Gazzola F., Serrin J. - Asymptotic behavior of ground states of quasilinear elliptic problems with...

7/25/2019 Gazzola F., Serrin J. - Asymptotic behavior of ground states of quasilinear elliptic problems with two vanishing par

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Asymptotic behavior of ground states

of quasilinear elliptic problemswith two vanishing parameters

Filippo GAZZOLA

Dipartimento di Scienze T.A. - via Cavour 84 - 15100 Alessandria (Italy)

James SERRIN

School of Mathematics - University of Minnesota - Minneapolis (Minnesota)

Abstract

We study the asymptotic behavior of radially symmetric ground state solutions of a quasilinearelliptic equation involving the m-Laplacian. The case of two vanishing parameters is considered: weshow that these parameters have opposite effects on the asymptotic behavior. Moreover the resultshighlight a surprising phenomenon: different asymptotic behaviors are obtained according to whethern > m2 orn m2, where n is the dimension of the underlying space.

1 Introduction

Let mu = div(|u|m2u) denote the degenerate m-Laplace operator and consider the quasilinearelliptic equation

mu= um1 + up1 in IRn, (Pp )wheren > m > 1, m < p < m, 0 and

m = nm

n m .

By the results in [Ci, GST] (see also [AP1, BL] for earlier results in the case m = 2) we know that (Pp )

admits a ground state for all p, in the given ranges. Here, by a ground statewe mean aC1(IRn) positive

distribution solution of (Pp ), which tends to zero as|x| . Since in this paper we only deal withradial solutions of (Pp ), from now on by a ground state we shall mean precisely a radial ground state. It

is known [PS, ST] moreover that radial ground states of (Pp ) are unique.

Equation (Pp ) is of particular interest because of the choice of the power m 1 for the lower orderterm: ifm= 2 (i.e. m = ) this is just the linear case, while for any m >1 the lower order term has

Supported by the Italian MURST project Metodi Variazionali ed Equazioni Differenziali non Lineari.

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the same homogeneity as the differential operator m, a fact which allows the use of rescaling methods.

Moreover, this case is precisely the borderline between compact support and positive ground states, see

[FLS, Section 1.3].

It is our purpose to study the behavior of (radial) ground states of (Pp ) as p m, 0. As faras we are aware, the asymptotic behavior of solutions of (Pp ) has been studied previously only for the

vanishing parameter = mpand only in the case ofbounded domains, see [AP2, GP, Ga, KP, R1, R2]and references therein.

Consider first the case when = 0. Then (Pp ) becomes

mu= up1 in IRn, (P0p )

which by [NS2, Theorem 5] admits no ground states (recall p < m). It is of interest therefore to study

the behavior of the ground states u of (Pp ) as 0 and p is fixed: in Theorem 1 below we prove inthis case that u 0 uniformly on IRn and moreover estimate the rate of convergence. As a side result,the arguments used in the proof of Theorem 1 allow us to show that the corresponding ground states u

converge to a Dirac measure concentrated at x = 0 when , see Theorem 9 in Section 4 below.Next, letp= m and >0; then (Pp ) becomes

mu= um1 + um1 in IRn, (Pm)

which by the results in [NS1] again admits no ground states. Thus we next study the behavior of ground

states uof (Pp ) as = m p 0 with >0 fixed. We prove in Theorem 2 that u then converges to a

Dirac measure concentrated at the origin, namely, u(0) and u(x)0 for all x= 0, while also, atthe same time,u converges strongly to 0 in any Lebesgue space Lq(IRn) withm 1 q < m. Our studyalso reveals a striking and unexpected phenomenon: the asymptotic behavior is different in the two cases

n m2

and n > m2

; for instance, in the case m = 2 (i.e. m = ) there is a difference of behaviorbetween the space dimensions n = 3,4 and n 5. More precisely, ifn > m2 we show that u(0) blowsup asymptotically like (nm)/m

2

while ifn m2 it blows up at a stronger rate. This phenomenonis closely related with the Lm summability of functions which achieve the best constant in the Sobolev

embeddingD1,m Lm , see [Ta] and (1) below for the explicit form of these functions.Finally, let both p= m and = 0; then equation (Pp ) reads

mu= um1 in IRn, (P0m)

which admits the one-parameter family of ground states

Ud(x) =d

1 + D

dm

nm |x|) mm1

nmm (d >0), (1)

where D = Dm,n = (m 1)/(n m)n1/(m1). Since the effects of vanishing m p and are in somesense opposite, it is reasonable to conjecture that there exists a continuous function h, with h(0) = 0,

such that if= h(), p= m , then ground states uof (Pp ) converge neither to a Dirac measure norto 0! In Theorem 4 below we prove the surprising fact that when n > m2 this equilibrium occurs exactly

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whenand are linearly related,h() Const. Moreover in this case the corresponding ground statesu then converge uniformly to a suitably concentrated ground state of (P0m), namely a function of the

family (1), with the parameter d = Ud(0) representing a measure of concentration and depending on

the limiting value of the ratio h()/.

