On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory
UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem $$ {-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega,&n...
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Institute of Mathematics, NAS of Ukraine
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| author | Taarabti, S. El Allali , Z. Ben Haddouch, K. Taarabti, S. El Allali , Z. Ben Haddouch, K. |
| author_facet | Taarabti, S. El Allali , Z. Ben Haddouch, K. Taarabti, S. El Allali , Z. Ben Haddouch, K. |
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| description | UDC 517.9
The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem
$$ {-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega, $$
$$ u = \Delta u = 0\quad  \mbox{on}\quad \partial\Omega.$$
By using variational approach and Krasnoselskii's genus theory, we prove the existence and multiplicity of solutions for the $p(x)$-Kirchhoff-type equation.  |
| doi_str_mv | 10.37863/umzh.v72i6.6019 |
| first_indexed | 2026-03-24T03:25:19Z |
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DOI: 10.37863/umzh.v72i6.6019
UDC 517.9
S. Taarabti (Nat. School Appl. Sci. Agadir Ibn Zohr Univ., Morocco),
Z. El Allali (Multidisciplinary Faculty of Nador, Mohammed First Univ., Oujda, Morocco),
K. Ben Haddouch (Nat. School Appl. Sci. Fes Sidi Mohammed Ben Abdellah Univ., Morocco)
ON \bfitp (\bfitx )-KIRCHHOFF-TYPE EQUATION
INVOLVING \bfitp (\bfitx )-BIHARMONIC OPERATOR VIA GENUS THEORY
ПРО \bfitp (\bfitx )-РIВНЯННЯ ТИПУ КIРХГОФА
IЗ \bfitp (\bfitx )-БIГАРМОНIЧНИМ ОПЕРАТОРОМ З ТОЧКИ ЗОРУ ТЕОРIЇ РОДУ
The paper deals with the existence and multiplicity of nontrivial weak solutions for the p(x)-Kirchhoff-type problem
- M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \Delta 2
p(x)u = f(x, u) in \Omega ,
u = \Delta u = 0 on \partial \Omega .
By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for
the p(x)-Kirchhoff-type equation.
Розглядаються проблеми iснування та множинностi нетривiальних слабких розв’язкiв p(x)-задачi типу Кiрхгофа
- M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \Delta 2
p(x)u = f(x, u) в \Omega ,
u = \Delta u = 0 на \partial \Omega .
Використовуючи варiацiйний пiдхiд та теорiю роду Красносельського, ми доводимо iснування та множиннiсть
розв’язкiв для p(x)-рiвняння типу Кiрхгофа.
1. Introduction. In this paper, we are interested in the following problem:
- M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \Delta 2
p(x)u = f(x, u) in \Omega ,
u = \Delta u = 0 on \partial \Omega ,
(1.1)
where \Omega is a bounded domain in \BbbR N , N \geq 2, with smooth boundary \partial \Omega , \Delta 2
p(x)u =
= \Delta
\bigl(
| \Delta u| p(x) - 2\Delta u
\bigr)
is the p(x)-biharmonic operator, p is a continuous function on \Omega with 1 <
< p(x) < N.
We assume that M(t) and f(x, t) satisfy the following assumptions:
(M1) M : \BbbR + \rightarrow \BbbR + is a continuous function and satisfies the (polynomial growth) condition
m1t
\beta - 1 \leq M(t) \leq m2t
\alpha - 1
for all t > 0 and m1, m2 real numbers such that 0 < m1 \leq m2 and \alpha \geq \beta > 1;
c\bigcirc S. TAARABTI, Z. EL ALLALI, K. BEN HADDOUCH, 2020
842 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING p(x)-BIHARMONIC OPERATOR VIA GENUS THEORY 843
(f1) f : \Omega \times \BbbR \rightarrow \BbbR is a continuous function such that
d1| t| s(x) - 1 \leq f(x, t) \leq d2| t| q(x) - 1
for all t \geq 0 and for all x \in \Omega , where d1, d2 are positive constants and s, q \in C(\Omega ) such that
1 < s(x) < q(x) < p\ast (x) <
Np(x)
N - p(x)
for all x \in \Omega ;
(f2) f is an odd function according to t, that is,
f(x, t) = - f(x, - t)
for all t \in \BbbR and for all x \in \Omega .
