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Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 147308, 19 pages
doi:10.1155/2009/147308
Research Article
On Multiple Solutions of Concave and Convex
Nonlinearities in Elliptic Equation on
R
N
Kuan-Ju Chen
Department of Applied Science, Naval Academy, 90175 Zuoying, Taiwan
Correspondence should be addressed to Kuan-Ju Chen,
Received 18 February 2009; Accepted 28 May 2009
Recommended by Martin Schechter
We consider the existence of multiple solutions of the elliptic equation on
R
N
with concave and
convex nonlinearities.
Copyright q 2009 Kuan-Ju Chen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
First, we look for positive solutions of the following problem:
−Δu u a
x
u
p−1
λb
x
u
q−1
, in R
N
,
u>0, in R
N
,
u ∈ H
1
R
N
,
1.1
where λ>0 is a real parameter, 1 <p<2 <q<2
∗
2N/N − 2, N ≥ 3. We will impose some
assumptions on ax and bx. Assume
a1 ax ≥ 0,ax ∈ L
α/α−1
R
N
∩ L
∞
R
N
, where 1 <α<2
∗
/p,
b1 bx ∈ CR
N
,bx → b
∞
> 0as|x|→∞,bx ≥ b
∞
for all x ∈ R
N
,
Such problems occur in various branches of mathematical physics and population
dynamics, and sublinear analogues or superlinear analogues of problem 1.1 have been
considered by many authors in recent years see 1–4. Little information is known about
the combination of sublinear and superlinear case of problem 1.1.In5, 6, they deal with
the analogue of problem 1.1 when R
N
is replaced by a bounded domain Ω. For the R
N
case,
the existence of positive solutions for problem 1.1 was proved by few people.
2 Boundary Value Problems
In the present paper, we discuss the Nehari manifold and examine carefully the
connection between the Nehari manifold and the fibrering maps, then using arguments
similar to those used in 7, we will prove the existence of the two positive solutions by
using Ekeland’s Variational Principle 8.
In 5, Ambrosetti et al. showed that for λ>0 small with respect t o μ>0 there exist
infinitely many solutions u ∈ H
1
0
Ω of the semilinear elliptic problem:
−Δu λ
|
u
|
p−2
u μ
|
u
|
q−2
u, in Ω,
u 0on∂Ω,
1.2
with negative energy:
ψ
u
1
2
Ω
|
∇u
|
2
−
λ
p
Ω
|
u
|
p
−
μ
q
Ω
|
u
|
q
, 1.3
and infinitely many solutions with positive energy, where Ω ⊂ R
N
is an open bounded
domain. In 9, Bartsch and Willem obtained infinitely many solutions of problem 1.2 with
negative energy for every λ>0. For the R
N
case, the existence of multiple solutions was
proved by few people.
Finally we propose herein a result similar to 9 or 10 for the existence of infinitely
many solutions possibly not positive of
−Δu u μa
x
|
u
|
p−2
u λb
x
|
u
|
q−2
u, in R
N
,
u ∈ H
1
R
N
,
1.4
by taking advantage of the oddness of the nonlinearity.
Our main results state the following.
Theorem 1.1. Under the assumptions (a1) and (b1), there exists λ
∗
> 0, such that for all λ ∈ 0,λ
∗
,
problem 1.1 has at least two positive solutions u
0
and u
1
, u
0
is a local minimizer of I
λ
and I
λ
u
0
< 0,
where I
λ
is the energy functional of problem 1.1.
Theorem 1.2. Under the assumptions (a1) and (b1), for every λ>0 and μ ∈ R, the problem 1.4
has infinitely many solutions with positive energy and for every μ>0 and λ ∈ R, infinitely many
solutions with negative energy.
2. The Existence of Two Positive Solutions
The variational functional of problem 1.1 is
I
λ
u
1
2
|
∇u
|
2
u
2
−
1
p
a
x
|
u
|
p
−
λ
q
b
x
|
u
|
q
, 2.1
here and from now on, we omit “dx”and“R
N
” in all the integrations if there is no other
indication.
Boundary Value Problems 3
Through this paper, we denote the universal positive constant by C unless some
special statement is given. Let ·, · denote the usual scalar product in H
1
R
N
.Easy
computations show that I
λ
is bounded from below on the Nehari manifold,
Λ
λ
u ∈ H
1
R
N
:
I
λ
u
,u
0
. 2.2
Thus u ∈ Λ
λ
if and only if
||
u
||
2
−
a
x
|
u
|
p
− λ
b
x
|
u
|
q
0. 2.3
In particular, on Λ
λ
, we have
I
λ
u
1
2
−
1
p
||
u
||
2
− λ
1
q
−
1
p
b
x
|
u
|
q
1
2
−
1
q
||
u
||
2
−
1
p
−
1
q
a
x
|
u
|
p
.