Let us heuristically describe the phenomena highlighted by our results. Whenp

m with fixed,

the mass of the ground stateu of (Pp ) tends to concentrate near the pointx = 0, that is, all other points

of the graph are attracted to this point: in order to let the other points fit near x= 0 the maximum

level u(0) is forced to blow up. When 0 with p fixed, the ground state spreads, since now x = 0behaves as a repulsive point, forcing the maximum level to blow down in order not to break the graph.

When both = m pand tend to 0 at the equilibrium velocity= h(), the pointx = 0 in neitherattractive nor repulsive: in this case, a further striking fact is that the exponential decay of the solution

u of (Pp ) at infinity reverts to a polynomial decay.

The outline of the paper is as follows. In the next section we state our main results, Theorems 1 5.

Then in Section 3 we present background material on radial ground states, including an estimate for the

asymptotic decay asr of ground states of (Pp ), see Theorem 8. This estimate, along with Theorems

6 and 7 in Section 3, seems to be new and may be useful in other contexts. These results allow us to give

a simple proof of Theorem 5 while the proofs of Theorems 1 4 are given in subsequent sections.

2 Main results

The existence and uniqueness of radial ground states for equation ( Pp ) is well known [GST, ST]. We

state this formally as

Proposition 1 For alln > m > 1, m < p < m and > 0 the equation (Pp ) admits a unique radial

ground stateu= u(r), r= |x|. Moreoveru

(r)< 0 forr >0.

We start the asymptotic analysis of (Pp ) by maintainingp fixed and letting 0. An important rolewill be played by the rescaled problem (= 1)

mv= vm1 + vp1 in IRn. (Qp)

By Proposition 1 there exists a unique (radial) ground state v of (Qp), so that the constant

= v(0) (2)

is a well-defined function of the parameters m, n, p.

Theorem 1 For all > 0, let u be the unique ground state of (Pp ) withm < p < m. Then u(0) =

1/(pm), while for fixedp andx = 0 there holds

u(x)

u(0) = 1 m 1

m

pm 1

n

1m1

|x| mm1 + o

1

m1 |x| mm1

as 0. (3)

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From Theorem 1 we can also obtain a result which, while slightly beyond the scope of the paper, is

nevertheless worth noting. It states that the unique solution of (Pp ) for fixed p < m tends to a Dirac

measure as , see Theorem 9 in Section 4.We now maintain >0 fixed and let pm. In order to state our main asymptotic result for this

case, it is convenient to introduce the beta function B(,) defined by

B(a, b) =

0

ta1

(1 + t)a+bdt, a, b >0.

Then we put

m,n =

n m

n m2 B n(m1)m , nm2m

Bn(m1)

m , nm

(nm)/m2

for n > m2

and

m,n = nm 1

m

n

n mm

1

m1n/mB

n(m 1)

m ,

n

m

(n = measure S

n1).

We also put Cm,n = D(m1)(nm)/m, where D= Dm,n is given in equation (1).

These coefficients allow us to describe the exact behavior of ground states when n > m2: in particular

note that m,n as m n.

Theorem 2 For allm < p < m, letu be the unique ground state for equation (Pp ) with fixed > 0.

Then, writing= m p, we have

lim0

(nm)/m2u(0)

=

m,n ifn > m2

ifn m2.(4)

Moreover for allx = 0lim0

u(0)um1(x)

Cm,n|x|(nm) (5)uniformly outside of any neighborhood of the origin, while also

lim0

IRn

um1 = 0, lim0

IRn

um

=m,n. (6)

Theorem 2 gives a complete description of the asymptotic behavior ofu when n > m2; it leaves open

only the exact behavior when nm2. This latter question is considered in more detail in Section 5.2.The results given there, while not as precise as in the casen > m2, nevertheless provide significant insight

into the behavior ofu(0) as 0 beyond that described in the second case of (4), see Lemma 8.Condition (6) shows that, as 0, not only does u approach a Dirac measure (u(0) and

u(|x|) 0 for|x| = 0), but also that the Lm norm of u approaches a non-zero finite limit. It is aremarkable fact, also, that the limit relation (6) is independent of the value of. (It may be worthwhile

to note as well that if the exponentm 1 in the first equation of (6) is replaced by a general value qthenthe limit remains 0 for m 1< q < m but becomes ifq > m).

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Remark. The constants in Theorem 2 in the important case m= 2, n >4 are given by

2,n =

4n

(n 2)2Bn2 ,

n42

Bn2 ,

n2

(n2)/4

, 2,n =n

2 [n(n 2)]n/2 B

n

2, n

2

,

and C2,n = [n(n 2)](n2)/2

.The results of Theorem 2 can be supplemented with the following asymptotic estimates for the gradient

u of a ground state.Theorem 3 For allm < p < m, letu be the unique ground state for equation (Pp ) with fixed > 0.

