Electronic Journal of Differential Equations, Vol. 2007(2007), No. 69, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
A FIBERING MAP APPROACH TO A SEMILINEAR ELLIPTIC
BOUNDARY VALUE PROBLEM
KENNETH J. BROWN, TSUNG-FANG WU
Abstract. We prove the existence of at least two positive solutions for the
semilinear elliptic boundary-value problem
−∆u(x) = λa(x)uq + b(x)up
for x ∈ Ω;
u(x) = 0
for x ∈ ∂Ω
on a bounded region Ω by using the Nehari manifold and the fibering maps
associated with the Euler functional for the problem. We show how knowledge
of the fibering maps for the problem leads to very easy existence proofs.
1. Introduction
We shall discuss the existence of positive solutions of the semilinear elliptic
boundary-value problem
−∆u(x) = λa(x)uq + b(x)up
for x ∈ Ω;
for x ∈ ∂Ω,
u(x) = 0
(1.1)
(1.2)
N +2
where Ω is a bounded region with smooth boundary in RN , 0 < q < 1 < p < N
−2 ,
λ > 0 and a, b : Ω → R are smooth functions which are somewhere positive but
which may change sign on Ω. Equation (1.1), (1.2) has been recently studied in [3]
by using the Mountain Pass Lemma and in [5] and [7] using the Nehari manifold.
In [4] and [2] it was shown that the Nehari manifold for an equation such as (1.1)
is closely related to the fibering maps for the problem. In this paper we show how
a fairly complete knowledge of all possible forms of the fibering maps provides a
very simple and comparatively elementary means of establishing results similar to
those proved in [5] and [7] on the existence of multiple solutions of (1.1), (1.2). In
section 2 we recall the properties which we shall require of fibering maps and of the
Nehari manifold. In section 3 we give a fairly complete description of the fibering
maps associated with (1.1) and in section 4 we use this information to give a very
simple variational proof of the existence of at least two positive solutions of (1.1),
(1.2) for sufficiently small λ.
We shall throughout use the function space W01,2 (Ω) with norm
Z
1/2
kuk =
|∇u|2 dx
Ω
2000 Mathematics Subject Classification. 35J20, 36J65.
Key words and phrases. Semilinear elliptic boundary value problem; variational methods;
Nehari manifold; fibering map.
c
2007
Texas State University - San Marcos.
Submitted Febraury 27, 2007. Published May 10, 2007.
1
2
K. J. BROWN, T.-F. WU
EJDE-2007/69
and the standard Lp (Ω) spaces whose norms we denote by kukp .
2. Fibering Maps and the Nehari manifold
The Euler functional associated with (1.1), (1.2) is
Z
Z
Z
1
λ
1
Jλ (u) =
|∇u|2 dx −
a(x)|u|q+1 dx −
b(x)|u|p+1 dx
2 Ω
q+1 Ω
p+1 Ω
for all u ∈ W01,2 (Ω).
As Jλ is not bounded below on W01,2 (Ω), it is useful to consider the functional
on the Nehari manifold
Mλ (Ω) = {u ∈ W01,2 (Ω) : hJλ0 (u), ui = 0}
where h, i denotes the usual duality. Thus u ∈ Mλ (Ω) if and only if
Z
Z
Z
|∇u|2 dx − λ
a(x)|u|q+1 dx −
b(x)|u|p+1 dx = 0
Ω
Ω
(2.1)
Ω
Clearly Mλ (Ω) is a much smaller set than W01,2 (Ω) and, as we shall show, Jλ is
much better behaved on Mλ (Ω). In particular, on Mλ (Ω) we have that
Z
Z
1
1
1
1
2
)
|∇u| + (
−
)
b(x)|u|p+1
Jλ (u) = ( −
2 q+1 Ω
q+1 p+1 Ω
Z
Z
(2.2)
1
1
1
1
)
|∇u|2 − λ(
−
)
a(x)|u|q+1
=( −
2 p+1 Ω
q+1 p+1 Ω
Theorem 2.1. Jλ is coercive and bounded below on Mλ (Ω).
