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Article ID jmaa.1999.6393, available online at http:rrwww.idealibrary.com on

Multiple Solutions to a Dirichlet Problem on Bounded Symmetric Domains

Francesca Alessio*

Dipartimento di Matematica ‘‘R. Caccioppoli,’’ Uni¨ersita di Napoli ‘‘Federico II’’ Via Cintia I-80126 Napoli, Italy

E-mail: alessiof@sissq.it

and Walter Dambrosio

Dipartimento di Matematica, Uni¨ersita di Torino, Via Carlo Alberto 10,` I-10123 Turin, Italy

E-mail: walterd@dm.unito.it Submitted by Neil S. Trudinger Received August 14, 1998

1. INTRODUCTION

In this paper we are concerned with a Dirichlet problem of the form

< <

y⌬u x s g x , u x , Ž . Ž Ž . . x g ⍀,

Ž P . u x Ž . s 0, x g ⭸ ⍀,

N

Ž .

where ⍀ is the unit ball in ⺢ N G 2 ; the nonlinearity g satisfies the following conditions:

Ž H

1

. g g C C

0,

Ž w 0, 1 x = y␧ , ␧ w

0 0

x. for some ␤ g 0, 1 and for some Ž .

Ž < < . Ž < < . w x

␧ - 0, g x , u s yg x , yu for all x g ⍀, u g y␧ , ␧ ;

0 0 0 Ž .

g r , u

Ž H

2

. lim

uª 0

u s q⬁ uniformly in r g 0, 1 . w x Ž .

In particular, condition H

2

means that the function g has a sublinear growth at u s 0. This kind of nonlinearity has not been widely studied in

* Supported by C.N.R., Consiglio Nazionale delle Ricerche, Italy and by MURST Project

‘‘Metoch Variazionali ed Equazioni Differenziali Non Lineari.’’

217

0022-247Xr99 $30.00

Copyright䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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the past; to the best of our knowledge, in literature it is possible to find only few papers on this subject. Among the others, we quote the earlier

w x w x

works by G. J. Butler 2, 3 and B. L. Shekhter 8 , for ordinary differential

w x

equations, the book of M. A. Krasnosel’skii, et al. 6, Sect. 22 and the w x

paper by E. W. C. Van Groesen 10 . In these works, an assumption on the Ž . behaviour of the nonlinearity g at infinity always follows condition H .

2

Ž .

The second author studied problems related to P , when, as in the present case, no assumptions at infinity are required. In particular, our

w x

work is strictly related and motivated by a previous paper 4 , where the Ž .

existence of infinitely radial solutions to P has been proven. More

w x Ž .

precisely, in 4 the authors show that, under H

2

and a technical assump- tion on g, there exists k

0

g ⺞ such that for every integer k G k Problem

0

Ž P has at least two radially symmetric solutions w , z .

k k

with k zeros in w 0, 1 . Moreover, the sequences w . Ž

k

. and z Ž

k

. converge uniformly to zero

w x in 0, 1 .

w x

The proof of the result in 4 , which is valid for problems involving operators more general than the Laplacian e.g., the p-Laplacian and the Ž mean curvature operator , is developed through a topological degree . approach together with a time-map technique. The crucial point is the fact

Ž .

that, when looking for radial solutions, P can be written as a boundary value for a second-order ordinary differential equation.

Ž . Ž .

The aim of this paper is to prove that, when H

2

holds, Problem P also admits nonradial solutions. To do this, we consider a modified problem of the form

< <

y⌬u x s g x , u x , Ž . ˜ Ž Ž . . x g ⍀,

P ˜ Ž . u x Ž . s 0, x g ⭸ ⍀,

w x

where g is any odd radial function extending g to 0, 1 ˜ = ⺢ in such a way that

g r , u Ž .

lim ˜ s 0, Ž 1.1 .

< <uªq⬁

u

w x Ž ˜ .

uniformly in r g 0, 1 . For Problem P we prove, by means of the direct method of the calculus of variation, that there exists a sequence u

k

of

Ž .

nonradial weak and then classical solutions for which 5 5

lim u

k

s 0. Ž 1.2 .

kªq⬁

Ž .

