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dimensional thermoelastic system. Working Paper of Systems Research Institute,

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theory for structural phase transitions in shape memory alloys. Physica D, 39 (1989),

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Semilinear elliptic boundary value problems with

critical Sobolev exponent

Catherine Bandle

Mathematisches Institut der Universitat Basel

Rheinsprung 21, CH-4051 Basel

Email : bandle@math.unibas.ch

Abstract

Problems of the form Lu + a(x)u + vP = 0 in D C M.

, N > 3, p = ^| with

Dirichlet boundary conditions are considered where L is a second order uniformly

elliptic differential operator in divergence form, in particular L = div (p

(x)V).

The question discussed in this paper is: What is the impact of p and a on the

existence of positive solutions ?

Some general existence criteria based on the concentration-compactness theorem

of Lions are discussed and applied to special cases. Nonexistence results follow from

the Pohozaev identity. A fairly complete picture is obtained for the Brezis-Nirenberg

problem in spaces of constant curvature. It can be used for comparison in more

general situations.

1 Introduction

Boundary value problems of the type

Att + u

= 0,inI)cK

:« = 0,onaD;ti>0,inD

where p > 1 and N >3,

have been studied extensively not only because they play a fondamental role in astro-

physics and chemistry but also because of their rich mathematical structure. They serve

as model equations in the study of semilineax elliptic boundary value problems. An im-

portant feature is that they are related to the following variational problem

SW{D)=inf

f\Vv\

dx;

where veV^iD), [\v\^

dx = l. (2)

D D

Here T>

(D) denotes the completion of CffiD) with respect to the norm |u|

1|2

Vu^da:}

If v* is a minimizer, then u = S(p, D)^

v* solves (1). To each solution

u of (1), we associate the "energy"

J(u)

(1)

All solutions of (1) correspond to critical points of J, in the sense that the Frechet

derivative vanishes.

In view of the embedding theorems, minimizers of S^(D) exist if :

P < ^zf and D is bounded or in cylindrical domains D = fi x K

, where s > 1 and

n is a bounded domain in 1"^' (Esteban and Lions [13]).

p = $±§ =:

- 1 and D = R

Further examples of unbounded domains for which a minimizer exists when p < ^|

can be derived from the embedding theorems in the book of Adams [1].

If p = 2* - 1 and D

=£

or if p > 2* - 1 the minimizing sequences for S^

(D)

concentrate and do not converge. Thus no minimizer exists. It should be pointed out

that even if the variational problem doesn't have a minimum in D

1,2

(D), (1) may still

have solutions. This is the case e.g. in annuli when p > ^|. They correspond to critical

points of J with larger energies than

S^^D).

In the following cases, no critical points and therefore no solutions to (1) exist :

p < jj±~ and D = R

- fi, where Q is starshaped (Reichel and Zou [15]).

P >

jjfz§

D, where D is starshaped (Pohozaev).

The reason for the failure of existence is that the Palais-Smale sequences for the

associated energy J are not compact.

The difference between the subcritical (p < 2*

—

I) and the supercritical case (p >

—

1) is also reflected in the behaviour of the Sobolev constants S^

'(D), namely :

{

positive constant if p < 2*

—

1 and D bounded,

S*>0

ifp = 2*-l,

0 ifp>2*-l.

Here 5* denotes the classical Sobolev constant, which can be computed explicitely in

terms of N.

In the subcritical case, S&^D) depends on the geometry of D. It decreases as D

increases and in the limit we have

(p)

(D)=0, asD^R

and S

{p)

(D) -> co, as D -> {z

Symmetrization applies to S^(D) and leads to the Rayleigh-Faber-Krahn type in-

equality

\D) > S^iBJvoliD)

^-^, where B

= {|z| < 1}.

The variational characterization is very useful for constructing solutions of problem

(1).

It is however restricted to the subcritical and partly to the critical cases. A different

approach to study the critical and the supercritical cases is via the nonlinear eigenvalue

problem

AM + \u +

= 0 ; u > 0, in D ; u = 0, on dD,

by means of bifurcation techniques ([11]). Up to now, most results available are for balls.

Motivated by the question whether a given compact Riemannian manifold is necessarily

conformal to one of positive constant scalar curvature, Brezis and Nirenberg [9] studied

the above problem for the critical exponent. This specific problem is communly called

the

Brezis-Nirenberg problem".