The problem (1.1) is related to the stationary problem of a model presented by Kirchhoff [16].
More precisely, Kirchhoff proposed a model given by the equation
\rho
\partial 2u
\partial t2
-
\left( \rho 0
h
+
E
2L
L\int
0
\bigm| \bigm| \bigm| \bigm| \partial u\partial x
\bigm| \bigm| \bigm| \bigm| 2 dx
\right) \partial 2u
\partial x2
= 0, (1.2)
which extends the classical D’Alembert’s wave equation, by considering the effect of the changing
in the length of the string during the vibration. A distinguishing feature of equation (1.2) is that
the equation contains a nonlocal coefficient
\rho 0
h
+
E
2L
\int L
0
\bigm| \bigm| \bigm| \bigm| \partial u\partial x
\bigm| \bigm| \bigm| \bigm| 2 dx which depends on the average
1
2L
\int L
0
\bigm| \bigm| \bigm| \bigm| \partial u\partial x
\bigm| \bigm| \bigm| \bigm| 2 dx, and hence the equation is no longer a pointwise identity. The parameters in (1.2)
have the following meanings: L is the length of the string, h is the area of the cross-section, E is
the Young modulus of the material, \rho is the mass density and \rho 0 is the initial tension.
The operator \Delta 2
p(x)u = \Delta
\bigl(
| \Delta u| p(x) - 2\Delta u
\bigr)
is said to be the p(x)-biharmonic, and becomes p-
biharmonic when p(x) = p (a constant). The study of problems involving variable exponent growth
conditions has a strong motivation due to the fact that they can model various phenomena which
arise in the study of elastic mechanics [19], electrorheological fluids [20] or image restoration [1].
In recent years, elliptic problems involving p-Kirchhoff-type operators have been studied in many
papers, we refer to [2, 4], in which the authors have used different methods to get the existence of
solutions for (1.1) in the case when p(x) = p is a constant.
The study of the Kirchhoff-type equations has already been extended to the case involving the
p-Laplacian operator given by the formula \Delta pu = \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| \Delta u| p - 2\Delta u
\bigr)
[8]
- M
\left( \int
\Omega
| \nabla u| p dx
\right) p - 1
\Delta pu = f(x, u) in \Omega ,
u = 0 on \partial \Omega ,
we point out that establishing conditions on M and f for which Kirchhoff-type equations possess
solutions is the key argument.
In the case p(x)-Laplacian operator, in [3], the authors studied the Kirchhoff-type equation
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
844 S. TAARABTI, Z. EL ALLALI, K. BEN HADDOUCH
- M
\left( \int
\Omega
1
p(x)
| \nabla u| p(x) dx
\right) \Delta p(x)u = f(x, u) in \Omega ,
u = 0 on \partial \Omega ,
(1.3)
by using the Krasnoselskii’s genus theory, they showed the existence and multiplicity of the solutions
of the the problem (1.3).
Motivated by the above papers and the results in [17, 18], we consider (1.1) to study the existence
and multiplicity of the solutions.
This paper is organized as follows. In Section 2, we present some necessary preliminary results
on variable exponent Sobolev spaces. Next, we give the main results and proofs about the existence
and multiplicity of the solutions.
2. Preliminaries. In order to deal with p(x)-biharmonic operator problems, we need some
results on spaces Lp(x)(\Omega ) and W k,p(x)(\Omega ) and some properties of p(x)-biharmonic operator, which
we will use later (for details see [21, 22]).
Define the generalized Lebesgue space by
Lp(x)(\Omega ) =
\left\{ u : \Omega - \rightarrow \BbbR , measurable and
\int
\Omega
| u(x)| p(x) dx < \infty
\right\} ,
where p \in C+(\Omega ) and
C+(\Omega ) =
\bigl\{
h \in C(\Omega ) : h(x) > 1 \forall x \in \Omega
\bigr\}
.