2.4
The Nehari manifold is closely linked to the behavior of the functions of the form
φ
u
: t → I
λ
tut>0. Such maps are known as fibrering maps and were introduced by
Dr
´
abek and Pohozaev in 11 and are discussed by Brown and Zhang 12.Ifu ∈ H
1
R
N
,
we have
φ
u
t
t
2
2
||
u
||
2
−
t
p
p
a
x
|
u
|
p
− λ
t
q
q
b
x
|
u
|
q
,
φ
u
t
t
||
u
||
2
− t
p−1
a
x
|
u
|
p
− λt
q−1
b
x
|
u
|
q
,
φ
u
t
||
u
||
2
−
p − 1
t
p−2
a
x
|
u
|
p
− λ
q − 1
t
q−2
b
x
|
u
|
q
.
2.5
Similarly to the method used in 7, we split Λ
λ
into three parts corresponding to local
minima, local maxima, and points of inflection, and so we define
Λ
λ
u ∈ Λ
λ
: φ
u
1
> 0
,
Λ
−
λ
u ∈ Λ
λ
: φ
u
1
< 0
,
Λ
0
λ
u ∈ Λ
λ
: φ
u
1
0
,
2.6
and note that if u ∈ Λ
λ
,thatis,φ
u
10, then
φ
u
1
2 − p
||
u
||
2
− λ
q − p
b
x
|
u
|
q
2 − q
||
u
||
2
−
p − q
a
x
|
u
|
p
.
2.7
4 Boundary Value Problems
This section will be devoted to prove Theorem 1.2. To prove Theorem 1.2, several preliminary
results are in order.
Lemma 2.1. Under the assumptions (a1), (b1), there exists λ
∗
> 0 such that when 0 <λ<λ
∗
,for
every u ∈ H
1
R
N
, u
/
≡ 0, there exist unique t
t
u > 0, t
−
t
−
u > 0 such that t
u ∈ Λ
−
λ
,
t
−
u ∈ Λ
λ
. In particular, one has
t
>
2 − q
u
2
p − q
a
x
|
u
|
p
1/p−2
t
max
>t
−
, 2.8
I
λ
t
−
umin
t∈0,t
I
λ
tu < 0 and I
λ
t
umax
t≥t
−
I
λ
tu.
Proof. Given u ∈ H
1
R
N
\{0},setϕ
u
tt
2−q
||u||
2
− t
p−q
ax|u|
p
. Clearly, for t>0, tu ∈ Λ
λ
if and only if t is a solution of
ϕ
u
t
λ
b
x
|
u
|
q
. 2.9
Moreover,
ϕ
u
t
2 − q
t
1−q
||
u
||
2
−
p − q
t
p−q−1
a
x
|
u
|
p
, 2.10
easy computations show that ϕ
u
is concave and achieves its maximum at
t
max
2 − q
||
u
||
2
p − q
a
x
|
u
|
p
1/p−2
. 2.11
If λ>0issufficiently large, 2.9 has no solution, and so φ
u
tI
λ
tu hasnocritical
points, in this case φ
u
is a decreasing function, hence no multiple of u lies in Λ
λ
.
If, on the other hand, λ>0issufficiently small, then there exist exactly two solutions
t
u >t
−
u > 0of2.9, where t
t
u, t
−
t
−
u, ϕ
u
t
−
> 0, and ϕ
u
t
< 0.
It follows from 2.7 and 2.10 that φ
tu
1t
q1
ϕ
u
t,andsot
u ∈ Λ
−
λ
, t
−
u ∈ Λ
λ
;
moreover φ
u
is decreasing in 0,t
−
,increasingint
−
,t
, and decreasing in t
, ∞.
Next, we will discussion the sufficiently small λ
∗
, such that when 0 <λ<λ
∗
, there
exist exactly two solutions of problem 2.9 for all u ∈ H
1
R
N
\{0},thatis,
λ
b
x
|
u
|
q
<ϕ
u
t
max
2 − q
p − q
2−q/p−2
p − 2
p − q
||
u
||
2p−2q/p−2
a
x
|
u
|
p
2−q/p−2
. 2.12
Boundary Value Problems 5
Since
a
x
|
u
|
p
≤
||
a
||
L
α/α−1
||
u
||
p
L
αp
≤
||
a
||
L
α/α−1
S
p
αp
||
u
||
p
, 2.13
where S
αp
denotes the Sobolev constant of the embedding of H
1
R
N
into L
αp
R
N
, hence,
ϕ
u
t
max
≥
2 − q
p − q
2−q/p−2
p − 2
p − q
||
u
||
2p−2q/p−2
||
a
||
L
α/α−1
S
p
αp
||
u
||
p
2−q/p−2
2 − q
p − q
2−q/p−2
p − 2
p − q
||
u
||
q
||
a
||
L
α/α−1
S
p
αp
2−q/p−2
,
2.14
and then
b
x
|
u
|
q
≤ M
||
u
||
q
L
q
≤ MS
q
q
||
u
||
q
≤ MS
q
q
p − q
2 − q
2−q/p−2
p − q
p − 2
||
a
||
L
α/α−1
S
p
αp
2−q/p−2
ϕ
u
t
max
cϕ
u
t
max
,
2.15
where S
q
denotes the Sobolev constant of the embedding of H
1
R
N
into L
q
R
N
, c is
independent of u, hence
ϕ
u
t
max
− λ
b
x
|
u
|
q
≥ ϕ
u
t
max
− λcϕ
u
t
max
ϕ
u
t
max
1 − λc
, 2.16
and so λ
bx|u|
q
<ϕ
u
t
max
for all u ∈ H
1
R
N
\{0} provided λ<1/2c λ
∗
.