From Relation 1.2 we then conclude that, for k sufficiently large, the Ž .

functions u are solutions of the original problem P .

k

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In order to state our main result, let us consider the standard

Ž . Ž .

parametrization of ⍀ in polar coordinates r,

i

i s 1, . . . , N y 1 :

w x w x

⍀ s r,  Ž ␪ , . . . , ␪

1 Ny1

. r g 0, 1 , ␪ g 0, ␲ ,

i

w x

i s 1, . . . , N y 2,

Ny1

g y␲ , ␲ 4 . Then, we will prove:

Ž . Ž .

T

HEOREM

1.1. If g satisfies H

1

and H , then there exists k

2 0

g ⺞ such

1

Ž . Ž .

that for all k G k there exists a nontri

0 ¨

ial solution u

k

g H ⍀ of P which

0 2

is

k

-periodic and odd in the

Ny1

component.

The assumption on the oddness of the nonlinearity g is crucial to show that the functions u are not radial. Moreover, as it is clear after the proof

k

Ž .

of Theorem 1.1, in order to obtain a multiplicity result for P in terms of functions periodic in the ␪

Ny1

component it is sufficient to require that the nonlinearity g does not depend on the angular variable

Ny1

.

This observation also shows the fact that when g is independent from the space variable x the existence of radial solutions can be obtained. We also stress the fact that no assumption on g at infinity is required;

moreover, the nonlinearity is defined only in a neighbourhood of u s 0.

Looking to the statement of Theorem 1.1, we understand that every solution u belongs to the set

k

2 ␲

E

k

s u g H ⍀ ½

01

Ž . u r , Ž ␪ , . . . , ␪

1 Ny2

, ⭈ is odd and . k -periodic . 5

w x

As observed in 10 , the subspaces E represent a natural constraint for

k

˜ Ž .

Problem P in the sense that every critical point of the energy functional

˜

1

Ž associated to P Ž .. on E is a critical point on the whole space H

k 0

Ž ⍀ . . Hence, by minimizing the energy functional on every set E , we prove

k

Ž ˜ .

the existence of infinitely many solutions to P . We point out that our multiplicity result follows from elementary arguments direct method of Ž the calculus of variations and regularity estimates on elliptic problems . In .

Ž w x.

literature see, e.g., 7 it is quite frequent to meet multiplicity theorems obtained through the application of more advanced theories as, e.g., the Ž index theory or the Ljusternik ᎐Schnirelmann theory . Most of these re- .

Ž w x.

sults see, e.g., 7, Theorem 9.38; 9, Theorem 6.6 are concerned with

Ž .

nonlinearities superlinear and subcritical at infinity; in this case the

existence of an unbounded sequence of solutions with arbitrarily large Ž

energy can be shown. Theorem 1.1 recovers the dual situation, in the .

sense that when a superlinear condition at infinity is assumed, then we

have an unbounded sequence of solutions, while when we require a

sublinear behaviour near zero we find an infinitesimal sequence of solu-

tions.

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The structure of the paper is as follows. In Section 2 we show that the

˜ Ž .

energy functional associated to P admits at least one global minimum on

Ž .

every set E

k

see Theorem 2.1 . Moreover, we will prove a global estimate

1

Ž . Ž ˜ .

for the H

0

⍀ -norm of the solutions u of P . Section 3 is devoted to

k 1

Ž .

regularity arguments proving that a global estimate for the H

0

⍀ -norm of the solutions u leads to an estimate of the corresponding C

k

C

1

-norm. This

Ž . will imply the crucial relation 1.2 .

5 5

In what follows, for every q G 1, we denote by ⭈

q

the usual norm in

q

Ž .

Ž .

the L ⍀ space. Analogously, the norm in the space L ⍀ will be

5 5

1,

Ž w x .

denoted by u . Finally, we denote by C

C 0, 1 = ⺢ the space of the

1

Ž .

C -functions with Holder exponent ␤, for ␤ g 0, 1 .

2. PRELIMINARY PROPERTIES

0,

Ž w x .

First of all, let us fix an extension g ˜ g C C 0, 1 = ⺢, ⺢ of g such that g r , u Ž .

lim inf ˜ s 0, Ž 2.1 .