Denote
p+ = \mathrm{m}\mathrm{a}\mathrm{x}
x\in \Omega
p(x), p - = \mathrm{m}\mathrm{i}\mathrm{n}
x\in \Omega
p(x),
and, for all x \in \Omega and k \geq 1,
p\ast (x) =
\left\{
Np(x)
N - p(x)
, if p(x) < N,
+\infty , if p(x) \geq N,
and
p\ast k(x) =
\left\{
Np(x)
N - kp(x)
, if kp(x) < N,
+\infty , if kp(x) \geq N.
One introduces in Lp(x)(\Omega ) the following norm:
| u| p(x) = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \mu > 0;
\int
\Omega
\bigm| \bigm| \bigm| \bigm| u(x)\mu
\bigm| \bigm| \bigm| \bigm| p(x) dx \leq 1
\right\} ,
and the space
\bigl(
Lp(x)(\Omega ), | .| p(x)
\bigr)
is a Banach.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING p(x)-BIHARMONIC OPERATOR VIA GENUS THEORY 845
Proposition 2.1 [9, 21]. The space
\bigl(
Lp(x)(\Omega ), | .| p(x)
\bigr)
is separable, uniformly convex, reflexive
and its conjugate space is Lq(x)(\Omega ), where q(x) is the conjugate function of p(x), i.e.,
1
p(x)
+
1
q(x)
= 1 \forall x \in \Omega .
For all u \in Lp(x)(\Omega ) and v \in Lq(x)(\Omega ), the H\"older’s type inequality\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
uv dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\biggl(
1
p -
+
1
q -
\biggr)
| u| p(x)| v| q(x)
holds true.
Furthermore, if we define the mapping \rho : Lp(x)(\Omega ) \rightarrow \BbbR by
\rho (u) =
\int
\Omega
| u| p(x) dx,
then the following relations hold.
Proposition 2.2 [21, 22].
(i) | u| p(x) < 1 (= 1, > 1) \leftrightarrow \rho (u) < 1 (= 1, > 1),
(ii) | u| p(x) > 1 \Rightarrow | u| p
-
p(x) \leq \rho (u) \leq | u| p
+
p(x),
(iii) | un - u| p(x) - \rightarrow 0 \leftrightarrow \rho (un - u) - \rightarrow 0.
The Sobolev space with variable exponent W k,p(x)(\Omega ) is defined by
W k,p(x)(\Omega ) =
\bigl\{
u \in Lp(x)(\Omega ) : D\alpha u \in Lp(x)(\Omega ), | \alpha | \leq k
\bigr\}
,
where
D\alpha u =
\partial | \alpha | u
\partial x\alpha 1
1 \partial x\alpha 2
2 . . . \partial x\alpha N
N
,
is the derivation in distribution sense, with \alpha = (\alpha 1, \alpha 2, . . . , \alpha N ) is a multiindex and | \alpha | =
=
\sum N
i=1
\alpha i.
The space W k,p(x)(\Omega ) is equipped with the norm
\| u\| k,p(x) =
\sum
| \alpha | \leq k
| D\alpha u| p(x),
also becomes a Banach, separable and reflexive space. For more details, we refer to [9, 10, 13, 15].
Proposition 2.3 [9]. Let p, r \in C+(\Omega ) such that r(x) \leq p\ast k(x) for all x \in \Omega . Then there is a
continuous embedding
W k,p(x)(\Omega ) \lhook \rightarrow Lr(x)(\Omega ).
If we replace \leq with <, the embedding is compact.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
846 S. TAARABTI, Z. EL ALLALI, K. BEN HADDOUCH
We denote by W
k,p(x)
0 (\Omega ) the closure of C\infty
0 (\Omega ) in W k,p(x)(\Omega ).