Hence when 0 <λ<λ
∗
, φ
u
has exactly two critical points—a local minimum at t
−
t
−
u and a local maximum at t
t
u; moreover I
λ
t
−
umin
t∈0,t
I
λ
tu < 0andI
λ
t
u
max
t≥t
−
I
λ
tu.
In particular, we have the following result.
Corollary 2.2. Under the assumptions (a1), (b1), when 0 <λ<λ
∗
, for every u ∈ Λ
λ
, u
/
≡ 0, one has
2 − q
||
u
||
2
−
p − q
a
x
|
u
|
p
/
≡ 0 2.17
(i.e., Λ
0
λ
∅).
6 Boundary Value Problems
Proof. Let us argue by contradiction and assume that there exists u ∈ Λ
λ
\{0} such that
2 − q||u||
2
− p − q
ax|u|
p
0, this implies
λ
b
x
|
u
|
q
||
u
||
2
−
a
x
|
u
|
p
p − 2
2 − q
a
x
|
u
|
p
p − 2
2 − q
a
x
|
u
|
p
p−q/p−2
a
x
|
u
|
p
q−2/p−2
p − 2
2 − q
1
p − q
p−q/p−2
p − q
a
x
|
u
|
p
p−q/p−2
a
x
|
u
|
p
q−2/p−2
p − 2
p − q
2 − q
p − q
2−q/p−2
||
u
||
2p−q/p−2
a
x
|
u
|
p
q−2/p−2
ϕ
u
t
max
2.18
which contradicts 2.12 for 0 <λ<λ
∗
.
As a consequence of Corollary 2.2, we have the following lemma.
Lemma 2.3. Under the assumptions (a1), (b1), if 0 <λ<λ
∗
, for every u ∈ Λ
λ
, u
/
≡ 0, then there exist
a >0 and a C
1
-map t tw > 0, w ∈ H
1
R
N
, ||w|| <satisfying that
t
0
1,t
w
u − w
∈ Λ
λ
, for
||
w
||
<,
t
0
,w
2
∇u∇w uw
− p
a
x
|
u
|
p−2
uw − λq
b
x
|
u
|
q−2
uw
2 − q
||
u
||
2
−
p − q
a
x
|
u
|
p
.
2.19
Proof. We define F : R × H
1
R
N
→ R by
F
t, w
t
||
u − w
||
2
− t
p−1
a
x
|
u − w
|
p
− λt
q−1
b
x
|
u − w
|
q
. 2.20
Since F1, 00andF
t
1, 0||u||
2
− p − 1
ax|u|
p
− λq − 1
bx|u|
q
2 − q||u||
2
− p −
q
ax|u|
p
/
≡ 0 by Corollary 2.2, we can apply the implicit function theorem at the point
1, 0 and get the result.
Apply Lemma 2.1, Corollary 2.2, Lemma 2.3, and Ekeland variational principle 8,we
can establish the existence of the first positive solution.
Proposition 2.4. If 0 <λ<λ
∗
, then the minimization problem:
c
0
inf I
λ
Λ
λ
inf I
λ
Λ
λ
2.21
Boundary Value Problems 7
is achieved at a point u
0
∈ Λ
λ
which is a critical point for I
λ
with u
0
> 0 and I
λ
u
0
< 0. Furthermore,
u
0
is a local minimizer of I
λ
.
Proof. First, we show that I
λ
is bounded from below in Λ
λ
. Indeed, for u ∈ Λ
λ
,from2.13,
we have
I
λ
u
1
2
||
u
||
2
−
1
p
a
x
|
u
|
p
−
λ
q
b
x
|
u
|
q
1
2
−
1
q
||
u
||
2
−
1
p
−
1
q
a
x
|
u
|
p
≥
1
2
−
1
q
||
u
||
2
−
1
p
−
1
q
||
a
||
L
α/α−1
S
p
αp
||
u
||
p
2.22
and so I
λ
is bounded from below in Λ
λ
.
Then we will claim that c
0
< 0, indeed if v ∈ H
1
R
N
\{0},fromLemma 2.1, there exist
0 <t
−
v <t
v such that t
−
vv ∈ Λ
λ
.Thus,
c
0
≤ I
λ
t
−
v
v
min
t∈0,t
v
I
λ
tv
< 0. 2.23
By Ekeland’s Variational Principle 8, there exists a minimizing sequence {u
n
}⊂Λ
λ
of the minimization problem 2.21 such that
c
0
≤ I
λ
u
n
<c
0
1
n
,
2.24
I
λ
v
≥ I
λ
u
n
−
1
n
||
v − u
n
||
, ∀ v ∈ Λ
λ
.