< <uª⬁

u

w x Ž . Ž . w x

uniformly in r g 0, 1 , and g r, u s yg r, yu for all r g 0, 1 and ˜ ˜

˜ Ž .

u

Ž < < .

u g ⺢. Moreover, let G x, u s H g x ,

0

˜

¨

d

¨

. We consider the energy functional defined by

1

< <

2

˜

␸ u s Ž .

2

H ⵜu dx y H G x , u dx, Ž .

1

Ž . 5 5 < <

2

on the Hilbert space H

0

⍀ endowed with the norm u s H ⵜu dx. It

Ž .

is standard to prove that, by 2.1 , ␸ is well defined and Frechet differen- ´

Ž w x.

tiable on E see, e.g., 7 . Moreover, critical points of ␸ are weak and then

Ž .

classical see Sect. 3 solutions of the problem

< <

y⌬u x s g x , u x , Ž . ˜ Ž Ž . . x g ⍀,

P ˜ Ž . u x Ž . s 0, x g ⭸ ⍀.

For all k g ⺞, let us consider the subspace

2 ␲

E

k

s u g H ⍀ ½

01

Ž . u r , Ž ␪ , . . . , ␪

1 Ny2

, ⭈ is odd and . k -periodic . 5

In the following lemma we give some useful properties of the sets E and

k

of the functional ␸:

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Ž .

L

EMMA

2.1. Under Assumption 2.1 we ha

¨

e:

1

Ž . 1. For e

¨

ery k g ⺞ the set E is weakly closed in H ⍀ .

k 0

2. The functional ␸ is sequentially weakly lower semicontinuous and coerci

¨

e on E , i.e.,

k

lim I u Ž . s q⬁.

5 5uªq⬁, ugEk

The proof of Statement 1 in Lemma 2.1 is standard and therefore it is Ž . Ž omitted; Statement 2 is an immediate consequence of Condition 2.1 see,

w x.

e.g., 7 .

w x

As in the paper by E. W. C. Van Groesen 10 , we observe that the set Ž ˜ .

E is a natural constraint for Problem P , in the sense that every critical

k

1

Ž . point of ␸ on E is a critical point of ␸ on the whole space H ⍀ . For

k 0

this reason, we look for minima of the functional ␸ restricted to E . Then,

k

according to this remark, let us denote m

k

s inf ␸.

Ek

The lemma gives a first estimate on m :

k

L

EMMA

2.2. For all k g ⺞ we ha

¨

e

y⬁ - m F m - 0.

1 k

Proof. Noting, by Lemma 2.1, that ␸ is coercive on E , we plainly

1

obtain that m

1

) y⬁. Moreover, because E ; E , we have m G m for

k 1 k 1

all k g ⺞. Finally, to prove that m - 0 for all k g ⺞, consider the

k

domain

⍀ s

k

½ Ž r , ␪ , . . . , ␪

1 Ny1

. g ⍀ ␪

Ny1

g 0, ␲ k / 5 .

k 1

Ž .

k

Let ␭ be the first eigenvalue of y⌬ on H ⍀ and

1 0 k ¨1

the correspond-

k

Ž .

ing eigenfunction; we first observe that

¨1

g L ⍀ . We can consider the

k

periodic and odd extension

¨k

and

¨1k

over ⍀. Then,

¨k

g E l L

k

and, by definition,

< ⵜ

¨

<

2

dx s 2k < ⵜ

¨

<

2

dx s 2k

k

< <

¨ 2

dx s ␭

k

< <

¨ 2

dx.

H

k

H

k 1

H

k 1

H

k

k k

˜

k 2

Ž . Ž . Ž . < <

Now, using H 2 , we can take ␧ g 0, ␧ to be such that G x, u G ␭ u

k 0 1

< < Ž 5 5

.

for all x g ⍀ and u g ⺢ with u F ␧ . Then, for all s g 0, ␧ r

k k ¨k L

we

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obtain s

¨k

g E and

k

1 2 2

2

< <

k

< <

␸ s Ž

¨k

. F s ž

2

H

¨k

dx y ␭

1

H

¨k

dx / - 0, from which we conclude that m

k

- 0.

