On the existence of solutions of one-dimensional fourth-order equations
UDC 517.9 Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. Multiplicity results are also pointed out.
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| author | Shokooh, S. Afrouzi , G. A. Hadjian, A. Shokooh, S. Afrouzi , G. A. Hadjian, A. |
| author_facet | Shokooh, S. Afrouzi , G. A. Hadjian, A. Shokooh, S. Afrouzi , G. A. Hadjian, A. |
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| description | UDC 517.9
Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. Multiplicity results are also pointed out. |
| doi_str_mv | 10.37863/umzh.v72i11.569 |
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DOI: 10.37863/umzh.v72i11.569
UDC 517.9
S. Shokooh (Gonbad Kavous Univ., Iran),
G. A. Afrouzi (Univ. Mazandaran, Babolsar, Iran),
A. Hadjian (Univ. Bojnord, Iran)
ON THE EXISTENCE OF SOLUTIONS
OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS
ПРО IСНУВАННЯ РОЗВ’ЯЗКIВ ОДНОВИМIРНИХ РIВНЯНЬ
ЧЕТВЕРТОГО ПОРЯДКУ
Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional
fourth-order equations. Multiplicity results are also pointed out.
За допомогою варiацiйних методiв та теореми про критичнi точки доведено iснування нетривiальних розв’язкiв
одновимiрних рiвнянь четвертого порядку. Також наведено вiдповiднi результати щодо кратностi.
1. Introduction. In this paper, we consider the following fourth-order boundary-value problem:
uivh(x, u\prime ) - u\prime \prime = [\lambda f(x, u) + g(u)]h(x, u\prime ) \mathrm{i}\mathrm{n} ] 0, 1 [,
u(0) = u(1) = 0 = u\prime \prime (0) = u\prime \prime (1),
(1.1)
where \lambda is a positive parameter, f : [0, 1]\times \BbbR \rightarrow \BbbR is an L1-Carathéodory function, g : \BbbR \rightarrow \BbbR is a
Lipschitz continuous function with the Lipschitz constant L > 0, i.e.,
| g(t1) - g(t2)| \leq L| t1 - t2|
for every t1, t2 \in \BbbR , with g(0) = 0, and h : [0, 1] \times \BbbR \rightarrow [0,+\infty [ is a bounded and continuous
function with m := \mathrm{i}\mathrm{n}\mathrm{f}(x,t)\in [0,1]\times \BbbR h(x, t) > 0.
Boundary-value problems for ordinary differential equations play a fundamental role both in the-
ory and applications. To establish the existence and multiplicity of solutions to nonlinear differential
problems is very important as well as the application of such results in the physical reality. In fact, it
is well-known that the mathematical modelling of important questions in different fields of research,
such as mechanical engineering, control systems, economics, computer science and many others,
leads naturally to the consideration of nonlinear differential equations. In particular, the deformations
of an elastic beam in an equilibrium state, whose two ends are simply supported, can be described by
fourth-order boundary-value problems. The work of Timoshenko [21] on elasticity, the monograph
by Soedel [19], the paper by Palamides [12] on deformation of elastic membrane, and the work of
Dulàcska [10] on the effects of soil settlement are rich sources of such applications. Pietramala [14]
presented some results on the existence of multiple positive solutions of a fourth-order differential
equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary
state of an elastic beam with nonlinear controllers.
For this reason, the existence and multiplicity of solutions for this kind of problems have been
widely investigated (see, for instance, [1, 4, 6 – 9, 18, 20] and references therein).
c\bigcirc S. SHOKOOH, G. A. AFROUZI, A. HADJIAN, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11 1575
1576 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
In the above papers, the right-hand side of equation is independent of u\prime . The main novelty of
our paper is the fact that we apply a recent critical-points result to a fourth-order equation given in
the form of (1.1), in which the right-hand side is dependent on u\prime . Such equations are also studied
by many authors, but most of them are of the second-order (see, for instance, [3, 11, 22]).
In the present paper, based on a recent critical point theorem of Bonanno (see Theorem 2.1
below), we obtain the existence of at least one solution for problem (1.1). It is worth noticing that,
usually, to obtain the existence of one solution, asymptotic conditions both at zero and at infinity on
the nonlinear term are requested, while, here, it is assumed only a unique algebraic condition (see
(A7) in Corollary 3.2). As a consequence, by combining with the classical Ambrosetti – Rabinowitz
condition (see [2]), the existence of two solutions is obtained (see Theorem 3.5).
As an example, we state here the following special case of our results.
Theorem 1.1. Let f : \BbbR \rightarrow \BbbR be a nonnegative continuous function such that
4
64\int
0
f(x) dx < 27
1\int
0
f(x) dx.
