Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 27, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
NON-TRIVIAL SOLUTIONS FOR TWO-POINT
BOUNDARY-VALUE PROBLEMS OF FOURTH-ORDER
STURM-LIOUVILLE TYPE EQUATIONS
SHAPOUR HEIDARKHANI
Abstract. Using critical point theory due to Bonanno [3], we prove the existence of at least one non-trivial solution for a class of two-point boundary-value
problems for fourth-order Sturm-Liouville type equations.
1. Introduction
In this note, we prove the existence of at least one non-trivial solution for the
two-point boundary-value problem of fourth-order Sturm-Liouville type:
(pi (x)u00i (x))00 − (qi (x)u0i (x))0 + ri (x)ui (x) = λFui (x, u1 , . . . , un ) x ∈ (0, 1),
ui (0) = ui (1) = u00i (0) = u00i (1) = 0
(1.1)
for 1 ≤ i ≤ n, where n ≥ 1 is an integer, pi , qi , ri ∈ L∞ ([0, 1]) with p−
i :=
ess inf x∈[0,1] pi (x) > 0 for 1 ≤ i ≤ n, λ is a positive parameter, F : [0, 1] × Rn → R
is a function such that F (., t1 , . . . , tn ) is measurable in [0, 1] for all (t1 , . . . , tn ) ∈ Rn ,
F (x, ., . . . , .) is C 1 in Rn for every x ∈ [0, 1] and for every % > 0,
sup
n
X
|Fti (x, t1 , . . . , tn )| ∈ L1 ([0, 1]),
|(t1 ,...,tn )|≤% i=1
and Fui denotes the partial derivative of F with respect to ui for 1 ≤ i ≤ n.
Due to importance of fourth-order two-point boundary-value problems in describing a large class of elastic deflection, many authors have studied the existence and multiplicity of solutions for such a problem; we refer the reader to
[1, 2, 4, 5, 6, 7, 11, 14, 17] and references therein.
In [4], the authors, employing a three critical point theorem due to Bonanno and
Marano [8, Theorem 2.6], determined an exact open interval of the parameter λ for
which system (1) in the case n = 1, admits at least three distinct weak solutions.
The aim of this article is to prove the existence of at least one non-trivial weak
solution for (1) for appropriate values of the parameter λ belonging to a precise
real interval, which extend the results in [7]. Our motivation comes from the recent
2000 Mathematics Subject Classification. 35J35, 47J10, 58E05.
Key words and phrases. Fourth-order Sturm-Liouville type problem; multiplicity results;
critical point theory.
c
2012
Texas State University - San Marcos.
Submitted September 12, 2011. Published February 15, 2012.
1
2
S. HEIDARKHANI
EJDE-2012/27
paper [5]. For basic notation and definitions, and also for a thorough account on
the subject, we refer the reader to [9, 12, 13].
2. Preliminaries and basic notation
First we recall for the reader’s convenience [16, Theorem 2.5] as given in [3,
Theorem 5.1] (see also [3, Proposition 2.1] for related results) which is our main
tool to transfer the question of existence of at least one weak solution of (1) to the
existence of a critical point of the Euler functional:
For a given non-empty set X, and two functionals Φ, Ψ : X → R, we define the
following two functions:
supu∈Φ−1 (]r1 ,r2 [) Ψ(u) − Ψ(v)
,
r2 − Φ(v)
v∈Φ−1 (]r1 ,r2 [)
Ψ(v) − supu∈Φ−1 (]−∞,r1 [) Ψ(u)
ρ(r1 , r2 ) =
sup
Φ(v) − r1
−1
v∈Φ (]r1 ,r2 [)
β(r1 , r2 ) =
inf
for all r1 , r2 ∈ R, r1 < r2 .
Theorem 2.1 ([3, Theorem 5.1]). Let X be a reflexive real Banach space, Φ : X →
R be a sequentially weakly lower semicontinuous, coercive and continuously Gâteaux
differentiable functional whose Gâteaux derivative admits a continuous inverse on
X ∗ and Ψ : X → R be a continuously Gâteaux differentiable functional whose
Gâteaux derivative is compact. Put Iλ = Φ − λΨ and assume that there are r1 , r2 ∈
R, r1 < r2 , such that
β(r1 , r2 ) < ρ(r1 , r2 ).