Now, we prove that for every k g ⺞ the infimum m is achieved;

k

Ž ˜ .

therefore Problem P admits a solution in E for all k

k

g N. Indeed, we have the following:

T

HEOREM

2.1. For all k g ⺞ there exists u g E _ 0 classical solution

k k

 4

˜

Ž . Ž .

of P such that ␸ u s m . Moreo

k k ¨

er, there exists M ) 0 such that

5 5 u

k

F M, ᭙k g ⺞. Ž 2.2 .

Proof. First of all, from Lemma 2.1, using the direct method of the calculus of variations, we immediately obtain that for all k g ⺞ there

 4 Ž .

exists u

k

g E _ 0 such that

k

␸ u s m . Moreover, as already noticed,

k k

˜ Ž .

u is a weak solution of problem P ; because u

k k

/ 0 and u is odd in

k

␪ , u cannot be radial.

N -1 k

Ž .

By regularity arguments see Sect. 3 , we can also prove that u

k

is a

Ž ˜ . Ž .

classical solution of P . Moreover, by Lemma 2.2, m

1

F ␸ u - 0 for all

k

Ž . k g ⺞: the coercivity of ␸ implies that there exists M ) 0 such that 2.2 holds.

Now, we state a crucial lemma whose proof is given in the next section:

˜

Ž . Ž .

L

EMMA

2.3. For the sequence u

k

of solutions of P gi

¨

en by Theorem 2.1 we ha

¨

e

5 5

lim u

k

s 0. Ž 2.3 .

kªq⬁

By means of Lemma 2.3 we can prove Theorem 1.1, stated in the Introduction.

Proof of Theorem 1.1. By Lemma 2.3, for ␧ given in Section 1, we can

0

5 5

find k

0

g ⺞ such that u

k

- ␧ for every k G k . Then

0 0

< < < <

g x , u Ž

k

Ž . x . s g x , u x , ˜ Ž

k

Ž . . ᭙k G k , ᭙ x g ⍀.

0

Ž 2.4 .

˜

Ž . Ž .

Because, by Theorem 2.1, u is a solution of P , Relation 2.4 implies

k

Ž .

that u is a solution of the original problem P .

k

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3. REGULARITY OF THE SOLUTIONS

In this section we study the regularity of the solutions to an elliptic boundary value problem of the form

y⌬u x s g x, u x , Ž . Ž Ž . . x g ⍀,

Ž 3.1 . u x Ž . s 0, x g ⭸ ⍀,

N

Ž .

where ⍀ is the open unit ball in ⺢ N G 2 . We prove two results, depending on the regularity of the nonlinearity g: ⍀ = ⺢ ª ⺢; as far as

Ž .

the first one is concerned, we assume that g g C ⍀ = ⺢ . Under this hypothesis, we have

Ž .

P

ROPOSITION

3.1. Assume that g g C ⍀ = ⺢ and that there exists C ) 0 such that

< g x , u Ž . < F C 1 q u , Ž < < . ᭙ x g ⍀, ᭙u g ⺢. Ž 3.2 .

1

Ž . Ž . 5 5

Then, for e

¨

ery weak solution u g H ⍀ to 3.1 , with u F M for some

0

Ž . Ž < < .

M ) 0, there exist ␣ g 0, 1 and K s K N, ⍀ , M ) 0 such that u g

1,

Ž .

C ⍀ and

5 5 u

C1 ,

F K. Ž 3.3 .

1

Ž . Ž . 5 5

Proof. Let u g H ⍀ be a solution to 3.1 with u F M. First note

0

Ž w x.

that, by standard Schauder estimates see 9, Appendix B , for every Ž < < .

q ) N there exists C s C q, ⍀ , N ) 0 such that

1 1

5 5 u

W2 , q

F C u q g u

1

Ž 5 5

q

5 Ž . 5

q

. . Ž 3.4 .

w x Ž .

Moreover, by 9, Lemma B3 , from Assumption 3.2 we deduce that there Ž < <.

exists C

2

s C C, q, ⍀ ) 0 such that

2

5 g u Ž . 5

q

F C 1 q u

2

Ž 5 5

q

. . Ž 3.5 .