2.25
Taking n large enough, from 2.7 we have
I
λ
u
n
1
2
−
1
q
||
u
n
||
2
−
1
p
−
1
q
a
x
|
u
n
|
p
<c
0
1
n
< 0, 2.26
from which we deduce that for n large
a
x
|
u
n
|
p
≥
pq
p − q
c
0
,
||
u
n
||
2
≤
2
q − p
p
q − 2
a
x
|
u
n
|
p
, 2.27
which yields
b
1
≤
||
u
n
||
≤ b
2
2.28
for suitable b
1
, b
2
> 0.
8 Boundary Value Problems
Now we will show that
I
λ
u
n
−→ 0asn −→ ∞ . 2.29
Since u
n
∈ Λ
λ
,byLemma 2.3, we can find a
n
> 0andaC
1
-map t
n
t
n
w > 0, w ∈ H
1
R
N
,
||w|| <
n
satisfying that
v
n
t
n
w
u
n
− w
∈ Λ
λ
, for
||
w
||
<
n
. 2.30
By the continuity of t
n
w and t
n
01, without loss of generality, we can assume
n
satisfies
that 1/2 ≤ t
n
w ≤ 3/2for||w|| <
n
.
It follows from 2.25 that
I
λ
t
n
w
u
n
− w
− I
λ
u
n
≥−
1
n
||
t
n
w
u
n
− w
− u
n
||
; 2.31
that is,
I
λ
u
n
,t
n
w
u
n
− w
− u
n
o
||
t
n
w
u
n
− w
− u
n
||
≥−
1
n
||
t
n
w
u
n
− w
− u
n
||
.
2.32
Consequently,
t
n
w
I
λ
u
n
,w
1 − t
n
w
I
λ
u
n
,u
n
≤
1
n
||
t
n
w
− 1
u
n
− t
n
w
w
||
o
||
t
n
w
u
n
− w
− u
n
||
.
2.33
By the choice of
n
,weobtain
I
λ
u
n
,w
≤
C
n
t
n
0
,w
o
||
w
||
C
n
||
w
||
o
t
n
0
,w
||
u
n
||
||
w
||
.
2.34
By Lemma 2.3, Corollary 2.2, and the estimate 2.28, we have
t
n
0
,w
2
∇u
n
∇w u
n
w
− p
a
x
|
u
n
|
p−2
u
n
w − λq
b
x
|
u
n
|
q−2
u
n
w
2 − q
||
u
n
||
2
−
p − q
a
x
|
u
n
|
p
≤ C
||
w
||
,
2.35
then from 2.34 we get
I
λ
u
n
,w
≤
C
n
||
w
||
C
n
||
w
||
o
||
w
||
, for
||
w
||
≤
n
. 2.36
Boundary Value Problems 9
Hence, for any ∈ 0,
n
, we have
I
λ
u
n
1
sup
||
w
||
I
λ
u
n
,w
≤
C
n
1
o
, 2.37
for some C>0 independent of and n. Taking → 0, we obtain 2.29.
Let u
0
∈ H
1
R
N
be the weak limit in H
1
R
N
of u
n
.From2.29,
I
λ
u
0
,w
0, ∀w ∈ H
1
R
N
; 2.38
that is, u
0
is a weak solution of problem 1.1 and consequently u
0
∈ Λ
λ
. Therefore,
c
0
≤ I
λ
u
0
≤ lim
n →∞
I
λ
u
n
c
0
; 2.39
that is,
c
0
I
λ
u
0
inf
Λ
λ
I
λ
. 2.40
Moreover, we have u
0
∈ Λ
λ
.Infact,ifu
0
∈ Λ
−
λ
,byLemma 2.1, there exists only one t
> 0
such that t
u
0
∈ Λ
−
λ
, we have t
t
u
0
1, t
−
t
−
u
0
< 1. Since
dI
λ
t
−
u
0
dt
0,
d
2
I
λ
t
−
u
0
dt
2
> 0,
2.41
there exists t
≥ t>t
−
such that I
λ
tu
0
>I
λ
t
−
u
0
.ByLemma 2.1,
I
λ
t
−
u
0
<I
λ
tu
0
≤ I
λ
t
u
0
I
λ
u
0
; 2.42
this is a contradiction.
To conclude that u
0
is a local minimizer of I
λ
, notice that for every u ∈ H
1
R
N
\{0},
we have from Lemma 2.1,
I
λ
su
≥ I
λ
t
−
u
∀0 <s<
2 − q
||
u
||
2
p − q
a
x
|
u
|
p
1/p−2
. 2.43
In particular, for u u
0
∈ Λ
λ
, we have
t
−
u
0
1 <
2 − q
||
u
0
||
2
p − q
a
x
|
u
0
|
p
1/p−2
. 2.44
10 Boundary Value Problems
Let >0sufficiently small to have
1 <
2 − q
||
u
0
− w
||
2
p − q
a
x
|
u
0
− w
|
p
1/p−2
, for
||
w
||
<. 2.45
From Lemma 2.3,lettw > 0satisfytwu
0
− w ∈ Λ
λ
for every ||w|| <. By the continuity
of tw and t01, we can always assume that
t
w
<
2 − q
||
u
0
− w
||
2
p − q
ax
|
u
0
− w
|
p
1/p−2
, for
||
w
||
<. 2.46
Namely, twu
0
− w ∈ Λ
λ
and for
0 <s<
2 − q
||
u
0
− w
||
2
p − q
a
x
|
u
0
− w
|
p
1/p−2
, 2.47
we have
I
λ
s
u
0
− w
≥ I
λ
t
w
u
0
− w
≥ I
λ
u
0
. 2.48
Taking s 1, we conclude
I
λ
u
0
− w
≥ I
λ
t
w
u
0
− w
≥ I
λ
u
0
, for
||
w
||
<, 2.49
which means that u
0
is a local minimizer of I
λ
.