2 The Brezis-Nirenberg problem with variable

coef-

ficients

If for subcritical exponents, the Laplacian is replaced by a more general uniformly elliptic

operator with smooth coefficients, the existence of minimizers is not affected. This is not

the case for the critical and supercritical exponents. In this section, we shall report on

the following weighted Brezis-Nirenberg problem

div (aWu) + \v(x)u + /i(x)u

= 0, in D ; u = 0, on dD, (3)

where n(x), v(x) e C°(D) and a(x) e C

(D) are given positive weights. The related

variational problem is

(D)=M\

a\Vv\

dx-\ vv

dx\,

U D J D J (4)

where v e

'\D),

I n\v\

'dx = 1.

J D

In the survey paper [8], Brezis observed that the weights can have an impact on the

compactness of the minimizing sequences. A general discussion was carried out by Egnell

[12],

for bounded domains in R

, N > 4. He extended Brezis-Nirenberg's results [9]

as follows. Let Ai be the first eigenvalue of v~

div (aV) in D, with Dirichlet boundary

conditions. Then

(i) Under very mild assumptions on the flatness of the coefficients, (3) has a positive

solution for all

6 (0, Ax), which is a minimizer for S\(D).

(ii) If D is starshaped with respect to the origin, if (x,Va) < 0, {x,Vn) < 0 and 2v +

(x, Vv) > 0, then under additional mild flatness conditions of a and

fj,

at the origin, the

boundary value problem (3) has no positive solution if

^ (0, A^.

By virtue of the variational characterization of Ai, we have

C > 0, if

< Ai,

{D)\

=0, if

= Ai,

( <0, if A> Ai.

Taking trial functions of the form u

(x) =

<j>(x)

+ \x

—

£o|

) ~ , where

(f>

is a

cut-off function with compact support in a small neighborhood of x

, it turns out that

A useful tool to establish the existence of a minimizer in (4) is the

Lemma 2.1 ('Concentration-Compactness Lemma

[14])-

If for all x G D

then S\(D) attains its minimum.

The minimizers are solutions of the Euler-Lagrange equation :

div (o-Vu) + \vu + SxWim

= 0,

u > 0, in D

;

u = 0, on 3D.

The lack of minimizers is due to the fact that the minimizing sequences concentrate,

i.e. they tend to Dirac functions. Some immediate consequences of the previous remarks

are collected in the next proposition.

Proposition 2.2 (i) If a minimizer exists for some Ao, it exists for all

> Ao-

(ii) Since 5,^ (D) = 0, the range of X-values for which a minimizer exists is an interval

(A*, oo), where

< Ai.

(Hi) For AG

(A*,

Ai), a multiple of the minimizers solves the generalized Brezis-Nirenberg

problem (3).

(iv) For

> \\, the minimizers are, up to a constant factor, solutions of the equation :

div (crVu) + \vu =

The maximum principle implies that there are no positive solutions of (3) if

> Ai.

For the critical value, we have

S,.(D)

=min-f§Ls*.

x€D

fj,(x)

In this case, there always exist minimizing sequences which concentrate at the points

where a(x)

jn(x)

takes its minimum. It is not possible to exclude minimizers a

priori.

For constant coefficients (a, fi, v = 1) Brezis and Nirenberg [9] observed that in R

N > 4, A* is zero for all domains, whereas in M

, A* is a positive number depending on

the domain. They were able to compute it for the balls BR(0) = {\x\ < R} and obtained

\*(BR)

= ^jj. For general domains, the critical value is unknown.

An upper bound for A* can be constructed by means of the Green's function G(x,y)

for A +

with Dirichlet boundary conditions

([17],

[10], [8]). Indeed this Green's function

can be decomposed in a regular and a singular part as follows:

G{x

4^hv\

In D C R

, the regular part h\(x, y) is continuous in D. By the maximum principle,

h\(x,y) satisfies the inequality : hy(x,y) < h\{x,y), for A' <

< A^ Hence the function

—>

max

h\(x,x) is increasing. Since max

(x,x) < 0 and max

hx(x,x) —> oo as

—>

Ai, there is a unique value v* such that max

(x, x) = 0. Schoen [17] has shown

that v* > A*. Based on numerical computations, Budd and Humphries [10] conjectured

that

v* =

A*.

