Existence of three weak solutions for fourth-order elastic beam equations on the whole space
UDC 517.9 Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, A critical point theorem via...
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| author | Tavani , M. R. H. Tavani , M. R. H. |
| author_facet | Tavani , M. R. H. Tavani , M. R. H. |
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| description | UDC 517.9
Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, 2992-3007 (2012)]. |
| doi_str_mv | 10.37863/umzh.v72i12.881 |
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DOI: 10.37863/umzh.v72i12.881
UDC 517.9
M. R. H. Tavani (Dep. Math., Ramhormoz Branch, Islamic Azad Univ., Ramhormoz, Iran)
EXISTENCE OF THREE WEAK SOLUTIONS
FOR FOURTH-ORDER ELASTIC BEAM EQUATIONS ON THE WHOLE SPACE
IСНУВАННЯ ТРЬОХ СЛАБКИХ РОЗВ’ЯЗКIВ РIВНЯНЬ
ПРУЖНОЇ БАЛКИ ЧЕТВЕРТОГО ПОРЯДКУ В УСЬОМУ ПРОСТОРI
Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one
real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained
in [G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, 2992 – 3007 (2012)].
Вивчено результати кратностi для збуреної задачi четвертого порядку на дiйснiй прямiй iз збуреним нелiнiйним
доданком, що залежить вiд одного дiйсного параметра. Пiдхiд базується на методах варiацiй та теорiї критичних
точок, що отриманi в [G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75,
2992 – 3007 (2012)].
1. Introduction. In this paper we consider the following problem:
ui\upsilon (x) +Au\prime \prime (x) +Bu(x) = \lambda \alpha (x).f(u(x)) a.e. x \in \BbbR , (P\lambda )
where A is a real negative constant and B is a real positive constant, \lambda is a positive parameter
and \alpha , f : \BbbR \rightarrow \BbbR are two functions such that \alpha \in L1(\BbbR ), \alpha (x) \geq 0, for a.e. x \in \BbbR , \alpha \not \equiv 0
and also f is continuous and nonnegative. It is known that fourth-order problems are important in
describing a large class of elastic deflections. Hence, many researchers have studied the existence and
multiplicity of solutions for fourth-order two-point boundary-value problems. We refer the reader to
[4 – 6, 8 – 10]. In [4], while A and B are real constants, using variational methods and critical point
theory, multiplicity results for the fourth-order elliptic problem,
ui\upsilon +Au\prime \prime +Bu = \lambda f(t, u), t \in [0, 1],
u(0) = u(1) = u\prime \prime (0) = u\prime \prime (1) = 0
(1.1)
by condition on the nonlinear term was established, while in [8], applying the Morse theory, the
existence of three solutions to problem (1.1), with A = B = 0, were discussed. Problems such as
(P\lambda ) that are discussed on the whole space, occur naturally in a variety of settings in physics and
material scinces, as in, for example, the study of mathematical models of deflection of beams. These
beams which appear in many structures, deflect under their own weight or under the influence of
some external forces.
Due to the lack of compactness of the operators on whole space, the study of such problems is
very important. Because, in such cases the operators which solve the problem are not regular enough
in comparison to operators which arise in problems on bounded domains. Due to this, for example,
we can not aplly [3] (Corollary 3.1) for the problem (P\lambda ). In the present paper, using one kind of
critical point theorem obtained in [1] which we recall in the next section (Theorem 2.1), we establish
the existence of at least three nonnegative weak solutions for the problem (P\lambda ). In fact, in presenting
c\bigcirc M. R. H. TAVANI, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12 1697
1698 M. R. H. TAVANI
Theorem 3.1, which one of the main results of this paper, we aplly the requirement ( non-standard
Palais – Smale condition for functional I\lambda which is the functional related to the problem (P\lambda )) based
on Theorem 2.1.
2. Preliminaries. Let us recall some basic concepts.
We denote by | .| t the usual norm on Lt(\BbbR ), for all t \in [1,+\infty ] and it is known that W 2,2(\BbbR ) is
continuously embedded in Lt(\BbbR ) for each t \in [2,+\infty ].
