Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics

300

J.

Serrin

and

H.

Zou

with

1

c=

(m

-

l)(nq

+

n

-

m)’

By

(5.2)

and Lemma

1

we have for suitably small

<

(depending on

q,

n

and

m)

which implies

K(r)

<

0

for

0

<

r

<

R.

If

R

<

00,

it follows that

Z,(R)

<

0.

But

also

u(R)

=

0,

so

that

Z,(R)

>

0,

a

contradiction.

Hence

R

=

00

and the theorem is proved.

hT

Iu’1

P+l

--m

<&,

o<~<R,

7

Non-existence

of

Ground States

If

p

is strictly greater than the critical exponent, there always exists

at least one ground state for (I)m. However, if

p

is subcritical we

know

so

far only that

(I)m

admits ground states

if

q

<

mp/(p

+

1).

In this section we shall use the main lemma established in Section

4

to prove non-existence for suitable pairs

(p,

q)

in the remaining part

of the subcritical region.

For

convenience, we denote

by

I1

and

1

the two critical values

11

=

,

1=

,

m<n.

(m

-

1).

(m

-

1).

+

nz

n-m

As

in the case

m

=

2

(see

[2],

Section

7),

it is important to study the

roots of the quadratic equation

D(t)

=

for fixed values

[2(m

-

1)t

+

(m

-

2)pt

-

m(m

-

1)p12

4(m

-

1)(n

-

m)

m(n

-

1)

-

(1

-

p)[(p

+

l)t2

-

mpt]

=

0

(7.1)

of

p

in

(11,1].

Ground States

of

Degenerate Quasilinear Equations

301

When

n

>

m/(m

-

l),

so

that

of

course also

n

>

2,

the situation

is simple:

for

p

E

(I1,/]

equation (7.1) has

a

unique root

tj

in the

interval

>.

(7.2)

($9

2(m

-

1)

+

(m

-

2)p

m(m

-

1)P

The

case

2

5

TI

<

rn/(m-

1)

is more delicate. In this case

m

<

2,

and moreover there

is

a

unique root

p

of

the quadratic equation

P(s)

=

[2(m

-

1)

-

(2

-

m)sI2

4(m

-

1)(n

-

m)

-

(I

-

s)(s

t

1)

=

0

(7.3)

m(n

-

1)

in the interval

I1

<

s

<

1;

to prove the existence

of

p,

note that

P(1)

=

[2(m

-

1)

-

(2

-

m)ZI2

>

0

and that

P(s)

is quadratic in

s.

Now consider (7.1) when

11

<

p

<

p.

Observe that

P(I1)

<

0

and

P(I)

>

0

while

P(p)

=

0,

so

P(p)

<

0

for

I1

<

p

<

p.

Since

P(p)

is

the coefficient

of

t2

in

D(t)

we now have

D(mp/(p+

1))

>

0,

D(t)

+

-00

as

t

+

00.

Hence (7.1) has exactly one root

tj

in the interval

(mp/(p

t

l),~)

since

D(t)

is

quadratic in

t.

Clearly

p

+

I

as

n

+

m/(m

-

1)

and

q+masp--+p.

When

n

=

m/(m

-

1)

then

m

<

2

and

n

>

2, and

2(m

-

1)

2-m

=

I,

P(I)

=

0.

In turn,

for

11

<

p

<

I

there

is

again exactly one root

tj

of

(7.1) in

the interval (7.2).

Theorem

C

Suppose

m

<

n

and

n

2

2.

Then equation

(I),,,

admits

no

positive

radial ground states

if

any

of

the following conditions

holds:

302

J.

Serrin

and

H.

Zou

(4

9

>

mP/(P

+

11,

m

-

1

L

P

L

11;

11

<

P

<

p,

11

<

P

<

1,

11

<

P

I

1,

(iii)

9

2

Q,

(iv)

9

2

Q,

(4

9

2

q,

nz

<

n/(n

-

1);

nz

=

n/(n

-

1);

m

>

n/(n

-

1).

Here

Q

(>

mp/(p

+

1))

satisfies

(7.1),

while

p

is the unique root of

(7.3)

in the interval

11

<

s

<

1.

