Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 389530, 15 pages
doi:10.1155/2012/389530
Research Article
Multiple Solutions for a Class of Multipoint
Boundary Value Systems Driven by
a One-Dimensional p1 , . . . , pn -Laplacian Operator
Shapour Heidarkhani
Department of Mathematics, Faculty of Sciences, Razi University, Kermanshash 67149, Iran
Correspondence should be addressed to Shapour Heidarkhani, sh.heidarkhani@yahoo.com
Received 20 November 2011; Accepted 11 December 2011
Academic Editor: Kanishka Perera
Copyright q 2012 Shapour Heidarkhani. 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.
Employing a recent three critical points theorem due to Bonanno and Marano 2010, the existence
of at least three solutions for the
following multipointboundary value system −|ui |pi −2 ui m
λFui x, u1 , . . . , un in 0, 1, ui 0 m
j1 aj ui xj , ui 1 j1 bj ui xj for 1 ≤ i ≤ n, is established.
1. Introduction
In this work, we consider the following multipoint boundary value system
p −2
− ui i ui λFui x, u1 , . . . , un ui 0 m
aj ui xj ,
j1
ui 1 m
in 0, 1,
bj ui xj ,
1.1
j1
for 1 ≤ i ≤ n, where pi > 1 for 1 ≤ i ≤ n, λ > 0, m, n ≥ 1, F : 0, 1 × Rn → R is a function
such that F·, t1 , . . . , tn is continuous in 0, 1 for all t1 , . . . , tn ∈ Rn , Fx, ·, . . . , · is C1 in Rn
for every x ∈ 0, 1 and Fx, 0, . . . , 0 0 for all x ∈ 0, 1, aj , bj ∈ R for j 1, . . . , m and
0 < x1 < x2 < x3 < · · · < xm < 1, and Fui denotes the partial derivative of F with respect to ui
for 1 ≤ i ≤ n.
The study of multiplicity of solutions is an important mathematical subject which is
also interesting from the practical point of view because the physical processes described
2
Abstract and Applied Analysis
by boundary value problems for differential equations exhibit, generally, more than one
solution. In 1–3, Ricceri proposed and developed an innovative minimal method for the
study of nonlinear eigenvalue problems. Following that, Bonanno 4 gave an application of
the method to the two-point problem
u λfu 0 in 0, 1,
u0 u1 0.
1.2
Bonanno also gave more precise versions of the three critical points of Ricceri in 5, 6. In
particular, in 5, an upper bound of the interval of parameters λ for which the functional
has three critical points is established. Candito 7 extended the main result of 4 to the
nonautonomous case
u λfx, u 0 in a, b,
ua ub 0.
1.3
In 8, He and Ge extended the main results of 4, 7 to the quasilinear differential equation
ϕp u λfx, u 0 in a, b,
ua ub 0.
1.4
In 9, the authors extended the main results of 4, 7, 9 to the quasilinear differential equation
with Sturm-Liouville boundary conditions
u p−2 u λfx, u 0
α1 ua − α2 u a 0,
in a, b,
β1 ub − β2 u b 0,
1.5
where p > 1 is a constant, λ is a positive parameter, a, b ∈ R; a < b. In particular, in 10, the
authors motivated by these works, established some criteria for the existence of three classical
solutions of the system 1.1, while in 11, based on Ricceri’s three critical points theorem 3,
the existence of at least three classical solutions to doubly eigenvalue multipoint boundary
value systems was established.
In the present paper, based on a three critical points theorem due to Bonanno and
Marano 12, we ensure the existence of least three classical solutions for the system 1.1.
Several results are known concerning the existence of multiple solutions for multipoint
boundary value problems, and we refer the reader to the papers 13–16 and the references
cited therein.