Furthermore, taking t
−
|u
0
| > 0witht
−
|u
0
||u
0
|∈Λ
λ
, therefore,
I
λ
u
0
≤ I
λ
t
−
|
u
0
|
|
u
0
|
≤ I
λ
|
u
0
|
≤ I
λ
u
0
. 2.50
So we can always take u
0
≥ 0. By the maximum principle for weak solutions see 13,we
can show that u
0
> 0inR
N
.
Since u
0
∈ Λ
λ
and c
0
inf
Λ
λ
I
λ
inf
Λ
λ
I
λ
, thus, in the search of our second positive
solution, it is natural to consider the second minimization problem:
c
1
inf
Λ
−
λ
I
λ
. 2.51
Let us now introduce the problem at infinity associated with 1.1:
−Δu u λb
∞
u
q−1
, in R
N
,
u>0, in R
N
,
u ∈ H
1
R
N
.
2.52
Boundary Value Problems 11
We state here some known results for problem 2.52. First of all, we recall that Lions 14
has studied the following minimization problem closely related to problem 2.52: S
∞
λ
inf{I
∞
λ
u : u ∈ H
1
R
N
,u
/
0 ,I
∞
λ
u0} > 0, where I
∞
λ
u1/2||u||
2
− 1/qλb
∞
|u|
q
.
For future reference, note also that a minimum exists and is realized by a ground state ω>0in
R
N
such that S
∞
λ
I
∞
λ
ωsup
s≥0
I
∞
λ
sω. Gidas et al. 15 showed that there exist a
1
,a
2
> 0
such that for all x ∈ R
N
,
a
1
|
x
|
1
−N−1/2
e
−
|
x
|
≤ ω
x
≤ a
2
|
x
|
1
−N−1/2
e
−
|
x
|
. 2.53
Lemma 2.5. Let ax ∈ L
α/α−1
R
N
∩ L
∞
R
N
,where1 <α<2
∗
/p and 1 <p<2.Ifu
n
u
weakly in H
1
R
N
, then a subsequence of {u
n
}, still denoted by {u
n
}, satisfies
lim
n →∞
a
x
|
u
n
− u
|
p
0. 2.54
Proof. Since ax ∈ L
α/α−1
R
N
, then for every >0, there exists R
0
> 0 such that
|
x
|
>R
0
|
a
x
|
α/α−1
dx
α−1/α
<. 2.55
Since u
n
uweakly in H
1
R
N
, u
n
→ u strongly in L
s
loc
R
N
,1≤ s<2N/N − 2, then we
have
|
x
|
≤R
0
|
u
n
− u
|
αp
dx
1/αp
<. 2.56
Observe that by H
¨
older inequality we have
a
x
|
u
n
− u
|
p
dx
|
x
|
≤R
0
a
x
|
u
n
− u
|
p
dx
|
x
|
>R
0
a
x
|
u
n
− u
|
p
dx ≤ C, 2.57
hence lim
n →∞
ax|u
n
− u|
p
0.
Our first task is to locate the levels free from this noncompactness effect.
Proposition 2.6. Every sequence {u
n
}⊂H
1
R
N
, satisfying
a I
λ
u
n
c o1 with c<c
0
S
∞
λ
,
b I
λ
u
n
o1 strongly in H
−1
R
N
,
has a convergent subsequence.
12 Boundary Value Problems
Proof. It is easy to see that {u
n
} is bounded in H
1
R
N
, so we can find a u ∈ H
1
R
N
such
that u
n
u weakly in H
1
R
N
, u
n
→ u almost every in R
N
, u
n
→ u strongly in L
s
loc
R
N
,
1 ≤ s<2N/N − 2. From condition b, we have
I
λ
u
,w
0, ∀ w ∈ H
1
R
N
; 2.58
that is,
u is a weak solution of problem 1.1 and u ∈ Λ
λ
.Setv
n
u
n
− u to get v
n
0 weakly
in H
1
R
N
, v
n
→ 0 almost every in R
N
, v
n
→ 0 strongly in L
s
loc
R
N
,1≤ s<2N/N − 2, we
can prove that there exists a subsequence of {v
n
} still denoted by {v
n
} satisfying v
n
→ 0
strongly in H
1
R
N
. Arguing by contradiction, we assume that there exists a constant β>0
such that ||v
n
|| ≥ β>0. Apply the Brezis-Lieb theorem see 16 and Lemma 2.5,
I
λ
u
n
1
2
||
u
n
||
2
−
1
p
a
x
|
u
n
|
p
−
λ
q
b
x
|
u
n
|
q
I
λ
u
1
2
||
v
n
||
2
−
1
p
a
x
|
v
n
|
p
−
λ
q
b
x
|
v
n
|
q
o
1
I
λ
u
1
2
||
v
n
||
2
−
λb
∞
q
|
v
n
|
q
−
λ
q
b
x
− b
∞
|
v
n
|
q
o
1
.