Then for allx = 0 we have

lim0

u(0)|u(x)|m1

n mm 1m1

Cm,n |x|1n (7)

and

lim0 IRn

|u|m =m,n. (8)

Finally, we may accurately describe the behavior of the ground states of (Pp ) when = m pand

approach zero simultaneously.

Theorem 4 For > 0 andm < p < m, letu be the unique ground state of (Pp ). Then for alld >0

there exists a positive continuous function() =(, d) such that

(i) () (d/m,n)m2/(nm) as 0 (whenn > m2), and() 0 as 0 (whenn m2).(ii) If= (), p= m , thenu(0) =d.

Moreover

u Ud as = m p 0

uniformly onIR

n

, whereUd is the function defined in(1).If, 0 without respecting the equilibrium behavior Const (in the case n > m2), the central

heightu(0) of the ground state may either converge to zero or diverge to infinity. We note finally that

as soon as the asymptotic behavior ofu(0) as p m is more accurately determined in the casen m2of (4), one also gets a more precise statement of (i): of course, the equilibrium behavior will no longer

be Const .To conclude the section, we supply two global estimates for u(0), supplementing the asymptotic

conditions (3) and (4).

Theorem 5 Letu be a ground state of(Pp ). Then

u(0)>

mp

mn p(n m) 1/(pm)

, (9)

and, provided thatp < n/(n 1),

u(0) m2;

see also Lemmas 58 in Sections 5.

Remark. The condition p < n/(n 1) implies p < m/(m 1), since n > m: therefore, the upper boundin (10) is obtained only for values m m) and values p far from the critical exponent

m, that ismp > n2(m 1)/(n m)(n 1). However, in the restricted range of values p < n/(n 1),the inequality (10) gives useful information aboutv(0) =; we quote here some numerical computations.

m n m p (m,n,p)

1.6 2 8 1.8 2.11<


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Proposition 2 A radial ground stateu= u(r) of(13) has the properties

|u(r)|m1r

f()n

as r 0, rn1|u(r)|m1 Finite limit as r ,

F() = (n

1)

0

|u(r)|m

r dr

and

E(r)> 0 r 0, E(r) 0 as r .

In the next result we recall a Pohozaev-type identity due to Ni and Serrin [NS1].

Proposition 3 Letu= u(r) be a radial ground state of(13), and put

Q(r) =nmF(u) (n m)uf(u). (16)

Then the functionsrn1Q(r) andrn1F(u(r)) are inL1(0, ), and moreover

0Q(r)rn1 dr= 0. (17)

Remark. In other terms, the result of Proposition 3 says that the functions Q(|x|) and F(u(|x|)) are inL1(IRn) and thatIRnQ(|x|) dx= 0.

For completeness we give a proof of Proposition 3. By direct calculation, using (13), one finds that

P(r) =

r0

Q(t)tn1dt, r >0,

where

P(r) = (n

m)rn1u(r)u(r)|u(r)

|m2 + mrnE(r).

Since E= m1m |u|m + F(u(r))> 0 and because f(s)< 0 for s near 0, we get

|F(u(r))|, E(r) m 1m

|u(r)|m

for all sufficiently large r . Using Proposition 2 then gives rn1|u|m1 Const. and

rn|F(u(r))|, rnE(r) Const. r(nm)/(m1) (18)

for sufficiently large r. Hence P(r) 0 as r , which yields

limr r0 Q(t)t

n1

dt= 0.

But from (18) we get rn1|F(u(r))| L1(0, ), while also uf(u)< 0 for all sufficiently larger. Thus theprevious equation together with the definition ofQ(r) shows in fact that rn1Q(r) is in L1(0, ) andthat (17) holds. This completes the proof.

Proposition 3 has the following important consequence.

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Theorem 6 Suppose there exists >0 such that

nmF(s) (n m)sf(s)< 0 f or 0< s < . (19)

Then > .

Proof. Suppose for contradiction that . Then sinceu 0, it follows that u(r)< for allr >0. In turn, by the hypothesis (19) we have Q(r) =nmF(u) (n m)uf(u)< 0 for all r >0, whichcontradicts Proposition 3.

An upper bound for u(0) can also be obtained in some circumstances, as in the following

Theorem 7 Supposef(s) 0 wheneverf(s)> 0 and that there exists >0 such that

nF(s) (n 1)sf(s) 0 for s . (20)

Then < .

Proof. We assert that the function r (r) = r1|u(r)|m1 is decreasing on (0, ). By directcalculation, using (13),

r(r) =f(u) n(r).Iff(u) 0 then < 0. On the other hand, for all r such that f(u) > 0, we have (f(u) n(r)) =f(u)u n(r) n(r), by hypothesis. Consequently

(r) n.

By integration this gives rn+1(r)rn+11 (r1) on any interval (r1, r) where f(u)> 0. The assertionnow follows by an easy argument, once one notes notes that rn+1(r) 0 as r 0.