Abstract and Applied Analysis
3
Here and in the sequel, X will denote the Cartesian product of n space
⎧
⎨
⎫
m
m
⎬
,
Xi ξ ∈ W 1,pi 0, 1; ξ0 aj ξ xj , ξ1 bj ξ xj
⎩
⎭
j1
j1
1.6
for i 1, . . . , n, that is, X X1 × · · · × Xn equipped with the norm
u1 , . . . , un n u ,
i pi
1.7
i1
where
u i pi
1
0
pi
u x dx
1/pi
1.8
i
for 1 ≤ i ≤ n.
We say that u u1 , . . . , un is a weak solution to 1.1 if u u1 , . . . , un ∈ X and
1 n
0 i1
pi −2 u x u xv xdx − λ
i
i
i
1 n
Fui x, u1 x, . . . , un xvi xdx 0,
1.9
0 i1
for every v1 , . . . , vn ∈ X.
A special case of our main result is the following theorem.
Theorem 1.1. Let f, g : R2 → R be two positive continuous functions such that the differential
1-form w : fξ, ηdξ gξ, ηdη is integrable, and let F be a primitive of w such that F0, 0 0.
Fix p, q > 2, 0 < x1 < x2 < 1, and assume that
F ξ, η
lim inf
q
p
ξ,η → 0,0 |ξ| /p η /q
lim sup
|ξ| → ∞, |η| → ∞
F ξ, η
0,
q
|ξ|p /p η /q
1.10
then there is λ∗ > 0 such that for each λ > λ∗ , the problem
p−2
− u1 u1 λfu1 , u2 in 0, 1,
q−2
− u2 u2 λgu1 , u2 in 0, 1,
ui 0 a1 ui x1 a2 ui x2 ,
1.11
ui 1 b1 ui x1 b2 ui x2 admits at least two positive classical solutions.
The main aim of the present paper is to obtain further applications of 12, Theorem 2.6
see Theorem 2.1 in the next section to the system 1.1, and the obtained results are strictly
4
Abstract and Applied Analysis
comparable with those of 9–11, and here we wil give the exact collocation of the interval of
positive parameters.
For other basic notations and definitions, we refer the reader to 17–26. We note that
some of the ideas used here were motivated by corresponding ones in 10.
2. Main Results
Our main tool is a three critical points theorem obtained in 12 see also 1, 2, 5, 27 for related
results, which is a more precise version of Theorem 3.2 of 28, to transfer the existence of
three solutions of the system 1.1 into the existence of critical points of the Euler functional.
We recall it here in a convenient form see 23.
Theorem 2.1 see 12, Theorem 2.6. Let X be a reflexive real Banach space, Φ : X → R be a
coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional
whose Gâteaux derivative admits a continuous inverse on X ∗ , Ψ : X → R be a continuously Gâteaux
differentiable functional whose Gâteaux derivative is compact such that Φ0 Ψ0 0. Assume
that there exist r > 0 and x ∈ X, with r < Φx such that
κ1 supΦx≤r Ψx/r < Ψx/Φx,
κ2 for each λ ∈ Λr :Φx/Ψx, r/supΦx≤r Ψx, the functional Φ − λΨ is coercive.
then, for each λ ∈ Λr , the functional Φ − λΨ has at least three distinct critical points in X.
Put
k max
sup
maxx∈0,1 |ui x|pi
ui ∈Xi \{0}
p
ui pii
; for 1 ≤ i ≤ n .
2.1
n
Since pi > 1 for 1 ≤ i ≤ n, and the embedding X X1 × · · · × Xn → C0 0, 1 is compact,
one has k < ∞. Moreover, from 10, Lemma 3.1, one has
⎛
m ⎞
m maxx∈0,1 |ux| 1
j1 aj
j1 bj
⎠.
sup
≤ ⎝1 m
m
2
u pi
u∈Xi \{0}
1 − j1 aj 1 − j1 bj 2.2
Put φpi s |s|pi −1 s for 1 ≤ i ≤ n. Let φp−1i denotes the inverse of φpi for 1 ≤ i ≤ n. then,
φp−1i t φqi t where 1/pi 1/qi 1. It is clear that φpi is increasing on R,
lim φpi t −∞,
t → −∞
lim φpi t ∞.
