Critical boundary constants and Pohozaev identity

A NNALES DE LA FACULTÉ DES SCIENCES DE T OULOUSE

O ULD A HMED -I ZID -B IH I SSELKOU

Critical boundary constants and Pohozaev identity

Annales de la faculté des sciences de Toulouse 6

^e

série, tome 10, n

^o

2 (2001), p. 347-359

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- 347 -

Critical boundary ^constants

and Pohozaev identity (*)

OULD AHMED-IZID-BIH ISSELKOU (1)

E-mail: [email protected]

Annales de la Faculte des Sciences de Toulouse Vol. X, ^n° 2, ²⁰⁰¹

pp. 347-359

a mes deux

filles

^{Kénizé et Maöna}

RESUME. - La premiere partie ^{de ce travail concerne} le probleme,

On démontre qu’il ^{existe une constante} critique

~*=(n(n-2) 4)n-2 4,

^telle

que le problème

(PE)

^admet exactement deux solutions u~1 et u~2

(u~1 u~2)

^{si 0 E} E*, ^{une solution} unique ^{si E = E* et n’admet} pas de solution si E > E* .Toutes ces solutions seront données explicitement. ^{Il est} démontré _que quand ^{E ~} 0,

Au cours de la seconde partie, ^on s’intéresse au probleme

ou Q est un domaine borne, regulier ^{et etoile} par rapport ^a l’origine, f

est continue et depend asymptotiquement ^{de u} "comme ua", ^{1 a et}

a ~ n+2 n-2. Différents résultats d’existence de constante au bord critique e* _pour le probleme

(Qe)

^{sont donnes.}

ABSTRACT. ^- The first part of this work deals with the problem

( ^_ ;~±2

( * ) > Recu ^{le 21 mai} 2000, accepte ^{le 29} janvier ²⁰⁰¹

(1) Faculte des Sciences et Techniques, B.P. 5026 Nouakchott, Mauritanie and The Abdus Salam ICTP, Trieste, Italy.

n. - 2

We show that there exists a critical constant

E* - ~ n~ 4 2~ ) 4 ,

^such

that the problem

(PE)

^{admits two solutions u~1 and u~2}

(u~1 u~2)

^if

0 E E*, only ^one solution if E = E* and no solution if E > E* .We give

all these solutions explicitly.We ^show that, ^{when E --~} 0,

The second part ^{is devoted to the} following problem

where Q is a regular bounded domain which is starshaped ^{about the} origin, f ^is continuous and behaves like u‘~ in the second variable, ^{1 a} and o; ~

’’~§-

^{We give} different existence results for a boundary ^critical datum e* for

(Qe).

1. The Sobolev

Exponent

^Growth

Let us consider the

problem

In

[9],

^{it is} shown that there exists a critical

boundary

^{datum E* such that}

V 0 E

E*, (PE)

^{admits -at least- one}

C2-solution.

There is no solution when E > 6*. It is known that

(PO)

^{does not admit a} nontrivial solution

(see [13]). According

^to

[5],

^every

^regular

^{solution of}

(PE)

^is

spherically symmetric. Let uE (E

^>

0),

^{be a solution of}

(PE),

^then

is a solution of

where

The maximum

principle implies

It is known

(see [11],[4]

^and

[3])

^{that there exists a constant A* such that}

(AB) admits just

^two

^{C3 -} spherically symmetric

solutions when 0 A

~ * , only

^{one solution for A = A* and no} solution if A > A*. We

give

^{here the}

value of A* and the

explicit

solutions for

every ~

^{A*. We show that when}

~ -~

0,

the "small" solution tends to v ⁼ 0 the trivial solution of

(AO)

^and

the

"big"

^{one tends to}

H(x)

⁼

^1,

^x

7~

^O.

PROPOSITION 1

Proo f.

^For

let uE be a

regular (Pe)

^{solution. As in}

[13],

^{let us}

^put

and use the

Divergence Theorem,

^to

_get

From the

identity

we infer that

Using again

^the

Divergence ^Theorem,

^we

^get

where v denotes the unit outward normal to

~B1.

