An eigenvalue of anisotropic discrete problem with three variable exponents
UDC 517.5We study the existence of a continuous spectrum of an anisotropic discrete problem, involving variable exponent.The proposed technical approach is based on the variational methods and critical point theory.
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Institute of Mathematics, NAS of Ukraine
2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507221913239552 |
|---|---|
| author | Ousbika, M. Allali, Z. El Ousbika, M. Allali, Z. El Allali, Z. El |
| author_facet | Ousbika, M. Allali, Z. El Ousbika, M. Allali, Z. El Allali, Z. El |
| author_sort | Ousbika, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:03:04Z |
| description | UDC 517.5We study the existence of a continuous spectrum of an anisotropic discrete problem, involving variable exponent.The proposed technical approach is based on the variational methods and critical point theory. |
| doi_str_mv | 10.37863/umzh.v73i6.860 |
| first_indexed | 2026-03-24T02:05:52Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i6.860
UDC 517.5
M. Ousbika, Z. El Allali (Oriental Appl. Math. Laboratory, Univ. Mohammed 1, Morocco)
AN EIGENVALUE OF ANISOTROPIC DISCRETE PROBLEM
WITH THREE VARIABLE EXPONENTS
ВЛАСНЕ ЗНАЧЕННЯ АНIЗОТРОПНОЇ ДИСКРЕТНОЇ ЗАДАЧI
З ТРЬОМА ЗМIННИМИ ЕКСПОНЕНТАМИ
We study the existence of a continuous spectrum of an anisotropic discrete problem, involving variable exponent.
The proposed technical approach is based on the variational methods and critical point theory.
Вивчається проблема iснування неперервного спектра анiзотропної дискретної задачi iз змiнною експонентою.
Запропонований пiдхiд базується на варiацiйних методах та теорiї критичних точок.
1. Introduction. Let T > 2 be a positive integer and [1, T ]\BbbZ = \{ 1, 2, 3, . . . , T\} . We consider the
discrete anisotropic problem
- \Delta
\Bigl(
| \Delta u(k - 1)| p(k - 1) - 2\Delta u(k - 1)
\Bigr)
+ | u(k)| p(k) - 2u(k) + | u(k)| q(k) - 2u(k) =
= \lambda | u(k)| r(k) - 2u(k) for k \in [1, T ]\BbbZ ,
u(0) = u(T + 1) = 0,
(1)
where \Delta denotes the forward difference operator defined by \Delta u(k) = u(k + 1) - u(k), \lambda > 0 is
a real parameter, p : [0, T ]Z \rightarrow [2,+\infty ) and q, r : [1, T ]Z \rightarrow [2,+\infty ) are given functions.
In the last years, the study of boundary-value problems for finite difference equations has captured
special attention. This type of problems have an important role in different domains of research, such
as control systems, economics, computer science, physics, artificial or biological neural networks,
cybernetics, ecology and many others. For example, view the recent results in the references [1 – 5,
17, 18]. The important tools employed to study this kind of problem are critical point theory and
variational methods.
However, there is an increasing interest to the existence results to boundary-value problems for
difference equations with p(k)-Laplacian operator, because of their applications in many fields. To
the best of our knowledge, discrete problems involving anisotropic exponents have been discussed for
the first time in [13, 16, 20], the authors proved the existence of a continuous spectrum of eigenvalues
for the problem
- \Delta
\Bigl(
| \Delta u(k - 1)| p(k - 1) - 2\Delta u(k - 1)
\Bigr)
= \lambda | u(k)| q(k) - 2u(k) for k \in [1, T ]\BbbZ ,
u(0) = u(T + 1) = 0,
(2)
In [8 – 11, 19], the authors have studied the existence of at least one solution, multiplicity of
solutions and a sequences of solutions for the problem
c\bigcirc M. OUSBIKA, Z. EL ALLALI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6 839
840 M. OUSBIKA, Z. EL ALLALI
- \Delta
\Bigl(
| \Delta u(k - 1)| p(k - 1) - 2\Delta u(k - 1)
\Bigr)
= \lambda f(k, u(k)) for k \in [1, T ]\BbbZ ,
u(0) = u(T + 1) = 0,
where f : [1, T ]\BbbZ \times \BbbR \mapsto - \rightarrow \BbbR is a continuous function.