The Sobolev space W 2,2(\BbbR ) is equipped with the following norm:
\| u\| W 2,2(\BbbR ) =
\left( \int
\BbbR
(| u\prime \prime (x)| 2 + | u\prime (x)| 2 + | u(x)| 2)dx
\right) 1/2
for all u \in W 2,2(\BbbR ). Also, we consider W 2,2(\BbbR ) with the norm
\| u\| =
\left( \int
\BbbR
(| u\prime \prime (x)| 2 - A| u\prime (x)| 2 +B| u(x)| 2)dx
\right) 1/2
for all u \in W 2,2(\BbbR ). According to
(\mathrm{m}\mathrm{i}\mathrm{n}\{ 1, - A,B\} )
1
2 \| u\| W 2,2(\BbbR ) \leq \| u\| \leq (\mathrm{m}\mathrm{a}\mathrm{x}\{ 1, - A,B\} )
1
2 \| u\| W 2,2(\BbbR ),
the norm \| .\| is equvalent to the norm \| .\| W 2,2(\BbbR ). Since embedding W 2,2(\BbbR ) \rightarrow L\infty (\BbbR ) is continu-
ous hence there exists a constant CA,B (depending on A and B) such that
| u| \infty \leq CA,B\| u\| \forall u \in W 2,2(\BbbR ).
In the following proposition, we provide an approximation for this constant.
Proposition 2.1. We have
| u| \infty \leq CA,B\| u\| (2.1)
where CA,B =
\biggl(
- 1
4AB
\biggr) 1
4
.
Proof. Let v \in W 1,1(\BbbR ), then from [7, p. 138] (formula (4.64)), one has
| v(x)| \leq 1
2
\int
\BbbR
| v\prime (t)| dt. (2.2)
Now if u \in W 2,2(\BbbR ), then v(x) = ( - AB)
1
2 | u(x)| 2 \in W 1,1(\BbbR ) and, thus, from (2.2) and Hölder’s
inequality one has
( - AB)
1
2 | u(x)| 2 \leq
\int
\BbbR
( - AB)
1
2 | u\prime (t)| | u(t)| dt \leq (( - A)
1
2 | u\prime | 2)(B
1
2 | u| 2),
that is,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF THREE WEAK SOLUTIONS FOR FOURTH-ORDER . . . 1699
| u(x)| \leq
\biggl(
- 1
AB
\biggr) 1
4
(( - A)
1
2 | u\prime | 2)
1
2 (B
1
2 | u| 2)
1
2 . (2.3)
Now according to xay1 - a \leq aa(1 - a)1 - a(x+ y), x, y \geq 0, 0 < a < 1 [7, p. 130] (formula (4.47))
and classical inequality a
1
p + b
1
p \leq 2
p - 1
p (a+ b)
1
p , from (2.3) one has
| u(x)| \leq
\biggl(
- 1
AB
\biggr) 1
4
\biggl(
1
2
\biggr) 1
2
\left[
\left( \int
\BbbR
- A| u\prime (t)| 2dt
\right) 1
2
+
\left( \int
\BbbR
B| u(t)| 2dt
\right) 1
2
\right] \leq
\leq
\biggl(
- 1
AB
\biggr) 1
4
\biggl(
1
2
\biggr) 1
2
\biggl(
1
2
\biggr) 1
2
(2)
1
2
\left( \int
\BbbR
( - A| u\prime (t)| 2 +B| u(t)| 2)dt
\right) 1
2
\leq
\leq
\biggl(
- 1
4AB
\biggr) 1
4
\left( \int
\BbbR
(| u\prime \prime (t)| 2 - A| u\prime (t)| 2 +B| u(t)| 2)dt
\right) 1
2
,
which means that | u| \infty \leq CA,B\| u\| .
Let \Phi ,\Psi : W 2,2(\BbbR ) \rightarrow \BbbR be defined by
(u) =
1
2
\| u\| 2 = 1
2
\int
\BbbR
(| u\prime \prime (x)| 2 - A| u\prime (x)| 2 +B| u(x)| 2)dx (2.4)
and
\Psi (u) =
\int
\BbbR
\alpha (x)F (u(x))dx (2.5)
for every u \in W 2,2(\BbbR ), where F (t) =
\int t
0
f(\xi )d\xi for all t \in \BbbR . Since F \prime (t) = f(t) \geq 0 for
all t \in \BbbR so F is an increasing function. The functional \Psi is well defined because for every
u \in W 2,2(\BbbR ) we have
| \Psi (u)| \leq
\int
\BbbR
\alpha (x)\mathrm{m}\mathrm{a}\mathrm{x}\{ - F ( - | u| \infty ), F (| u| \infty )\} dx < +\infty .