2.59
Moreover, taking into account 2.58,
o
1
I
λ
u
n
,u
n
||
u
n
||
2
−
a
x
|
u
n
|
p
− λ
b
x
|
u
n
|
q
I
λ
u
, u
||
v
n
||
2
−
a
x
|
v
n
|
p
− λ
b
x
|
v
n
|
q
o
1
||
v
n
||
2
− λb
∞
|
v
n
|
q
− λ
b
x
− b
∞
|
v
n
|
q
o
1
.
2.60
By b1, for any >0, there exist: R
0
> 0 such that |bx − b
∞
| <for |x|≥R
0
. Since v
n
→ 0
strongly in L
s
loc
R
N
for 1 ≤ s<2N/N − 2, {v
n
} is a bounded sequence in H
1
R
N
, therefore,
bx − b
∞
|v
n
|
q
≤ C
B
R
0
|v
n
|
q
C. Setting n →∞, then → 0, we have
b
x
− b
∞
|
v
n
|
q
o
1
. 2.61
Combining 2.60 and 2.59,weobtain
||
v
n
||
2
− λb
∞
|
v
n
|
q
o
1
,I
λ
u
n
≥ c
0
1
2
||
v
n
||
2
−
λ
q
b
∞
|
v
n
|
q
o
1
. 2.62
Boundary Value Problems 13
Since ||v
n
|| ≥ β>0, we can find a sequence {s
n
}, s
n
> 0, s
n
→ 1asn →∞, such that t
n
s
n
v
n
satisfying ||t
n
||
2
− λb
∞
|t
n
|
q
0. Hence
I
λ
u
n
≥ c
0
1
2
||
t
n
||
2
−
λ
q
b
∞
|
t
n
|
q
o
1
≥ c
0
S
∞
λ
o
1
; 2.63
that is, c lim
n →∞
I
λ
u
n
≥ c
0
S
∞
λ
, contradicting condition a. Consequently, u
n
→ u
strongly.
Let e 1, 0, ,0 be a fixed unit vector in R
N
and ω be a ground state of problem
2.52. Here we use an interaction phenomenon between u
0
and ω.
Proposition 2.7. Under the assumptions (a1) and (b1), Then
I
λ
u
0
tω
<c
0
I
∞
λ
ω
∀t>0. 2.64
Proof.
I
λ
u
0
tω
1
2
||
u
0
tω
||
2
−
1
p
a
x
|
u
0
tω
|
p
−
λ
q
b
x
|
u
0
tω
|
q
<I
λ
u
0
1
2
||
tω
||
2
−
λ
q
t
q
b
x
|
ω
|
q
≤ I
λ
u
0
1
2
||
tω
||
2
−
λ
q
b
∞
|
tω
|
q
I
λ
u
0
I
∞
λ
tω
≤ c
0
I
∞
λ
ω
.
2.65
Proposition 2.8. If 0 <λ<λ
∗
,forc
1
inf
Λ
−
λ
I
λ
, one can find a minimizing sequence {u
n
}⊂Λ
−
λ
such that
a I
λ
u
n
c
1
o1,
b I
λ
u
n
o1 strongly in H
−1
R
N
,
c c
1
<c
0
S
∞
λ
.
Proof. Set Σ{u ∈ H
1
R
N
: ||u|| 1} and define the map Ψ : Σ → Λ
−
λ
given by Ψu
t
uu. Since the continuity of t
u follows immediately from its uniqueness and extremal
property, thus Ψ is continuous with continuous inverse given by Ψ
−1
uu/||u||. Clearly Λ
−
λ
disconnects H
1
R
N
in exactly two components:
U
1
u 0oru :
||
u
||
<t
u
||
u
||
,
U
2
u :
||
u
||
>t
u
||
u
||
,
2.66
and Λ
λ
⊂ U
1
.
14 Boundary Value Problems
We will prove that there exists t
1
such that u
0
t
1
ω ∈ U
2
. Denote t
0
t
u
0
tω/||u
0
tω||. Since t
u
0
tω/||u
0
tω||u
0
tω/||u
0
tω|| ∈ Λ
−
λ
, we have
t
2
0
−
t
q
0
λ
b
x
|
u
0
tω
|
q
||
u
0
tω
||
q
t
p
0
a
x
|
u
0
tω
|
p
||
u
0
tω
||
p
≥ 0. 2.67
Thus
t
0
≤
||
u
0
tω
||
λ
bx
|
u
0
tω
|
q
1/q
q/q−2
||
u
0
/t ω
||
λ
bx
|
u
0
/t ω
|
q
1/q
q/q−2
≤
||
u
0
/t ω
||
λ
b
∞
|
u
0
/t ω
|
q
1/q
q/q−2
−→
||
ω
||
< ∞ as t −→ ∞ .