More recently, in [6, 7, 12, 14, 15, 21] the authors have been investigated the existence and
multiplicity of solutions for nonlinear discrete boundary-value problems involving p(.)-Laplacian
operator using variational methods.
Our analysis mainly concern the existence and the nonexistence of a weak solutions to problem
(1) more general than (2), with three variable exponents under appropriate assumptions (4) below,
between the functions exponents p(k), q(k) and r(k). Our aim is to determine the concrets intervals
for the parameter \lambda for which problem (1) has, or not has, a nontrivial solutions. More precisely,
we prove the existence of two positive constants \lambda \ast and \lambda \ast with \lambda \ast \leq \lambda \ast such that for each
\lambda \in [\lambda \ast ,+\infty ) the problem (1) has at least one nontrivial solution, while for any \lambda \in (0, \lambda \ast )
problem (1) has no nontrivial solution. For these results, we use some known tools such as the direct
variational methods and the critical point theory.
This paper is organized as follows. The second section is devoted to mathematical preliminaries
and statement of main results. In the third section we give the mains results and thier proofs.
2. Framework and preliminary results. Solutions to boundary-value problem (1) will be
investigated in the space
E = \{ u : [0, T + 1]\BbbZ \rightarrow \BbbR , u(0) = u(T + 1) = 0\} ,
which is a T -dimensional Hilbert space [1], with the inner product
(u, v) =
T\sum
k=0
\Delta u(k)\Delta v(k) \forall u, v \in E.
The associated norm is defined by
\| u\| =
\Biggl(
T\sum
k=0
| \Delta u(k)| 2
\Biggr) 1
2
.
Moreover, it is useful to introduce other norm on E :
| u| m =
\Biggl(
T\sum
k=1
| u(k)| m
\Biggr) 1
m
for m \geq 2. (3)
For any function h : [0, T ]Z \rightarrow [2,+\infty ), we use the following notations:
h - = \mathrm{m}\mathrm{i}\mathrm{n}
k\in [0,T ]\BbbZ
h(k) and h+ = \mathrm{m}\mathrm{a}\mathrm{x}
k\in [0,T ]\BbbZ
h(k).
In this paper, we study the boundary-value problem (1) assuming that the functions p, q and r
satisfy the following assumptions:
2 \leq p - \leq p+ < r - \leq r+ < q - \leq q+. (4)
We start with the following auxillary result, will be are used later.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
AN EIGENVALUE OF ANISOTROPIC DISCRETE PROBLEM . . . 841
Lemma 2.1 [20]. (a) For any m \geq 2 there exists a positive constant Cm such that
T\sum
k=1
| u(k)| m \leq Cm
T+1\sum
k=1
| \Delta u(k - 1)| m \forall u \in E.
(b) There exist two positive constants C1 and C2 such that
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) \geq C1\| u\| p
- - C2 \forall u \in E with \| u\| > 1.
(c) There exists a positive constant C3 such that
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) \geq C3\| u\| p
+ \forall u \in E with \| u\| < 1.
(d)
\sum T+1
k=1
| \Delta u(k - 1)| p(k - 1) \leq (T + 1)
\Bigl(
\| u\| p+ + 1
\Bigr)
\forall u \in E.
Definition 2.1. We say that \lambda > 0 is an eigenvalue of problem (1) if there exists u \in E such
that u \not = 0 and
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) - 2\Delta u(k - 1)\Delta v(k - 1) +
T\sum
k=1
| u(k)| p(k) - 2u(k)v(k)+
+
T\sum
k=1
| u(k)| q(k) - 2u(k)v(k) = \lambda
T\sum
k=1
| u(k)| r(k) - 2u(k)v(k)
for any v \in E.
If \lambda > 0 is an eigenvalue of problem (1), then the corresponding eigenfunction u\lambda \in E is a weak
solution for the problem (1).