It is known that \Psi is a differentiable functional whose differential at the point u \in W 2,2(\BbbR ) is
\Psi \prime (u)(v) = \mathrm{l}\mathrm{i}\mathrm{m}
s\rightarrow 0
\Psi (u+ sv) - \Psi (u)
s
=
d
ds
\Psi (u+ sv)
\bigm| \bigm| \bigm|
s=0
=
\int
\BbbR
\alpha (x)f(u(x))v(x)dx,
and \Phi is a continuously Gâteaux differentiable functional and in a similar way, whose differential at
the point u \in W 2,2(\BbbR ) is
\Phi \prime (u)(v) =
\int
\BbbR
(u\prime \prime (x)v\prime \prime (x) - Au\prime (x)v\prime (x) +Bu(x)v(x))dx
for every v \in W 2,2(\BbbR ).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1700 M. R. H. TAVANI
Definition 2.1. Let \Phi and \Psi be defined as above. Put I\lambda = \Phi - \lambda \Psi , \lambda > 0. We say that
u \in W 2,2(\BbbR ) is a critical point of I\lambda when I \prime \lambda (u) = 0\{ W 2,2(\BbbR )\ast \} , that is, I \prime \lambda (u)(v) = 0 for all
v \in W 2,2(\BbbR ).
Definition 2.2. A function u : \BbbR \rightarrow \BbbR is a weak solution to the problem (P\lambda ) if u \in W 2,2(\BbbR )
and \int
\BbbR
(u\prime \prime (x)v\prime \prime (x) - Au\prime (x)v\prime (x) +Bu(x)v(x) - \lambda \alpha (x)f(u(x))v(x))dx = 0
for all v \in W 2,2(\BbbR ).
Remark 2.1. We clearly observe that the weak solutions of the problem (P\lambda ) are exactly the
solutions of the equation I \prime \lambda (u)(v) = \Phi \prime (u)(v) - \lambda \Psi \prime (u)(v) = 0. Also if \alpha is, in addition, a
continuous function on \BbbR then each weak solution of (P\lambda ) is a classical solution.
Lemma 2.1. If u0 \not \equiv 0 is a weak solution for problem (P\lambda ), then u0 is nonnegative.
Proof. From Remark 2.1 one has, I \prime \lambda (u0)(v) = 0 for all v \in W 2,2(\BbbR ). Choose v(x) = \=u0 =
= \mathrm{m}\mathrm{a}\mathrm{x}\{ - u0(x), 0\} and let M = \{ x \in \BbbR : u0(x) < 0\} . Then we have\int
M\cup Mc
(u\prime \prime 0(x)\=u
\prime \prime
0(x) - Au\prime 0(x)\=u
\prime
0(x) +Bu0(x)\=u0(x))dx =
\int
M\cup Mc
\lambda \alpha (x)f(u0(x))\=u0(x)dx,
that is,
-
\int
M
(| \=u\prime \prime 0(x)| 2 - A| \=u\prime 0(x)| 2 +B| \=u0(x)| 2)dx =
\int
M
\lambda \alpha (x)f(u0(x))\=u0(x)dx \geq 0
which means that - \| \=u0\| 2 \geq 0 and one has \=u0 = 0. Hence - u0 \leq 0, that is, u0 \geq 0 and the proof
is complete.
Definition 2.3. A Gâtuax differentiable function I from Banach space X to \BbbR satisfies the
Palais – Smale condition (in short (PS)-condition) if any sequence \{ un\} such that
(\mathrm{a}) \{ I(un)\} is bounded,
(\mathrm{b}) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| I \prime (un)\| X\ast = 0 \forall n \in \BbbN ,
has a convergent subsequence.
Below, we will present a non-standard state of the Palais – Smale condition that is intro-
duced in [1].
Definition 2.4 (see [1]). Fix r \in ] - \infty ,+\infty ]. A Gâtuax differentiable function I from Banach
space X to \BbbR satisfies the Palais – Smale condition cut off upper at r (in short (PS)[r]-condition) if
any sequence \{ un\} such that
(\mathrm{a}) \{ I(un)\} is bounded,
(\mathrm{b}) \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| I \prime (un)\| X\ast = 0,
(\mathrm{c}) \Phi (un) < r \forall n \in \BbbN ,
has a convergent subsequence.
Our main tool is the following critical point theorem.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF THREE WEAK SOLUTIONS FOR FOURTH-ORDER . . . 1701
Theorem 2.1 ([1], Theorem 7.3). Let X be a real Banach space, and let \Phi ,\Psi : X - \rightarrow \BbbR be
two continuously Gâteaux differentiable functions with \Phi bounded from below and convex such that
\mathrm{i}\mathrm{n}\mathrm{f}
X
\Phi = \Phi (0) = \Psi (0) = 0.