Now by Proposition 2 and the assertion, we have

F() = (n 1)0

|u(r)|mr

dr= (n 1)0

(r)|u(r)| dr


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Moreover

nF(s) (n 1)sf(s) = n m(n 1)m

sm +n p(n 1)

p sp.

Thus we can take

= p

m

n m(n 1)

n p(n 1)

1/(pm)

in (20), giving the second conclusion as a consequence of Theorem 7.

We conclude the section by showing that radial ground states u = u(r) of (P ) have exponential decay

as r approaches infinity. This is well-known in the case m= 2, see [BL, Theorem 1 (iv)]: we give here a

different proof in the general case m >1.

Theorem 8 Suppose that there exist constants, , >0 such thatf satisfies the inequality

sm1 f(s) sm1 f or 0< s < . (21)

Then there exist constants0, 1, 2>0, (depending on m,n,,) such that, forr suitably large,

u(r) 0er |u(r)| 1er |u(r)| 2er . (22)

Remark. For general nonlinearities f in (13), one usually expects polynomial decay at infinity, see

[NS1, Lemma 5.1] [ST, Proposition 2.2]. Nevertheless, Theorem 8 is not entirely unexpected, since the

nonlinearity (21) has borderline behavior which separates compact support and positive ground states,

see [FLS, Section 1.3].

Proof of Theorem 8. Obviously u = u(r) satisfies (13). LetR 0 be such that u(r) when r R.Since u

0 as r

, it is clear that such a value R exists. By Proposition 2 and the right hand

inequality of (21) we thus obtain

m 1m

|u(r)|m > F(u(r)) m

um(r)

for r R. Therefore, u

(r)

u(r) >

m 11/m

r R. (23)

integrating this inequality on the interval [R, r] yields the first part of the result, with

0 = eR , = (/(m 1))1/m. (24)

For the other estimates, we rewrite (13) in the form

(rn1|u(r)|m1) =rn1f(u(r)). (25)

Since f(u) < 0 for u near 0, it follows that rn1|u(r)|m1 is ultimately decreasing, clearly to a non-negative limit as r (this is the first result of Proposition 2). By the exponential decay proved

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above, the limit must be 0. Therefore we can integrate (25) on [r, ) forr Rto obtain, with the helpof (21),

rn1|u(r)|m1 = r

tn1f(u(t)) dt <

r

tn1um1(t)dt m10r

tn1e(m1)tdt.

With n 1 integrations by parts, this proves that|u(r)| 1er r R.

Finally, we write (13) as

(m 1)|u(r)|m2u(r) = n 1r

|u(r)|m1 f(u).

From the right hand inequality of (21) we get f(u)0 for r R, which shows that u(r) > 0 for allr R. Further, from the left hand inequality,

u(r)< n 1(m 1)R |u(r)| + m 1 u

m1

(r)|u(r)|m2 .

Hence by (23) and by the exponential decay ofu and u, this yields

0< u(r)< n 1(m 1)R |u

(r)| + m 1

m 1

(m2)/mu(r) 2er r R.

The proof of Theorem 8 is now complete.

Remarks. The first estimate of (22) requires only the right hand inequality of (21) for its validity.

It almost goes without saying that the function f(u) = um1 + up1 satisfies (21) for suitable ,.

4 Proof of Theorem 1

Letu = u(r) be a ground state of (Pp ). Define v= v(r) by means of the rescaling

v(r) =1/(pm) u

r

1/m

, (26)

so thatv is the unique ground state of the rescaled equation (Qp). By the definition (2) and by (26) one

has u(0) =1/(pm).

Next, from (Qp) we find, as in (25),

|v(r)|m1 = 1rn1

r0

sn1{vm1(s) + vp1(s)} ds

= 1

rn1

r0

sn1{m1 + p1 + o(1)} ds

= r

n{m1 + p1 + o(1)}

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as r 0. Taking the 1/(m 1) root and integrating from 0 to r then gives

v(r) = m 1m

p1 m1

n

1/(m1)rm/(m1) + o(rm/(m1)) as r 0. (27)

This, together with (26), yields (3) and completes the proof of Theorem 1.

When we can obtain a partial companion result to (3) in Theorem 1.

Theorem 9 For fixedx = 0 we haveu(x) =o(e

1/m|x|)

as , whereis any (positive) number less than1/(m 1)1/m.

Proof. We apply Theorem 7 for ground states of (Qp). Heref(s) = sm1 + sp1, so that one can take to be any number less than 1 in (21), provided that is chosen appropriately near 0. Thus by Theorem

8 we have

v(r) 0er

for all sufficiently large r , where, see (24), is any number less than 1/(m 1)1/m. Hence, by (26),

u(x) =1/(pm) v(1/m|x|) 0 1/(pm)e1/m|x|

for all fixed x = 0 and sufficiently large . Finally, taking = , with small, we get

u(x) 0 1/(pm)e1/m|x| e1/m|x| =o(e1/m|x|)

as . The conclusion now follows at once, since clearly by appropriate choice of and we canassume that is any number less than 1/(m 1)

1/m

.