Then, for each
\lambda \in
\right] 21327
(\pi 4 + \pi 2 + 1)
\pi 4
\int 1
0
f(x) dx
,
211(\pi 4 + \pi 2 + 1)
\pi 4
\int 64
0
f(x) dx
\left[ ,
the problem
uiv - u\prime \prime + u = \lambda f(u) \mathrm{i}\mathrm{n} ] 0, 1 [,
u(0) = u(1) = 0 = u\prime \prime (0) = u\prime \prime (1),
admits at least one positive classical solution \=u such that \=u(x) < 64 for all x \in [0, 1].
2. Preliminaries. Our main tool is Ricceri’s variational principle [17] (Theorem 2.5) as given
in [5] (Theorem 5.1) which is below recalled (see also [5], Proposition 2.1).
For a given nonempty set X, and two functionals \Phi , \Psi : X \rightarrow \BbbR , we define the following
functions:
\beta (r1, r2) = \mathrm{i}\mathrm{n}\mathrm{f}
v\in \Phi - 1( ]r1,r2[)
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1( ]r1,r2[)\Psi (u) - \Psi (v)
r2 - \Phi (v)
and
\rho (r1, r2) = \mathrm{s}\mathrm{u}\mathrm{p}
v\in \Phi - 1( ]r1,r2[)
\Psi (v) - \mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1])\Psi (u)
\Phi (v) - r1
for all r1, r2 \in \BbbR with r1 < r2.
Theorem 2.1 ([5], Theorem 5.1). Let X be a reflexive real Banach space; \Phi : X \rightarrow \BbbR be a
sequentially weakly lower semicontinuous, coercive and continuously Gâteaux differentiable func-
tion whose Gâteaux derivative admits a continuous inverse on X\ast ; \Psi : X \rightarrow \BbbR be a continuously
Gâteaux differentiable function whose Gâteaux derivative is compact. Assume that there are r1,
r2 \in \BbbR , with r1 < r2, such that
\beta (r1, r2) < \rho (r1, r2). (2.1)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1577
Then setting I\lambda := \Phi - \lambda \Psi for each \lambda \in
\biggr]
1
\rho (r1, r2)
,
1
\beta (r1, r2)
\biggl[
there is u0,\lambda \in \Phi - 1
\bigl(
]r1, r2[
\bigr)
such
that I\lambda (u0,\lambda ) \leq I\lambda (u) for all u \in \Phi - 1
\bigl(
]r1, r2[
\bigr)
and I \prime \lambda (u0,\lambda ) = 0.
Let us introduce some notation which will be used later. Define
H1
0 ([0, 1]) :=
\bigl\{
u \in L2([0, 1]) : u\prime \in L2([0, 1]), u(0) = u(1) = 0
\bigr\}
,
H2([0, 1]) :=
\bigl\{
u \in L2([0, 1]) : u\prime , u\prime \prime \in L2([0, 1])
\bigr\}
.
Let X := H2([0, 1]) \cap H1
0 ([0, 1]) be the Sobolev space endowed with the usual norm defined as
follows:
\| u\| :=
\left( 1\int
0
| u\prime \prime (t)| 2 dt
\right) 1/2
.
We recall the following Poincaré type inequalities (see, for instance, [13], Lemma 2.3):
\| u\prime \| 2L2([0,1]) \leq
1
\pi 2
\| u\| 2, (2.2)
\| u\| 2L2([0,1]) \leq
1
\pi 4
\| u\| 2 (2.3)
for all u \in X. For the norm in C1([0, 1]),
\| u\| \infty := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
\mathrm{m}\mathrm{a}\mathrm{x}
x\in [0,1]
| u(x)| , \mathrm{m}\mathrm{a}\mathrm{x}
x\in [0,1]
| u\prime (x)|
\biggr\}
,
we have the following relation.
Proposition 2.1. Let u \in X. Then
\| u\| \infty \leq 1
2\pi
\| u\| . (2.4)
Proof. Taking (2.2) into account, the conclusion follows from the known inequality \| u\| \infty \leq
\leq 1
2
\| u\prime \| L2([0,1]).
Let g : \BbbR \rightarrow \BbbR is a Lipschitz continuous function with the Lipschitz constant L > 0, i.e.,
| g(t1) - g(t2)| \leq L| t1 - t2|
for every t1, t2 \in \BbbR , and g(0) = 0, h : [0, 1]\times \BbbR \rightarrow [0,+\infty ] is a bounded and continuous function
with m := \mathrm{i}\mathrm{n}\mathrm{f}(x,t)\in [0,1]\times \BbbR h(x, t) > 0, and f : [0, 1]\times \BbbR \rightarrow \BbbR be an L1-Carathéodory function.
