class="bi x0 y0 w0 h1"
Spectral Theory and
Nonlinear Analysis with
Applications to Spatial Ecology
This page intentionally left blankThis page intentionally left blank
Spectral Theory and
Nonlinear
Applications
to
Spatial Ecology
Universidad Complutense de Madrid, Spain
14-15
June
2004
S.
Cano-Casanova
Universidad Pontificia Comillas de Madrid, Spain
J.
Lopez-Gomez
Universidad complutense de Madrid, Spain
C.
Nlora-Corral
university of Oxford,
UK
editors
K
World
Scientific
NEW JERSEY
*
LONDON

for
this
book is available
from
the British Library
SPECTRAL THEORY AND NONLINEAR ANALYSIS WITH APPLICATIONS
TO SPATIAL ECOLOGY
Copyright
0
2005 by World Scientific Publishing Co. Re. Ltd.
All
rights reserved. This book, or parts thereof; may not be reproduced in any form or by any means,
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ISBN 981-256-514-0
Printed in Singapore by Mainland Press
Preface
This volume collects the Proceedings of the Complutense International
Seminar

friend and colleague
J.
Esquinas, born
in
March 27th 1960 at Ocaiia
(Toledo, Spain), who suddenly died in August 11th 2003 at Covadonga
National Park (Asturias, Spain). J. Esquinas was a tremendously gifted
mathematician who did some seminal contributions to the theory of gen-
eralized algebraic multiplicites in the context of bifurcation theory. His
scientific carrier was
as
short
as
intense, since he dedicated many of his
efforts to the defense of the rights of the workers in Spanish Universities,
becoming a renowned very popular personality in both issues.
Besides collecting most of the contributions delivered by the participants
in this Complutense International Seminar
,
this volume also includes a
number
of
contributions by well recognized experts in Spectral Theory,
Differential Equations and Nonlinear Analysis whose mathematical work
is closely related to the one developed by
J.
Esquinas. The editors are
delighted to thank all of them for their contributions to this
so
special

31
Well Posedness and Asymptotic Behaviour of a Closed Loop
Thermosyphon
A.
Jimknez-Casas

59
Uniqueness of Large Solutions for
a
Class of Radially Symmetric
Elliptic Equations
J. Ldpez-Gdmez

75
Cooperation and Competition, Strategic Alliances, and the
Cambrian Explosion
J.
Ldpez-Gdmez and
M.
Molina-Meyer

111
Local Smith Form and Equivalence for One-parameter Families
of
F'redholm Operators of Index Zero
J. Lbpez-Gdmez and
C.
Mora-Corral

127

229
L.
Vega

247
J.
M.
Vegas

257
Some Recent Results on Periodic, Jumping Nonlinearity Problems
Some Remarks about the Cubic Schrodinger Equation on the Line
Some Remarks on the Invariance
of
Level Sets in Dynamical Systems
ON THE POSITIVE SOLUTIONS
OF
THE LOGISTIC
WEIGHTED ELLIPTIC BVP WITH SUBLINEAR
MIXED BOUNDARY CONDITIONS*
S.
CANO-CASANOVA
Departamento de Matemdtica Aplicada
y
Computacidn
Escuela Te'cnica Superior de Ingenieria
Uniuersidad Pontificia Comillas de Madrid,
2801
5-Madrid, SPA IN
E-mail: [email protected]

positive solutions of the following
nonlinear weighted elliptic boundary value problem of Logistic type, with
sublinear weighted mixed boundary conditions given by
15
=
XW(z)u
-
a(z)uT
&u
=
V(z)u
-
b(z)uq
=
0
on
rl,
q
>
1,
in
R,
T
>
1,
{
u=O
on
ro,
(1)

a.e.
in
R
for the value
X
of the parameter. The results obtained in this work
are an extension of the previous one, obtained by
S.
Cano-Casanovag for
Problem
(1)
in the particular case when
W
=
1
in
R
and
V
=
0
on
rl.
Throughout this work we make the following assumptions:
(a)
The domain
R
is
a
bounded domain of

with
ayij
=
aji
E
C'(fi),
ai
E
C(fi),
a0
E
&(a),
1
5
i,j
5
N.
(c)
The potential
a(.)
E
L,(R)
is
a
nonnegative bounded measurable
real weight function in
R
for which there exists an open subset
R:
of

a
bounded measurable real weight function
in
$2
with arbitrary sign.
(e)
As
far
as
the nonlinear mixed boundary conditions,
b(x)
E
C(rl)
is
a
positive function on
I'l
which is bounded away from zero on
I'l
n
do:,
V(x)
E
C(F1)
with arbitrary sign in
rl,
u
:=
(~1,.
. .