Proof. It follows from (2.2) and the Sobolev embedding theorems that there exist
positive constants c1 , c2 and c3 such that
Z
2
Jλ (u) ≥ c1 kuk − c2
|u|q+1 dx ≥ c1 kuk2 − c3 kukq+1
Ω
and so Jλ is coercive and bounded below on Mλ (Ω).
The Nehari manifold is closely linked to the behaviour of the functions of the
form φu : t → Jλ (tu) (t > 0). Such maps are known as fibering maps and were
introduced by Drabek and Pohozaev in [4] and are also discussed in Brown and
Zhang [2]. If u ∈ W01,2 (Ω), we have
Z
Z
Z
1
tq+1
tp+1
|∇u|2 − λ
a|u|q+1 −
b|u|p+1
(2.3)
φu (t) = t2
2
q
+
1
p
+
1
Ω
Ω
Ω
Z
Z
Z
φ0u (t) = t
|∇u|2 − λ tq
a|u|q+1 − tp
b|u|p+1
(2.4)
Ω
Ω
Ω
Z
Z
Z
φ00u (t) =
|∇u|2 − λqtq−1
a|u|q+1 − ptp−1
b|u|p+1
(2.5)
Ω
Ω
Ω
φ0u (1)
It is easy to see that u ∈ Mλ (Ω) if and only if
= 0 and, more generally,
that φ0u (t) = 0 if and only if tu ∈ Mλ (Ω), i.e., elements in Mλ (Ω) correspond to
stationary points of fibering maps. Thus it is natural to subdivide Mλ (Ω) into sets
EJDE-2007/69
A FIBERING MAP APPROACH
3
corresponding to local minima, local maxima and points of inflection and so we
define
Mλ+ (Ω) = {u ∈ Mλ (Ω) : φ00u (1) > 0},
Mλ− (Ω) = {u ∈ Mλ (Ω) : φ00u (1) < 0},
Mλ0 (Ω) = {u ∈ Mλ (Ω) : φ00u (1) = 0},
and note that if u ∈ Mλ (Ω), i.e., φ0u (1) = 0, then
Z
Z
00
2
φu (1) = (1 − q)
|∇u| dx − (p − q) b(x)|u|p+1 dx
Ω
Z
Z
= (1 − p)
|∇u|2 dx − λ(q − p) a(x)|u|q+1 dx .
(2.6)
Ω
Also, as proved in Binding, Drabek and Huang [1] or in Brown and Zhang [2], we
have the following lemma.
Lemma 2.2. Suppose that u0 is a local maximum or minimum for Jλ on Mλ (Ω).
Then, if u0 6∈ Mλ0 (Ω), u0 is a critical point of Jλ .
3. Analysis of the Fibering Maps
In this section we give a fairly complete description of the fibering maps associated with the problem. RAs we shall see theR essential nature of the maps is
determined by the signs of a(x)|u|q+1 dx and Ω b(x)|u|p+1 dx. We will find it
useful to consider the function
Z
Z
1−q
2
p−q
mu (t) = t
|∇u| dx − t
b(x)|u|p+1 dx.
Ω
Ω
Clearly, for t > 0, tu ∈ Mλ (Ω) if and only if t is a solution of
Z
mu (t) = λ
a(x)|u|q+1 dx.
(3.1)
Ω
Morever,
m0u (t) = (1 − q)t−q
Z
|∇u|2 dx − (p − q)tp−q−1
Ω
Z
b(x)|u|p+1 dx.
(3.2)
Ω
It
to see that mu is a strictly increasing function for t ≥ 0 whenever
R is easyp+1
b(x)|u|
dx ≤ 0 and mu is initially increasing
and eventually decreasing with
Ω
R
a single turning point as in Figure 1(b) when Ω b(x)|u|p+1 dx > 0.