Assume that there are two positive constants r1, r2 and u \in X, with 2r1 < \Phi (u) <
r2
2
, such
that
(\mathrm{b}1)
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1[)\Psi (u)
r1
<
2
3
\Psi (u)
\Phi (u)
;
(\mathrm{b}2)
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r2[)\Psi (u)
r2
<
1
3
\Psi (u)
\Phi (u)
.
Assume also that, for each
\lambda \in \Lambda =
\right] 3
2
\Phi (u)
\Psi (u)
, \mathrm{m}\mathrm{i}\mathrm{n}
\left\{ r1
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1[)\Psi (u)
,
r2
2
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r2[)\Psi (u)
\right\}
\left[ ,
the functional \Phi - \lambda \Psi satisfies the (PS)[r2]-condition and
\mathrm{i}\mathrm{n}\mathrm{f}
t\in [0,1]
\Psi (tu1 + (1 - t)u2) \geq 0
for each u1, u2 \in X which are local minima for the functional \Phi - \lambda \Psi and such that \Psi (u1) \geq 0
and \Psi (u2) \geq 0.
Then, for each \lambda \in \Lambda , the functional \Phi - \lambda \Psi admits at least three critical points which lie in
\Phi - 1(] - \infty , r2[).
Now we present one proposition that will be needed to prove the main theorem of this paper.
Proposition 2.2. Take \Phi and \Psi as in the Definition 2.1 and fix \lambda > 0. Then I\lambda = \Phi - \lambda \Psi
satisfies the (PS)[r]-condition for any r > 0.
Proof. Consider sequence \{ un\} \subseteq W 2,2(\BbbR ) such that \{ I\lambda (un)\} is bounded,
\mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| I \prime \lambda (un)\| W 2,2(\BbbR )\ast = 0 and\Phi (un) < r \forall n \in \BbbN . Since \Phi (un) < r, we have
1
2
\| un\| 2 < r
and so \{ un\} is bounded in W 2,2(\BbbR ). Therefore passing to a subsequence if necessary we can as-
sume that un(x) \rightarrow u(x), x \in \BbbR (from the compact embedding W 2,2(\BbbR ) \rightarrow C([ - T, T ]), T > 0)
and \{ un\} weakly converges to u in L\infty (\BbbR ) (from the continuous embedding W 2,2(\BbbR ) \rightarrow L\infty (\BbbR ))
and, hence, there is s > 0 such that | un(x)| \leq s for a.e. x \in \BbbR and for all n \in \BbbN . Here, it
is useful to note that the subsequence \{ un\} converges weakly to u in W 2,2(\BbbR ) and we want to
show that this subsequence is strongly converging to u in W 2,2(\BbbR ). For this purpose, according to
Lebesque’s dominated convergence theorem since \alpha f(un(x)) \leq \alpha .\mathrm{m}\mathrm{a}\mathrm{x}| \xi | \leq s f(\xi ) \in L1(\BbbR ) for all
n \in \BbbN and f(un(x)) \rightarrow f(u(x)) for a.e. x \in \BbbR (f is continuous function), one has \alpha f(un) is
strongly converging to \alpha f(u) in L1(\BbbR ). Now since un \rightharpoonup u in L\infty (\BbbR ) and \alpha f(un) \rightarrow \alpha f(u) in
L1(\BbbR ) \subseteq (L\infty (\BbbR ))\ast then from [2] (Proposition 3.5(iv)), one has
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\int
\BbbR
\alpha (x)f(un(x))(un(x) - u(x))dx = 0. (2.6)
From \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| I \prime \lambda (un)\| W 2,2(\BbbR )\ast = 0, there exists a sequence \{ \varepsilon n\} , with \varepsilon n \rightarrow 0+, such that
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1702 M. R. H. TAVANI\bigm| \bigm| \bigm| \bigm| \bigm|
\int
\BbbR
(u\prime \prime n(x)v
\prime \prime (x) - Au\prime n(x)v
\prime (x) +Bun(x)v(x) - \lambda \alpha (x)f(un(x))v(x))dx
\bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon n (2.7)