2.68
Therefore, there exists t
2
> 0 such that t
0
t
u
0
tω/||u
0
tω|| < 2||ω||,fort ≥ t
2
.Set
t
1
>t
2
2, then
||
u
0
t
1
ω
||
2
||
u
0
||
2
t
2
1
||
ω
||
2
2t
1
∇u
0
∇ω u
0
ω
||u
0
||
2
t
2
1
||
ω
||
2
2t
1
λb
∞
|
ω
|
q−1
u
0
>t
2
1
||
ω
||
2
> 4
||
ω
||
2
>t
2
0
,
2.69
hence u
0
t
1
ω ∈ U
2
.
However, Λ
−
λ
disconnects H
1
R
N
in exactly two components, so we can find a s ∈
0, 1 such that u
0
st
1
ω ∈ Λ
−
λ
. Therefore, c
1
≤ I
λ
u
0
st
1
ω <c
0
S
∞
λ
, which follows from
Proposition 2.7.
Analogously to the proof of Proposition 2.4, one can show that the Ekeland variational
principle 8 gives a sequence {u
n
}⊂Λ
−
λ
satisfying the conditions a, b,andc.
Proposition 2.9. If 0 <λ<λ
∗
, then the minimization problem c
1
inf
Λ
−
λ
I
λ
is achieved at a point
u
1
∈ Λ
−
λ
which is a critical point for I
λ
and u
1
> 0.
Proof. Applying Propositions 2.6 and 2.8, we have u
n
→ u
1
strongly in H
1
R
N
.
Consequently, u
1
is a critical point for I
λ
, u
1
∈ Λ
−
λ
since Λ
−
λ
is closed and I
λ
u
1
c
1
.
Let t
|u
1
| > 0satisfyt
|u
1
||u
1
|∈Λ
−
λ
. Since u
1
∈ Λ
−
λ
, t
u
1
1. From Lemma 2.1,we
conclude that
t
|
u
1
|
≥ t
max
|
u
1
|
t
max
u
1
, 2.70
c
1
I
λ
u
1
max
t≥t
max
u
1
I
λ
tu
1
≥ I
λ
t
|
u
1
|
u
1
≥ I
λ
t
|
u
1
|
|
u
1
|
≥ c
1
.
2.71
Boundary Value Problems 15
Hence, It
|u
1
||u
1
|c
1
, So we can always take u
1
≥ 0. By standard regularity method and
the maximum principle for weak solutions see 13, we can show that u
1
> 0inR
N
.
Proof of Theorem 1.1. Applying Propositions 2.4 and 2.9, we can obtain the conclusion of
Theorem 1.1.
3. Proof of Theorem 1.2
In the sequel, X : H
1
R
N
, e
k
denotes an orthonormal base of X,
X
j
: span
e
1
, ,e
j
,X
k
: ⊕
j≥k
X
j
,X
k
: ⊕
j≤k
X
j
, 3.1
and C, C
1
,C
2
, , denote possibly different positive constants.
If u ∈ X, we let the variational functional of problem 1.4 be
I
u
1
2
|
∇u
|
2
u
2
−
μ
p
a
x
|
u
|
p
−
λ
q
b
x
|
u
|
q
. 3.2
Proposition 3.1. Under the assumptions (a1) and (b1), for every λ>0 and μ ∈ R, the problem 1.4
has infinitely many solutions with positive energy Iu.
Proof. We will show that the energy functional Iu satisfies the assumptions of Fountain
theorem in 17. These assumptions are as follows.
A1 The energy functional I ∈ C
1
X, R and is even.
A2 Every sequence u
n
∈ X with C : sup
n
Iu
n
< ∞ and I
u
n
→ 0asn →∞has a
convergent subsequence.
A3 inf
ρ>0
sup
u∈X
k
,||u||≥ρ
Iu ≤ 0, for every k ∈ N.
A4 sup
r>0
inf
u∈X
k
,||u||r
Iu →∞as k →∞.
We define
λ
k
sup
u∈X
k
−{0}
b
x
|
u
|
q
1/q
||
u
||
, 3.3
then
λ
k
−→ 0, as k −→ ∞ . 3.4
Indeed, clearly we have
0 <λ
k1
≤ λ
k
. 3.5
16 Boundary Value Problems
Assume that λ
k
→ λ
0
> 0, as k →∞. Then for every k ≥ 1, there exists u
k
∈ X
k
such that
||u
k
|| 1and
λ
0
2
<
b
x
|
u
k
|
q
. 3.6
By definition, u
k
0inX, this contradicts with λ
0
> 0. Now, let us prove A1–A4.The
PS-condition A2 has be shown as in Proposition 2.6. In order to prove A3,sincethe
subspace X
k
is finite dimensional, all norms on X
k
are equivalent, hence, we obtain
I
u
≤
1
2
||
u
||
2
− C
1
||
u
||
p
− C
2
λ
||
u
||
q
≤
1
2
||
u
||
2
− C
2
λ
||
u
||
q
. 3.7
Therefore, the term −C
2
λ||u||
q
dominates for ||u|| sufficiently large, and A3 follows. To show
A4,sincep<2, there exists R>0 large enough so that
μ
p
||
a
||
L
α/α−1
S
p
αp
||
u
||
p
≤
1
4
||
u
||
2
3.8
for ||u|| ≥ R. Then, for u ∈ X
k
, it follows from 2.13, 3.8,and3.3 that
I
u
1
2
||
u
||
2
−
μ
p
a
x
|
u
|
p
−
λ
q
b
x
|
u
|
q
≥
1
4
||
u
||
2
−
λ
q
λ
q
k
||
u
||
q
.