(a)
(b)
Figure 1. Possible forms of m(u)
4
K. J. BROWN, T.-F. WU
EJDE-2007/69
Suppose tu ∈ Mλ (Ω). It follows from (2.6) and (3.2) that φ00tu (1) = tq+2 m0u (t)
and so tu ∈ Mλ+ (Ω)(Mλ− (Ω)) provided m0u (t) > 0 (< 0).
now describe
the nature of Rthe fibering maps for allR possible signs of
R We shall
R
p+1
q+1
b(x)|u|
dx
and
a(x)|u|
dx. If Ω b(x)|u|p+1 dx ≤ 0 and Ω a(x)|u|q+1 dx ≤
Ω
Ω
0, clearly φu is an increasing function of t and so has graph
as shown in Figure
R
p+1
2(a);
thus
in
this
case
no
multiple
of
u
lies
in
M
(Ω).
If
b(x)|u|
dx ≤ 0 and
λ
Ω
R
q+1
a(x)|u|
dx
>
0,
then
m
has
graph
as
in
Figure
1(a),
and
it
is
clear
that there
u
Ω
is exactly one solution of (3.1). Thus there is a unique value t(u) > 0 such that
t(u)u ∈ Mλ (Ω). Clearly m0u (t(u)) > 0 and so t(u)u ∈ Mλ+ (Ω). Thus the fibering
map φu has a unique critical point at t = t(u) which is a local minimum. Since
limt→∞ φu (t) = ∞,
R it follows that φu hasRgraph as shown in Figure 2(c).
Suppose now Ω b(x)|u|p+1 dx > 0 and Ω a(x)|u|q+1 dx ≤ 0. Then mu has graph
as shown in Figure 1(b) and it is clear that there is exactly one positive solution
of (3.1). Thus there is again a unique value t(u) > 0 such that t(u)u ∈ Mλ (Ω) and
since m0u (t(u)) < 0 in this case t(u)u ∈ Mλ− (Ω). Hence the fibering map φu has
a unique critical point which is a local maximum. Since limt→∞ φu (t) = −∞, it
follows that φu has graph as shown
in Figure 2(b).
R
R
Finally we consider the case Ω b(x)|u|p+1 dx > 0 and Ω a(x)|u|q+1 dx > 0 where
the situation is more complicated. As in the previous case mu has a graph as shown
in Figure 1(b). If λ > 0 is sufficiently large, (3.1) has no solution and so φu has
no critical points - in this case φu is a decreasing function. Hence no multiple
of u lies in Mλ (Ω). If, on the other hand, λ > 0 is sufficiently small, there are
exactly two solutions t1 (u) < t2 (u) of (3.1) with m0u (t1 (u)) > 0 and m0u (t2 (u)) < 0.
Thus there are exactly two multiples of u ∈ Mλ (Ω), namely t1 (u)u ∈ Mλ+ (Ω)
and t2 (u)u ∈ Mλ− (Ω). It follows that φu has exactly two critical points - a local
minimum at t = t1 (u) and a local maximum at t = t2 (u); moreover φu is decreasing
in (0, t1 ), increasing in (t1 , t2 ) and decreasing in (t2 , ∞) as in Figure 2(d).
The following result ensures that when λ is sufficiently small the graph of φu
must be as shown in Figure 2(d) for all non-zero u.
Lemma 3.1. There exists λ1 > 0 such that, when λ < λ1 , φu takes on positive
values for all non-zero u ∈ W01,2 (Ω).
R
Proof. If Ω b(x)|u|p+1 dx ≤ 0, then φu (t) > 0 for t sufficiently large. Suppose
R
u ∈ W01,2 (Ω) and Ω b(x)|u|p+1 dx > 0. Let
Z
Z
tp+1
t2
2
|∇u| dx −
b(x)|u|p+1 dx.
hu (t) =
2 Ω
p+1 Ω
Then elementary calculus shows that hu takes on a maximum value of
R
1
1
R |∇u|2 dx p−1
p − 1 n ( Ω |∇u|2 dx)p+1 o p−1
R
R Ω
when
t
=
t
=
.
max
2(p + 1) ( Ω b(x)|u|p+1 dx)2
b(x)|u|p+1 dx
Ω
However
R
( Ω |∇u|2 dx)p+1
1
R
≥ 2(p+1)
p+1
2
( Ω |u|
dx)
Sp+1
where Sp+1 denotes the Sobolev constant of the embedding of W01,2 (Ω) into Lp+1 (Ω).