for all n \in \BbbN and for all v \in W 2,2(\BbbR ) with \| v\| \leq 1. Putting v(x) =
un(x) - u(x)
\| un - u\|
, from (2.7)
one has \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\BbbR
(u\prime \prime n(x)(u
\prime \prime
n(x) - u\prime \prime (x)) - Au\prime n(x)(u
\prime
n(x) - u\prime (x)) +Bun(x)(un(x) - u(x)) -
- \lambda \alpha (x)f(un(x))(un(x) - u(x)))dx
\bigm| \bigm| \bigm| \bigm| \bigm| \leq \varepsilon n\| un - u\| (2.8)
for all n \in \BbbN . Now according to inequality | a| | b| \leq 1
2
| a| 2 + 1
2
| b| 2 we have\int
\BbbR
(u\prime \prime n(x)(u
\prime \prime
n(x) - u\prime \prime (x)) - Au\prime n(x)(u
\prime
n(x) - u\prime (x)) +Bun(x)(un(x) - u(x)))dx =
=
\int
\BbbR
(| u\prime \prime n(x)| 2 - A| u\prime n(x)| 2 +B| un(x)| 2)dx -
-
\int
\BbbR
(u\prime \prime n(x)u
\prime \prime (x) - Au\prime n(x)u
\prime (x) +Bun(x)u(x))dx \geq
\geq \| un\| 2 -
\int
\BbbR
\biggl(
1
2
| u\prime \prime n(x)| 2 +
1
2
| u\prime \prime (x)| 2 - 1
2
A| u\prime n(x)| 2 -
1
2
A| u\prime (x)| 2 + 1
2
B| un(x)| 2+
+
1
2
B| u(x)| 2
\biggr)
dx = \| un\| 2 -
1
2
\| un\| 2 -
1
2
\| u\| 2 = 1
2
\| un\| 2 -
1
2
\| u\| 2.
Hence from (2.8), we obtain
1
2
\| un\| 2 -
1
2
\| u\| 2 \leq \lambda
\int
\BbbR
\alpha (x)f(un(x))(un(x) - u(x))dx+ \varepsilon n\| un - u\| ,
that is,
1
2
\| un\| 2 \leq
1
2
\| u\| 2 + \lambda
\int
\BbbR
\alpha (x)f(un(x))(un(x) - u(x))dx+ \varepsilon n\| un - u\| . (2.9)
Taking into account (2.6), from (2.9) when \varepsilon n \rightarrow 0+, we have
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
n\rightarrow +\infty
\| un\| \leq \| u\| .
Thus, [2] (Proposition 3.32) ensures that un \rightarrow u in W 2,2(\BbbR ).
Proposition 2.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF THREE WEAK SOLUTIONS FOR FOURTH-ORDER . . . 1703
3. Main results. Before presenting the main theorems of this section, we introduce notations
that are related to some constants that will appear in the main results of this section. Put
k =
\biggl(
2048
27
- 32
9
A+
13
40
B
\biggr) - 1
,
\alpha 0 =
5
8\int
3
8
\alpha (x)dx,
E =
\int 5
8
3
8
\alpha (x)dx\int
\BbbR
\alpha (x)dx
=
\alpha 0
| \alpha | 1
, and, hence, E \leq 1,
h = CA,B
\Biggl(
2
k
\Biggr) 1
2
and I =
E
h2
,
where CA,B is given in Proposition 2.1. Now we express the main results.
Theorem 3.1. Assume that there exist three positive constants \eta , \theta 1 and \theta 2 with 2\theta 1 <
\surd
2\eta h <
< \theta 2 such that
(\mathrm{i})
F (\theta 1)
\theta 1
2 <
2
3
I
F (\eta )
\eta 2
,
(\mathrm{i}\mathrm{i})
F (\theta 2)
\theta 2
2 <
1
3
I F (\eta )
\eta 2
.
Then, for each
\lambda \in \Lambda \prime =
\Biggr]
3
4
1
| \alpha | 1C2
A,B
1
I
\eta 2
F (\eta )
, \mathrm{m}\mathrm{i}\mathrm{n}
\Biggl\{
1
2| \alpha | 1C2
A,B
\theta 1
2
F (\theta 1)
,
1
4
1
| \alpha | 1C2
A,B
\theta 2
2
F (\theta 2)
\Biggr\} \Biggl[
,
the problem (P\lambda ) admits at least three distinct nonnegative weak solutions ui \in W 2,2(\BbbR ) such that
| ui| \infty < \theta 2, i = 1, 2, 3.
Proof. Our aim is to apply Theorem 2.1, to problem (P\lambda ). Fix \lambda , as in the conclusion. Take
X = W 2,2(\BbbR ) and \Phi and \Psi as in the previous section. We observe that the regularity assumptions
of Theorem 2.1 on \Phi and \Psi are satisfied and also according to Proposition 2.2, the functional I\lambda
satisfies the (PS)[r]-condition for all r > 0.