2.3
t → ∞
Lemma 2.2 see 10, Lemma 3.3. For fixed λ ∈ R and u u1 , . . . , un ∈ C0, 1n , define
αi t; u : R → R by
αi t; u 1
0
φp−1i
t−λ
δ
0
Fui ξ, u1 ξ, . . . , un ξdξ dδ m
j1
m
aj ui xj −
bj ui xj .
j1
2.4
Abstract and Applied Analysis
5
then the equation
αi t; u 0
2.5
has a unique solution tu,i .
Direct computations show the following.
Lemma 2.3 see 10, Lemma 3.4. The function u u1 , . . . , un is a solution of the system 1.1
if and only if ui x is a solution of the equation
m
ui x aj ui xj x
0
j1
φp−1i
tu,i − λ
δ
Fui ξ, u1 ξ, . . . , un ξdξ dδ,
2.6
0
for 1 ≤ i ≤ n, where tu,i is the unique solution of 2.5.
Lemma 2.4. A weak solution to the systems 1.1 coincides with classical solution one.
Proof. Suppose that u u1 , . . . , un ∈ X is a weak solution to 1.1, so
1 n
0 i1
1 n
Fui x, u1 x, . . . , un xvi xdx,
φpi ui x vi xdx − λ
2.7
0 i1
for every v1 , . . . , vn ∈ X. Note that, in one dimension, any weakly differentiable function is
absolutely continuous, so that its classical derivative exists almost everywhere, and that the
classical derivative coincides with the weak derivative. Now, using integration by part, from
2.7, we obtain
n 1
0
i1
φpi ui x λFui x, u1 x, . . . , un x vi xdx 0,
2.8
and so for 1 ≤ i ≤ n,
φpi ui x λFui x, u1 x, . . . , un x 0,
2.9
for almost every x ∈ 0, 1. Then, by Lemmas 2.2 and 2.3, we observe
ui x m
j1
aj ui xj x
0
φp−1i
tu,i − λ
δ
Fui s, u1 s, . . . , un sds dδ,
2.10
0
for 1 ≤ i ≤ n, where tu,i is the unique solution of 2.5. Hence, ui ∈ C1 0, 1 and φpi ui x ∈
C1 0, 1 for 1 ≤ i ≤ n, namely u u1 , . . . , un is a classical solution to the system 1.1.
6
Abstract and Applied Analysis
For all γ > 0, we denote by Kγ the set
t1 , . . . , tn ∈ R :
n
n
|ti |pi
i1
pi
≤γ .
2.11
Now, we formulate our main result as follows.
Theorem 2.5. Assume that there exist 2m constants aj , bj for 1 ≤ j ≤ m with m
1 and
j1 aj /
m
b
,
.
.
.
,
w
∈
X
such
that
1,
a
positive
constant
r
and
a
function
w
w
1
n
j1 j /
n
p
A1 i1 wi pii /pi > r,
1
1
<
r ni1 pi 0 Fx, w1 x, . . . , wn xdx/
A2 0 supt1 ,...,tn ∈Kkr Fx, t1 , . . . , tn dx
n n
n
pi
pi
i1
j1,j / i pj wi pi where Kkr {t1 , . . . , tn |
i1 |ti | /pi ≤ kr}see 2.11,
1
A3 lim sup|t1 | → ∞,..., |tn | → ∞ Fx, t1 , . . . , tn / ni1 |ti |pi /pi < 0 supt1 ,...,tn ∈Kkr × Fx, t1 ,
. . . , tn dx/kr uniformly with respect to x ∈ 0, 1.