(5)

If this conjecture were true, it would yield a remarkable connection between a quantity

arising in a linear problem and a nonlinear phenomenon.

Remark 2.3 As stated

before

in the Proposition 2.2 (Hi), the range of \-values for which

a minimizer exists is an interval

(X*,\x).

This doesn't exclude solutions of (3) for A ^

(A*,A!).

They correspond to higher energy levels. Examples will be given in the next

section.

The tool to establish nonexistence is Pohozaev's identity and its variants. It is obtained

by testing (3) with suitable functions. A different approach which leads to a better

understanding of the role of this identity in the calculus of variations is proposed by

Reichel [16].

3 Problems with critical exponents in spaces of con-

stant curvature

As an application of the previous section, we shall study the Brezis-Nirenberg problem in

spaces of constant curvature. This topic has been studied in several papers [7, 5, 6, 4].

The iV-dimensional spherical space S

is described by {x e R

N+1

: \x\ = 1}. The TV-

dimensional hyperbolic space M

is given by {x £ K

w+1

N+l

l +

Eili^

We shall

restrict ourselves to the spherical case and refer to [18] for the discussion of the hyperbolic

space. The stereographic projection maps S

into R

, with the Riemannian metric given

by the line element ds = j^ =:

p\dx\,

x e K

The corresponding variational problem follows immediately from (4), if we replace V

by V/p and the volume element dx by

dx.

The Laplace-Beltrami operator assumes the

form A

= p

div(p

JV_2

V). Hence the Brezis-Nirenberg problem for S^ reads as

div(p

Vu) + \u + u

= 0, in D ; u = 0, on dD. (6)

The corresponding variational problem assumes the form

{D, p) = inf ( / | Vv\

dx -X [

dx\

VD Jp J (

)

where v e

{D),

/ \v\

dx = 1.

J D

The problems (6) and (7) can easily be treated with the same arguments developed in

[5] for the special case

= 0.

Theorem 3.1 (i) If N > 4 and -N{N - 2)/4 <

< Ai, then S

{D,p) is attained for

any domain DcS".

(ii) If\< -N(N - 2)/4 and D ^ S

, then S\{D,p) is never attained.

(Hi) If

< —N(N

—

2)/4 and if D is a starshaped domain in Bi(0) (hemisphere), then

(6) has no nontrivial positive solution.

Proof.

If in (7) we replace v by v

p 5~, we get

S,(D,

p) =

inf U

\Vv

-f\+

(

J ^g^J .

where v

e V

and / |ii

*da; = 1.

J D

(i) follows from Egnell's theorem in Section 2. The second assertion is a consequence

of the results for the problem in R". In order to prove nonexistence, we observe that

Egnell's condition : 2v + (x, V^) > 0 becomes :

(Ar(JV-2)/4 + A)(l-r

)

(l + r^

Open Problem Do there exist "large" domains in S

and A < —N(N

—

2)/4, for

which (6) has a solution ? They must necessarily correspond to critical points of higher

energy. Some numerical computations indicate that this is the case in balls containing

the hemisphere.

In S

, the picture is more involved. The following result has been obtained in [7].

Theorem 3.2 For \*{D) < A < A

there exists a solution of (6) which is a minimizer

(D,p) = inf j J \Vv\

pdx -\ I v

dx\ ,

where v G D

(D) and / \v\

dx = 1.

J D

A*(D) is negative for "large" domains

whereas

for "small" domains \*(D) is positive.

As in R

, the picture is more complete if D is a ball BR :

< A

Theorem 3.3 (i) If

40?

^ " ^

~ 0? '

where 0i is the geodesic radius of the ball defined by R = cot 0^2, then there is a

unique solution. In this case S\ < S*.

(ii) If D is contained in the hemisphere, i.e. 0i < 7r/2, and if

< *

, then there is

no solution to (6).

(iii) (a) For balls containing the hemisphere, there is a function v(8i) such that there is

no solution for v(8i) <

^a'.

(/3) Numerical computations indicate that, for

< v{di), there are solutions. They

correspond to a higher energy level.

Remark 3.4 Elementary bounds for the critical

are obtained as follows :

A*(D) depends monotonically on the domain, i.e. \*(D) <

\*(D

C D.

If D has the same volume with respect to the spherical metric as the ball BR, then

\*(B

)<\*(D).