We recall that f : [0, 1]\times \BbbR \rightarrow \BbbR is an L1-Carathéodory function if
(a) the mapping x \mapsto \rightarrow f(x, \xi ) is measurable for every \xi \in \BbbR ;
(b) the mapping \xi \mapsto \rightarrow f(x, \xi ) is continuous for almost every x \in [0, 1];
(c) for every \rho > 0 there exists a function l\rho \in L1([0, 1]) such that
\mathrm{s}\mathrm{u}\mathrm{p}
| \xi | \leq \rho
| f(x, \xi )| \leq l\rho (x)
for almost every x \in [0, 1].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1578 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
Corresponding to f, g and h we introduce the functions F : [0, 1] \times \BbbR \rightarrow \BbbR , G : \BbbR \rightarrow \BbbR and
H : [0, 1]\times \BbbR \rightarrow [0,+\infty ], respectively, as follows:
F (x, t) :=
t\int
0
f(x, \xi ) d\xi , G(t) := -
t\int
0
g(\xi ) d\xi ,
H(x, t) :=
t\int
0
\left( \tau \int
0
1
h(x, \delta )
d\delta
\right) d\tau
for all x \in [0, 1] and t \in \BbbR .
In the following, suppose that the Lipschitz constant L > 0 of the function g satisfies the
condition L < \pi 4.
We say that a function u \in X is a weak solution of problem (1.1) if
1\int
0
u\prime \prime (x)v\prime \prime (x) dx+
1\int
0
\left( u\prime (x)\int
0
1
h(x, \tau )
d\tau
\right) v\prime (x) dx -
- \lambda
1\int
0
f(x, u(x))v(x) dx -
1\int
0
g(u(x))v(x) dx = 0
holds for all v \in X.
By standard regularity results, if f is continuous in [0, 1] \times \BbbR , then weak solutions of prob-
lem (1.1) belong to C2([0, 1]), thus they are classical solutions.
3. Main results. In this section we present our main results. Put
A :=
\pi 4 - L
2\pi 4
, B :=
\pi 2 +m(\pi 4 + L)
2m\pi 4
,
and suppose that B \leq 4\pi 2A. Given a nonnegative constant c1 and two positive constants c2 and d
with c21 <
4096
27
d2 < c22, put
a(c2, d) :=
\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c2 F (x, t) dx -
\int 5/8
3/8
F (x, d) dx
27Bc22 - 4096Bd2
,
b(c1, d) :=
\int 5/8
3/8
F (x, d) dx -
\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c1 F (x, t) dx
4096Bd2 - 27Ac21
.
Theorem 3.1. Assume that there exist a nonnegative constant c1 and two positive constants c2
and d, with c21 <
4096
27
d2 < c22, such that
(A1) F (x, t) \geq 0 for all (x, t) \in
\biggl( \biggl[
0,
3
8
\biggr] \bigcup \biggl[ 5
8
, 1
\biggr] \biggr)
\times [0, d];
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1579
(A2) a(c2, d) < b(c1, d).
Then, for each \lambda \in
\biggr]
1
27 b(c1, d)
,
1
27 a(c2, d)
\biggl[
, problem (1.1) admits at least one nontrivial weak
solution \=u \in X such that
A
B
c21 < \| \=u\| 2 < B
A
c22.
Proof. Our aim is to apply Theorem 2.1 to our problem. To this end, for every u \in X, we
introduce the functionals \Phi , \Psi : X \rightarrow \BbbR by setting
\Phi (u) :=
1
2
\| u\| 2 +
1\int
0
H(x, u\prime (x)) dx+
1\int
0
G(u(x)) dx,
\Psi (u) :=
1\int
0
F (x, u(x)) dx
and put
I\lambda (u) := \Phi (u) - \lambda \Psi (u) \forall u \in X.
Note that the weak solutions of (1.1) are exactly the critical points of I\lambda . The functionals \Phi and \Psi
satisfy the regularity assumptions of Theorem 2.1. Indeed, by standard arguments, we have that \Phi
is Gâteaux differentiable and sequentially weakly lower semicontinuous and its Gâteaux derivative is
the functional \Phi \prime (u) \in X\ast , given by
\Phi \prime (u)(v) =
1\int
0
u\prime \prime (x)v\prime \prime (x) dx+
1\int
0
\left( u\prime (x)\int
0
1
h(x, \tau )
d\tau
\right) v\prime (x) dx -
1\int
0
g(u(x))v(x) dx
for any v \in X. Furthermore, the differential \Phi \prime : X \rightarrow X\ast is a Lipschitzian operator. Indeed, taking
(2.2) and (2.3) into account, for any u, v \in X, there holds
\| \Phi \prime (u) - \Phi \prime (v)\| X\ast = \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
| (\Phi \prime (u) - \Phi \prime (v), w)| \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
1\int
0
| u\prime \prime (x) - v\prime \prime (x)| | w\prime \prime (x)| dx+ \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
1\int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
v\prime (x)\int
u\prime (x)
1
h(x, \tau )
d\tau
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| | w\prime (x)| dx+
+ \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
1\int
0
| g(u(x)) - g(v(x))| | w(x)| dx \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
\| u - v\| \| w\| + 1
m
\mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
\| u\prime - v\prime \| L2([0,1])\| w\prime \| L2([0,1])+
+L \mathrm{s}\mathrm{u}\mathrm{p}
\| w\| \leq 1
\| u - v\| L2([0,1])\| w\| L2([0,1]) \leq
\leq
\biggl(
1 +
1
m\pi 2
+
L
\pi 4
\biggr)
\| u - v\| = 2B\| u - v\| .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1580 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
In particular, we derive that \Phi is continuously differentiable. Also, for any u, v \in X, we have
(\Phi \prime (u) - \Phi \prime (v), u - v) = \| u - v\| 2 +
1\int
0
\left( v\prime (x)\int
u\prime (x)
1
h(x, \tau )
d\tau
\right) (u\prime (x) - v\prime (x)) dx -
-
1\int
0
(g(u(x)) - g(v(x)))(u(x) - v(x)) dx \geq
\geq \| u - v\| 2 + 1
M
\| u\prime - v\prime \| 2L2([0,1]) - L\| u - v\| 2L2([0,1]) \geq
\geq \| u - v\| 2 - L
\pi 4
\| u - v\| 2 = 2A\| u - v\| 2.