The Maximum

Principle implies

^that

As Vf;

^is

spherically symmetric

^{and vanishs on}

~B1,

we

get

where l is a constant ^on

~B1. Using

^{the fact that x.v = 1 on}

aBl,

^{we obtain}

from (*1

This

equation

^is

equivalent

^to

where Bl is

^the

^Lebesgue

^{measure of the unit} ball of Rn and

|S1|

⁼

is the surface ^measure of the unit

sphere.

We obtain the

following equation

^{in l =}

When

this

equation

^{admits two}

negative

^solutions

When

equation (1)

^{admits a}

unique negative

^solution

and no real solution if

So it is clear that

The

proof

^{will be}

complete,

^{if one shows that}

there exists

just

^two

regular

solutions of

(PE).

^{Let us recall that the}

problem

admits the radial solutions

(see [12])

Let us

put

It is immediate to

verify

^that

In

particular,

As the function

is continuous on with

we obtain two distinct solutions of

(PE),

^when

and ^one solution when

which is

~i.

So the

proof

^of

Proposition

^{1 is}

complete.

Let ^us

study

^the behavior of solutions when Let _uEl and uE2 be the two solutions of

(PE),

^with

and

such that

Let us

put

PROPOSITION 2

Proof.

^{Let us remark the}

following

We

give

^{here a direct}

proof, using

^the

explicit knowledge

^of

i E ~1, 2~.

As we have _seen, for _every 0 E

E*,

there exists a

unique

such that we have

We

finally get

It is immediate to

verify

^that

and

Remark 1. ^-

According

^{to Theorem 1.1 in it}

is,

ⁱⁿ

general, false

that _every

positive

^{solution u in}

B1 of

^{Du + ~ca =}

0,

^{is a} restriction

of

^a

positive

^{solution v}

of

^this

problem

^{in ~’~ .}

2. Nonlinearities with Noncritical Growth 2.1. The Subcritical Behavior

We deal here with the

following problem

Let us suppose the

following

(i)

^{SZ is a bounded}

regular

^{domain of}

~n,

^{which is}

starshaped

^about

the

origin.

(ii) (’0 x ~+, ~J2+) ,

(iii)

^{there exist}

^positive

^{constants and a}

^positive

^{function a E}

such that

lim

t)

=

a(x)

^and

f(x, t)

⁼

o(t)

^{near t =}

^0, uniformly

^{in x E 03A9.}

to.

PROPOSITION 3.2014 Under the

previous hypotheses

^{on SZ and}

f , there

exists a

positive

^constant

E* (SZ, f ),

^{such that}

for every_0 ^{E E* (SZ, f),}

^the

problem (QE) admits,

^at

least,

^one

solution u~ ~ C1,03B4 (03A9)

^,

^{0 03B4}

^{1. . There}

is no bounded solution

of (QE) if

^{E >}

E* (SZ, f ) .

Proof.

^The

proof

^is

nearly

^{the same as in}

(Theorem

^{1 in}

[9]).

^The

only

difference is that the subsolutions and

supersolutions

^are considered as

elements of ⁿ and the

inequalities

^{are in the sense of}

duality

H-1(~) ~ Ho (~)~

^°

Let us recall the main

steps

^{for this}

proof.

1. We use the

hypothesis (iii)

and Theoreme 3.1 in

[2],

^to show that the

problem (QE)

^{admits -at least- one} solution u E when E is "small"

enough.Using

the LP-estimates

(see [1]),

^{we infer that u E}

W 2~p (SZ), Vp

^{> 1.}

One can use

embedding

^results

(see [8])

^to deduce that u E

2. We show that if

(QE)

^{admits a}

^solution,

^{so does}

(QE)

^{for every E E.}

3.Using

^{the a}

priori

^{estimate in}

[6],

^{we show that}

(QE)

^{does not admit a}

solution,

^{if E is}

great enough.

From these

steps,

^{we infer that the set I of E, for which}

(QE)

^{admits a}

solution,

^{is a bounded interval.}

4. Let

f )

^{be the upper bound of I. The}

blow-up argument

^{used in}

[6],

^{can be}

^applied

^{to show that there exists no}

increasing

sequence in

I,

^{such that}

This last a

priori

~°°-estimate of the solutions _uE near

E* (SZ, f ),

^{leads to a}

solution of

(QE* (S~, f ) )

^.

Remark 2. When SZ ⁼

Br

^-

~x

^E

~’~; r ~

^and

f(x,u)

^-

ua,

then

l. every solution

of (QE)

^is

spherically symmetric (see f5J),

2.