Then, for each λ ∈] ρ(r11,r2 ) , β(r11,r2 ) [ there is u0,λ ∈ Φ−1 (]r1 , r2 [) such that Iλ (u0,λ ) ≤
Iλ (u) ∀u ∈ Φ−1 (]r1 , r2 [) and Iλ0 (u0,λ ) = 0.
Let us introduce notation that will be used later. Assume that
q− r− q−
r− min i2 , i4 , i2 + i4 > −p−
i ,
π π π
π
where
p−
i := ess inf x∈[0,1] pi (x) > 0,
qi− := ess inf x∈[0,1] qi (x),
(2.1)
ri− := ess inf x∈[0,1] ri (x),
for 1 ≤ i ≤ n. Moreover, set
q
qi− ri− qi−
ri− p−
,
,
+
,
0
,
δ
:=
i
i + σi ,
π2 π4 π2
π4
for 1 ≤ i ≤ n. Let Y := H 2 ([0, 1]) ∩ H01 ([0, 1]) be the Sobolev space endowed
with the usual norm. We recall the following Poincaré type inequalities (see, for
instance, [15, Lemma 2.3]):
σi := min
1 00 2
ku k 2
,
(2.2)
π 2 i L ([0,1])
1
kui k2L2 ([0,1]) ≤ 4 ku00i k2L2 ([0,1])
(2.3)
π
for all ui ∈ Y for 1 ≤ i ≤ n. Therefore, taking into account (2.1)-(2.3), the norm
Z 1
1/2
kui k =
(pi (x)|u00i (x)|2 + qi (x)|u0i (x)|2 + ri (x)|ui (x)|2 )dx
ku0i k2L2 ([0,1]) ≤
0
EJDE-2012/27
NON-TRIVIAL SOLUTIONS
3
for 1 ≤ i ≤ n is equivalent to the usual norm and, in particular, one has
1
ku00i kL2 ([0,1]) ≤ kui k
(2.4)
δi
for 1 ≤ i ≤ n. We need the following proposition in the proof of Theorem 3.1.
Proposition 2.2 ([4, Proposition 2.1]). Let ui ∈ Y for 1 ≤ i ≤ n. Then
kui k∞ ≤
1
kui k
2πδi
for 1 ≤ i ≤ n.
Put
1/2
1
1
ki := kpi k∞ + 2 kqi k∞ + 4 kri k∞
π
π
for 1 ≤ i ≤ n. It is easy to see that ki > 0 and δi < ki for 1 ≤ i ≤ n. Set
δ := min{δi ; 1 ≤ i ≤ n} and k := max{ki ; 1 ≤ i ≤ n}. Here and in the sequel,
X := Y × · · · × Y .
We say that u = (u1 , . . . , un ) is a weak solution to the (1) if u = (u1 , . . . , un ) ∈ X
and
n Z 1
X
(pi (x)u00i (x)vi00 (x) + qi (x)u0i (x)vi0 (x) + ri (x)ui (x)vi (x)) dx
i=1
−λ
0
n Z
X
i=1
1
Fui (x, u1 , . . . , un )vi (x)dx = 0
0
for every v = (v1 , . . . , vn ) ∈ X. For γ > 0 we denote the set
n
X
K(γ) = (t1 , . . . , tn ) ∈ Rn :
|ti | ≤ γ .
(2.5)
i=1
3. Results
2
For a given non-negative constant ν and a positive constant τ with 2( δπν
n ) 6=
put
R1
R5
sup(t1 ,...,tn )∈K(ν) F (x, t1 , . . . , tn )dx − 38 F (x, τ, . . . , τ )dx
0
8
aτ (ν) :=
2 − 4096 n(kτ )2
2( δπν
)
n
54
Pn
n
where K(ν) = (t1 , . . . , tn ) ∈ R : i=1 |ti | ≤ ν (see (2.5)).