.
Throughout this work, for each
X
E
R,
we will denote by
C(X)
the differen-
tial operator
C(X)
:=
C
-
XW(x)
,
Positive Solutions
of
the Logistic Weighted Elliptic
BVP
3
which is uniformly strongly elliptic in
R
with the same ellipticity constant
as
the operator
L.
The following theorem collects the main results of this work.
Theorem
1.1.
Problem

in
this case, the positive solution
of
Problem
(1)~
is unique and
if
we denote
it
by
ux,
then
ux
is strongly positive
in
R
in
the sense that
.A(.)
>
0
for each
~t:
E
s2
U
rl
and
@ux(z)
<

ux
is a.e.
in
52
twice continuously differentiable.
2.
Preliminaries, Definitions and Main Notations
In the sequel a function
u
E
Wi(R)
is said to be
strongly positive
in
R
if
u(z)
>
0
for
each
z
E
R
Url
and
apu(z)
<
0
for

x
wi-+rl)
defined by
4
S.
Cano-Casanova
where
v
:=
(~1,.
. .
,VN)
E
C1(I'l;RN)
is any outward pointing nowhere
tangent vector field to
I'l.
We want to point out that using the boundary
operator
B(k(z))
just defined, Problem (1)~ can be written in the form
{
B(-V(z)
+
b(~)uP-~)u
=
0
on
dR
(6)

that there exists
a
least real eigenvalue of Problem
(7),
denoted in the
sequel by
@[L,
B(k(z))],
and called
principal eigenvalue
of
(L,
B(k(s)),
0).
The principal eigenvalue is simple and associated with it there is
a
positive
eigenfunction, unique up to multiplicative constants and called
principal
eigenfunction
of
(L,
B(k(z)),
0).
Thanks to Theorem 12.1 of Amann3, the
principal eigenfuntion of
(L,
B(k(z)),
R)
belongs to

of
R
of class
C2
with
dist
(rl,
dRonR)
>
0,
we will denote by
B(k(z),
no)
the boundary operator
build up from
B(k(z))
by
and by
I$"
[C,
B(k(z),
Ro)],
the principal eigenvalue of
(C,
B(k(z),
Ro),
00).
To
develop the mathematical analysis
of

(l)~,
if it satisfies
(L(X)u
+
u(z)uT,B(-V(z)
+
b(z)u"-1>u)
2
0
(I
0)
Positive Solutions
of
the Logistic Weighted Elliptic BVP
5
and it will be said that it is
a
strict supersolution (subsolution), if the
respective inequality is strict, where
2
stands for the natural product order
on
Lp(R)
x
L,(aR).
3.
On the Uniqueness and Regularity of Positive Solutions
of Problem
(1)~
In this section we prove the uniqueness of positive solution of Problem

C'+"(fi)
for all
0
<
Q
<
1.
Moreover,
ux
is a.e.
in
R
twice continuously differen-
tiable.
Proof.
Let
u~
be
a
positive solution of Problem
(1)~.
Then,
u~
E
Wz(R)
for some
p
>
N,
satisfies Eq.

and
UX,
b(z),
V(z)
E
C(rl), and therefore, Prob-
lem
(6)~
is well defined in the sense of Problem
(7)
and Eq.
(5).
Now, since
u~
is
a
positive solution of Problem
(6)~,
we have that
0
is an eigenvalue of
Problem
(6)~
and
u~
is
a
positive eigenfunction of Problem
(6)~
associated

Thanks to the previous result, if we set
and we denote by
U+
the cone of non-negative functions of
U
and by
F:
RXU+
-
v,
0"
:
u+
-
w,
6
S.
Cano-Casanova
the nonlinear operators defined by
3(A,
u)
:=
L(A)u
+
a(x)uT,
(A,
u)
E
IW
x

satisfying the system
F(A,ux)
=
0
in
R
{
Gv(ux)
=
0
on
dR.
On the other hand,
D,Gv(ux)
can be written in terms of the boundary
opertor
B(lc(x))
defined in
Eq.
(5),
by
D,Gv(ux)u
:=
B(-V(x)
+
qb(x)u;-l)u.
Theorem
3.2.
For
each

solutions of Problem
(l)~,
such that u1
#
212.
Then, thanks to
Eq.
(S),
O~[L(A)
+
a(z)u~-l,
~(-v(x)
+
b(x)uf-l)]
=
0,
i
=
1,2,
(10)
and the following problem is satisfied
(L(4
+
a(z)F(x))(.u1
-
U2)
=
0
in
R

Positive Solutions
of
the Logistic Weighted Elliptic BVP
7
By construction,
F(.)
2
TuY-'
>
u;-',
G(-)
2
qu;-l
>
.;-I.
(1'4
Now, using the boundary operator
B(k(z))
defined by
Eq.
(5), Problem
(11)~
can be written in the form
{
B(-V(z)
+
b(z)G(z))(ul
-
ua)
=