Consider the function space X defined by
X = W 2,p(x)(\Omega ) \cap W
1,p(x)
0 (\Omega ).
Then X is a separable and reflexive Banach space equipped with the norm
\| u\| = \| u\| 1,p(x) + \| u\| 2,p(x).
Remark 2.1. According to [7], the norm \| u\| 2,p(x) is equivalent to the norm | \Delta u| p(x) in the space
W 2,p(x)(\Omega ) \cap W
1,p(x)
0 (\Omega ). Consequently, the norms \| .\| 2,p(x), \| .\| and | \Delta .| p(x) are equivalent.
Proposition 2.4 [14]. If we put
J(u) =
\int
\Omega
| \Delta u| p(x)dx,
then, for all u, un \in X, the following relations hold true:
(i) \| u\| < 1 (= 1; > 1) \Leftarrow \Rightarrow J(u) < 1 (= 1; > 1),
(ii) \| u\| > 1 =\Rightarrow \| u\| p - \leq J(u) \leq \| u\| p+ ,
for all un \in X, we have
(iii) \| un\| - \rightarrow 0 \Leftarrow \Rightarrow J(un) - \rightarrow 0,
(iv) \| un\| - \rightarrow \infty \Leftarrow \Rightarrow J(un) - \rightarrow \infty .
Proposition 2.5 [12]. Let X be a Banach space and
\Lambda (u) =
\int
\Omega
1
p(x)
| \Delta u| p(x) dx.
The functional \Lambda : X - \rightarrow \BbbR is convex. The mapping \Lambda \prime : X - \rightarrow X \prime (\Lambda \prime is the Fréchet derivative of
\Lambda ) is a strictly monotone, bounded homeomorphism and of (S+), namely,
un \rightharpoonup u (weakly) and \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\langle \Lambda \prime (un), un - u\rangle \leq 0 implies un - \rightarrow u (strongly),
where X = W 2,p(x)(\Omega ) \cap W
1,p(x)
0 (\Omega ).
Definition 2.1. We say that u \in X is a weak solution of (1.1) if
M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \int
\Omega
| \Delta u| p(x) - 2\Delta u\Delta \varphi dx =
\int
\Omega
f(x, u)\varphi dx
for all \varphi \in X.
We associate to the problem (1.1) the energy functional, defined as I : X - \rightarrow \BbbR ,
I(u) = \widehat M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) -
\int
\Omega
F (x, u) dx,
where \widehat M(t) =
\int t
0
M(s) ds and F (x, u) =
\int u
0
f(x, t) dt.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING p(x)-BIHARMONIC OPERATOR VIA GENUS THEORY 847
Standard arguments show that I \in C1(X,\BbbR ) and
\langle I \prime (u), v\rangle = \mathrm{l}\mathrm{i}\mathrm{m}
h\rightarrow 0
I(u+ hv) - I(u)
h
=
= M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \int
\Omega
| \Delta u| p(x) - 2\Delta u\Delta v dx -
\int
\Omega
f(x, u)v dx
for any u, v \in X.
Hence, we can notice that critical points of functional I are the weak solutions for problem (1.1).
For simplicity, we use di, to denote the general nonnegative or positive constant (the exact value
may change from line to line).
3. Main results and proofs. We present some basic notions on the Krasnoselskii’s genus (see
[5, 6]) that we will use in the proof of our main results.
Let Y be a real Banach space. Set
\Re =
\bigl\{
E \subset Y \setminus \{ 0\} : E is compact and E = - E
\bigr\}
.
Definition 3.1 [6, 23]. Let E \in \Re and Y = \BbbR k. The genus \gamma (E) of E is defined by
\gamma (E) = \mathrm{m}\mathrm{i}\mathrm{n}
\bigl\{
k \geq 1; there exists an odd continuous mapping \phi : E \rightarrow \BbbR k\setminus \{ 0\}
\bigr\}
.
If such a mapping does not exist for any k > 0, we set \gamma (E) = \infty . Note also that if E is a subset,
which consists of finitely many pairs of points, then \gamma (E) = 1. Moreover, from definition, \gamma (\varnothing ) = 0.