We formulate our main result as follows:
4096
2
54 n(kτ ) ,
Theorem 3.1. Assume that there
constant ν1 and two positive
q
qexist a non-negative
constants ν2 and τ with πν1 < n
4096
108 nτ
and n
4096
108 nkτ
< δπν2 such that
(A1) F (x, t1 , . . . , tn ) ≥ 0 for each (x, t1 , . . . , tn ) ∈ ([0, 3/8] ∪ [5/8, 1]) × [0, τ ]n ;
(A2) aτ (ν2 ) < aτ (ν1 ).
1
1
, aτ (ν
[, system (1) admits at least one non-trivial weak
Then, for each λ ∈] aτ (ν
1)
2)
solution u0 = (u01 , . . . , u0n ) ∈ X such that
n
4(
δπν1 2 X
δπν2 2
) <
ku0i k2 < 4(
) .
n
n
i=1
4
S. HEIDARKHANI
EJDE-2012/27
Proof. To apply Theorem 2.1 to our problem, arguing as in [7, 10], we introduce
the functionals Φ, Ψ : X → R for each u = (u1 , . . . , un ) ∈ X, as follows
Φ(u) =
n
X
kui k2
i=1
Z
Ψ(u) =
2
,
1
F (x, u1 (x), . . . , un (x))dx.
0
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
Φ0 (u), Ψ0 (u) ∈ X ∗ , given by
n Z 1
X
0
Φ (u)(v) =
(pi (x)u00i (x)vi00 (x) + qi (x)u0i (x)vi0 (x) + ri (x)ui (x)vi (x)) dx,
i=1
0
Ψ0 (u)(v) =
Z
0
n
1X
Fui (x, u1 (x), . . . , un (x))vi (x)dx
i=1
for every v = (v1 , . . . , vn ) ∈ X, respectively. Moreover, Φ is sequentially weakly
lower semicontinuous, Φ0 admits a continuous inverse on X ∗ as well as Ψ is sequentially weakly upper semicontinuous. Furthermore, Ψ0 : X → X ∗ is a compact
operator. Indeed, it is enough to show that Ψ0 is strongly continuous on X. For
this, for fixed (u1 , . . . , un ) ∈ X let (u1m , . . . , unm ) → (u1 , . . . , un ) weakly in X as
m → +∞, then we have (u1m , . . . , unm ) converges uniformly to (u1 , . . . , un ) on
[0, 1] as m → +∞(see [18]). Since F (x, ., . . . , .) is C 1 in Rn for every x ∈ [0, 1], the
derivatives of F are continuous in Rn for every x ∈ [0, 1], so for 1 ≤ i ≤ n,
Fui (x, u1m , . . . , unm ) → Fui (x, u1 , . . . , un ) strongly as m → +∞ which follows
Ψ0 (u1m , . . . , unm ) → Ψ0 (u1 , . . . , un ) strongly as m → +∞. Thus we proved that
Ψ0 is strongly continuous on X, which implies that Ψ0 is a compact operator by
Proposition 26.2 of [18]. Set w(x) = (w1 (x), . . . , wn (x)) such that for 1 ≤ i ≤ n,

3
64τ
2

x ∈ [0, 38 [,
− 9 (x − 4 x)
wi (x) = τ
x ∈ [ 38 , 85 ],

 64τ 2 5
1
− 9 (x − 4 x + 4 ) x ∈] 85 , 1],
δπν2 2
1 2
r1 = 2( δπν
n ) and r2 = 2( n ) . It is easy to verify that w = (w1 , . . . , wn ) ∈ X,
and in particular, one has
4096 2 2
4096 2 2
δ τ ≤ kwi k2 ≤
k τ
27 i
27 i
for 1 ≤ i ≤ n. So, from the definition of Φ, we have
n
n
X
X
4096
4096 2 2
4096 2 2
4096
n(δτ )2 ≤
δi τ ≤ Φ(w) ≤
ki τ ≤
n(kτ )2 .
54
27
27
54
i=1
i=1
q
q
4096
From the conditions πν1 < n 4096
108 nτ and n
108 nkτ < δπν2 , we obtain
r1 < Φ(w) < r2 .