Ž . Ž .

Hence, 3.4 and 3.5 imply that

5 5 u

W2 , q

F C 1 q u

3

Ž 5 5

q

. , Ž 3.6 .

N

w x

for some constant C

3

) 0. Now, let ␣ s 1 y ; by 5, Theorem 7.26 , from

q

Ž 3.6 , we infer that there exists C .

4

s C q, ⍀ , N ) 0 such that

4

Ž < < .

5 5 u

C1 ,

F C 1 q u

4

Ž 5 5

q

. . Ž 3.7 .

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Ž . Now, we perform some Sobolev estimates in order to obtain, from 3.7 , a

5 5

1,

5 5

relation between u

C

and u . We distinguish two cases, according to the value of N:

Ž w

N s 2: in this case it is well known see for instance 1, Corollary

x. Ž < <.

IX.14 that for every q G 2 there exists K s K q, ⍀ ) 0 such that

1 1

5 5 u

q

F K u ;

1

5 5 Ž 3.8 .

5 5 Ž . Ž .

now, using the fact that u F M, from 3.7 and 3.8 we deduce that 5 5 u

C1 ,

F K , Ž 3.9 .

Ž . < <

where K s C 1 q K M . We also observe that K only depends on N, ⍀ ,

4 1

and M: this completes the proof.

Ž . w

N G 3: let us fix q ) 2* in 3.7 . The Sobolev imbedding theorem 1,

x Ž < <.

Corollary IX.14 ensures that there exists K

2

s K N, ⍀ ) 0 such that

2

5 5 u

2*

F K u .

2

5 5 Ž 3.10 . Using the fact that u g C

1,

, we can write

5 5 u

qq

s H < u x Ž . <

qy2*

< u x Ž . <

2*

dx F u 5 5

Cqy2*1 ,

5 5 u

2*2*

. Ž 3.11 .

Ž . 5 5 Ž .

Now, from 3.10 and the fact that u F M, 3.11 becomes

2*rq 1y2*r q1 ,

5 5 u

q

F u 5 5

C

Ž K M

2

. . Ž 3.12 . Ž .

Therefore, from 3.7 we can conclude that

5 5 u

C1 ,

F C 1 q K u

4

Ž

4

5 5

1Cy2*r q1 ,

. , Ž 3.13 . Ž < < .

for some K

4

s K N, ⍀ , M ) 0. Because 0 - 1 y 2*rq - 1, Condition

4

Ž 3.13 implies that there exists K . s K N, ⍀ , M ) 0 such that u Ž < < . 5 5

C1,

F K and this completes the proof.

Using Proposition 3.1 and some standard arguments based on a greater

Ž w x.

regularity of g see, e.g., 9, Appendix 2 we obtain the following result:

0,

Ž . Ž .

P

ROPOSITION

3.2. Assume that g g C ⍀ = ⺢ for some ␤ g 0, 1 Ž .

and that there exists C ) 0 such that 3.2 holds. Then, for e

¨

ery weak

1

Ž . Ž . 5 5

solution u g H ⍀ to 3.1 , with u F M for some M ) 0, there exists

0

Ž . Ž < < .

1,

Ž .

␣ g 0, 1 and K s K N, ⍀ , M ) 0 such that u g C ⍀ and

5 5 u

C1 ,

F K. Ž 3.14 .

2

Ž . Ž .

Moreo

¨

er, u g C ⍀ and u is a classical solution to 3.1 .

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In particular, we observe that Proposition 3.2 applies to the solutions of

˜ Ž .

the boundary value problem P considered in Section 2. Hence, all the

˜

Ž . Ž .

solutions u

k

k g ⺞ given in Theorem 2.1, are classical solutions to P ;

5 5 Ž Ž ..

moreover, because u

k

F M for every k g ⺞ see Condition 2.2 , from Proposition 3.2 we deduce that there exists K ) 0 such that

5 5 u

k C1 ,

F K , ᭙k g ⺞. Ž 3.15 . This relation is crucial to prove Lemma 2.3, stated at the end of the previous section:

Ž .