1
1
Then, for each λ ∈ Λr : ni1 wi ppii /pi / 0 Fx, w1 x, . . . , wn xdx, r/ 1 supt1 ,...,tn ∈Kkr
Fx, t1 , . . . , tn dx, the system 1.1 admits at least three distinct classical solutions in X.
Proof. In order to apply Theorem 2.1 to our problem, we introduce the functionals Φ, Ψ :
X → R for each u u1 , . . . , un ∈ X, as follows:
Φu Ψu 1
p
n u i
i pi
i1
pi
,
2.12
Fx, u1 x, . . . , un xdx.
0
n
Since pi > 1 for 1 ≤ i ≤ n, X is compactly embedded in C0 0, 1 and it is well known that
Φ and Ψ are well defined and continuously differentiable functionals whose derivatives at
the point u u1 , . . . , un ∈ X are the functionals Φ u, Ψ u ∈ X ∗ , given by
Φ uv 1 n
0 i1
pi −2 u x u xv xdx,
i
i
i
1 n
Ψ uv Fui x, u1 x, . . . , un xvi xdx
2.13
0 i1
for every v v1 , . . . , vn ∈ X, respectively, as well as Ψ is sequentially weakly upper
semicontinuous. Furthermore, Lemma 2.6 of 11 gives that Φ admits a continuous inverse on
X ∗ , and since Φ is monotone, we obtain that Φ is sequentially weakly lower semiacontinuous
see 29, Proposition 25.20. Moreover, Ψ : X → X ∗ is a compact operator. From assumption
A1, we get 0 < r < Φw. Since from 2.1 for each ui ∈ Xi ,
p
sup |ui x|pi ≤ kui i
x∈0,1
2.14
Abstract and Applied Analysis
7
for i 1, . . . , n, we have
sup
n
|ui x|pi
≤k
pi
x∈0,1 i1
p
n u i
i pi
pi
i1
2.15
,
for each u u1 , . . . , un ∈ X, and so using 2.15, we observe
Φ−1 −∞, r {u1 , . . . , un ∈ X; Φu1 , . . . , un ≤ r}
⎫
⎧
p
n u i
⎬
⎨
i pi
≤r
u1 , . . . , un ∈ X;
⎭
⎩
pi
i1
⊆
u1 , . . . , un ∈ X;
n
|ui x|pi
pi
i1
2.16
≤ kr ∀x ∈ 0, 1 ,
and it follows that
sup
u1 ,...,un ∈Φ−1 −∞,r
Ψu1 , . . . , un ≤
1
sup
u1 ,...,un ∈Φ−1 −∞,r
1
sup
0 t1 ,...,tn ∈Kkr
Fx, u1 x, . . . , un xdx
0
2.17
Fx, t1 , . . . , tn dx.
Therefore, owing to assumption A2, we have
sup
u∈Φ−1 −∞,r
Ψu1 , . . . , un ≤
1
sup
u1 ,...,un ∈Φ−1 −∞,r
1
sup
0 t1 ,...,tn ∈Kkr
<
r
n
1
pi
i1
1
<r
r
0
0
Fx, u1 x, . . . , un xdx
0
Fx, t1 , . . . , tn dx
Fx, w1 x, . . . , wn xdx
pi
n n
i1
j1,j / i pj wi pi
2.18
Fx, w1 x, . . . , wn xdx
n w pi /pi
i1
i
Ψw
.
Φw
Furthermore, from A3, there exist two constants γ, υ ∈ R with
1
0<γ <
0
supt1 ,...,tn ∈Kkr Fx, t1 , . . . , tn dx
r
,
2.19
8
Abstract and Applied Analysis
such that
kFx, t1 , . . . , tn ≤ γ
n
|ti |pi
i1
pi
υ,
∀x ∈ 0, 1, ∀t1 , . . . , tn ∈ Rn .
2.20
Fix u1 , . . . , un ∈ X, Then
1
Fx, u1 x, . . . , un x ≤
k
γ
n
|ui x|pi
i1
pi
υ
∀x ∈ 0, 1.