Moreover, from Proposition 2.2 one has
n
X
sup
|ui (x)|2 ≤
x∈[0,1] i=1
n
1 X
kui k2
(2πδ)2 i=1
EJDE-2012/27
NON-TRIVIAL SOLUTIONS
5
for each u = (u1 , . . . , un ) ∈ X, so from the definition of Φ, we observe that
Φ−1 (] − ∞, r2 [) = {(u1 , . . . , un ) ∈ X; Φ(u1 , . . . , un ) < r2 }
n
X
= (u1 , . . . , un ) ∈ X;
kui k2 < 2r2
i=1
⊆ (u1 , . . . , un ) ∈ X;
n
X
|ui (x)|2 <
i=1
⊆ (u1 , . . . , un ) ∈ X;
n
X
ν22
for all x ∈ [0, 1]
2
n
|ui (x)| < ν2 for all x ∈ [0, 1] ,
i=1
from which it follows
Z
sup
Ψ(u) =
(u1 ,...,un )∈Φ−1 (]−∞,r2 [)
(u1 ,...,un )∈Φ−1 (]−∞,r2 [)
Z
≤
1
F (x, u1 (x), . . . , un (x))dx
sup
0
1
sup
F (x, t1 , . . . , tn )dx.
0 (t1 ,...,tn )∈K(ν2 )
Since for 1 ≤ i ≤ n, 0 ≤ wi (x) ≤ τ for each x ∈ [0, 1], the condition (A1) ensures
that
Z 38
Z 1
F (x, w1 (x), . . . , wn (x))dx ≥ 0.
F (x, w1 (x), . . . , wn (x))dx +
5
8
0
So, one has
supu∈Φ−1 (]−∞,r2 [) Ψ(u) − Ψ(w)
r2 − Φ(w)
R1
sup(t1 ,...,tn )∈K(ν2 ) F (x, t1 , . . . , tn )dx − Ψ(w)
≤ 0
r2 − Φ(w)
≤ aτ (ν2 ).
β(r1 , r2 ) ≤
On the other hand, by similar reasoning as before, one has
Ψ(w) − supu∈Φ−1 (]−∞,r1 [) Ψ(u)
Φ(w) − r1
R1
Ψ(w) − 0 sup(t1 ,...,tn )∈K(ν1 ) F (x, t1 , . . . , tn )dx
≥
Φ(w) − r1
≥ aτ (ν1 ).
ρ(r1 , r2 ) ≥
Hence, from Assumption (A2), one has β(r1 , r2 ) < ρ(r1 , r2 ). Therefore, from Theorem 2.1, taking into account that the weak solutions of the system (1) are exactly
the solutions of the equation Φ0 (u) − λΨ0 (u) = 0, we have the conclusion.
Now we point out the following consequence of Theorem 3.1.
Theorem 3.2. Assume that Assumption (A1) inqTheorem 3.1 holds. Suppose that
there exist two positive constants ν and τ with n 4096
108 nkτ < δπν such that
6
S. HEIDARKHANI
EJDE-2012/27
(A3)
R1
R 58
sup(t1 ,...,tn )∈K(ν) F (x, t1 , . . . , tn )dx
108 δπ 2 38 F (x, τ, . . . , τ )dx
<
( )
;
ν2
4096n3 k
τ2
(A4) F (x, 0, . . . , 0) = 0 for every x ∈ [0, 1]
Then, for each
4096
h
i
2
n(kτ )2
2( δπν
n )
, R1
,
λ ∈ R 5 54
8
sup
F
(x,
t
,
.
.
.
,
t
)dx
1
n
(t
,...,t
)∈K(ν)
3 F (x, τ, . . . , τ )dx
1
n
0
0
8
system (1)Padmits at least one non-trivial weak solution u0 = (u01 , . . . , u0n ) ∈ X
n
such that i=1 kui k∞ < ν.
Proof. The conclusion follows from Theorem 3.1, by taking ν1 = 0 and ν2 = ν.
Indeed, owing to our assumptions, one has
4096n3 τ k 2 R
( )
1
1 − 108 ν 2 δπ
sup(t1 ,...,tn )∈K(ν) F (x, t1 , . . . , tn )dx
0
aτ (ν2 ) <
4096
2
2
2( δπν
n ) − 54 n(kτ )
2
4096
R1
n(τ k)
1 − 54 δπν 2
sup(t1 ,...,tn )∈K(ν) F (x, t1 , . . . , tn )dx
0
2( n )
=
δπν 2
2
2( n ) − 4096
54 n(kτ )
R1
sup(t1 ,...,tn )∈K(ν) F (x, t1 , . . . , tn )dx
= 0
.