Hence
1
p−1
p−1 1
hu (tmax ) ≥
=δ
2(p + 1) kb+ k2∞ S 2(p+1)
p+1
EJDE-2007/69
A FIBERING MAP APPROACH
5
φu
φu
t
t
(b)
(a)
φu
φu
t
t
(d)
(c)
Figure 2. Possible forms of fibering maps
where δ is independent of u.
We shall now show that there exists λ1 > 0 such that φu (tmax ) > 0, i.e.,
Z
λ (tmax )q+1
hu (tmax ) −
a(x)|u|q+1 dx > 0
q+1
Ω
for all u ∈ W01,2 (Ω) − {0} provided λ < λ1 . We have
Z
(tmax )q+1
a(x)|u|q+1 dx
q+1
Ω
q+1 Z
q+1
R |∇u|2 dx p−1
1
2
q+1
2
Ω
R
≤
kak∞ Sq+1
|∇u|
dx
p+1 dx
q+1
b(x)|u|
Ω
Ω
p+1 o q+1
R
n
2
|∇u| dx
1
2(p−1)
q+1
Ω
=
kak∞ Sq+1
2
R
p+1
q+1
b(x)|u|
dx
Ω
q+1
q+1
q+1
1
q+1 2(p + 1) 2
=
kak∞ Sq+1
hu (tmax ) 2 = c hu (tmax ) 2
q+1
p−1
where c is independent of u. Hence
φu (tmax ) ≥ hu (tmax ) − λchu (tmax )
q+1
2
= hu (tmax )
q+1
2
hu (tmax )
1−q
2
− λc
and so, since hu (tmax ) ≥ δ for all u ∈ W01,2 (Ω) − {0}, it follows that φu (tmax ) > 0
1−q
for all non-zero u provided λ < δ 2 /2c = λ1 . This completes the proof.
R
from the above lemma that when λ < λ1 , Ω a(x)|u|q+1 dx > 0 and
R It follows
p+1
b(x)|u|
dx > 0 then φu must have exactly two critical points as discussed in
Ω
the remarks preceding the lemma.
6
K. J. BROWN, T.-F. WU
EJDE-2007/69
Thus when λ < λ1 we have obtained a complete knowledge of the number of
critical points of φu , of the intervals on which φu is increasing and decreasing
and
of the multiples Rof u which lie in Mλ (Ω) for every possible choice of signs of
R
p+1
b(x)|u|
dx and Ω a(x)|u|q+1 dx. In particular we have the following result.
Ω
Corollary 3.2. Mλ0 (Ω) = ∅ when 0 < λ < λ1 .
Corollary 3.3. If λ < λ1 , then there exists δ1 > 0 such that Jλ (u) ≥ δ1 for all
u ∈ Mλ− (Ω).
Proof.
Consider u ∈ Mλ− (Ω). Then φu has a positive global maximum at t = 1 and
R
b(x)|u|p+1 dx > 0. Thus
Jλ (u) = φu (1) ≥ φu (tmax )
q+1
1−q
≥ hu (tmax ) 2 hu (tmax ) 2 − λc
≥δ
q+1
2
(δ
1−q
2
− λc)
and the left hand side is uniformly bounded away from 0 provided that λ < λ1 . 4. Existence of Positive Solutions
In this section using the properties of fibering maps we shall give simple proofs
of the existence of two positive solutions, one in Mλ+ (Ω) and one in Mλ− (Ω).
Theorem 4.1. If λ < λ1 , there exists a minimizer of Jλ on Mλ+ (Ω).
Proof. Since Jλ is bounded below on Mλ (Ω) and so on Mλ+ (Ω), there exists a
minimizing sequence {un } ⊆ Mλ+ (Ω) such that
lim Jλ (un ) =
n→∞
inf
u∈Mλ+ (Ω)
Jλ (u).
Since Jλ is coercive, {un } is bounded in W01,2 (Ω). Thus we may assume, without
loss of generality, that un * u0 in W01,2 (Ω) and un → u0 in Lr (Ω) for 1 < r < N2N
−2 .