By the assumption L < \pi 4, it turns out that \Phi \prime is a strongly monotone operator. So, by applying
Minty – Browder theorem (Theorem 26.A of [23]), \Phi \prime : X \rightarrow X\ast admits a Lipschitz continuous
inverse. On the other hand, the fact that X is compactly embedded into C0([0, 1]) implies that
the functional \Psi is well defined, continuously Gâteaux differentiable and with compact derivative,
whose Gâteaux derivative is given by
\Psi \prime (u)(v) =
1\int
0
f(x, u(x))v(x) dx
for any v \in X.
Since g is Lipschitz continuous and satisfies g(0) = 0, while h is bounded away from zero, the
inequalities (2.2) and (2.3) yield for any u \in X the estimate
A\| u\| 2 \leq \Phi (u) \leq B\| u\| 2. (3.1)
Now, put
r1 := Ac21, r2 := Bc22
and
w(x) =
\left\{
- 64d
9
\biggl(
x2 - 3
4
x
\biggr)
, x \in
\biggr]
0,
3
8
\biggl[
,
d, x \in
\biggr]
3
8
,
5
8
\biggl[
,
- 64d
9
\biggl(
x2 - 5
4
x+
1
4
\biggr)
, x \in
\biggr]
5
8
, 1
\biggl[
.
It is easy to verify that w \in X and, in particular,
\| w\| 2 = 4096
27
d2.
So, taking (3.1) into account, we deduce
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1581
4096
27
Ad2 \leq \Phi (w) \leq 4096
27
Bd2.
From the condition c21 <
4096
27
d2 < c22, we obtain r1 < \Phi (w) < r2. Since B \leq 4\pi 2A, for all u \in X
such that u \in \Phi - 1
\bigl(
] - \infty , r2[
\bigr)
, from (2.4), one has | u(x)| < c2 for all x \in [0, 1], which implies
\mathrm{s}\mathrm{u}\mathrm{p}
u\in \Phi - 1(] - \infty ,r2[)
\Psi (u) = \mathrm{s}\mathrm{u}\mathrm{p}
u\in \Phi - 1(] - \infty ,r2[)
1\int
0
F (x, u(x)) dx \leq
1\int
0
\mathrm{m}\mathrm{a}\mathrm{x}
| t| \leq c2
F (x, t) dx.
Arguing as before, we obtain
\mathrm{s}\mathrm{u}\mathrm{p}
u\in \Phi - 1(] - \infty ,r1])
\Psi (u) \leq
1\int
0
\mathrm{m}\mathrm{a}\mathrm{x}
| t| \leq c1
F (x, t) dx.
Since 0 \leq w(x) \leq d for each x \in [0, 1], assumption (A1) ensures that
3/8\int
0
F (x,w(x)) dx+
1\int
5/8
F (x,w(x)) dx \geq 0,
and
\Psi (w) \geq
5/8\int
3/8
F (x, d) dx.
Therefore, one has
\beta (r1, r2) \leq
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r2[)\Psi (u) - \Psi (w)
r2 - \Phi (w)
\leq
\leq
27
\Biggl( \int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c2 F (x, t) dx -
\int 5/8
3/8
F (x, d) dx
\Biggr)
27Bc22 - 4096Bd2
=
= 27a(c2, d).