2
2( δπν
n )
On the other hand, taking Assumption (A4) into account, one has
R 58
3 F (x, τ, . . . , τ )dx
8
= aτ (ν1 ).
4096
2
54 n(kτ )
Moreover, since
n
X
n
1 X
sup
|ui (x)| ≤
kui k2
(2πδ)2 i=1
x∈[0,1] i=1
2
Pn
for each u = (u1 , . . . , un ) ∈ X, an easy computation ensures that i=1 kui k∞ < ν
whenever Φ(u) < r2 . Now, owing to Assumption (A3), it is sufficient to invoke
Theorem 3.1 for concluding the proof.
Now, we point out a simple version of Theorem 3.1 in the case n = 1. Let
p1 = p, q1 = q, r1 = r, δ1 = δ and k1 = k. Let f : [0, 1] × R → R be an L2 Rt
Carathéodory function. Let F be the function defined by F (x, t) = 0 f (x, s)ds for
each (x, t) ∈ [0, 1] × R. For a given non-negative constant ν and a positive constant
2
τ with 2(δπν)2 6= 4096
54 (kτ ) , put
R1
R5
sup|t|≤ν F (x, t)dx − 38 F (x, τ )dx
0
8
bτ (ν) :=
.
2(δπν)2 − 4096
(kτ
)2
54
Then, we have the following result.
Theorem 3.3. Assume that there
constant ν1 and two positive
q exist a non-negative
q
constants ν2 and τ with πν1 <
4096
108 τ
and
4096
108 kτ
< δπν2 such that
EJDE-2012/27
NON-TRIVIAL SOLUTIONS
7
(B1) F (x, t) ≥ 0 for each (x, t) ∈ ([0, 3/8] ∪ [5/8, 1]) × [0, τ ];;
(B2) bτ (ν2 ) < bτ (ν1 ).
1
1
Then, for each λ ∈] bτ (ν
, bτ (ν
[, the problem
1)
2)
(p(x)u00 (x))00 − (q(x)u0 (x))0 + r(x)u(x) = λf (x, u)
00
x ∈ (0, 1),
00
u(0) = u(1) = u (0) = u (1) = 0
(3.1)
admits at least one non-trivial weak solution u0 ∈ Y such that 2δπν1 < ku0 k <
2δπν2 .
We remark that if p(x) = 1, q(x) = −A and r(x) = B for every x ∈ [0, 1], then
Theorem 3.3 gives [7, Theorem 3.1].
The following result gives the existence of at least one non-trivial weak solution
in Y to the problem (3.1) in the autonomous case. Let f : R → R be a continuous
Rt
function. Put F (t) = 0 f (ξ)dξ for all t ∈ R. We have the following result as a
direct consequence of Theorem 3.3.
Theorem 3.4. Assume that there
constant ν1 and two positive
q
q exist a non-negative
constants ν2 and τ with πν1 <
4096
108 τ
and
4096
108 kτ
< δπν2 such that
(C1) f (t) ≥ 0 for each t ∈ [−ν2 , max{ν2 , τ }];
F (ν )− 41 F (τ )
F (ν )− 14 F (τ )
(C2) 2(δπν )22 − 4096
< 2(δπν )12 − 4096
.
(kτ )2
(kτ )2
2
Then, for each λ
54
1
54
2
2(δπν2 )2 − 4096
2(δπν )2 − 4096 (kτ )2
54 (kτ )
[,
∈] F (ν1 )− 154F (τ ) ,
1
F
(τ
)
F
(ν
)−
1
2
4
4
the problem
(p(x)u00 (x))00 − (q(x)u0 (x))0 + r(x)u(x) = λf (u)
00
x ∈ (0, 1),
00
u(0) = u(1) = u (0) = u (1) = 0
(3.2)
0k
admits at least one non-trivial weak solution u0 ∈ Y such that ν1 < ku
2δπ < ν2 .