3.9
Now we set r
k
8λλ
q
k
/q
1/2−q
so that
1
8
r
2
k
λ
q
λ
q
k
r
q
k
. 3.10
Clearly
r
k
−→ ∞ as k −→ ∞ , 3.11
A4 follows. Since the energy functional Iu is even, then, by Fountain theorem, there exist
a sequence of critical points v
k
such that Iv
k
→∞as k →∞.
Proposition 3.2. Under the assumptions (a1) and (b1), for every μ>0 and λ ∈ R, the problem 1.4
has infinitely many solutions with negative energy Iu.
Boundary Value Problems 17
Proof. We will show that the energy functional Iu satisfies the assumptions of in 9,
Theorem 2. These assumptions are as follows.
B1 The energy functional I ∈ C
1
X, R and is even.
B2 There exists k
0
such that for every k ≥ k
0
there exists R
k
> 0 such that Iu ≥ 0for
every u ∈ X
k
with ||u|| R
k
.
B3 b
k
: inf
B
k
Iu → 0ask →∞, where B
k
{u ∈ X
k
: ||u|| ≤ R
k
}.
B4 For every k ≥ 1, there exist r
k
∈ 0,R
k
and d
k
< 0 such that Iu ≤ d
k
for every
u ∈ X
k
with ||u|| r
k
.
B5 Every sequence u
n
∈ X
n
−n
: ⊕
n
j−n
Xj with Iu
n
< 0 bounded and I|
X
n
−n
u
n
→ 0
as n →∞has a subsequence which converges to a critical point of I.
We define
μ
k
sup
u∈X
k
−{0}
a
x
|
u
|
p
1/p
||
u
||
. 3.12
By Lemma 2.5,
μ
k
−→ 0, as k −→ ∞ . 3.13
Indeed, clearly we have
0 <μ
k1
≤ μ
k
. 3.14
Assume that μ
k
→ μ
0
> 0, as k →∞. Then for every k ≥ 1, there exists u
k
∈ X
k
such that
||u
k
|| 1and
μ
0
2
<
a
x
|
u
k
|
p
. 3.15
By definition, u
k
0inX.ByLemma 2.5, this contradicts 3.15. Now, let us prove B1–B5.
Since q>2, there exists R>0 small enough so that
λ
q
b
∞
S
q
q
||
u
||
q
≤
1
4
||
u
||
2
3.16
for ||u|| ≤ R. Then, for u ∈ X
k
, it follows from 3.16 and 3.12 that
I
u
≥
1
2
||
u
||
2
−
μ
p
a
x
|
u
|
p
−
λ
q
b
∞
S
q
q
||
u
||
q
≥
1
4
||
u
||
2
−
μ
p
μ
p
k
||
u
||
p
.
3.17
18 Boundary Value Problems
Now we set R
k
4μμ
p
k
/p
1/2−p
so that
1
4
R
2
k
μ
p
μ
p
k
R
p
k
. 3.18
Clearly
R
k
−→ 0ask −→ ∞ , 3.19
so there exists k
0
such that R
k
≤ R when k ≥ k
0
.Thusifu ∈ X
k
, k ≥ k
0
satisfies ||u|| R
k
,we
have
I
u
≥
1
4
||
u
||
2
−
μ
p
μ
p
k
||
u
||
p
0. 3.20
This proves B2. Next, B3 follows immediately from 3.19. On the other hand, since the
subspace X
k
is finite dimensional, all norms on X
k
are equivalent, hence, we obtain
I
u
≤
1
2
||
u
||
2
− C
1
μ
||
u
||
p
− C
2
||
u
||
q
≤
1
2
||
u
||
2
− C
1
μ
||
u
||
p
. 3.21
Therefore, the term −C
1
μ||u||
p
dominates near 0, and B4 follows. This is precisely the point
where μ>0 enters. Finally, the PS condition B5 has been shown as in Proposition 2.6.
Since the energy functional Iu is even, all the assumptions of in 9, Theorem 2 are satisfied.
Then, there exists k
0
such that for each k ≥ k
0
,
I
u
has a critical value c
k
∈
b
k
,d
k
, so that c
k
−→ 0ask −→ ∞ . 3.22
This completes the proof of Theorem 1.2, since observe t hat B3 and B4 imply b
k
≤ d
k
<
0.
Proof of Theorem 1.2. The proof follows from Propositions 3.1 and 3.2.
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