On the other hand, we have
\rho (r1, r2) \geq
\Psi (w) - \mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1])\Psi (u)
\Phi (w) - r1
\geq
\geq
27
\Biggl( \int 5/8
3/8
F (x, d) dx -
\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c1 F (x, t) dx
\Biggr)
4096Bd2 - 27Ac21
=
= 27b(c1, d).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1582 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
So, from assumption (A2), it follows that \beta (r1, r2) < \rho (r1, r2). Therefore, from Theorem 2.1, for
each \lambda \in
\biggr]
1
27b(c1, d)
,
1
27a(c2, d)
\biggl[
, the functional I\lambda admits at least one critical point \=u such that
r1 < \Phi (\=u) < r2,
that is,
A
B
c21 < \| \=u\| 2 < B
A
c22.
Theorem 3.1 is proved.
Now, we point out the following consequence of Theorem 3.1.
Theorem 3.2. Assume that there exist two positive constants c and d, with 64 d < 3
\surd
3 c, such
that assumption (A1) in Theorem 3.1 holds. Furthermore, suppose that
(A3)
\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c F (x, t) dx
c2
<
27
4096
\int 5/8
3/8
F (x, d) dx
d2
.
Then, for each
\lambda \in
\right] 409627
Bd2\int 5/8
3/8
F (x, d) dx
,
Bc2\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c F (x, t) dx
\left[ ,
problem (1.1) admits at least one nontrivial weak solution \=u \in X such that | \=u(x)| < c for all
x \in [0, 1].
Proof. The conclusion follows from Theorem 3.1, by taking c1 = 0 and c2 = c. Indeed, owing
to assumption (A3), one has
a(c, d) =
\int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c F (x, t) dx -
\int 5/8
3/8
F (x, d) dx
27Bc2 - 4096Bd2
<
<
\biggl(
1 - 4096d2
27c2
\biggr) \int 1
0
\mathrm{m}\mathrm{a}\mathrm{x}| t| \leq c F (x, t) dx
B(27c2 - 4096d2)
=
1
27Bc2
1\int
0
\mathrm{m}\mathrm{a}\mathrm{x}
| t| \leq c
F (x, t) dx.
On the other hand,
b(0, d) =
\int 5/8
3/8
F (x, d) dx
4096Bd2
.
Now, owing to assumption (A3) and (2.4), it is sufficient to invoke Theorem 3.1 for concluding the
proof.
The following result gives the existence of at least one nontrivial weak solution in X to problem
(1.1) in the autonomous case. Let f : \BbbR \rightarrow \BbbR be a continuous function and put F (t) :=
\int t
0
f(\xi ) d\xi
for all t \in \BbbR .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1583
Corollary 3.1. Assume that there exist a nonnegative constant c1 and two positive constants c2
and d, with c21 <
4096
27
d2 < c22, such that
(A4) f(t) \geq 0 for all t \in [ - c2,\mathrm{m}\mathrm{a}\mathrm{x}\{ c2, d\} ];
(A5)
F (c2) -
1
4
F (d)
27Bc22 - 4096Bd2
<
F (c1) -
1
4
F (d)
27Ac21 - 4096Bd2
.
Then, for each
\lambda \in
\right] 1
27
27Ac21 - 4096Bd2
F (c1) -
1
4
F (d)
,
1
27
27Bc22 - 4096Bd2
F (c2) -
1
4
F (d)
\left[ ,
the problem
uivh(x, u\prime ) - u\prime \prime = [\lambda f(u) + g(u)]h(x, u\prime ) \mathrm{i}\mathrm{n} ] 0, 1 [,
u(0) = u(1) = 0 = u\prime \prime (0) = u\prime \prime (1),
admits at least one nontrivial weak solution \=u \in X such that
A
B
c21 < \| \=u\| 2 < B
A
c22.
Proof. From the condition c21 <
4096
27
d2 < c22, we obtain c1 < c2. Therefore, assumption (A4)
means f(t) \geq 0 for each t \in [ - c1, c1] and f(t) \geq 0 for each t \in [ - c2, c2], which implies
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [ - c1,c1]
F (t) = F (c1) and \mathrm{m}\mathrm{a}\mathrm{x}
t\in [ - c2,c2]
F (t) = F (c2).
So, from assumptions (A4) and (A5), we arrive at assumptions (A1) and (A2), respectively. Hence,
Theorem 3.1 yields the conclusion.
Here, we point out a special situation of our main result when the nonlinear term has separable
variables. To be precise, let \alpha \in L1([0, 1]) be such that \alpha (x) \geq 0 a.e. x \in [0, 1], \alpha \not \equiv 0, and let \gamma :
\BbbR \rightarrow \BbbR be a nonnegative continuous function. Consider the following Dirichlet boundary-value
problem:
uivh(x, u\prime ) - u\prime \prime = [\lambda \alpha (x)\gamma (u) + g(u)]h(x, u\prime ) \mathrm{i}\mathrm{n} ] 0, 1 [,
u(0) = u(1) = 0 = u\prime \prime (0) = u\prime \prime (1).