q
q
4096
4096
Proof. Since δ < k, from the conditions πν1 <
108 τ and
108 kτ < δπν2 ,
4096
2
2
2
2
we obtain 2(δπν1 )2 < 4096
54 (δτ ) and 54 (kτ ) < 2(δπν2 ) , and so 2(δπν1 ) <
2
2(δπν2 ) , and so ν1 < ν2 . Therefore, Assumptions (C1) means f (t) ≥ 0 for each t ∈
[−ν1 , ν1 ] and f (t) ≥ 0 for each t ∈ [−ν2 , ν2 ], which follow maxt ∈ [−ν1 , ν1 ]F (t) =
F (ν1 ) and maxt ∈ [−ν2 , ν2 ]F (t) = F (ν2 ). So, from Assumptions (C1) and (C2)
we arrive at assumptions (B1) and (B2), respectively. Hence, we achieve the stated
assertion by applying Theorem 3.3 observing that the problem (1) reduces to the
problem (3.2).
As an example, we point out the following special case of our main result.
Theorem 3.5. Let g : R → R be a nonnegative continuous function such that
2
2
R ν
limt→0+ g(t)
t = +∞. Then, for each λ ∈]0, 2(δπ) supν>0 ν g(ξ)dξ [, the problem
0
(p(x)u00 (x))00 − (q(x)u0 (x))0 + r(x)u(x) = λg(u)
00
x ∈ (0, 1),
00
u(0) = u(1) = u (0) = u (1) = 0
admits at least one non-trivial weak solution in Y .
Proof. For fixed λ as in the conclusion, there exists positive constant ν such that
λ < 2(δπ)2 R ν
0
ν2
.
g(ξ)dξ
8
S. HEIDARKHANI
EJDE-2012/27
Rt
g(t)
t
g(ξ)dξ
= +∞ implies limt→0+ 0 t2
= +∞. Therefore, a posiq
tive constant τ satisfying 4096
108 kτ < δπν can be chosen such that
Moreover, limt→0+
1 4 × 4096 2
(
k )<
λ
54
Rτ
0
g(ξ)dξ
.
τ2
Hence, arguing as in the proof of Theorem 3.2, the conclusion follows from Theorem
3.4 with ν1 = 0, ν2 = ν and f (t) = g(t) for every t ∈ R.
2
ν
Remark 3.6. For fixed ρ put λρ := 2(δπ)2 supν∈]0,ρ[ R ν g(ξ)dξ
. The result of Theo0
rem 3.5 for every λ ∈]0, λρ [ holds with |u0 (x)| < ρ for all x ∈ [0, 1] where u0 is the
ensured non-trivial weak solution in Y (see [7, Remark 4.3]).
We close this article by presenting the following examples to illustrate our results.
example 3.7. Consider the problem
+
(3ex u00 )00 − ((x2 − π 2 )u0 )0 + (x2 − π 4 )u = λ(1 + e−u (u+ )2 (3 − u+ )) x ∈ (0, 1),
u(0) = u(1) = 0,
u00 (0) = u00 (1) = 0
(3.3)
where u+ = max{u, 0}. Let
+
g(t) = 1 + e−t (t+ )2 (3 − t+ )
for all t ∈ R where t+ = max{t, 0}. It is clear that limt→0+ g(t)
t = +∞. Note that
p− = 3, q − = −π 2 and r− = −π 4 , we have σ = −2, and so δ = 1. Hence, taking
2
e
ν2
, by
Remark 3.6 into account, since supν∈]0,1[ R ν g(ξ)dξ
= supν∈]0,1[ ν+eν−ν ν 3 = 1+e
0
2
e
applying Theorem 3.5, for every λ ∈]0, 2π
1+e [ the problem (3.3) has at least one
non-trivial classical solution u0 ∈ Y such that ku0 k∞ < 1.
example 3.8. Put p(x) = 1, q(x) = π 2 , r(x) = x − π for all x ∈ [0, 1] and
1
g(t) = (1 + t)et for every t ∈ R. Clearly, one has σ = (1 − π13 ) 2 . Hence, since
sup R ν
ν∈]0,1[
0
ν2
ν2
1
= sup
= ,
e
g(ξ)dξ ν∈]0,1[ νeν
from Theorem 3.5, taking Remark 3.6 into account, for every λ ∈]0,
problem
uiv − π 2 u00 + (x − π)u = λ(1 + u)eu
u(0) = u(1) = 0,
00
2π 2 (1− π13 )
[
e
x ∈ (0, 1),
00
u (0) = u (1) = 0
the
(3.4)
has at least one non-trivial classical solution u0 ∈ Y such that ku0 k∞ < 1.