A typical example of a set of genus k is a set, which is homeomorphic to a (k - 1)-dimensional
sphere via an odd map.
Now, we will give some results of Krasnoselskii’s genus, which are necessary throughout the
present paper.
Theorem 3.1 [6, 23]. Let Y = \BbbR N and \partial \Omega be the boundary of an open, symmetric, and
bounded subset \Omega \subset \BbbR N with 0 \in \Omega . Then \gamma (\partial \Omega ) = N.
Corollary 3.1 [6, 23]. \gamma (SN - 1) = N (recall the notation SN - 1 which stands for the unit sphere
in \BbbR N ).
Remark 3.1 [6, 23]. If Y is of infinite dimension and separable and S is the unit sphere in Y,
then \gamma (S) = \infty .
Definition 3.2 [6, 23]. We say that the functional satisfies the Palais – Smale condition (PS) if
every sequence (un) \subset Y such that
| I(un)| \leq C and I \prime (un) - \rightarrow 0 as n - \rightarrow \infty
contains a convergent subsequence in the norm of Y.
The first result of the present paper is the following theorem.
Theorem 3.2. Suppose (M1), (f1), and (f2) hold. If p(x) < q(x) < p\ast (x) for all x \in \Omega and
q+ < \beta p - , then the problem (1.1) has infinitely many solutions.
The following result obtained by Clark in [11] is the main idea, which we use in the proof of
Theorem 3.2.
Theorem 3.3. Let J \in C1(X,\BbbR ) be a functional satisfying the (PS) condition. Furthermore, let
us suppose that:
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
848 S. TAARABTI, Z. EL ALLALI, K. BEN HADDOUCH
(i) J is bounded from below and even,
(ii) there is a compact set K \in \Re such that \gamma (K) = k and \mathrm{s}\mathrm{u}\mathrm{p}x\in K J(x) < J(0).
Then J possesses at least k pairs of distinct critical points, and their corresponding critical values
are less than J(0).
Lemma 3.1. Suppose (M1), (f1), and q+ < \beta p - hold. Then I is bounded from below.
Proof. From (M1) and (f1), we have
I(u) = \widehat M
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) -
\int
\Omega
F (x, u) dx \geq
\geq m1
\int
\Omega
1
p(x)
| \Delta u| p(x) dx\int
0
\rho \beta - 1d\rho - d2
q -
\int
\Omega
| u| q(x) dx =
=
m1
\beta
\left( \int
\Omega
1
p(x)
| \Delta u| p(x) dx
\right) \beta
- d2
q -
\int
\Omega
| u| q(x) dx
and by Propositions 2.2, 2.3, and 2.4, for all u \in X, we get
I(u) \geq m1
\beta (p+)\beta
\bigl(
\alpha (\| u\| )
\bigr) \beta - d2C
q+
q -
\| u\| q+ , (3.1)
where \alpha : [0,+\infty [ - \rightarrow \BbbR is defined by
\alpha (t) =
\left\{ tp
+
, if t \leq 1,
tp
-
, if t > 1.
As \beta p+ \geq \beta p - > q+, I is bounded from below.
Lemma 3.2. Suppose (M1), (f1), and q+ < \beta p - hold. Then I satisfies the (PS) condition.
Proof. Let (un) in X be a sequence such that
I(un) - \rightarrow c and I \prime (un) - \rightarrow 0 as n \rightarrow \infty . (3.2)
From (3.2), we have | I(un)| \leq d3. This fact, combined with (3.1), implies that
d3 \geq I(un) \geq
m1
\alpha (p+)\alpha
\| un\| \beta p
- - d4
q -
\| un\| q
+
,
where \| un\| > 1. Because q+ < \beta p - , I is coercive, we deduce that (un) is bounded in X. Hence,
there exists a subsequence, still denoted by (un) \subset X and u \in X such that
un \rightharpoonup u as n \rightarrow \infty in X.