The argument is delicate, covering a number of pages. For the proof of (4) we need to distinguish the

two cases n > m2 and m < n m2; this is done in Sections 5.1 and 5.2 below. The proof of (5) and (6)is given in Section 5.3.

We shall prove (4) first, for the case = 1, and then obtain the general estimate by means of the

rescaling (26).

Thus we assume that u= u(r) satisfies (13) with f(s) = sm1 + sp1, namely

(|u|m2u) +n 1r

|u|m2u um1 + up1 = 0 (28)

with u(0) = . From the estimate (9) in Theorem 5 we have always > 1 (since p > m) and, more

precisely,

>

mp

(n m)1/(pm)

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where

p= m .Hence

> m2

n m

1

1/(pm)

,

which gives the important condition

pm K (0, m m), (29)

whereK=m2/(n m).We make a second rescaling

w(r) = 1

u

(pm)/m r

, (30)

so that ifu= u(r) solves (28), then w= w(r) satisfies

(|w

|m2

w

)

+

n1

r |w

|m2

w

wm1

+ wp1

= 0w(0) = 1, w(0) = 0

(31)

where = (pm). Note that < 1 since > 1, and also, by (29), 0 as 0. Now define themodified nonlinearity

f(s) = sm1 + sp1

and the corresponding functions (see (14) and (16))

F(s) = m

sm +1

psp, Q(r) = mwm(r) + (n m)

p wp(r). (32)

Also, for r 0 let us define the function

z(r) =

1 + (1 )1/(m1)Drm/(m1)(nm)/m

(33)

where the constant D = Dm,n is given in (1).

We can now prove the following comparison result:

Lemma 1 We have

w(r)< z(r) r >0. (34)

Proof. We make use of the function Hintroduced in Lemma 2.1 in [KP]: here however it will be applied

without a previous Emden-Fowler inversion. Thus set

H(r) = (m 1)rn|w(r)|m (n m)rn1w(r)|w(r)|m1 +n mn

rnw(r)f(w(r)).

Then by using the fact that w solves (31) we obtain

H(r) =rn

n

m2wm1(r) (n m)wp1(r)

w(r).

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LetRbe the unique value ofr where

w(R) =

m2

(n m)

1/(pm) (0, 1);

see (29) and recall from Proposition 2 that w

0 r >0. (35)Consider the function

(r) = |w(r)|m1

r wn(m1)/(nm)(r)=

(r)

wn(m1)/(nm)(r),

where (r) = |w(r)|m1/r (see the proof of Theorem 7). By using (31) again we find that

(r) = n

n m1

rn+1 wm(n1)/(nm)(r) H(r).

From (35) it follows that is strictly increasing on [0, ). Therefore, by Proposition 2 we have

(r)> limt0

(t) =f(1)

n =

1 n

;

hence|w(r)|

wn/(nm)(r)>

1

n

1/(m1)r1/(m1) r >0.

The conclusion (34) follows upon integration, and the proof is complete.

For later use we observe that the function z = z(r) defined in (33) satisfies the equation

(|z|m2z) + n 1r

|z|m2z + (1 )zm1 = 0. (36)

Now let

C1 = C1() =

n m

m2

/(pm). (37)

Then by differential calculus (recalling that p = m and = (pm)) we find without difficulty that

f(s) C1sm1 s >0 and lim0

C1= 1. (38)

This allows us to obtain the following partial converse of Lemma 1.

Lemma 2 There exists a positive functionC2= C2() such that lim0 C2= 1 and

w(r)> C2/(m1)z(r) (C2/(m1) 1) r >0. (39)

MoreoverC2/(m1) >1.

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Proof. Equation (31) may be rewritten as

(rn1|w|m1) =rn1f(w). (40)

Integrating on [0, r], and taking into account (38) and Lemma 1, yields

rn1|w(r)|m1 = r

0tn1f(w(t))dt < C1

r

0tn1zm

1(t)dt= C11

rn1|z(r)|m1,

the last equality being obtained by a similar integration of (36) on [0, r]. Therefore,

|w(r)| < C2/(m1)|z(r)| r >0, (41)

where

C2=

C11 1

m1

.

Integrating (41) on [0, r] then gives (39).

Finally, from (38) one sees that C2 1 as 0, while by (34) and (39) we infer thatC2

/(m1) 1

(z(r) 1)< 0 r >0,

that is,C2/(m1) 1> 0 since z (r)< 1 for r >0 by (34) and the fact that


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Lemma 4 We have IRn

wp (C)(nm)/m,IRn

|w|m (C)(nm)/m,

whereCis a Sobolev constant for the embedding ofD1,m(IRn) into Lm(IRn).