C2)

⁼

r 1 2a

E*

(B1, cx), (see ~10~~.

2.2. The

Supercritical

^{Growth Case}

Let us consider the

following problem

We _suppose that

(i)

^{Q is a bounded}

regular domain,

^{which is}

starshaped

about the ori-

gin.

(ii)

^{a E}

R*+) ^{and 03B2}

^>

n+2 n-2

^.

Under

appropriate hypotheses,

^the

following problem

admits solutions

(see [14]).

PROPOSITION 4. ^- Let us suppose that the

problem (P)

^{admits a so-}

lution,

then under

hypotheses (i)

^and

(ii),

^{there exists a}

positive

^constant

E*(SZ, a)

^{such that}

(TE) admits,

^at

least,

^{one solution} uE E ,

0

^b

1,

^{when 0 E}

E*(SZ, a).

^{There is no} LOO-solution

of (TE) for

^{E >}

E*(S2, a).

Proof.

^{- The}

proof

is similar to the demonstration of Theorem 2 in

[9].

^.

Remark 3. ^- The

hypothesis concerning

the existence

of

^{a solution}

of

(P) is justified by

^{the critical}

^growth

^case

^(see

^section

^1).

The

E* (SZ, a)-limit

^case.

Before

dealing

^{with this case, let us state the}

following

^lemma.

LEMMA 1.2014 Under

^the

hypotheses (i)

^and

(ii)

^{, assume that is a}

sequence

of ^C2 (03A9) - ^functions

^{and is a real sequence, such that}

Then, if

^{the real sequence is bounded in}

~,

^{so is in}

Ho (SZ) . Proof. Using

^Pohozaev

Identity,

^we

get

Using

^the Green’s first

identity,

^we

get

So we infer that

Using

the maximum

principle,

and the fact

that uj

^{= , on}

^8Q,

^{we obtain}

As f2 is

regular

^and

starshaped,

^we

get

where,

As,

we

get

where c2 0 _c3 and _ci, i ⁼

2,

^{..4 are} constants not

depending

^on

j. Using

Holder’s

Inequality,

^{we obtain}

Let us

put vj

⁼

then vj

^{E and}

Using

^Poincaré

Inequality,

^we

get

As

t - J

and E~ ^{is bounded in}

~,

^this

completes

^the

proof

^{of Lemma 1. .}

Remark 4. ^- The a

priori

estimate in Lemma 1 remains true

for

^non-

linearities such

that,

^there exist constants c

and 03B3,

^with

PROPOSITION 5. Under the

hypotheses of Proposition ,~, if

a E

C°~~ (SZ),

⁰

^{~ 1,}

^then

(TE* (SZ, a))

^{admits a solution.}

Proof.

^{Let be an}

increasing

^real sequence such that

For

every

^{let u~} be the solution of

(see Proposition 4).

^As

u~ ^G

C2(O),

one can use Lemma 1 to obtain

Then,

up to ^a

subsequence,

^{u in - u in}

L2 (SZ) -

strong

^and _uj ^~ u, ^a.e. in SZ. One can

multiply (Pj) ^by

^{uj to}

^verify

^that

By using

the Lp-estimates and a

bootstrap argument,

^{one can show that u} is a solution of

(TE* (SZ, a) )

^.

PROPOSITION 6. ^- Let u be a

spherically symmetric L~ (~n)-

^solution

of

then u E and u E _,

Vp

^>

ny 1 .

^.

Proof.-

^{Let us choose r >}

0,

^{such that}

u(r)

^{oo. As u G}

LOO(Br),

^one

can use the LP-estimates

(see [1])

^{to infer that u E V} p ^> 1. We infer that

(see [8])

From the

previous line,

^{we see that}

u{3

E

C°~~(Br),

^{V 0 ~}

1,

^{So we}

can use the Schauder Estimates to deduce that u E We can use

Proposition 4,

^{to infer that}

It is easy to

verify(

^see

[10])

^that

so we deduce

that, if p > n ~’ ~ 1 ^{, then}

^{u E .}

Acknowledgments.

^The author would like to thank Prof. A. Bahri and the Mathematics

Department

^of

Rutgers University (USA)

^for

hospitality

during

the fall 1998.

- 359 -

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