(3.2)
Put
\Gamma (t) :=
t\int
0
\gamma (\xi ) d\xi for all t \in \BbbR ,
and set
\| \alpha \| 1 :=
1\int
0
\alpha (x) dx, \alpha 0 :=
5/8\int
3/8
\alpha (x) dx.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1584 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
Theorem 3.3. Assume that there exist a nonnegative constant c1 and two positive constants c2
and d, with c21 <
4096
27
d2 < c22, such that
(A6)
\Gamma (c2)\| \alpha \| 1 - \Gamma (d)\alpha 0
27Bc22 - 4096Bd2
<
\Gamma (d)\alpha 0 - \Gamma (c1)\| \alpha \| 1
4096Bd2 - 27Ac21
.
Then, for each
\lambda \in
\biggr]
1
27
4096Bd2 - 27Ac21
\Gamma (d)\alpha 0 - \Gamma (c1)\| \alpha \| 1
,
1
27
27Bc22 - 4096Bd2
\Gamma (c2)\| \alpha \| 1 - \Gamma (d)\alpha 0
\biggl[
,
problem (3.2) admits at least one positive weak solution \=u \in X, such that
A
B
c21 < \| \=u\| 2 < B
A
c22.
Proof. Put f(x, \xi ) := \alpha (x)\gamma (\xi ) for all (x, \xi ) \in [0, 1] \times \BbbR . Clearly, F (x, t) = \alpha (x)\Gamma (t) for
all (x, t) \in [0, 1]\times \BbbR . Therefore, applying Theorem 3.1, problem (3.2) admits at least one nontrivial
weak solution \=u \in X such that
A
B
c21 < \| \=u\| 2 < B
A
c22.
Now, we prove here that the attained solution is positive. Arguing by contradiction, if we assume
that \=u is negative at a point of [0, 1], the set
\Omega := \{ x \in [0, 1] : \=u(x) < 0\}
is nonempty and open. Moreover, let us consider \=v := \mathrm{m}\mathrm{i}\mathrm{n}\{ \=u, 0\} , one has \=v \in X. So, taking into
account that \=u is a weak solution and by choosing v = \=v, from our assumptions, one has
0 \geq \lambda
\int
\Omega
\alpha (x)\gamma (\=u(x))\=u(x) dx =
=
\int
\Omega
| \=u\prime \prime (x)| 2 dx+
\int
\Omega
\left( \=u\prime (x)\int
0
1
h(x, \tau )
d\tau
\right) \=u\prime (x) dx -
\int
\Omega
g(\=u(x))\=u(x) dx \geq
\geq \pi 4 - L
\pi 4
\| \=u\| 2H2(\Omega )\cap H1
0 (\Omega ).
Therefore, \| \=u\| H2(\Omega )\cap H1
0 (\Omega ) = 0 which is absurd. Hence, owing to the strong maximum principle
(see, e.g., [15], Theorem 11.1) the weak solution \=u, being nontrivial, is positive and the conclusion
is achieved.
Theorem 3.3 is proved.
An immediate consequence of Theorem 3.3 is the following.
Corollary 3.2. Assume that there exist positive constants c and d, with 64 d < 3
\surd
3 c, such that
(A7)
\Gamma (c)\| \alpha \| 1
c2
<
27
4096
\Gamma (d)\alpha 0
d2
.
Then, for each
\lambda \in
\biggr]
4096
27
Bd2
\Gamma (d)\alpha 0
,
Bc2
\Gamma (c)\| \alpha \| 1
\biggl[
,
problem (3.2) admits at least one positive weak solution \=u \in X, such that \=u(x) < c for all x \in [0, 1].
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ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1585
Proof. This follows directly from Theorem 3.2.
Remark 3.1. Theorem 1.1 in the introduction is an immediate consequence of Corollary 3.2, on
choosing g(u) = - u, h \equiv 1, c = 64 and d = 1.
Here, we point out another relevant consequence of Corollary 3.2.
Theorem 3.4. Assume that
(A8) \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0+
\gamma (t)
t
= +\infty ,
and put \lambda \star :=
B
\| \alpha \| 1
\mathrm{s}\mathrm{u}\mathrm{p}c>0
c2
\Gamma (c)
. Then, for each \lambda \in ]0, \lambda \star [ , problem (3.2) admits at least one
positive weak solution.
Proof. For fixed \lambda as in the conclusion, there exists a positive constant c such that
\lambda <
Bc2
\Gamma (c)\| \alpha \| 1
. (3.3)
Moreover, assumption (A8) implies that \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0+
\Gamma (t)
t2
= +\infty . Therefore, there is d <
3
\surd
3
64
c such
that
27
4096
\Gamma (d)\alpha 0
Bd2
>
1
\lambda
.
Hence, Corollary 3.2 implies the conclusion.