To study the boundary-value problem (1), we define the following functionals, for u \in E :
\varphi 0(u) =
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) +
T\sum
k=1
| u(k)| p(k) +
T\sum
k=1
| u(k)| q(k), (5)
\psi 0(u) =
T\sum
k=1
| u(k)| r(k), (6)
\varphi 1(u) =
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1)
p(k - 1)
+
T\sum
k=1
| u(k)| p(k)
p(k)
+
T\sum
k=1
| u(k)| q(k)
q(k)
, (7)
\psi 1(u) =
T\sum
k=1
| u(k)| r(k)
r(k)
, (8)
and, for any \lambda > 0 and u \in E, we define the functional I\lambda as follows:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
842 M. OUSBIKA, Z. EL ALLALI
I\lambda (u) = \varphi 1(u) - \lambda \psi 1(u). (9)
With any fixed \lambda > 0 the functionals I\lambda is differentiable [11, 20], and its derivatives at u reads
\Bigl(
I
\prime
\lambda (u), v
\Bigr)
=
\Bigl(
\varphi
\prime
1(u), v
\Bigr)
- \lambda
\Bigl(
\psi
\prime
1(u), v
\Bigr)
, (10)
for any v \in E, where
\Bigl(
\varphi
\prime
1(u), v
\Bigr)
=
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) - 2\Delta u(k - 1)\Delta v(k - 1)+
+
T\sum
k=1
\Bigl(
| u(k)| p(k) - 2 + | u(k)| q(k) - 2
\Bigr)
u(k)v(k) (11)
and \Bigl(
\psi
\prime
1(u), v
\Bigr)
=
T\sum
k=1
| u(k)| r(k) - 2u(k)v(k). (12)
Remark 2.1. According to equalities (10) – (12) and the Definition 2.1, it follows that \lambda is an
eigenvalue of problem (1) if and only if there exists u\lambda \in E such that u\lambda \not = 0 is a critical point of
the funtional I\lambda .
3. Main results and thier proof. In this paper, we study the boundary-value problem (1)
assuming that the functions p, q and r satisfy the hypothesis given in (4).
Theorem 3.1. Assume that the hypothesis (4) holds, then there exists a positive constant \lambda \star such
that any \lambda \in (0, \lambda \star ) is not an eigenvalue of the problem (1).
Proof. Put
\lambda \star = \mathrm{i}\mathrm{n}\mathrm{f}
u\in E - \{ 0\}
\varphi 0(u)
\psi 0(u)
, (13)
where \varphi 0 and \psi 0 are given by (5) and (6).
Firstly, we show that \lambda \star > 0. From (4) we infer that, for all k \in [1, T ]\BbbZ ,
p(k) < r(k) < q(k),
then, for any u \in E and k \in [1, T ]\BbbZ , we have
| u(k)| r(k) \leq | u(k)| p(k) + | u(k)| q(k). (14)
Then
T\sum
k=1
\Bigl(
| u(k)| p(k) + | u(k)| q(k)
\Bigr)
\geq
T\sum
k=1
| u(k)| r(k),
and we deduce that
\varphi 0(u) \geq \psi 0(u) \forall u \in E.
Therefore,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
AN EIGENVALUE OF ANISOTROPIC DISCRETE PROBLEM . . . 843
\lambda \star \geq 1 > 0.
Secondly, we show that any \lambda \in (0, \lambda \star ) is not an eigenvalue of the boundary-value problem (1).
To do this, assuming by contradiction that there is \lambda \in (0, \lambda \star ) an eigenvalue of problem (1), then by
Remark 2.1, we deduce that there exists u\lambda \in E such that u\lambda \not = 0 and I
\prime
\lambda (u\lambda ) = 0. So,\Bigl(
\varphi
\prime
1(u\lambda ), v
\Bigr)
= \lambda
\Bigl(
\psi
\prime
1(u\lambda ), v
\Bigr)
\forall v \in E.
In particular, for v = u\lambda , we get
\varphi 0(u\lambda ) = \lambda \psi 0(u\lambda ).
Since u\lambda \not = 0, it follows that \varphi 0(u\lambda ) > 0 and \psi 0(u\lambda ) > 0. Then from (13) and the fact that
\lambda < \lambda \star , we deduce that
\varphi 0(u\lambda ) \geq \lambda \star \psi 0(u\lambda ) > \lambda \psi 0(u\lambda ) = \varphi 0(u\lambda ).
This inequality is absurd, then the proof is completed.
Theorem 3.2. Assume that the hypothesis (4) holds, then there exists a positive constant \lambda \star such
that \lambda \star \leq \lambda \star and each \lambda \in [\lambda \star ,+\infty ) is an eigenvalue of the problem (1).
We need to prove the following lemmas which will be used to show the Theorem 3.2.
Lemma 3.1. If the condition (4) is true, then
\mathrm{l}\mathrm{i}\mathrm{m}
\| u\| \rightarrow 0
\varphi 0(u)
\psi 0(u)
= +\infty .