If

m

<

n

<

2

then

P(1)

>

0.

Clearly cases

(iv)

and

(v)

cannot

occur, but otherwise the results continue to apply when

P(11)

<

0.

On the other hand, if

P(l1)

2

0,

that is if

m(m

-

1)(n

-

2)2

2

4(n

-

m),

then equation

(7.3)

has no root in

(II,~].

In this case part (iii) of the

above theorem

also

does not apply, and the whole strip

9>-

mp

11<p<l

p+

1’

is left undetermined. We state this as Theorem

C’.

Theorem

C’

Suppose

m

<

n

<

2,

and let

p

=

4(n

-

m)

-

m(m

-

I>(.

-

212.

If

p

>

0

then the first three parts

of

Theorem

C

continue to apply,

while

if

p

5

0

just the first two parts

of

Theorem

C

are valid.

The proof of the first two parts

of

Theorem

C

only involves the

asymptotic estimates given in Section

5.

To

prove the remaining

parts we use the identity in Section

4

and the following lower estimate

for ground states.

Lemma

6

Suppose that u(r) is

a

solution

of

(IVP),

satisfying

(I,,,.

Ground States

of

Degenerate Quasilinear Equations

303

and assume that any

of

the conditions (iii),

(iv)

or

(v)

holds. Then

liminf

T+oO

rkG(r)

>

0.

(7.6)

The proof of this lemma and of parts (iii)-(v) of the theorem is

exactly parallel to that for the case

m

=

2

in Section

7

of

[2].

Remark.

When

m

-

1

<

p

5

11

and

q

>

mp/(p

+

l),

equation

(I)m

does not even admit singular radial ground states, that is, non-

negative solutions

of

(1)m

on

R"

\

0

which tend to infinity at the

origin. Indeed,

for

this range the argument only depends on the

asymptotic behavior of solutions at infinity, having nothing to do

with their behavior at the origin. However, when

n-m

n-1

q=-

P,

o<p<m,

(I)m

does admit singular solutions of the form

cr-a.

The nonexistence results above do not cover the supercritical

range

mp/(p

+

1)

<

q

<

a

if

11

<

p

<

jj

and

mp/(p

+

1)

<

q

if

jj

5

p

<

1,

since

D(t,p)

>

0

in this region and the proof does not ap-

ply. Indeed we do not know whether

or

not existence holds for these

parameter values. However, we can show that any radial ground

state with

(p,q)

in this range must have

a

suitably large central

value (depending only on

p,

q,

n

and

m).

Theorem

4

Suppose

that either

mp

<q<q,

11<p<p

Of

-

mp

<

q,

p

5

p

<

1.

(7.7)

P+l P+l

Then there exists

a

constant

(0

=

<o(p,q,n,m)

such

that

if

u

=

u(r)

is a radial

ground

state

of

(I)m,

then necessarily

When

n

5

m,

both the critical exponents

I1

and

l

are infinity

so

that every

p

is subcritical. In this case the following non-existence

theorem holds,

a

simple extension of Theorem

C.

304

J.

Serrin

and

H.

Zou

Theorem

D

Suppose

n

5

m

and

Then equation

(I)m

admits no positive radial ground states.

Theorem

D

involves only the first two cases

of

Theorem

C,

the

proof of which only used the asymptotic behavior of solutions. Since

these estimates apply

for

all

m

>

1

and

n

>

1,

Theorem

D

follows

directly from Theorem

3

and the corollary in Section

5.

8

Compact Support Ground

States

We have concentrated on

positive

radial ground states in the earlier

results of the paper. However, equation

(I)m

can admit

compact

support

ground states, these being positive for

r

<

a

(0

<

a

<

w)

and identically zero for

r

2

a.

This section contains two results about

radial compact support ground states, the first concerning existence,

the second non-existence.

We first note,

as

shown in the following lemma, that radial com-

pact support ground states can only exist when

q

<

m-

1

and

p

<

1.

Lemma

7

Suppose that q

2

m

-

1

or

p

2

1.

Then equation

(I)m

cannot admit any mdid compact support ground state.

Proof.

The case

p

2

I

follows from the proof in Section

3.