2.21
So, for any fixed λ ∈ Λr , from 2.15 and 2.21, we have
Φu − λΨu p
n u i
i pi
pi
pi
n u i pi
−λ
1
Fx, u1 x, . . . , un xdx
0
i1
λγ
−
k
n
1 1
λυ
|ui x| dx −
p
k
i
0
i1
⎛
pi ⎞
n u λγ ⎝ i pi
⎠ − λυ
k
≥
−
p
k
p
k
i
i
i1
i1
pi
pi
n u n u i pi
i pi
λυ
− λγ
−
p
p
k
i
i
i1
i1
⎛
⎞ pi
n u i pi
r
λυ
⎠
,
≥ ⎝1 − γ 1
−
p
k
i
sup
Fx,
t
,
.
.
.
,
t
dx
i1
1
n
t1 ,...,tn ∈Kkr
0
≥
pi
i1
pi
n u i pi
pi
2.22
and thus,
lim
Φu1 , . . . , un − λΨu1 , . . . , un ∞,
u1 ,...,un → ∞
2.23
which means that the functional Φ − λΨ is coercive. So, all assumptions of Theorem 2.1 are
satisfied. Hence, from Theorem 2.1 with x w, taking into account that the weak solutions of
the system 1.1 are exactly the solutions of the equation Φ u1 , . . . , un − λΨ u1 , . . . , un 0
and using Lemma 2.4, we have the conclusion.
Now we want to present a verifiable consequence of the main result where the test
function w is specified.
Put
pi
pi ⎞⎤1/pi
m
m
1−p
σi ⎣2pi −1 ⎝x1 i 1 − aj 1 − xm 1−pi 1 − bj ⎠⎦
j1 j1 ⎡
⎛
for 1 ≤ i ≤ n.
2.24
Abstract and Applied Analysis
9
Define
⎤n
⎧⎡
m
⎪
⎪
⎪⎣x a , x⎦
⎪
j
⎪
⎪
⎪
⎨ j1
⎤n
B1,n x ⎡
m
⎪
⎪
⎪
⎪⎣x, x aj ⎦
⎪
⎪
⎪
⎩
j1
⎤n
⎧⎡
m
⎪
⎪
⎪
⎪⎣x bj , x⎦
⎪
⎪
⎪
⎨ j1
⎤n
B2,n x ⎡
m
⎪
⎪
⎪
⎪
⎣x, x bj ⎦
⎪
⎪
⎪
⎩
j1
if
m
aj < 1,
j1
if
m
aj > 1,
j1
2.25
m
if
bj < 1,
j1
if
m
bj > 1,
j1
where ·, ·n ·, · × · · · × ·, ·, then we have the following consequence of Theorem 2.5.
Corollary 2.6. Assume that there exist 2m constants aj , bj for 1 ≤ j ≤ m with m
1 and
j1 aj /
n
n
m
pi
1 and two positive constants θ and τ with i1 σi τ /pi > θ/k i1 pi such that
j1 bj /
B1 Fx, t1 , . . . , tn ≥ 0 for each x ∈ 0, x1 /2 ∪ 1 xm /2, 1 and t1 , . . . , tn ∈ B1,n τ ∪
B2,n τ,
1
1x /2
n
B2 i1 σi τpi /pi 0 supt1 ,...,tn ∈Kθ/n pi Fx, t1 , . . . , tn dx < θ/k ni1 pi x1 /2 m
i1
n
Fx, τ, . . . , τdx, where σi is given by 2.24 and Kθ/ i1 pi {t1 , . . . , tn |
n
n
pi
i1 |ti | /pi ≤ c/
i1 pi } (see 2.11);
n
B3 lim sup|t1 | → ∞,...,|tn | → ∞ Fx, t1 , . . . , tn / ni1 |ti |pi /pi <
×
i1 pi /θ
1
supt1 ,...,tn ∈Kθ/ n pi Fx, t1 , . . . , tn dx uniformly with respect to x ∈ 0, 1.