Proof. For any k \in [1, T ]\BbbZ , we have r - \leq r(k) \leq r+. Then, for any u \in E, we get
| u(k)| r(k) \leq | u(k)| r - + | u(k)| r+ .
Summing for k from 1 to T, we obtain, for any u \in E,
\psi 0(u) \leq
\Biggl(
T\sum
k=1
| u(k)| r - +
T\sum
k=1
| u(k)| r+
\Biggr)
.
By using Lemma 2.1(a), we infer that
\psi 0(u) \leq
\Biggl(
Cr -
T+1\sum
k=1
| \Delta u(k - 1)| r - + Cr+
T+1\sum
k=1
| \Delta u(k - 1)| r+
\Biggr)
.
Again by Lemma 2.1(d), we deduce that
\psi 0(u) \leq (1 + T )
\Bigl(
Cr - (1 + \| u\| r - ) + Cr+(1 + \| u\| r+)
\Bigr)
. (15)
Next, for any u \in E, with \| u\| < 1, from (5) and Lemma 2.1(c), we have
\varphi 0(u) \geq C3\| u\| p
+
. (16)
Then, for any u \in E with \| u\| < 1, small enough, from the inequalities (15) and (16), we get
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
844 M. OUSBIKA, Z. EL ALLALI
\varphi 0(u)
\psi 0(u)
\geq C3
(1 + T )
\| u\| p+
Cr - (1 + \| u\| r - ) + Cr+(1 + \| u\| r+)
.
Since r+ \geq r - > p+, passing to the limit as \| u\| \rightarrow 0, in the above inequality we prove that
\mathrm{l}\mathrm{i}\mathrm{m}\| u\| \rightarrow 0
\varphi 0(u)
\psi 0(u)
= +\infty .
Lemma 3.1 is proved.
Lemma 3.2. If the condition (4) is true, then, for any \lambda > 0, I\lambda is coercive, i.e.,
\mathrm{l}\mathrm{i}\mathrm{m}
\| u\| \rightarrow \infty
(\varphi 1(u) - \lambda \psi 1(u)) = +\infty .
Proof. For any u \in E, from (7) we have
\varphi 1(u) =
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1)
p(k - 1)
+
T\sum
k=1
| u(k)| p(k)
2p(k)
+
T\sum
k=1
\Biggl(
| u(k)| p(k)
2p(k)
+
| u(k)| q(k)
q(k)
\Biggr)
\geq
\geq 1
p+
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) +
1
\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
T\sum
k=1
\Bigl(
| u(k)| p(k) + | u(k)| q(k)
\Bigr)
. (17)
Let s fix such that r+ < s < q - , then, for any u \in E and k \in [1, T ]\BbbZ , we get
| u(k)| p(k) + | u(k)| q(k) \geq | u(k)| s,
and, by (17), we obtain
\varphi 1(u) \geq
1
p+
T+1\sum
k=1
| \Delta u(k - 1)| p(k - 1) +
1
\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
| u| ss. (18)
Next, since | u(k)| r(k) \leq (| u(k)| r - + | u(k)| r+), then, from (8), we have
\psi 1(u) \leq
1
r -
\Biggl(
T\sum
k=1
| u(k)| r - +
T\sum
k=1
| u(k)| r+
\Biggr)
. (19)
By using Hölder’s inequality, we prove that, for any u \in E,
T\sum
k=1
| u(k)| r - \leq T
s - r -
s
\Biggl(
T\sum
k=1
\Bigl(
| u(k)| r -
\Bigr) s
r -
\Biggr) r -
s
= A| u| r - s (20)
and
T\sum
k=1
| u(k)| r+ \leq T
s - r+
s
\Biggl(
T\sum
k=1
\Bigl(
| u(k)| r+
\Bigr) s
r+
\Biggr) r+
s
= B| u| r+s , (21)
where
A = T
s - r -
s > 0 and B = T
s - r+
s > 0. (22)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
AN EIGENVALUE OF ANISOTROPIC DISCRETE PROBLEM . . . 845
Therefore, for any u \in E with \| u\| > 1, from (11), inequalities (18) – (21) and Lemma 2.1(b),
we deduce that, for any \lambda > 0,
I\lambda (u) \geq
1
p+
\Bigl(
C1\| u\| p
- - C2
\Bigr)
+
1
\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
| u| ss - \lambda
1
r -
\Bigl(
A| u| r - s +B| u| r+s
\Bigr)
\geq
\geq C1\| u\| p
- - C2
p+
+
| u| ss
2\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
- \lambda
A| u| r - s
r -
+
+
| u| ss
2\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
- \lambda
B| u| r+s
r -
,
so,
I\lambda (u) \geq
C1\| u\| p
- - C2
p+
-
\Bigl(
\alpha | u| r - s - \beta | u| ss
\Bigr)
- (\gamma | u| r+s - \beta | u| ss), (23)
where \alpha =
A\lambda
r -
> 0, \gamma =
B\lambda
r -
> 0 and \beta =
1
2\mathrm{m}\mathrm{a}\mathrm{x}(2p+, q+)
> 0.
