Av, in Lp(fR”). Multiplying (3.5) by vq and integrating over B(0, R,), -+ v, in ,Y’*(R”), Av4 --t pq and letting q + p* we get lim MqmP* = 1. Q

4+p*

Thus v, is a positive

radial

solution

of the problem

h(]h~]~-~ V(O)= withOsu,<

(3.13)

as 1x1 -+ 00

1,&x)+0

1.

LEMMA3.2. Problem Proof.

in R”

Av) = S]C/~~*-’

(3.13) possesses

We consider

the following

a unique

initial

radial

solution.

value problem

n-l __v71’- 7

V = /y/I”@-‘)

n-l _v/” - -y-

v/’ = sl$Oy&

!

sign w 9

v(O) = 1, v’(O) = 0, y(O) = d, q/(O) = 0.

For any given d E R, similar to the argument of Ni and Nussbaum (see proposition 2.35 of [lo]) we see that there is a unique solution (9, u/) of (3.14). For any fixed d, and d,, let (pt, ty,) and (v)~, w2) be the solutions of (3.14) with d = d, and d = d2 respectively. We claim that if d, > d2, then v1 - v)~ is strictly decreasing and I+U~- I,V~is strictly increasing. Indeed, if d, > d2, then for r > 0 sufficiently small, we have w,(r) > ty2(r), hence

-(R

-

v)z)”

By the maximum

+(p,

-

principle

-

=

jW,]l’(p-l)sign

v1 - ]~2(1’@-1)sign~2

we thus infer that CJI,- p’z is strictly

V,(r) small enough.

cp2)’

v2(r)

<

v*(O)

-

v)2(0)

=

0

decreasing

> 0.

(3.15)

and so

for r > 0

Next by -(WI

-

w2)”

-

?f2 (WI- WI)’= N%lp*-2% - I~21p*-2v72) <0

(3.16)

we infer that I,V, - ‘c/~is strictly increasing for r > 0 small. Hence for all r > 0 the right-hand side of (3.15) is always positive and the right-hand side of (3.16) is always negative, and consequently our claim follows. From the assertion above we immediately obtain the uniqueness of positive radial solutions of (3.14), this completes the proof. n

XU-JIA WANG

268

When p = 2 the radial

solution

of (3.13) is given by p(x) = (1 + cI.X\2)(4-n)‘2,

and the best constant S is given by (1.3), where c = [(n + 2)n(n - 2)(n - 4)]-1’2. From the above argument we thus obtain the results of theorem 1.1. But for general p E (1, n/2), we have not found the explicit expression of the solution. We remark that the above method can also be used to derive the best Sobolev constant s^ in (1.3) without any new difficulty. We conclude this paper with the following corollary. COROLLARY

3.1. For

any bounded s*

Proof.

Since

domain

= UE $f”cn,

W,2Tp(sZ) > w,2rp(Q),

i!;.

Q c

I?“, we have

l*ul”dxl[

jn

lUIQ_qp’p~

we have S* 5 S. Hence

= S.

we only need to prove S* 2 S.

For 4 E [I?, P*I, let

then S,(Q) is continuous for q E [p,p*]. For any given q < p*, by the compactness of the embedding W2,p(sZ) + Lq(Q), we see that the infimum s,(Q) is achieved by some function qq. Let uq be the solution of -Au,

= IAqql

in Q,

Au, = 0

in

B,\Q,

u,=O

ondB,,

where BR is any ball containing ~2. Then Il~~qllL~pcB,, = IlA~,ll~~~~,, II~qllL~~BR~ 2 IIID~IL~~~~~ by (3.4) that S* z S,.(B,) = S. Thus hence S,(Q) 2 S,(B,). Let q + p* we obtain s*=s. n Acknowledgement-The author wishes to thank the referee for his helpful of China and NSF of Zhejiang province.

comments.

This work is supported

by NNSF

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

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