Proof. If we multiply (42) by w and integrate by parts, we obtainIRn

|w|m = IRn

wm +

IRn

wp m2

By (33) we see that z(|x|) |x|(nm)/(m1) as|x| , so z Lm(IRn) if and only ifn > m2. Thisallows us to derive

Lemma 5 Letn > m2. Then there existsA > 0 (depending only onm, n) such that

A

(nm)/mfor all

0, m 1

n

m2

n m

, (45)

Proof. Define z(|x|) to be the function given by (33) with the parameter fixedat the value

=(m 1)(n m)

n2 m(m 1).

Using (9) with = 1, an easy calculation shows that for in the range stated in the lemma we have

= (pm) (0,). Hence, for the given range of, we infer from (34) thatIRn

wm IRn

zm IRn

zm c

(recall n > m2, and observe specifically that c= c(m, n)).

On the other hand, by Lemmas 3 and 4,IRn

wm c1IRn

wp c1(C)(nm)/m.

Combining the two previous lines, and remembering that = pm, p = m , we obtain

mm n

m A/, (46)

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whereA (c/c1)C(nm)/m depends only on m, n. Finally, using the given restriction

0< m 1n

m2

n m (47)

(note m

m= m2/(n

m)), one derives from (46) that

m/(nm) A

;

(45) now follows immediately, and the proof is complete.

Together with the inequality >1, Lemma 5 implies the important conclusion

1 as 0. (48)

Lemma 6 Letn > m2. Then there existsK >0 (depending only onm, n) such that

= pm

K

for all 0, m

1

n

m2

n m ,Proof. We have

pm =mm n

m nmm A

A

(nmm )2,

by (45) and (46). Hence

= pm A

A

(nmm )2.

It remains to show that the right side is bounded, but this follows directly from the fact that (1 /s)s is

bounded (

e1/e) on (0,

). The proof is complete.

Remark. A short calculation, taking into account the restriction (47), shows that in fact we can choose

K =Am(nm+1)/n e(nm)2/em2.

We can now complete the proof of (4). Here it is convenient to revert to the original understanding

that w = w(r) and z = z(r), We first rewrite the results of Lemmas 1, 2 as

0< z w < C3 1 for all r 0, (49)

whereC3= C3() =C2/(m1) 1 as 0; of course also C3> 1 by Lemma 2.

Now by (29) and Lemma 6 we know that /= [K, K]. Then, since w 1, it follows from (32)

that |Q(r)| Const m wm Constm zm,see the proof of Lemma 5. Thus, recalling that zm L1(IRn) and thatQ(r)< 0 for large r, we see thatfor every constant >0 there exists a suitably large value R= R() such that

R

Q(r)rn1 dr

m.16


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In turn, using Proposition 3 we get R0

wm(r) n m

mp wp(r)

rn1 dr

.Here we let

0. Clearly converges to some limit 0

[K, K], up to a subsequence; since we will

determine a unique possible value for0, this shows that 0 on the continuum >0. Then by (49)and the obvious fact that (recall 0 as 0)

z(r) z0(r)

1 + Drm/(m1)(nm)/m

uniformly for all r 0, there results R0

zm0 (r) 0

(n m)2nm2

zm

0 (r)

rn1 dr

. (50)Both zm0 r

n1 and zm

0 rn1 are in L1(0, ) since n > m2, so we can let 0 and R in (50); this

gives finally 0

zm0 (r)rn1 dr= 0

(n m)2nm2

0

zm

0 (r)rn1 dr.

By means of the change of variables s = Drm/(m1) one obtains

0

zm0 (r)rn1 dr=

m 1m

Dm1m nB

n(m 1)

m ,

n m2m

(51)

and 0

zm

0 (r)rn1 dr=

m 1m

Dm1m nB

n(m 1)

m ,

n

m

. (52)

Hence,

0= n

m

n m2 Bn(m1)m , nm

2

m

Bn(m1)

m , nm

.We can now prove the asymptotic relation (4). Indeed,

(nm)/m2

= ( )(nm)/m2 (nm)/m20 =m,n

as 0 (recall 1), which is just (4) for the case = 1. Since for general one has u(0) =1/(pm) (nm)/m2, relation (4) is proved (case n > m2).

5.2 The case n

m

2

Here z Lm(IRn) and the crucial Lemma 6 does not hold; nevertheless, we can prove the followingweaker result.

Lemma 7 Assume thatn m2. Then then there existsK =K(m, n)> 0 such that

m2/(n+m2m) K.

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Proof. We argue as in the proof of Lemmas 5 and 6, with a few changes. First, from (34) we haveIRn

wm IRn

zm IRn

zm

= d < (53)

since z Lm(IRn). On the other hand, by Lemmas 3 and 4 we findIRn

wm c1IRn

wp c1(C)(nm)/m. (54)

Next, integrating (40) over (0, ) and taking into account the exponential decay ofw andw , as well as(34), we get

IRnwm1 =pmIRn

wp1 pmIRn

zp1 = d1 pm, (55)

where we have used the fact that z Lp1(IRn) (for < m/(n m)).By Holder interpolation,

IRn

wm

IRn wm1

1

IRn

wm

,

where = (n m)/(n+m2 m) (0, 1). Inserting (53), (54), (55) into this inequality gives, after alittle calculation,

m2/(n+m2m)(n+m2)/m A1

whereA1= A1(m, n). For < m2/(n m)(n + m2) one finds in turn (compare Lemma 5)

A1

(n+m2m)/m(m1). (56)

As before this implies that is bounded, from which the lemma follows at once, subject of course to

the restriction on noted above.