Let h1, h2 : ]0,+\infty [\rightarrow \BbbR two real functions, given by
h1(t) = \alpha tr
- - \beta ts and h2(t) = \gamma tr
+ - \beta ts \forall t > 0.
It is easy to show that h1 and h2 achieves its positive global maximums M1 = h1(t1) and M2 =
= h2(t2), where
t1 =
\biggl(
\alpha r -
\beta s
\biggr) 1
s - r -
> 0 and t2 =
\biggl(
\gamma r+
\beta s
\biggr) 1
s - r+
> 0.
Then we infer that h1(t) \leq M1 and h2(t) \leq M2 \forall t > 0.
Therefore, for any u \in E with \| u\| > 1 and \lambda > 0, from (23), we get that
I\lambda (u) \geq
C1\| u\| p
- - C2
p+
- M1 - M2. (24)
Passing to the limit as \| u\| \rightarrow \infty in (24), we complete the proof of Lemma 3.2.
Proof of Theorem 3.2. Put
\lambda \star = \mathrm{i}\mathrm{n}\mathrm{f}
u\in E - \{ 0\}
\varphi 1(u)
\psi 1(u)
. (25)
Step 1. We show that \lambda \star > 0.
By (14) and from (4), we infer that, for any u \in E,
| u(k)| p(k)
p(k)
+
| u(k)| q(k)
q(k)
\geq | u(k)| r(k)
q(k)
\geq | u(k)| r(k)
q+
.
Then
T\sum
k=1
| u(k)| p(k)
p(k)
+
T\sum
k=1
| u(k)| q(k)
q(k)
\geq r -
q+
T\sum
k=1
| u(k)| r(k)
r(k)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
846 M. OUSBIKA, Z. EL ALLALI
and
\varphi 1(u) \geq
r -
q+
\psi 1(u) \forall u \in E.
So,
\lambda \star \geq r -
q+
> 0.
Thus, step 1 is verified.
Step 2. We show that each \lambda \in (\lambda \star ,+\infty ) is an eigenvalue of the problem (1).
We fix \lambda \in (\lambda \star ,+\infty ). According to Lemma 3.2, we have I\lambda is coercive and is weakly lower
semicontinuous. Applying Theorem 1.2 in [22] in order to prove that there exists u\lambda \in E as a global
minimum point of I\lambda and, thus, as a critical point of I\lambda .
In order to finish the proof of step 2, it is enough to prove that u\lambda is nontrivial. Indeed, since
\lambda > \lambda \star and from (13) there exists v\lambda \in E such that
\varphi 1(v\lambda ) < \lambda \psi 1(v\lambda ),
that is,
I\lambda (v\lambda ) < 0,
Then u\lambda \not = 0E , and we conclude that there exists u\lambda \in E with u\lambda \not = 0E , which is a critical point of
I\lambda or \lambda is an eigenvalue of the problem (1). Thus, step 2 is true.
Step 3. We show that \lambda \star is an eigenvalue of problem (1). For this we will prove that there exists
u \star \in E such that u \star \not = 0 and I
\prime
\lambda \star (u \star ) = 0.