Remark 3.2. Taking (A8) into account, fix \rho > 0 such that \gamma (t) > 0 for all t \in ]0, \rho [. Then put
\lambda \rho :=
B
\| \alpha \| 1
\mathrm{s}\mathrm{u}\mathrm{p}
c\in ]0,\rho [
c2
\Gamma (c)
.
Theorem 3.4 for every \lambda \in ]0, \lambda \rho [ holds with \=u(x) < \rho for all x \in [0, 1], where \=u is the ensured
positive weak solution in X.
Here, we present the following example to illustrate the applicability of our results.
Example 3.1. Let \alpha (x) = 1 + x, \gamma (t) = et, g(t) = - t and h(x, t) = (2 + x + \mathrm{c}\mathrm{o}\mathrm{s} t) - 1 for all
x \in [0, 1] and t \in \BbbR . It is clear that \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow 0+ \gamma (t)/t = +\infty . Pick \rho = 1. Hence, taking Remark 3.2
into account, by applying Theorem 3.4, since
B =
\pi 4 + 4\pi 2 + 1
2\pi 4
,
for every
\lambda \in
\biggr]
0,
\pi 4 + 4\pi 2 + 1
3\pi 4(e - 1)
\biggl[
,
problem
uiv - u\prime \prime (2 + x+ \mathrm{c}\mathrm{o}\mathrm{s}u\prime ) + u = \lambda eu(1 + x) \mathrm{i}\mathrm{n} ] 0, 1 [,
u(0) = u(1) = 0 = u\prime \prime (0) = u\prime \prime (1),
has at least one positive weak solution \=u \in X such that \| \=u\| \infty < 1.
Next, as consequence of Theorem 3.1, taking into account the classical theorem of Ambrosetti
and Rabinowitz, we have the following multiplicity result.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
1586 S. SHOKOOH, G. A. AFROUZI, A. HADJIAN
Theorem 3.5. Let the assumptions of Theorem 3.1 be satisfied, and f(\cdot , 0) \not = 0 in ] 0, 1 [. More-
over, let
(A9) there exist positive constants \nu and R such that \nu A > 2B, and for all | t| \geq R and for all
x \in [0, 1], one has
0 < \nu F (x, t) \leq t \cdot f(x, t).
Then, for each \lambda \in
\biggr]
1
27b(c1, d)
,
1
27a(c2, d)
\biggl[
, problem (1.1) admits at least two nontrivial weak
solutions \=u1, \=u2, such that
A
B
c21 < \| \=u1\| 2 <
B
A
c22. (3.4)
Proof. Fix \lambda as in the conclusion. So, Theorem 3.1 ensures that problem (1.1) admits at
least one nontrivial weak solution \=u1 satisfying the condition (3.4) which is a local minimum of the
functional I\lambda .
Now, we prove the existence of the second solution distinct from the first one. To this end, we
must show that the functional I\lambda satisfies the hypotheses of the mountain pass theorem.
Clearly, the functional I\lambda is of class C1 and I\lambda (0) = 0.
We can assume that \=u1 is a strict local minimum for I\lambda in X. Therefore, there is \rho > 0 such that
\mathrm{i}\mathrm{n}\mathrm{f}\| u - \=u1\| =\rho I\lambda (u) > I\lambda (\=u1), so condition [16] ((I1), Theorem 2.2) is verified.
From (A9), by standard computations, there is a positive constant C such that
F (x, t) \geq C| t| \nu (3.5)
for all x \in [0, 1] and | t| > R. In fact, setting a(x) := \mathrm{m}\mathrm{i}\mathrm{n}| \xi | =R F (x, \xi ) and
\varphi t(s) := F (x, st) \forall s > 0, (3.6)
by (A9), for every x \in [0, 1] and | t| > R one has
0 < \nu \varphi t(s) = \nu F (x, st) \leq st \cdot f(x, st) = s\varphi \prime
t(s) \forall s > R
| t|
.
Therefore,
1\int
R/| t|
\varphi \prime
t(s)
\varphi t(s)
ds \geq
1\int
R/| t|
\nu
s
ds.
Then
\varphi t(1) \geq \varphi t
\biggl(
R
| t|
\biggr)
| t| \nu
R\nu
.
Taking into account of (3.6), we obtain
F (x, t) \geq F
\biggl(
x,
R
| t|
t
\biggr)
| t| \nu
R\nu
\geq a(x)
| t| \nu
R\nu
\geq C| t| \nu ,
where C > 0 is a constant. Thus, (3.5) is proved. Now, choosing any u \in X \setminus \{ 0\} , one has
I\lambda (tu) = (\Phi - \lambda \Psi )(tu) \leq
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
ON THE EXISTENCE OF SOLUTIONS OF ONE-DIMENSIONAL FOURTH-ORDER EQUATIONS 1587
\leq Bt2\| u\| 2 - \lambda t\nu C
1\int
0
| u(x)| \nu dx \rightarrow - \infty
as t \rightarrow +\infty (since \nu > 2). So, the functional I\lambda is unbounded from below and condition [16] ((I2),
Theorem 2.2) is verified. Therefore, I\lambda satisfies the geometry of mountain pass.