R
1,2
q+1
If we choose u ∈ W0 (Ω) such that Ω a(x)|u|
dx > 0, then the graph of
the fibering map φu must be of one of the forms shown in Figure 2(c) or (d)
and so there exists t1 (u) such that t1 (u)u ∈ Mλ+ (Ω) and Jλ (t1 (u)u) < 0. Hence,
inf u∈M + (Ω) Jλ (u) < 0. By (2.2),
λ
Z
Z
1
1
1
1
2
Jλ (un ) = ( −
)
|∇un | dx − λ (
−
)
a(x)|un |q+1 dx
2 p+1 Ω
q+1 p+1 Ω
and so
Z
Z
1
1
1
1
q+1
λ(
−
)
a(x)|un |
dx = ( −
)
|∇un |2 dx − Jλ (un ).
q+1 p+1 Ω
2 p+1 Ω
R
Letting n → ∞, we see that Ω a(x)|u0 |q+1 dx > 0.
Suppose un 6→ u0 in W0R1,2 (Ω). We shall obtain a contradiction by discussing the
fibering map φu0 . Since Ω a(x)|u0 |q+1 dx > 0, the graph of φu0 must be either
of the form shown in Figure 2(c) or (d). Hence there exists t0 > 0 such that
t0 u0 ∈ Mλ+ (Ω) and φu0 is decreasing on (0, t0 ) with φ0u0 (t0 ) = 0.
R
R
Since un 6→ u0 in W01,2 (Ω), Ω |∇u0 |2 dx < lim inf n→∞ Ω |∇un |2 dx. Thus, as
Z
Z
Z
0
2
q
q+1
p
φun (t) = t
|∇un | dx − λt
a(x)|un |
dx − t
b(x)|un |p+1 dx
Ω
Ω
Ω
EJDE-2007/69
A FIBERING MAP APPROACH
7
and
φ0u0 (t) = t
Z
|∇u0 |2 dx − λtq
Z
Ω
a(x)|u0 |q+1 dx − tp
Z
Ω
b(x)|u0 |p+1 dx,
Ω
it follows that φ0un (t0 ) > 0 for n sufficiently large. Since {un } ⊆ Mλ+ (Ω), by
considering the possible fibering maps it is easy to see that φ0un (t) < 0 for 0 < t < 1
and φ0un (1) = 0 for all n. Hence we must have t0 > 1. But t0 u0 ∈ Mλ+ (Ω) and so
Jλ (t0 u0 ) < Jλ (u0 ) < lim Jλ (un ) =
n→∞
inf
Jλ (u)
u∈Mλ+ (Ω)
and this is a contradiction. Hence un → u0 in W01,2 (Ω) and so
Jλ (u0 ) = lim Jλ (un ) =
n→∞
inf
u∈Mλ+ (Ω)
Jλ (u).
Thus u0 is a minimizer for Jλ on Mλ+ (Ω).
Theorem 4.2. If λ < λ1 , there exists a minimizer of Jλ on Mλ− (Ω).
Proof. By Corollary 3.3 we have Jλ (u) ≥ δ1 > 0 for all u ∈ Mλ− (Ω) and so
inf u∈M − (Ω) Jλ (u) ≥ δ1 . Hence there exists a minimizing sequence {un } ⊆ Mλ− (Ω)
λ
such that
lim Jλ (un ) =
n→∞
inf
u∈Mλ− (Ω)
Jλ (u) > 0.
As in the previous proof, since Jλ is coercive, {un } is bounded in W01,2 (Ω) and we
may assume, without loss of generality, that un * u0 in W01,2 (Ω) and un → u0 in
Lr (Ω) for 1 < r < N2N
−2 . By (2.2)
1
1
Jλ (un ) = ( −
)
2 q+1
Z
1
1
|∇un | dx + (
−
)
q
+
1
p
+
1
Ω
2
Z
b(x)|un |p+1 dx
Ω
and, since limn→∞ Jλ (un ) > 0 and
Z
b(x)|un |p+1 dx =
lim
n→∞
Ω
Z
b(x)|u0 (x)|p+1 dx,
Ω
we must have that Ω b(x)|u0 (x)|p+1 dx > 0. Hence the fibering map φu0 must
have graph as shown in Figure 2(b) or (d) and so there exists t̂ > 0 such that
t̂u0 ∈ Mλ− (Ω).