Let \lambda n > 0 be a minimizing sequence for \lambda \star (\mathrm{i}.\mathrm{e}., \lambda n > \lambda \star ). From step 2, we deduce that for
each n there exists a sequence \{ un\} \in E such that un \not = 0 and I
\prime
\lambda n
(un) = 0. So,\Bigl(
\varphi
\prime
1(un), v
\Bigr)
= \lambda n
\Bigl(
\psi
\prime
1(un), v
\Bigr)
\forall v \in E. (26)
For v = un, we find that
\varphi 0(un) - \lambda n\psi 0(un) = 0, (27)
and passing to the limit as n\rightarrow +\infty in relation (27), we have
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
(\varphi 0(un) - \lambda n\psi 0(un)) = 0. (28)
On the other hand, a similar argument as those used in proof of Lemma 3.2, we show that
\mathrm{l}\mathrm{i}\mathrm{m}
\| un\| \rightarrow +\infty
(\varphi 0(un) - \lambda n\psi 0(un)) = +\infty . (29)
Then, from (28) and (29) we show that the sequence \{ un\} is bounded in E. Since E is a finite
dimensional Hilbert space, then there exists a subsequence, still denoted by \{ un\} and u \star \in E, such
that un \rightarrow u \star as n\rightarrow +\infty .
Therefore, passing to the limit as n\rightarrow +\infty in relation (26), we get that\Bigl(
\varphi
\prime
1(u
\star ), v
\Bigr)
= \lambda \star
\Bigl(
\psi
\prime
1(u
\star ), v
\Bigr)
\forall v \in E
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
AN EIGENVALUE OF ANISOTROPIC DISCRETE PROBLEM . . . 847
or \Bigl(
I
\prime
\lambda \star (u \star ), v
\Bigr)
= 0 \forall v \in E.
So, u \star is a critical point of I\lambda \star .
It remains to show that u \star is nontrivial. In fact, if not we have un \rightarrow 0 in E as n \rightarrow +\infty or
\| un\| \rightarrow 0, then Lemma 3.1 implies that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\biggl(
\varphi 0(un)
\psi 0(un)
\biggr)
= +\infty .
From the equality (27), we deduce that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\biggl(
\varphi 0(un)
\psi 0(un)
\biggr)
= \lambda \star ,
which is a contradiction. Consequently, u \star \not = 0 and, thus, \lambda \star is an eigenvalue of the problem (1).
Step 4. We prove that \lambda \star \leq \lambda \star . Since \lambda \star is an eigenvalue of the problem (1), so Theorem 3.1
implies that
\lambda \star /\in ]0;\lambda \star [.
Since 0 < \lambda \star , therefore, \lambda \star \leq \lambda \star .
Theorem 3.2 is proved.
Remark 3.1. We are not able deduce whether \lambda \star = \lambda \star or \lambda \star < \lambda \star . In the latter case, an intersting
open problem consern the existence of eigenvalue of problem (1) in the interval [\lambda \star < \lambda \star ).
References
1. R. P. Agarwal, K. Perera, D. O’Regan, Multiple positive solutions of singular and nonsingular discrete problems via
variational methods, Nonlinear Anal., 58, 69 – 73 (2004).
2. R. P. Agarwal, K. Perera, D. O’Regan, Multiple positive solutions of singular p-Laplacian discrete problems via
variational methods, Adv. Difference Equat., 2, 93 – 99 (2009).
3. A. Cabada, A. Iannizzotto, S. Tersian, Multiple solutions for discrete boundary-value problems, J. Math. Anal. and
Appl., 356, 418 – 428 (2009).
4. G. Bonanno, P. Candito, Infinitely many solutions for a class of discrete nonlinear boundary-value problems, Appl.
Anal., 884, 605 – 616 (2009).
5. G. Bonanno, P. Candito, G. Dágui, Variational methods on finite dimensional Banach spaces and discrete problems,
Adv. Nonlinear Stud., 14, 915 – 939 (2014).
6. G. Bonanno, G. Dágui, Two non-zero solutions for elliptic Dirichlet problems, Z. Anal. und Anwend., 35, 449 – 464
(2016).
7. M. Galewski, R. Wieteska, Existence and multiplicity results for boundary-value problems connected with the discrete
p(.)-Laplacian on weighted finite graphs, Appl. Math. and Comput., 290, 376 – 391 (2016).
8. M. Galewski, R. Wieteska, On the system of anisotropic discrete BVPs, J. Difference Equat. and Appl., 19, № 7,
1065 – 1081 (2013).
9. M. Galewski, R. Wieteska, Existence and multiplicity of positive solutions for discrete anisotropic equations, Turkish
J. Math., 38, 297 – 310 (2014).
10. M. Galewski, R. Wieteska, Positive solutions for anisotropic discrete boundary-value problems, Electron. J. Different.