Put
r1 :=
1
2
\Biggl(
\theta 1
CA,B
\Biggr) 2
, r2 :=
1
2
\Biggl(
\theta 2
CA,B
\Biggr) 2
and
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1704 M. R. H. TAVANI
w(x) :=
\left\{
- 64\eta
9
\biggl(
x2 - 3
4
x
\biggr)
, if x \in
\biggl[
0,
3
8
\biggr]
,
\eta , if x \in
\biggr]
3
8
,
5
8
\biggr]
,
- 64\eta
9
\biggl(
x2 - 5
4
x+
1
4
\biggr)
, if x \in
\biggr]
5
8
, 1
\biggr]
,
0 otherwise.
(3.1)
We clearly observe that w \in X and, in particular,
\Phi (w) =
1
2
\| w\| 2 = 1
2
\int
\BbbR
(| w\prime \prime (x)| 2 - A| w\prime (x)| 2 +B| w(x)| 2)dx =
= \eta 2
\biggl(
2048
27
- 32
9
A+
13
40
B
\biggr)
=
\eta 2
k
=
1
2
\biggl(
\eta h
CA,B
\biggr) 2
.
Therefore, using the condition 2\theta 1 <
\surd
2\eta h < \theta 2, one has, 2r1 < \Phi (w) <
r2
2
.
Now for each u \in X and bearing (2.1) in mind, we see that
\Phi - 1(] - \infty , ri[) = \{ u \in X; \Phi (u) < ri\} =
=
\left\{ u \in X;
1
2
\| u\| 2 < 1
2
\Biggl(
\theta i
CA,B
\Biggr) 2
\right\} =
= \{ u \in X;CA,B\| u\| < \theta i\} \subseteq \{ u \in X; | u| \infty < \theta i\} ,
and it follows that
\mathrm{s}\mathrm{u}\mathrm{p}
u\in \Phi - 1(] - \infty ,ri[)
\Psi (u) = \mathrm{s}\mathrm{u}\mathrm{p}
u\in \Phi - 1(] - \infty ,ri[)
\int
\BbbR
\alpha (x) F (u(x))dx \leq
\leq
\int
\BbbR
\alpha (x) \mathrm{s}\mathrm{u}\mathrm{p}
| \xi | <\theta i
F (\xi )dx = | \alpha | 1 F (\theta i).
Hence, we have
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1[)\Psi (u)
r1
\leq | \alpha | 1F (\theta 1)
1
2
\biggl(
\theta 1
CA,B
\biggr) 2 = 2| \alpha | 1C2
A,B
F (\theta 1)
\theta 1
2 <
1
\lambda
. (3.2)
On the other hand, one has
2
3
\Psi (w)
\Phi (w)
=
2
3
\int
\BbbR
\alpha (x)F (w(x))dx
1
2
\biggl(
\eta h
CA,B
\biggr) 2 \geq 2
3
\int 5
8
3
8
\alpha (x)F (\eta )dx
1
2
\biggl(
\eta h
CA,B
\biggr) 2 =
2
3
\alpha 0F (\eta )
1
2
\biggl(
\eta h
CA,B
\biggr) 2 =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF THREE WEAK SOLUTIONS FOR FOURTH-ORDER . . . 1705
=
4
3
| \alpha | 1C2
A,B
E
h2
F (\eta )
\eta 2
=
4
3
| \alpha | 1C2
A,BI
F (\eta )
\eta 2
>
1
\lambda
. (3.3)
Now from (3.2) and (3.3) we have
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r1[)\Psi (u)
r1
<
2
3
\Psi (w)
\Phi (w)
.
Analogously, from (3.3) we get
2 \mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r2[)\Psi (u)
r2
\leq 2| \alpha | 1F (\theta 2)
1
2
\biggl(
\theta 2
CA,B
\biggr) 2 = 4| \alpha | 1C2
A,B
F (\theta 2)
\theta 2
2 <
1
\lambda
<
2
3
\Psi (w)
\Phi (w)
(3.4)
which means that
\mathrm{s}\mathrm{u}\mathrm{p}u\in \Phi - 1(] - \infty ,r2[)\Psi (u)
r2
<
1
3
\Psi (w)
\Phi (w)
.
Hence, (\mathrm{b}1) and (\mathrm{b}2) of Theorem 2.1 are established.