Acknowledgements. The author expresses his sincere gratitude to the referees for
reading this paper very carefully and specially for valuable suggestions concerning
improvement of the manuscript. This research was partially supported by a grant
90470020 from IPM.
EJDE-2012/27
NON-TRIVIAL SOLUTIONS
9
References
[1] G. A. Afrouzi, S. Heidarkhani, D. O’Regan; Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem, Taiwanese J. Math. 15, No. 1 (2011) 201-210.
[2] Z. Bai; Positive solutions of some nonlocal fourth-order boundary value problem, Appl. Math.
Comput. 215 (2010) 4191-4197.
[3] G. Bonanno; A critical point theorem via the Ekeland variational principle, preprint.
[4] G. Bonanno, B. Di Bella; A boundary value problem for fourth-order elastic beam equations,
J. Math. Anal. Appl. 343 (2008) 1166-1176.
[5] G. Bonanno, B. Di Bella; A fourth-order boundary value problem for a Sturm-Liouville type
equation, Appl. Math. Comput. 217 (2010) 3635-3640.
[6] G. Bonanno, B. Di Bella; Infinitely many solutions for a fourth-order elastic beam equation,
Nonlinear Differential Equations and Applications NoDEA, DOI 10.1007/s00030-011-0099-0.
[7] G. Bonanno, B. Di Bella, D. O’Regan; Non-trivial solutions for nonlinear fourth-order
elastic beam equations, Computers and Mathematics with Applications, doi: 10.1016/
j.camwa.2011.06.029.
[8] G. Bonanno, S. A. Marano; On the structure of the critical set of non-differentiable functionals with a weak compactness condition, Appl. Anal. 89 (2010) 1-10.
[9] G. Bonanno, G. Molica Bisci, D. O’Regan; Infinitely many weak solutions for a class of
quasilinear elliptic systems, Math. Comput. Modelling 52 (2010) 152-160.
[10] G. Bonanno, P. F. Pizzimenti; Neumann boundary value problems with not coercive potential,
Mediterr. J. Math., DOI 10.1007/s00009-011-0136-6.
[11] G. Chai; Existence of positive solutions for fourth-order boundary value problem with variable
parameters, Nonlinear Anal. 66 (2007) 870-880.
[12] M. Ghergu, V. Rădulescu; Singular Elliptic Problems. Bifurcation and Asymptotic Analysis,
Oxford Lecture Series in Mathematics and Its Applications, vol. 37, Oxford University Press,
2008.
[13] A. Kristály, V. Rădulescu, C. Varga; Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems,
Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press,
Cambridge, 2010.
[14] Y. Li; On the existence of positive solutions for the bending elastic beam equations, Appl.
Math. Comput. 189 (2007) 821-827.
[15] L. A. Peletier, W. C. Troy, R. K. A. M. Van der Vorst; Stationary solutions of a fourth order
nonlinear diffusion equation, (Russian) Translated from the English by V. V. Kurt. Differentsialnye Uravneniya, 31 no. 2, (1995) 327-337. English translation in Differential Equations,
31, no. 2, (1995) 301-314.
[16] B. Ricceri; A general variational principle and some of its applications, J. Comput. Appl.
Math. 113 (2000) 401-410.
[17] Y.-H. Wang, S. Heidarkhani, L. Li; Multiple solutions for a fourth-order Sturm–Liouville
type boundary value problem, preprint.
[18] E. Zeidler; Nonlinear functional analysis and its applications, Vol. II, III. Berlin-HeidelbergNew York 1985.
Shapour Heidarkhani
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.
Box: 19395-5746, Tehran, Iran
E-mail address: sh.heidarkhani@yahoo.com, s.heidarkhani@razi.ac.ir