0
i1
1x /2
1
n
pi
then, for each λ ∈ i1 σi τ /pi / x1 /2 m Fx, τ, . . . , τdx,θ/k ni1 pi / 0supt1 ,...,tn ∈Kθ/n pi i1
Fx, t1 , . . . , tn dx the systems 1.1 admits at least three distinct classical solutions.
Proof. Set wx w1 x, . . . , wn x such that for 1 ≤ i ≤ n,
⎞
⎧ ⎛
m
⎪
m
a
2
1
−
j
⎪
j1
⎪
⎟
⎜
⎪
⎪
x⎠
τ ⎝ aj ⎪
⎪
x
⎪
1
j1
⎪
⎪
⎪
⎪
⎪
⎪
⎨τ
wi x ⎪
⎛
⎞
⎪
m
⎪
m
m
⎪
⎪
b
2
1
−
2
−
b
−
x
b
j
⎪
j1
j
m
j
j1
j1
⎪ ⎜
⎟
⎪
⎪
τ⎝
−
x⎠
⎪
⎪
1
−
x
1
−
x
⎪
m
m
⎪
⎪
⎩
and r θ/k
n
i1 pi .
x 1
,
if x ∈ 0,
2
&
%
x1 1 xm
,
,
if x ∈
2
2
%
if x ∈
2.26
&
1 xm
,1 ,
2
It is easy to see that w w1 , . . . , wn ∈ X, and, in particular, one has
pi
w σi τpi ,
i pi
2.27
10
Abstract and Applied Analysis
for 1 ≤ i ≤ n, which, employing the condition
n
i1 σi τ
p
n w i
i pi
pi
i1
pi
/pi > θ/k
n
i1 pi ,
gives
2.28
> r.
Since for 1 ≤ i ≤ n,
τ
m
m
x 1
if
aj < 1,
for each x ∈ 0,
2
j1
aj ≤ wi x ≤ τ
j1
τ ≤ wi x ≤ τ
m
aj
j1
τ
m
m
x 1
if
for each x ∈ 0,
aj > 1,
2
j1
& m
1 xm
, 1 if
bj < 1,
for each x ∈
2
j1
%
bj ≤ wi x ≤ τ
j1
τ ≤ wi x ≤ τ
m
bj
%
for each x ∈
j1
2.29
& m
1 xm
, 1 if
bj > 1,
2
j1
the condition B1 ensures that
x1 /2
Fx, w1 x, . . . , wn xdx 1
0
1xm /2
Fx, w1 x, . . . , wn xdx ≥ 0.
2.30
Therefore, owing to assumption B2, we have
1
sup
0 t1 ,...,tn ∈Kkr
θ
n pi
k i1 pi
σi τ /pi
Fx, t1 , . . . , tn dx < n i1
1
1xm /2
Fx, τ, . . . , τdx
x1 /2
Fx, w1 x, . . . , wn xdx
pi
n n
i1
j1,j / i pj wi pi
n 1
Fx, w1 x, . . . , wn xdx
r
pi 0 n n
,
p
pj w i
i1
θ
≤
k
0
i1
j1,j / i
i pi
2.31
where Kθ/ ni1 pi {t1 , . . . , tn | ni1 |ti |pi /pi ≤ θ/ ni1 pi }. Moreover, from assumption
B3 it follows that assumption A3 is fulfilled. Hence, taking into account that
⎤
n pi
i1 σi τ /pi
⎦
, 1
1xm /2
Fx, τ, . . . , τdx 0
x1 /2
θ/k
n
i1 pi
supt1 ,...,tn ∈Kθ/n
using Theorem 2.5, we have the desired conclusion.
i1
pi Fx, t1 , . . . , tn dx
⎡
⎣ ⊆ Λr ,
2.32
Abstract and Applied Analysis
11
Let us present an application of Corollary 2.6.