Now, to verify the Palais – Smale condition it is sufficient to prove that any sequence of Palais –
Smale is bounded. To this end, taking into account (A9) one has
\nu I\lambda (un) - \| I \prime \lambda (un)\| X\ast \| un\| \geq \nu I\lambda (un) - I \prime \lambda (un)(un) =
= \nu \Phi (un) - \lambda \nu \Psi (un) - \Phi \prime (un)(un) + \lambda \Psi \prime (un)(un) \geq
\geq (\nu A - 2B) \| un\| 2 - \lambda
1\int
0
[\nu F (x, un(x)) - f(x, un(x))un(x)] dx \geq
\geq (\nu A - 2B) \| un\| 2. (3.7)
If \{ un\} is not bounded, from (3.7) we have a contradiction. Thus, I\lambda satisfies the Palais – Smale
condition.
Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point \=u2 of I\lambda such
that I\lambda (\=u2) > I\lambda (\=u1). So, \=u1 and \=u2 are two distinct weak solutions of (1.1).
Theorem 3.5 is proved.
Corollary 3.3. Assume that there exist two positive constants c, d, with 64d < 3
\surd
3c, such that
(A7) holds. Assume also that
(A10) there exist positive constants \nu and R such that \nu A > 2B, and for all | t| \geq R, one has
0 < \nu \Gamma (t) \leq t \cdot \gamma (t).
Then, for each
\lambda \in
\biggr]
4096
27
Bd2
\Gamma (d)\alpha 0
,
Bc2
\Gamma (c)\| \alpha \| 1
\biggl[
,
problem (3.2) admits at least two nonnegative weak solutions \=u1, \=u2, such that \=u1(x) < c for all
x \in [0, 1].
Corollary 3.4. Assume that (A8) and (A10) are satisfied. Then, for each \lambda \in ] 0, \lambda \star [, problem (3.2)
admits at least two nonnegative weak solutions.
Remark 3.3. If \gamma (0) \not = 0, Corollaries 3.3 and 3.4 ensure two positive weak solutions (see proof
of Theorem 3.3).
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Received 24.11.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 11
|
| id | umjimathkievua-article-569 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:07Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d1/029df460711d642b45cad76600f9a4d1.pdf |
| spelling | umjimathkievua-article-5692025-03-31T08:49:35Z On the existence of solutions of one-dimensional fourth-order equations On the existence of solutions of one-dimensional fourth-order equations Shokooh, S. Afrouzi , G. A. Hadjian, A. Shokooh, S. Afrouzi , G. A. Hadjian, A. Non-trivial solution Variational methods Dirichlet problem Non-trivial solution Variational methods Dirichlet problem UDC 517.9 Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. Multiplicity results are also pointed out. УДК 517.9 Про існування розв'язків одновимірних рівнянь четвертого порядку За допомогою варіаційних методiв та теореми про критичні точки&nbsp; доведено існування нетривіальних розв'язків одновимірних рівнянь четвертого порядку.&nbsp;Також наведено відповідні результати щодо кратності. Institute of Mathematics, NAS of Ukraine 2020-11-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/569 10.37863/umzh.v72i11.569 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 11 (2020); 1575-1588 Український математичний журнал; Том 72 № 11 (2020); 1575-1588 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/569/8785 |
| spellingShingle | Shokooh, S. Afrouzi , G. A. Hadjian, A. Shokooh, S. Afrouzi , G. A. Hadjian, A. On the existence of solutions of one-dimensional fourth-order equations |
| title | On the existence of solutions of one-dimensional fourth-order equations |
| title_alt | On the existence of solutions of one-dimensional fourth-order equations |
| title_full | On the existence of solutions of one-dimensional fourth-order equations |
| title_fullStr | On the existence of solutions of one-dimensional fourth-order equations |
| title_full_unstemmed | On the existence of solutions of one-dimensional fourth-order equations |
| title_short | On the existence of solutions of one-dimensional fourth-order equations |
| title_sort | on the existence of solutions of one-dimensional fourth-order equations |
| topic_facet | Non-trivial solution Variational methods Dirichlet problem Non-trivial solution Variational methods Dirichlet problem |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/569 |
| work_keys_str_mv | AT shokoohs ontheexistenceofsolutionsofonedimensionalfourthorderequations AT afrouziga ontheexistenceofsolutionsofonedimensionalfourthorderequations AT hadjiana ontheexistenceofsolutionsofonedimensionalfourthorderequations AT shokoohs ontheexistenceofsolutionsofonedimensionalfourthorderequations AT afrouziga ontheexistenceofsolutionsofonedimensionalfourthorderequations AT hadjiana ontheexistenceofsolutionsofonedimensionalfourthorderequations |