~2)
=
0
in
R
.T[.c(W
+
a(z)F(z),
B(-V(z)
+
b(z)G(z))l
>
>
CTf[L(X)
+
a(z)u;-l,
B(-V(z)
+
b(z)u;-l)]
=
0.
Then, Problem
(13)~
is invertible and hence
u1
=
u2,
which gives the
contradiction. This completes the proof of the uniqueness of the positive
solution of Problem

1,
a
>
0,
b
>
0
and that
u~
is strongly positive in
R,
it follows from the monotonicity
of
the principal eigenvalue with respect to the potential and with respect to
the weight on the boundary (cf.
S.
Cano-Casanova and
J.
L6pez-G6mez5),
that
CJ?[DuF(X,
4,
DuGV('ILX)I
>
>
CT?[C(X)
+
a(z)ul-l,
B(-V(z)
+

then
Eq.
(2)
is satisfied, where
B(-V(z))
and
2)
are the boundary operators defined
by
(3).
Proof.
Let
UA
be
a
positive solution of Problem (1)~. Then, thanks to
Eq.
(8),
and owing to the monotonicity of the principal eigenvalue with
respect to the domain (cf. Proposition
3.2
of
S.
Cano-Casanova and
J.
L6pez-G6mez5) and the dominance of the principal eigenvalue under Dirich-
let boundary conditions (cf. Proposition
3.1
of
S.

Cano-Casanova and
J.
L6pez-G6mez5), it follows from
Eq.
(8)
that
0
=
@[L(X)
+
a(z>u;-l,
B(-V(z)
+
b(z)uI-')]
>
o~[L(X),B(-V(Z)
+
b(~>~:-l)]
>
cF[L(X),
B(-V(z))]
since
b
>
0
on
rl
and
UA
is strongly positive in

Theorem
4.1.
Let
cA
be a subsolution and
?j,
be a supersolution
of
Prob-
lem
(l)~,
such that
gA
5
ii~.
Then, Problem
(1)~
has at least one solution
in
the order interval
[cA,
FA].
More precisely, there exists a minimal solution
every solution
ux
E
[cA,iiA]
of
Problem
(1)A

satisfying
.?IW),f?(-V(z))l
<
0
7
(14)
Problem
(1)~
possesses
a
positive strict subsolution arbitrarily small and
strongly positive
in
0.
Proof.
Let
X
satisfy
Eq.
(14).
Then, thanks to the monotonicity and the
continuous dependence
of
CTF[L,
B(lc(x))]
with respect to the weight
k(z)
on the boundary, recently proved in Proposition
3.5
and Theorem

and
(16),
it is
possible to take
p
>
0
such that
4V(X),
f?(-V(z))l
<
4[W,f?(-V(4
+
PI1
<
0
*
(17)
Fix
p
>
0
satisfying
Eq.
(17).
Now, let
us
consider the positive function
2
=

E(P>cp(P)(4v(X),
f?(-V(z>
+
PI1
+
a(+'-l(p)(P'-l(p))
7
on
l?l
the following estimate is satisfied
&u-
V(~)G+
b(z)gq
=
E(P)V(P)(-P
+
~(~)E"-'(P)P"~(P))
(19)
g=~(p)cp(p)
=O
on
ro.
(20)
and
Thus, thanks to
Eqs.
(17), (18), (19)
and
(20),
21

0
small enough. The remaining assertion follows
from the fact that the principal eigenfunction
p(p)
is strongly positive in
R
(cf.
H.
Amann3). This completes the proof.
10
S.
Cano-Casanova
Proposition
4.3.
For
each
X
E
R
satisfying
0::
[L(X),
D]
>
0,
(21)
Problem
(1)~
possesses a positive strict supersolution, arbitrarily large and
bounded away

l?:
U
rq,
where
r:,
i
=
1,2
are two components
of
rl,
and that
R:
=
0;
U
Ri
E
C2,
being
R6,
i
=
1,2
two components satisfying
Q;nQi=0,
fiicn,
m;=r;uF,
FCR
(22)

J.
L6pez-G6mez5,
for all
n
E
N
and
lim
0:'
[L(x),
~(n,
R;)]
=
0::
[L(x),
DI
.
0
<
07:[L(A),B(n,n;)]
<
.3L(X),D].
(25)
ntm
Thus, owing to
Eqs.
(23),
(24) and (25), there exists
no
E