Example 2.7. Let F : 0, 1 × R3 → R be the function defined as
Fx, t1 , t2 , t3 ⎧
⎪
0
⎪
⎪
⎪
⎪
⎪
2 100 −t
⎪
⎪ x t1 e 1
⎪
⎪
⎪
⎨x2 t100 e−t2
∀ti < 0, i 1, 2,
for t1 ≥ 0, t2 < 0, t3 < 0,
for t1 < 0, t2 ≥ 0, t3 < 0,
2
⎪
−t3
⎪
x2 t100
⎪
⎪
3 e
⎪
⎪
⎪
3
⎪
⎪
⎪
2
−ti
⎪
t100
⎩x
i e
for t1 < 0, t2 < 0, t3 ≥ 0.
2.33
for ti ≥ 0, i 1, 2, 3,
i1
for each x, t1 , t2 , t3 ∈ 0, 1 × R3 . In fact, by choosing p1 p2 p3 3 and a1 b1 x1 1/2,
by a simple calculation, we obtain that k 27/8 and σ1 σ2 σ3 41/3 , and so with θ 9
and τ 100, we observe that the assumptions B1 and B3 in Corollary 2.6 are satisfied. For
B2,
3
σi τpi 1
i1
sup
0 t1 ,t2 ,t3 ∈K θ/3 pi i1
pi
4100
3
≤ 4100
3
1
Fx, t1 , t2 , t3 dx
Fx, t1 , t2 , t3 dx
sup
0 t1 ,t2 ,t3 ∈K1/3
41003
1
x2
sup
0 t1 ,t2 ,t3 ∈K1/3
max
3
t1 ,t2 ,t3 ∈K1/3
3
−ti
t100
i e dx
i1
−ti
t100
i e
i1
'
(
4
3
100 −t
≤ 100 3maxt e
3
|t|≤1
1
x2 dx
0
2.34
41003 e
13
100100 e−100
4 × 34
1x1 /2
θ
3
Fx, τ, τ, τdx.
k i1 pi x1 /2
<
So, for every
)
λ∈
44 1003
8
*
,
,
26100100 e−100 35 1003 e
2.35
12
Abstract and Applied Analysis
Corollary 2.6 is applicable to the system
99 − u1 u1 λx2 u1 e−u1 100 − u1
in 0, 1,
99
in 0, 1,
− u2 u2 λx2 u2 e−u2 100 − u2
99
in 0, 1,
− u3 u3 λx2 u3 e−u3 100 − u3
' (
1
1
for i 1, 2, 3,
ui 0 ui 1 ui
2
2
2.36
where t max{t, 0}.
Here is a remarkable consequence of Corollary 2.6.
Corollary 2.8. Let F : Rn → R be a C1 function in Rn such that F0, . . . , 0 0. Assume that there
1 and m
1 and two positive constants
exist 2m constants aj , bj for 1 ≤ j ≤ m with m
j1 aj /
j1 bj /
n
θ and τ with i1 σi τpi /pi > θ/k ni1 pi such that
C1 Ft1 , . . . , tn ≥ 0 for each t1 , . . . , tn ∈ B1,n τ ∪ B2,n τ,
C2 ni1 σi τpi /pi maxt1 ,...,tn ∈Kθ/ni1 pi Ft1 , . . . , tn < θ1xm −x1 /2k ni1 pi Fτ, . . . ,
τ where σi is given by 2.24,
C3 lim sup|t1 | → ∞,..., |tn | → ∞ Ft1 , . . . , tn / ni1 |ti |pi /pi <
ni1 pi /θ
maxt1 ,...,tn ∈Kθ/ni1 pi Ft1 , . . . , tn ,
Then, for each λ
∈ ni1 σi τpi /pi /1 xm − x1 /2Fτ, . . . , τ, θ/k ni1 pi /
maxt1 ,...,tn ∈Kθ/ni1 pi Ft1 , . . . , tn , the systems
p −2
− ui i ui λFui u1 , . . . , un ui 0 m
aj ui xj ,
ui 1 j1
in 0, 1,
m
bj ui xj ,
2.37
j1
for 1 ≤ i ≤ n, admits at least three distinct classical solutions.