Now, if u1, u2 \in W 2,2(\BbbR ) be two local minima of the functional I\lambda = \Phi - \lambda \Psi , with \Psi (u1) \geq 0
and \Psi (u2) \geq 0, then according to Lemma 2.1, u1 and u2 are nonnegative, and we get
\mathrm{i}\mathrm{n}\mathrm{f}
t\in [0,1]
\Psi (tu1 + (1 - t)u2) \geq 0.
Finally, for every \lambda \in \Lambda \prime \subseteq \Lambda (see (3.2) – (3.4)), Theorem 2.1 (with u = w) and Lemma 2.1
guarantee the conclusion.
Theorem 3.1 is proved.
Now, we present the following example to illustrate Theorem 3.1.
Example 3.1. Suppose that f : \BbbR \rightarrow \BbbR is continuous and nonnegative function and
\alpha (x) :=
\left\{ 1, if x \in
\biggl[
3
8
,
5
8
\biggr]
,
0 otherwise.
Let A = - 1 and B = 1, then we have
k =
\biggl(
86111
1080
\biggr) - 1
, CA,B =
\surd
2
2
, h =
\biggl(
86111
1080
\biggr) 1
2
, E = 1, I =
1080
86111
.
Also let \eta = 1, \theta 1 = 0.001 and \theta 2 = 20. So the condition 2\theta 1 <
\surd
2 \eta h < \theta 2 is satisfied.
Now if
1000000
0.001\int
0
f(\xi )d\xi <
720
86111
1\int
0
f(\xi )d\xi
and
1
400
20\int
0
f(\xi )d\xi <
360
86111
1\int
0
f(\xi )d\xi ,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
1706 M. R. H. TAVANI
then according to Theorem 3.1 for each
\lambda \in
\right] 86111
180
\int 1
0
f(\xi )d\xi
,\mathrm{m}\mathrm{i}\mathrm{n}
\left\{
0.000004\int 0.001
0
f(\xi )d\xi
,
800\int 20
0
f(\xi )d\xi
\right\}
\left[ ,
problem
uiv(x) - u\prime \prime (x) + u(x) = \lambda \alpha (x)f(u(x)), x \in \BbbR ,
u( - \infty ) = u(+\infty ) = 0
(3.5)
has at least three nonnegative weak solutions ui such that | ui| \infty < 20 , i = 1, 2, 3.
Remark 3.1. For example, in problem (3.5) we can consider
f(t) :=
\left\{
18000 t2, if t \leq 1,
- 1800000 t+ 1818000, if 1 < t \leq 1.01,
0, if t > 1.01.
Now, we point out the following existence result, as consequence of Theorem 3.1.
Corollary 3.1. Let f : \BbbR \rightarrow [0,+\infty [ be a continuous and nonzero function such that
\mathrm{l}\mathrm{i}\mathrm{m}
\xi \rightarrow 0+
f(\xi )
\xi
= \mathrm{l}\mathrm{i}\mathrm{m}
\xi \rightarrow +\infty
f(\xi )
\xi
= 0.
Then, for each \lambda > \lambda \ast , where
\lambda \ast = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{
3
4| \alpha | 1C2
A,BI
\eta 2\int \eta
0
f (\xi ) d\xi
: \eta > 0,
\eta \int
0
f (\xi ) d\xi > 0
\right\}
the problem (P\lambda ) admits at least three distinct nonnegative weak solutions.
Proof. Suppose that \lambda > \lambda \ast is fixed. Let \eta > 0 such that
\int \eta
0
f (\xi ) d\xi > 0, and
\lambda >
3
4| \alpha | 1C2
A,BI
\eta 2\int \eta
0
f (\xi ) d\xi
.