The first claim follows from the monotonicity of S\ (D) and the second assertion is

obtained by means of symmetrization, see [5].

3.1 Qualitative properties

In this section, we consider a minimizer u\, \*{D) < A < Ai of (7). As it was already

observed by Egnell [12], we have

lim KIL =

In order to discuss the behaviour of u\ as A

—>

A*, we argue as follows. If we use u\

as a trial function for the variational characterization of S\» (D) we get

&.(£>) < S

(D) -

(A*

u\dx.

Note that S

-{D) > S

(D): Holder's inequality implies

/ \

u\dx<

I [

Hence

Sx* >S

Sx- + {\*-\)J

Letting

tend to A*, we conclude that {ux} is a minimizing sequence for Sx*. If

Sx*

not attained, then there exists a subsequence of {ux} which converges weakly in measure

to Sx'S{

}(x).

Problem Are there cases for which

Sx*

(D) is attained ?

The question arises what happens in Theorem 3.3 (iii), with ux, as

A —>

v(8i) ? The

answer is not known. Numerical computations indicate that the solution blows up at the

boundary.

Remark 3.5 Consider (7) in balls B

in S

. In this case, the Green's function G(x,Q)

for the operator A

is radial and has the form

(

G(r,0) =

2 cos ( Vl +

arccot

47rr

—c

sin I y/1 +

arccot

where c^ = cos (71VI + A/2) and

is such that G(R,0) = 0. Setting R = cot a and

observing that

we find that

' 1

—

arccot I — ) = arccot(—2a) =

—

arccot(2a) = ir

—

2a,

hxi

o,o)

= -

cos{Vrrx{

2a))

47rsin(\/l + A(7r - 2a))

h\(0,0) vanishes if and only if y/1 + A(7r

—

2a) = n/2. Since the angle

IT —

2a corre-

sponds to the geodesic radius 6\ of BR, it follows that 2A =

/A0\

—

1, which is the same

value as

\*{BR)

in Theorem 3.3 (i). By a rather involved symmetry argument, it can be

shown that for

< Ai, h\{x,x) takes its maximum at the center. Hence we have found a

further example for which the Budd-Humphries conjecture (5) holds.

Estimates for Sobolev constants

In this section, we shall describe a method to estimate Sobolev constants Sx{D,p) with

general weights p. For this purpose, we shall use the following lemma.

Lemma 4.1 Let a =

<j\Oi,

<t>

CQ°(D)

and £ =

^/a2<p.

Then

a\Vct>\

= / a

\VZ\

dx+ J (div

(CT!V^))

-^=dx.

Proof.

We have

^i|V£|

<T|V^|

+ a^lV^I

+ <r

/5£(V^, V^)

and

div(o-

/^0

VV^2) = ffi(V(

/^0

),V

/^) +

div(cr

V^)

= o-i0

|Vv^|

+ o-

^(V</>

, V^) +

^'?i

div(cr

Vy^).

Integration of the two identities yields the assertion. •

Applying this lemma to S\(D,p) with p = p

[cf. (7)] and keeping in mind that

= p~

div

V) we get

(D,p)

= inf ||

|V<f pf~

J^ [A

p^ - Apf^) p^pf v

dx}

where t; 6 2>

ll2

(D) and / pf \v\

'dx = 1.

If p

satisfies the inequality

N-2 JV-2 N±2

A«P

+o;p

<Ap

,mD, (8)

then we get

Sx(D,p)<S

{D,pi).

If we choose

(D), then (8) holds for all

such that :

A>inf(A,

+A;

(

D)p

«=2

fiz2\

_N±1

ft ' •

Hence

/ JV-2 N-2\

iV+2

(

JV-2 N-2\ _,

+A;

(.D)P

Similar transformations were used in geometry in connection with the Yamabe problem

[3].

References

[1] R.A. Adams, Sobolev spaces. Academic Press, New York (1975).

[2] F.V. Atkinson, L.A. Peletier, Emden-Fowler equations involving critical exponents.

Nonlinear Analysis TMA 10 (1986), 755-776.

[3] T. Aubin, Some nonlinear problems in riemannian geometry. Springer-Verlag, Berlin

(1998).

[4] C. Bandle, M. Flucher, Harmonic radius and concentration of energy, hyperbolic

radius and Liouville's equations Au = e" and

(

)/(

). SI AM Review 38

(1996),

191-238.