Proof. Set Fx, t1 , . . . , tn Ft1 , . . . , tn for all x ∈ 0, 1 and ti ∈ R for 1 ≤ i ≤ n. From
the hypotheses, we see that all assumptions of Corollary 2.6 are satisfied. So, we have the
conclusion by using Corollary 2.6.
Example 2.9. Let p1 p2 3, m 2, x1 1/3, x2 2/3 and ai bi 1/3, i 1, 2. Consider
the system
in 0, 1,
− u1 u1 λ e−u1 u11
1 12 − u1 ,
− u2 u2 λ e−u2 u92 10 − u2 , in 0, 1,
' (
' (
' (
' (
1
1
1
2
1
2
1
1
u1 0 u1 1 u1
u1
, u2 0 u2 1 u2
u2
.
3
3
3
3
3
3
3
3
2.38
Abstract and Applied Analysis
13
Clearly, H1 and H2 hold. A simple calculation shows that k 125/8 and σ1 σ2 8/31/3 . So, by choosing θ 3 and τ 10, we observe that the assumptions C1 and C3 in
−t1
−t2
t10
for every t1 , t2 ∈ R2 , we have
Corollary 2.8 hold. For C3, since Ft1 , t2 t12
2 e
1 e
max
t1 ,t2 ∈Kθ/
2
i1 pi Ft1 , t2 max
Ft1 , t2 t1 ,t2 ∈K1/3
−t1
−t2
t12
t10
1 e
2 e
t1 ,t2 ∈K1/3
&
%
−t1
10 −t2
e
max
t
e
≤ max t12
1
2
max
|t1 |≤1
|t2 |≤1
2e
2.39
1
1012 e−10 1010 e−10
3
125 · 10
θ1 x2 − x1 2 Fτ, τ.
2
pi
2k i1 pi
i1 τσi /pi
<
Note that lim|t1 | → ∞,|t2 | → ∞ Ft1 , t2 /
2
i1 |ti |
pi
/pi 0. We see that for every
*
8
4
,
λ ∈ 9 −10
,
3 10 e 107 e−10 375e
)
2.40
Corollary 2.8 is applicable to the system 2.38.
Finally, we prove the theorem in the introduction.
Proof of Theorem 1.1. Since f and g are positive, then F is nonnegative in R2 . Fix λ >
λ∗ : σ1 τp /p σ2 τq /q/1 x2 − x1 /2Fτ, τ for some τ > 0. Note that
lim infξ,η → 0,0 Fξ, η/|ξ|p /p |η|q /q 0, and there is {θn }n∈N ⊆0, ∞such that
limn → ∞ θn 0 and
maxξ,η∈Kθn /pq F ξ, η
lim
0.
n → ∞
θn
2.41
In fact, one has limn → ∞ maxξ,η∈Kθn /pq Fξ, η/θn limn → ∞ Fξθn , ηθn /|ξθn |p /p |ηθn |q /q · |ξθn |p /p |ηθn |q /q/θn 0, where Fξθn , ηθn supξ,η∈Kθn /pq Fξ, η. Hence, there
is θ > 0 such that
maxξ,η∈Kθ/pq F ξ, η
θ
1
1 x2 − x1 < min ,
Fτ, τ;
λpqk
2 qσ1 τp pσ2 τq k
and θ < kqσ1 τp pσ2 τq .
from Corollary 2.8, with n 2 follows the conclusion.
2.42
14
Abstract and Applied Analysis
Acknowledgment
The author would like to thank Professor Juan J. Nieto for valuable suggestions and reading
this paper carefully.
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