From Proposition 2.3, we obtain
un - \rightarrow u in Lq(x)(\Omega ),
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING p(x)-BIHARMONIC OPERATOR VIA GENUS THEORY 849
un - \rightarrow u a.e. \Omega .
Then by (3.2), we have \langle I \prime (un), un - u\rangle - \rightarrow 0. Thus,
\bigl\langle
I \prime (un), un - u
\bigr\rangle
= M
\left( \int
\Omega
1
p(x)
| \Delta un| p(x) dx
\right) \int
\Omega
| \Delta un| p(x) - 2\Delta un(\Delta un - \Delta u) dx -
-
\int
\Omega
f(x, un)(un - u) dx - \rightarrow 0.
By (f1) and Proposition 2.1, it follows that\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
\Omega
f(x, un)(un - u) dx
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq d2
\int
\Omega
| un| q(x) - 1| un - u| dx \leq
\leq d6
\bigm| \bigm| | un| q(x) - 1
\bigm| \bigm|
q\prime (x)
| un - u| q(x).
Since (un) converges strongly to u in Lq(x)(\Omega ), that is, | un - u| q(x) \rightarrow 0 as n \rightarrow \infty , we get\int
\Omega
f(x, un)(un - u) dx \rightarrow 0.
Hence,
M
\left( \int
\Omega
1
p(x)
| \Delta un| p(x) dx
\right) \int
\Omega
| \Delta un| p(x) - 2\Delta un(\Delta un - \Delta u) \rightarrow 0.
From (M1), it follows \int
\Omega
| \Delta un| p(x) - 2\Delta un(\Delta un - \Delta u) \rightarrow 0.
By Proposition 2.5, we get that un \rightarrow u in X.
Proof of Theorem 3.2. We consider (see [5])
\Re k = \{ E \subset \Re : \gamma (E) \geq k\} ,
ck = \mathrm{i}\mathrm{n}\mathrm{f}
E\in \Re k
\mathrm{s}\mathrm{u}\mathrm{p}
u\in E
I(u), k = 1, 2, . . . ,
then we have
- \infty < c1 \leq c2 \leq . . . \leq ck \leq ck+1 \leq . . . .
Now, we will prove that ck < 0 for every k \in \BbbN . Since X is a separable Banach space, for any
k \in \BbbN , we can choose a k-dimensional linear subspace Xk of X such that Xk \subset C\infty
0 (\Omega ). As the
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
850 S. TAARABTI, Z. EL ALLALI, K. BEN HADDOUCH
norms on Xk are equivalent, there exists rk \in (0, 1) such that u \in Xk with \| u\| \leq rk implies
| u| L\infty \leq \delta .
Set Sk
rk
= \{ u \in Xk : \| u\| = rk\} . By the compactness of Sk
rk
and condition (f1), there exists a
constant \eta k > 0 such that\int
\Omega
F (x, u) dx \geq d1
s+
\int
\Omega
| u| s(x) dx \geq \eta k \forall u \in Sk
rk
. (3.3)
From (M1) and (f1), for u \in Sk
rk
and t \in (0, 1), we have
I(tu) = \widehat M
\left( \int
\Omega
| \Delta tu| p(x)
p(x)
dx
\right) -
\int
\Omega
F (x, tu) dx \leq
\leq m2
\left( \int
\Omega
| \Delta tu| p(x)
p(x)
\right) - d1
s+
\int
\Omega
| tu| s(x) dx \leq
\leq m2
\alpha (p - )\alpha
t\alpha p
-
r\alpha p
-
k - ts
+
\eta k. (3.4)
Since s+ < q - \leq q+ < \beta p - \leq \alpha p - , we can find tk \in (0, 1) and \varepsilon k > 0 such that
I(tku) \leq - \varepsilon k < 0 \forall u \in Sk
rk
,
that is,
I(u) \leq - \varepsilon k < 0 \forall u \in Sk
tkrk
.