(r:
+
B~)
n
R,
@
:=
(ro
+
B6),
(27)
where
Bg
stands for the ball of radius
6
>
0
centered at the origin. Under
the general assumptions, and thanks to
Eq.
(22), it follows the existence of
60
>
0
small enough such that for each
0
<
6
<
60

is
a
proper subdomain of
R;,
and
lim
a;
=
at,
in the sense
of
Definition
6.1
of
S.
Cano-Casanova and
J.
L6pez-G6mez5.
Thus, it follows from Theorem
4.2
of
J.
L6pez-G6mez1O, and Theorem
7.1
of
S.
Cano-Casanova and
J.
L6pez-G6mez5, that
6-0+

of
S.
Cano-Casanova and
J.
L6pez-G6mez5, the existence
of
0
<
61
<
60
such that
0
<
[L(X),
B(n,
R;)]
<
c?
[L(X),
B(n,
RA)]
<
a?[L(X),
D]
,
(31)
0
<
c::[L(X),D]

6
<
61,
let
cptl
cpf,,,
cp;
and
cp!
denote the principal eigenfunc-
tions associated with
cF[L(X),
D],
0:"
[L(X),
B(n,
at)]
,
0:'
[L(X),
D]
and
OF'^
[L(X),
B(-V
-
6)],
respectively, and consider the function
RUG.
-

n
R)
U
R\
U
R?
U
Nlf2
to
d,
which is bounded away from zero
in
d
\
(fl
U
Ri
U
0:
U
Nil2). Note that
$6
exists, since the functions
CP~I~M:
nn,
(Pi,nIan\nn,
'~2Ian2,
and
(~3Ia~1.2nn
are positive and bounded

I
93
2
5
2
2
T
6
6 6
z
2
I
4
5
(35)
No-
L(X)Ti
+
u(z)~
=
M~;((T,
'
[L(X),D]
+
a(z)M'-'((~;)'-~),
in
52;
the following holds:
T
L(X)S+a(z)Ti'

(33),
we infer from
Eqs.
(35),
(36)
and
(37)
that for each
M
>
0
L(X)E
+-
a(z)~i'
>
o
in
(N$'
5
n
R)
u
R:
52
u
R:
.
(38)
Moreover, in
Ntp2

in
Nil2
5
(39)
On the other hand, in
\
(fl
u
R\
U
R$
UNls2) we have that
BTI
2
L(X)Ti
+
a(s)E'
=
M(L@
+
a(z)M'-'(@)'),
Positive Solutions
of
the Logistic Weighted Elliptic
BVP
13
and since by construction, the functions
u(x),
$J~
are bounded away from

(40)
in
ST
\
(N,"
u
u
UJV:~~).
2
523
Finally, on the boundary, we have by construction that
;ii=~p;
>O
on
ro,
(41)
that on
I?:,
3,ii
-
V(z)U
+
b(z)$
=
Mpi(6
+
Mq-lb(x)(pi)q-l)
2
M6p;
>

>
0,
such that for each
M
2
M3
>
0,
a,~
-
V(Z)E
+
~(x)P
>
o
on
r:
.
(43)
Thus, for each
X
E
R
satisfying
Eq.
(21),
it follows from
Eqs.
(38),
(39),

2
M3
>
0
is
a
positive strict superso-
lution bounded away from zero in
fi
of Problem
(1)~.
This completes the proof.
0
Now, we are ready to prove Theorem
1.1.
Proof
of
Theorem
1.1.
The necessary condition for the existence of
positive solution of Problem
(1)~
is
Proposition
4.1.
We now prove the
sufficient condition for the existence of positive solution of Problem
(1)~.
Let
X

and therefore, the sufficient condition for the exis-
tence of positive solution of Problem
(1)~
follows from Theorem 4.1. Now,
0
The following result is the adapted version of Theorem 1.1, for the
Theorem 3.1 and Theorem 3.2 complete the proof of the result.
particular case when
a
=
0
in
R,
i.e.,
52;
=
R.
Corollary
4.1.
Under the general assumptions of Sec.
1,
let
us
consider
the
BVP
cu
=
XW(z)u
in

YN)
E
C1(rl,
IRN)
is any outward pointing nowhere tangent
vector field to
rl.
Then, Problem
(44)
possesses a positive solution,
if
and
only
if
~?[c(~),,13(-v(~c>)l
<
0
<
.?[c(4,q
,
(45)
ant the remaining assertions of Theorem
1.1
are satisfied.
Proof.
To
build the positive strict supersolution bounded away from zero
in
0
of Problem (44)x, for each

is
a
tubular neighbourhood of
ro,
and
L(X)
is an adequate regular
extension of the operator
L(X)
from
R
to
0.
The remainder of the
proof
is obtained, arguing
as
in the
case
of
Problem
(1)x.
This completes the proof.
0
In order to complete the exposition of this work, we include the follow-
ing result, which is the counterpart of Theorem 1.1 for the case when the
potential
a(.)
is
bounded away from zero on any compact subset of