In the

case

q

2

m

-

1,

suppose for contradiction that

(1)m

admits radial

compact support ground states and let

u

be such

a

solution. Then

there exists

a

finite number

R

>

0

such that

0

bviously

since

q

2

m

-

1.

Using

(IVP),,

(8.1)

and

(8.2),

we have

Ground States

of

Degenerate Quasilinear Equations

305

for

r

E

(O,R),

since

u'

<

0

by Lemma

1.

Clearly this is impossible

since the left side is negative, while the right positive. This completes

the proof.

Theorem

5

Suppose p

<

q

<

mp/(p

+

l),

0

<

p

<

m

-

1.

Then

equation

(I)m

admits infinitely many compact support ground states.

The proof

is

a

combination of those for Theorem

1

and Theorem

C. Indeed by Theorem C we have

R

<

00.

Then u'(R)

=

0

if the

initial value

(

is small enough (see Theorem

5

of

[2]).

Extending

u

by

defining

u

=

0

for

T

2 R

gives the desired compact support ground

state.

Theorem

6

Suppose q

2

mp/(p

+

1)

and

p

>

0.

(I)m

admits

no

radial compact support ground states.

Then equation

The proof is exactly parallel to that

for

Theorem

6

in

[2].

Bibliography

[l]

M. Chipot and

F.

Weissler,

Some blow up results for nonlinear

evolution equations with a gradient term.

SIAM

J.

Math. Anal.

20

(1989), p. 886-907.

[2]

J.

Serrin and

H.

Zou,

Existence and non-existence for ground

states

of

quasilinear elliptic equations.

Arch. Rational Mech.

Anal., to appear.

This page intentionally left blank

Gradient Estimates,

Rearrangements

and

Symmetries

Giorgio Talenti

Department

of

Mathematics

University

of

Florence

viale Morgagni

67A,

50134

Florence, Italy

Consider

a

Dirichlet boundary value problem for

a

second-order

partial differential equation

of

elliptic type and suppose that esti-

mates

of

the gradient

of

relevant solutions are in demand.

By

way

of

example, consider the following archetype

u

=

0

on the boundary of

G

(W

Here

G

is an open subset of euclidean n-dimensional space

Rn;

coef-

ficients

aij

are real-valued, measurable and bounded; ellipticity reads

n

C

aij(z)titj

2

t;

t

* *

*

t

ti

(2)

i,j=l

for

every

x

in

G

and cvery

t

from

Rn;

f

is real-valued and belongs to

an appropriate Lebesgue space.

If

u

is

a

weak

or

variational

solution

to problem

(l),

then the following inequality

Differential Equations

with

Copyright

@

1993

by

Academic Press, Inc.

Applications to Mathematical

All

rights

of

reproduction

in

any

form

reserved.

Physics

ISBN

0-12-056740-7

307

308

Giorgio

Talen

ti

holds, where

q

=

np/(n

+p)

and

n/(n

-

1)

5

p

5

2.

The inequality is

sharp:

some

coefficients

a;j,

some domain

G,

and special

f

and

u

exist which

render

(3a)

an equality. Inequality (3) is an easy corollary of the

following theorem.

Theorem

1

Consider the variational problem:

lgrad

ulpdx

=

maximum,

Jl

where u is

a

weak

solution to problem

(l),

and domain

G,

coefficients

a;j, and right-hand side

f

are the competing variables. Assume the

measure

of

G

is given; coefficients a;j satisfy ellipticity condition

(2);

f

belongs to

L2n/(n+2)(G)

and is equidistributed with

a

given

function.

In other words, the collection

of

those points x from

G

such that

If

(.)I

exceeds t

-

a

level set

of

f

-

has

a

prescribed

measure for every nonnegative t. (There are some minor alterations

if

the dimension,

n,

is

2.)

Assume

0

<

p

5

2.

Then the maximum

in

question is acheived when

G

is

a

ball

-

centered at the origin,

say;

a;j

=

6;j,

the coefficients

of

Laplace operator;

f

is nonnegative,

spherically symmetric

-

i.e., invariant under rotations about the

origin

-

and radially decreasing.

Inequality (3) and Theorem

1

appeared in

[GI.

Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics

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