From (56) it follows that 1 as 0, just as in the case n > m2. In turn (49) holds exactly asbefore, with C3 1 as 0.

For the next conclusion, we shall need a sharper form for the behavior ofC3. First, it is not difficult

to verify that the function C1= C1() defined in (37) satisfies C1 1 + c| log | for some constantc >0;we understand here and in what follows that c denotes a generic positive constant, depending only on

m and n. Moreover, by (29) we have < c, so the function C2 = C2() defined in (41) also satisfies

C2 1 + c| log |. FinallyC3= C2

/(m1) 1 + c| log | (57)

for sufficiently small .Next, let R >0 denote the unique value ofr where z (R) =| log |, where >0 is a constant to be

determined later; note in particular that R as 0. Now, arguing from (39) and the fact that1< C3< 1 + c | log |, we infer

w(r)> C3z(r) (C3 1)z(r)z(R)

=

1 C3 1

| log |

z(r)

1 c

z(r) r [0, R].

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In turn, fixing sufficiently large,

w(r) 12

z(r) r [0, R]. (58)We can now prove a companion result to (29); in particular, it shows that Lemma 6 does nothold when

n m2.

Lemma 8 There existsK1> 0 such that for sufficiently small

m/(m1) K1 | log |1 when m < n < m2

and

m/(m1) K1 | log | when n= m2.

Proof. Assume first that n < m2. Then for sufficiently small there holds

d1 = IRn

zp

IRnwp by (34)

c

IRn

wm by Lemma 3

c

|x| m.

Remark. As already mentioned in the introduction, more precision in the asymptotic behavior ofu(0)is needed in the case nm2. We conjecture that also in this case there exists a continuous increasingfunction gm,n defined on [0, ) such that gm,n(0) = 0 and lim0[gm,n()u(0)] = 1. (gm,n (m1)/mwhen n < m2?)

19


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5.3 Dirac limits

Here we shall complete the demonstration of Theorem 2 by proving conditions (5) and (6). It will be

convenient here and in the sequel not to makethe initial assumption = 1, though we continue to write

u(0) =.

From Section 5.1 we recall the basic estimate (49); with the help of (57) this can be rewritten in theform

0< z w < c| log |. (59)Here we wish to scale back to the original function u, this being accomplished by means of (26) and (30).

More specifically, in (30) it is necessary to replace u and respectively by v and (as in (2)) because

of the initial assumption in Section 5 that = 1. The required rescaling is therefore given by

w(r) = 1

1/(pm) u

r

1/m(pm)/m

=

1

u

r

(pm)/m

(60)

where from Theorem 1 we have 1/(pm) = . After a little calculation, (59) then leads to the basic

formula

0< z u c| log |, (61)where

z= z(x) =z((pm)/m|x|) =/

1 + (1 )1/(m1)(pm)/(m1)D|x|m/(m1)(nm)/m

(62)

and (33) is used at the last step.

Observe from the left hand inequality of (61) that (recall 0 as 0)

1/(m1)u(x)< 1/(m1)z(x) Dnmm |x|nmm1 as 0,

which immediately yields (5).

To prove (6), let X=XR denote the Lebesgue space Lm over the domain{|x|< R}, and similarly

let X =XR be the space Lm over the domain{|x| R}. By Minkowskis inequality and (61),uX zX u zX c | log | 1X. (63)

In particular, let us make the new choice

R= m/(nm)+

where > 0 is a positive constant to be determined later. Then with the obvious change of variables

s= (pm)/m

r, we find

zmX =nn/m /m+0

sn1 ds1 + (1 )1/(m1)Dsm/(m1)n m,n (64)

as 0, see (52) and (48) (which as shown in Section 5.2 is valid for all n > m). By the same calculation

zmX 0 (65)

20


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as 0, since the integration is now over the interval (/m+,) and the integral is convergent.Next, one calculates that

1X= nn

Rn/m

=n

n1+(nm)/m

in view of the definition ofR. We can now determine the limit as 0 of the quantity

| log | 1X= (n/n)(nm)/m| log |.

From Lemmas 6 and 7 it is evident that, whatever the case considered, there exists >0 (depending

only on m, n) such that < c, provided is small. (One can check that = (n m)/m2 + 1 in factsuffices). Hence

(nm)/m| log | c1(nm)/m| log |,which tends to 0 as 0 if is chosen small enough. It now follows at once from (63) and (64) thatumX m,n as 0.

We observe finally from the left hand inequality of (61) that

umX < zm

X 0

by (65). Hence

umm = um

X + um

X m,n,proving the second part of (6).