Equat. and Appl., 2013, № 32, 1 – 9 (2013).
11. M. Galewski, Sz. Glab, On the discrete boundary-value problem for anisotropic equation, J. Math. Anal. and Appl.,
386, 956 – 965 (2012).
12. M. Galewski, G. Molica Bisci, R. Wieteska, Existence and multiplicity of solutions to discrete inclusions with the
p(k)-Laplacian problem, J. Difference Equat. and Appl., 21, № 10, 887 – 903 (2015).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
848 M. OUSBIKA, Z. EL ALLALI
13. B. Kone, S. Ouaro, Weak solutions for anisotropic discrete boundary-value problems, J. Difference Equat. and Appl.,
16, № 2, 1 – 11 (2010).
14. M. Khaleghi Moghadam, J. Henderson, Triple solutions for a dirichlet boundary-value problem involving a perturbed
discrete p(k)-Laplacian operator, Open Math., 15, 1075 – 1089 (2017).
15. M. Khaleghi Moghadam, M. Avci, Existence results to a nonlinear p(k)-Laplacian difference equation, J. Difference
Equat. and Appl., 23, № 10, 1652 – 1669 (2017).
16. B. Kone, S. Ouaro, Weak solutions for anisotropic discrete boundary-value problems, J. Difference Equat. and Appl.,
17, № 10, 1537 – 1547 (2011).
17. A. Kristaly, M. Mihăilescu, V. Rădulescu, S. Tersian, Spectral estimates for a nonhomogeneous difference problem,
Commun. Contemp. Math., 12, 1015 – 1029 (2010).
18. G. Molica Bisci, D. Repovs, Existence of solutions for p-Laplacian discrete equations, Appl. Math. and Comput.,
242, 454 – 461 (2014).
19. G. Molica Bisci, D. Repovs, On sequences of solutions for discrete anisotropic equations, Expo. Math., 32, 284 – 295
(2014).
20. M. Mihăilescu, V. Rădulescu, S. Tersian, Eigenvalue problems for anisotropic discrete boundary-value problems, J.
Difference Equat. and Appl., 15, 557 – 567 (2009).
21. J. Smejda, R. Wieteska, On the dependence on parameters for second order discrete boundary-value problems with
the p(k)-Laplacian, Opuscula Math., 344, 851 – 870 (2014).
22. M. Struwe, Variational methods, applications to nonlinear partial differential equations and Hamiltonian systems,
Springer-Verlag, Berlin (1986).
Received 06.10.17,
after revision — 25.03.21
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 6
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| id | umjimathkievua-article-860 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:52Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/82/e9676be45022fbeef8bf20f9676ff782.pdf |
| spelling | umjimathkievua-article-8602022-03-26T11:03:04Z An eigenvalue of anisotropic discrete problem with three variable exponents An eigenvalue of anisotropic discrete problem with three variable exponents Ousbika, M. Allali, Z. El Ousbika, M. Allali, Z. El Allali, Z. El anisotropic discrete problem Eigenvalue critical points theory anisotropic discrete problem Eigenvalue critical points theory UDC 517.5We study the existence of a continuous spectrum of an anisotropic discrete problem, involving variable exponent.The proposed technical approach is based on the variational methods and critical point theory. УДК 517.5 Власне значення анiзотропної дискретної задачiз трьома змiнними експонентами Вивчається проблема iснування неперервного спектра анiзотропної дискретної задачi iз змiнною експонентою.Запропонований пiдхiд базується на варiацiйних методах та теорiї критичних точок. Institute of Mathematics, NAS of Ukraine 2021-06-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/860 10.37863/umzh.v73i6.860 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 6 (2021); 839 - 848 Український математичний журнал; Том 73 № 6 (2021); 839 - 848 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/860/9029 |
| spellingShingle | Ousbika, M. Allali, Z. El Ousbika, M. Allali, Z. El Allali, Z. El An eigenvalue of anisotropic discrete problem with three variable exponents |
| title | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_alt | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_full | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_fullStr | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_full_unstemmed | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_short | An eigenvalue of anisotropic discrete problem with three variable exponents |
| title_sort | eigenvalue of anisotropic discrete problem with three variable exponents |
| topic_facet | anisotropic discrete problem Eigenvalue critical points theory anisotropic discrete problem Eigenvalue critical points theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/860 |
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