Suppose un 6→ u0 in W01,2 (Ω). Using the facts that
R
Z
Z
2
|∇u0 | dx < lim inf
Ω
n→∞
Ω
|∇un |2 dx
8
K. J. BROWN, T.-F. WU
EJDE-2007/69
and that, since un ∈ Mλ− (Ω), J(un ) ≥ J(sun ) for all s ≥ 0, we have
Z
Z
Z
1 2
λt̂q+1
t̂p+1
2
q+1
J(t̂u0 ) = t̂
|∇u0 | dx −
a(x)|u0 |
dx −
b(x)|u0 |p+1 dx
2
q+1 Ω
p+1 Ω
Ω
Z
h1 Z
λt̂q+1
2
2
t̂
|∇un | dx −
a(x)|un |q+1 dx
< lim
n→∞ 2
q+1 Ω
Ω
Z
i
t̂p+1
−
b(x)|un |p+1 dx
p+1 Ω
= lim J(t̂un )
n→∞
≤ lim J(un ) =
n→∞
inf
u∈Mλ− (Ω)
Jλ (u)
which is a contradiction. Hence un → u0 in W01,2 (Ω) and the proof can be completed
as in the previous theorem.
Corollary 4.3. Equation (1.1), (1.2) has at least two positive solutions whenever
0 < λ < λ1 .
Proof. By Theorems 4.1 and 4.2 there exist u+ ∈ Mλ (Ω) and u− ∈ Mλ− (Ω)
such that J(u+ ) = inf u∈M + (Ω) J(u) and J(u− ) = inf u∈M − (Ω) J(u). Moreover
λ
λ
J(u± ) = J(|u± |) and |u± | ∈ Mλ± (Ω) and so we may assume u± ≥ 0. By Lemma
2.2 u± are critical points of J on W01,2 (Ω) and hence are weak solutions (and so
by standard regularity results classical solutions) of (1.1), (1.2). Finally, by the
Harnack inequality due to Trudinger [6], we obtain that u± are positive solutions
of (1.1), (1.2).
Acknowledgement. We would like to thank the referee for bringing [5] to our
attention and for making the important observation that the fact that Mλ0 (Ω) = ∅
follows from sufficient knowledge of the fibering maps.
References
[1] P. A. Binding, P. Drabek and Y. X. Huang; On Neumann boundary value problems for some
quasilinear elliptic equations, Electron. J. Differential Equations, 1997, 1997, no. 5, 1-11.
[2] K. J. Brown and Y. Zhang; The Nehari manifold for a semilinear elliptic problem with a
sign changing weight function , J. Differential Equations, 193 , 2003, 481-499.
[3] D. G. de Figueiredo, J. P. Gossez and P. Ubilla; Local superlinearity and sublinearity for
indefinite semilinear elliptic problems, J. Funct. Anal., 199, 2003, 452-467.
[4] P. Drabek and S. I. Pohozaev; Positive solutions for the p-Laplacian: application of the
fibering method, Proc. Royal Soc. Edinburgh Sect A, 127, (1997), 703-726.
[5] Y. Il’yasov; On non-local existence results for elliptic operators with convex-concave nonlinearities, Nonlinear Analysis, 61, 2005 211-236.
[6] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic
equations, Comm. Pure Applied Math. 20 (1967) 721-747.
[7] T.-F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight
function, to appear in Rocky Mountain J. Math.
Kenneth J. Brown
School of Mathematical and Computer Sciences and the Maxwell Institute, HeriotWatt University, Riccarton, Edinburgh EH14 4AS, UK
E-mail address: K.J.Brown@hw.ac.uk
EJDE-2007/69
A FIBERING MAP APPROACH
9
Tsung-Fang Wu
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811,
Taiwan
E-mail address: tfwu@nuk.edu.tw