It is clear that \gamma (Sk
tkrk
) = k, so ck \leq - \varepsilon k < 0. Therefore, by Lemma 3.1, Lemma 3.2 and above
results, we can apply Theorem 3.3 to obtain that the functional I admits at least k pairs of distinct
critical points, and since k is arbitrary, we obtain infinitely many critical points of I.
Theorem 3.4. Suppose (M1), (f1), and (f2) hold. If q(x) < p(x) < p\ast (x) for all x \in \Omega , then
the problem (1.1) has a sequence of solution \{ \pm uk : k = 1, 2, . . .\} such that I(\pm uk) < 0.
Proof. We follow the same steps applied in the proof of the Lemma 3.1, and the fact q+ < p - ,
we prove that I is coercive. Because I is weak lower semicontinuous, I attaints its minimum on X,
that is, (1.1) has a solution. By the coercivity of I, we know that I satisfies (PS) condition on X.
And from condition (f2) , I is even.
In the rest of the proof, since we use the same arguments which we used in the proof of the
Theorem 3.2, we omit the discussions here.
Hence, if we follow the similar processes as we did in (3.3) and (3.4), and the fact s+ < q - \leq
\leq q+ < p - < \alpha p - , we can find tk \in (0, 1) and \varepsilon k > 0 such that
I(u) \leq - \varepsilon k < 0 \forall u \in Sk
tkrk
.
Clearly, \gamma (Sk
tkrk
) = k, so ck \leq - \varepsilon k < 0. By Krasnoselskii’s genus, each ck is a critical value of I,
then there is a sequence of solutions \{ \pm uk : k = 1, 2, . . .\} such that I(\pm uk) < 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
ON p(x)-KIRCHHOFF-TYPE EQUATION INVOLVING p(x)-BIHARMONIC OPERATOR VIA GENUS THEORY 851
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ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
|
| id | umjimathkievua-article-6019 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:19Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b7/87e6f8bfa02377661eabf5cf63c822b7.pdf |
| spelling | umjimathkievua-article-60192022-03-26T11:01:51Z On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory Taarabti, S. El Allali , Z. Ben Haddouch, K. Taarabti, S. El Allali , Z. Ben Haddouch, K. UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the $p(x)$-Kirchhoff-type problem $$&nbsp;{-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega,&nbsp;$$ $$ u = \Delta u = 0\quad&nbsp; \mbox{on}\quad \partial\Omega.$$ By using variational approach and Krasnoselskii's genus theory, we prove the existence and multiplicity of solutions for the $p(x)$-Kirchhoff-type equation.&nbsp; УДК 517.9 Розглядаються проблеми iснування та множинностi нетривiальних слабких розв’язкiв $\mathcal{p}(x)$-задачi типу Кiрхгофа $$&nbsp;{-M}\!\left(\displaystyle\int\limits_{\Omega}\frac{1}{p(x)}|\Delta u|^{p(x)}\,dx\right)\!\Delta_{p(x)}^{2} u = f(x,u)\quad \mbox{in}\quad \Omega, $$ $$ u = \Delta u = 0\quad&nbsp; \mbox{on}\quad \partial\Omega.$$ Використовуючи варiацiйний пiдхiд та теорiю роду Красносельського, ми доводимо iснування та множиннiсть розв’язкiв для $\mathcal{p}(x)$-рiвняння типу Кiрхгофа. Institute of Mathematics, NAS of Ukraine 2020-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6019 10.37863/umzh.v72i6.6019 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 6 (2020); 842-851 Український математичний журнал; Том 72 № 6 (2020); 842-851 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6019/8719 |
| spellingShingle | Taarabti, S. El Allali , Z. Ben Haddouch, K. Taarabti, S. El Allali , Z. Ben Haddouch, K. On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_alt | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_full | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_fullStr | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_full_unstemmed | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_short | On $\mathcal{p}(x)$-Kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| title_sort | on $\mathcal{p}(x)$-kirchhoff-type equation involving $\mathcal{p}(x)$-biharmonic operator via genus theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6019 |
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