Proof of Lemma 2.3. From Condition 3.15 and the Ascoli ᎐Arzela ` theorem, we deduce that there exists a subsequence which is still denoted Ž

.

0

Ž .

by u

k

such that u

k

ª u in C ⍀ .

First, we will prove that u ' 0. To this aim, we recall that for every

2 Ny2

w x w x Ž .

d g 0, 1 = 0, ␲ the function u d,

k

⭈ is periodic of period

k

. Now,

w x w x

Ny2

w x

suppose that there exist d

0

g 0, 1 = 0, ␲ and ␪ g y␲ , ␲ such

0

Ž . Ž .

that ␧ [ u d , ␪ / 0 and suppose, e.g., that ␧ ) 0. We denote

0 0 ¨k

␪ s

Ž . Ž . Ž .

u d ,

k 0

␪ and

¨

␪ s u d , ␪ .

0

w x < <

By the continuity of

¨

there exists I

0

; y␲ , ␲ with I ) 0 such that

0

Ž .

¨

␪ G for every ␪ g I . Therefore, by the uniform convergence of

2 0 ¨k

to Ž .

¨

, we infer that there exists k

0

g ⺞ such that

¨k

␪ G ␧r4 for every ␪ g I

0

< <

and for every k G k . But if k I ) 2␲ , I must contain at least one zero

0 0 0

of

¨k

, a contradiction. This shows that u ' 0.

The argument developed above gives us a more precise information:

Ž .

indeed, it shows that every convergent in the uniform norm subsequence of u

k

must converge to zero. This is sufficient to ensure that all the

Ž .

sequence u

k

converges to zero in the uniform norm and the lemma is proved.

REFERENCES

1. H. Brezis, ‘‘Analyse Fonctionelle,’’ Masson, Paris, 1983.´

2. G. J. Butler, Rapid oscillation, nonextendability and the existence of periodic solutions to second order nonlinear ordinary differential equations, J. Differential Equations 22 Ž1976 , 467. ᎐477.

3. G. J. Butler, Periodic solutions of sublinear second order differential equations, J. Math.

Ž .

Anal. Appl. 62 1978 , 676᎐690.

4. A. Capietto, W. Dambrosio, and F. Zanolin, Infinitely many radial solutions to a

Ž .

boundary value problem in a ball, Quad. Dip. Mat., Uni¨. Torino 20 1998 , 1᎐29.

5. D. Gilbarg and N. S. Trudinger, ‘‘Elliptic Partial Differential Equations of Second Order,’’ 2nd ed., Springer-Verlag, Berlin᎐Heidelberg, 1983.

6. M. A. Krasnosel’skii, A. I. Perov, A. I. Povolotskii, and P. P. Zabreiko, ‘‘Plane Vector Fields,’’ Academic Press, New York, 1966.

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7. P. H. Rabinowitz, ‘‘Minimax Methods in Critical Point Theory with Applications to Differential Equations,’’ Conf. Board Math. Sci., Vol. 65, Providence, RI, 1986.

8. B. L. Shekhter, On existence and zeros of solutions of a nonlinear two-point boundary

Ž .

value problem, J. Math. Anal. Appl. 97 1983 , 1᎐20.

9. M. Struwe, ‘‘Variational Methods,’’ Springer-Verlag, Berlin᎐Heidelberg, 1990.

10. E. W. C. Van Groesen, Applications of natural constraints in critical point theory to

Ž .

boundary value problems on domains with rotation simmetry, Arch. Math. 44 1985 , 171᎐179.

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Although in many cases it has regular solutions, in order to minimize the theory it is convenient to introduce the concept of measure-valued solution t ; moreover we restrict to

Zhou, Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators, Indiana Univ.. Rubio de Francia, Weighted norm inequalities and related

Szulkin, Multiple solutions to a nonlinear Schr¨ odinger equation with Aharonov- Bohm magnetic potential, Nonlinear Differ.. Lenzmann, Mean-field limit of quantum Bose gases

We study the nonlinear Neumann problem (1) involving a critical Sobolev expo- nent and a nonlinearity of lower order... The assumption (f 1 ) replaces the usual