To obtain the first part, note that integration of (Pp ) over IRn and use of Theorem 8 yields

IRn

um1 =

IRn

up1. (66)

But, as in the calculation (64) we haveIRn

up1 IRn

zp1 =n1+(nm)/m 0

sn1 ds1 + (1 )1/(m1)Dsm/(m1)(nm)(p1)/m .

Since the integral is uniformly convergent for any less than m/2(n m), we then getIRn

up1 0 as 0.

With the help of (66) this completes the proof of (6), and therefore of Theorem 2.


First we prove (8). Multiplying the equation (Pp ) byu and integrating over IRn gives

IRn

|u|m = IRn

um +

IRn

up. (67)

We now let 0. The first term on the right approaches 0 by (6) (and a trivial interpolation).

21


22/24

To treat the second term on the right side of (67), we slightly modify the space X from its meaning

in the previous subsection, so that now it represents the Lebesgue space Lp over the domain{|x| < R},and similarly for the space X. Then as in (64) there holds

z

pX=n

(nm)/m /m+

0

sn1 ds1 + (1 )1/(m1)Dsm/(m1)n(nm)/m ,

the integral being convergent when < m/(n m). To evaluate the limit of the right side, note first thaton the interval 0< s < there holds (for small )

1 m2 (case= 1)

(nm)/m2

m,n as 0,

so that

() =

d(nm)/m2pm (nm)/m2 dm,n

m2/(nm)

as 0; similarly, when nm2, by Theorem 2 we infer that ()0 as 0. Statement (i) is soproved.

To prove the final statement of the theorem, we first use (61), together with the fact that in the present

case = u(0) =d, to infer the fundamental relation

|u zd)| cd | log |. (69)

But by (62), and since 0 as 0, it now follows that

zd(x) d

1 + D

dm

nm |x| mm1

n

mm Ud(x)

uniformly forx in IRn; see (1) in the introduction. Together with (69) this completes the proof of (ii).

An easy consequence of the above argument is the following companion result for Theorem 4.

Corollary. Letn > m2. In place of the condition=(), suppose that=a, wherea is a positive

constant. Thenu Ud uniformly on IRn as= p m 0, whered= a(nm)/m2m,n.Acknowledgement. The second author wishes to thank Prof. Grozdena Todorova for many valuable and

helpful conversations during the preparation of the paper.

References

[AP1] F.V. Atkinson, L.A. Peletier,Ground states ofu= f(u)and the Emden-Fowler equation, Arch. RationalMech. Anal. 93, 1986, 103-127

[AP2] F.V. Atkinson, L.A. Peletier,Elliptic equations with nearly critical growth, J. Diff. Eq. 70, 1987, 349-365

[BL] H. Berestycki, P.L. Lions, Nonlinear scalar field equations, I, Existence of a ground state, Arch. RationalMech. Anal. 82, 1983, 313-345

[Ci] G. Citti,Positive solutions of quasilinear degenerate elliptic equations inIRn, Rend. Circolo Mat. Palermo 35,1986, 364-375

[FLS] B. Franchi, E. Lanconelli, J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear equa-tions in IRn, Advances in Math. 118, 1996, 177-243

[GP] J.P. Garca Azorero, I. Peral Alonso, On limits of solutions of elliptic problems with nearly critical exponent,Comm. Part. Diff. Eq. 17, 1992, 2113-2126

[Ga] F. Gazzola, Critical growth quasilinear elliptic problems with shifting subcritical perturbation, to appear, Diff.Int. Eq.

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[GST] F. Gazzola, J. Serrin, M. Tang,Existence of ground states and free boundary problems for quasilinear ellipticoperators, Adv. Diff. Eq. 5, 2000, 1-30

[KP] M.C. Knaap, L.A. Peletier, Quasilinear elliptic equations with nearly critical growth, Comm. Part. Diff. Eq.14, 1989, 1351-1383

[NS1] W.M. Ni, J. Serrin, Nonexistence theorems for quasilinear partial differential equations, Rend. Circolo Mat.Palermo (Centenary Supplement), Series II, 8, 1985, 171-185

[NS2] W.M. Ni, J. Serrin,Existence and nonexistence theorems for ground states of quasilinear partial differentialequations. The anomalous case, Accad. Naz. dei Lincei, Atti dei Convegni 77, 1986, 231-257

[PS] P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47,1998, 501-528

[R1] O. Rey, Proof of two conjectures of H. Brezis and L.A. Peletier, Manuscripta Math. 65, 1989, 19-37

[R2] O. Rey, The role of Greens function in a nonlinear elliptic equation involving the critical Sobolev exponent,J. Funct. Anal. 89, 1990, 1-52

[ST] J. Serrin, M. Tang,Uniqueness of ground states for quasilinear elliptic equations, to appear, Indiana Univ.Math. J.

[Ta] G. Talenti,Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110, 1976, 353-372

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