Then from \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow 0+
f(\xi )
\xi
= 0 we have \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow 0+
\int \xi
0
f(t)dt
\xi 2
= 0 and there is \theta 1 > 0 such
that 2\theta 1 <
\surd
2\eta h, and
\int \theta 1
0
f(t)dt
\theta 1
2 <
1
2| \alpha | 1C2
A,B\lambda
. Also from \mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow +\infty
f(\xi )
\xi
= 0 we have
\mathrm{l}\mathrm{i}\mathrm{m}\xi \rightarrow +\infty
\int \xi
0
f(t)dt
\xi 2
= 0 and there is \theta 2 > 0 such that
\surd
2 \eta h < \theta 2 and
\int \theta 2
0
f(t)dt
\theta 2
2 <
<
1
4| \alpha | 1C2
A,B\lambda
. Now we can apply Theorem 3.1 and the conclusion follows.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
EXISTENCE OF THREE WEAK SOLUTIONS FOR FOURTH-ORDER . . . 1707
Example 3.2. Let \eta = 1, A = - 1 and B = 1 and so CA,B =
\surd
2
2
and I =
1080
86111
. Also
suppose that f(x) = x2e - x3
and \alpha (x) =
1
1 + x2
and hence | \alpha | 1 =
\int +\infty
- \infty
1
1 + x2
dx = \pi . Therefore
according to Corollary 3.1 for each \lambda >
86111
240\pi
\biggl(
1 - 1
e
\biggr) problem
uiv(x) - u\prime \prime (x) + u(x) = \lambda
u(x)2e - u(x)3
1 + x2
, x \in \BbbR ,
u( - \infty ) = u(+\infty ) = 0
admits at least three nonnegative classical solutions.
References
1. G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, 2992 – 3007 (2012).
2. H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer Science+Business Media,
LLC (2011), DOI: 10.1007/978-0-387-70914-7.
3. G. Bonanno, P. Candito, Non-differentiable functionals and applications to elliptic problems with discontinuous
nonlinearities, J. Different. Equat., 244, 3031 – 3059 (2008).
4. G. Bonanno, B. Di Bella, A boundary-value problem for fourth-order elastic beam equations, J. Math. Anal. and
Appl., 343, 1166 – 1176 (2008).
5. G. Bonanno, B. Di Bella, Infinitely many solutions for a fourth-order elastic beam equations, Nonlinear Different.
Equat. and Appl., 18, 357 – 368 (2011).
6. G. Bonanno, B. Di Bella, D. O’Regan, Non-trivial solutions for nonlinear fourth-order elastic beam equations,
Comput. Math. and Appl., 62, 1862 – 1869 (2011).
7. V. I. Burenkov, Sobolev spaces on domains, Vol. 137, Teubner, Leipzig (1998).
8. G. Han, Z. Xu, Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear Anal., 68, 3646 – 3656
(2008).
9. X.-L. Liu, W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary-values problems with parameters,
J. Math. Anal. and Appl., 327, 362 – 375 (2007).
10. F. Wang, Y. An, Existence and multiplicity of solutions for a fourth-order elliptic equation, Boundary Value Problems,
№ 6 (2012).
Received 26.12.17,
after revision — 27.10.19
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 12
|
| id | umjimathkievua-article-881 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:56Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b1/ce94092d7817bc4a0a2b9e0867ee89b1.pdf |
| spelling | umjimathkievua-article-8812025-03-31T08:49:28Z Existence of three weak solutions for fourth-order elastic beam equations on the whole space Existence of three weak solutions for fourth-order elastic beam equations on the whole space Tavani , M. R. H. Tavani , M. R. H. Multiplicity results Non-trivial solution Critical point theory Variational methods Multiplicity results Non-trivial solution Critical point theory Variational methods UDC 517.9 Multiplicity results for a perturbed fourth-order problem on the real line with a perturbed nonlinear term depending on one real parameter is investigated. Our approach is based on variational methods and critical point theory which are obtained in [G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, 2992-3007 (2012)]. УДК 517.9 Iснування трьох слабких розв’язкiв рiвнянь пружної балки четвертого порядку у всьому просторiВивчено результати кратності для збуреної задачі четвертого порядку на дійсній прямій із збуреним нелінійним доданком, що залежить від одного дійсного параметра. Підхід базується на методах варіацій та теорії критичних точок, що отримані в [G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, 2992-3007 (2012)]. Institute of Mathematics, NAS of Ukraine 2020-12-24 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/881 10.37863/umzh.v72i12.881 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 12 (2020); 1697-1707 Український математичний журнал; Том 72 № 12 (2020); 1697-1707 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/881/8877 |
| spellingShingle | Tavani , M. R. H. Tavani , M. R. H. Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_alt | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_full | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_fullStr | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_full_unstemmed | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_short | Existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| title_sort | existence of three weak solutions for fourth-order elastic beam equations on the whole space |
| topic_facet | Multiplicity results Non-trivial solution Critical point theory Variational methods Multiplicity results Non-trivial solution Critical point theory Variational methods |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/881 |
| work_keys_str_mv | AT tavanimrh existenceofthreeweaksolutionsforfourthorderelasticbeamequationsonthewholespace AT tavanimrh existenceofthreeweaksolutionsforfourthorderelasticbeamequationsonthewholespace |