Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type
UDC 517.9 We use variational methods to study the existence and multiplicity of solutions for an nonhomogeneous $p$-Kirchhoff equation involving the critical Sobolev exponent.
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| author | Benaissa, A. Matallah, A. Benaissa, A. Matallah, A. |
| author_facet | Benaissa, A. Matallah, A. Benaissa, A. Matallah, A. |
| author_sort | Benaissa, A. |
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| description | UDC 517.9
We use variational methods to study the existence and multiplicity of solutions for an nonhomogeneous $p$-Kirchhoff equation involving the critical Sobolev exponent. |
| first_indexed | 2026-03-24T02:22:24Z |
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UDC 517.9
A. Benaissa (Laboratory of Analysis and Control of PDEs, Djillali Liabes Univ., Sidi Bel Abbes, Algeria),
A. Matallah (Ecole Préparatoire en Sciences Economiques, Commerciales et Sciences de Gestion, Tlemcen, Algeria)
NONHOMOGENEOUS ELLIPTIC KIRCHHOFF EQUATIONS
OF THE \bfitp -LAPLACIAN TYPE
НЕОДНОРIДНI ЕЛIПТИЧНI РIВНЯННЯ КIРХГОФФА ТИПУ \bfitp -ЛАПЛАСIАНА
We use variational methods to study the existence and multiplicity of solutions for an nonhomogeneous p-Kirchhoff
equation involving the critical Sobolev exponent.
Варiацiйнi методи застосовуються для вивчення iснування та кратностi розв’язкiв неоднорiдного елiптичного p-
рiвняння Кiрхгоффа з критичним показником Соболева.
1. Introduction. This paper deals with the existence and multiplicity of solutions to the following
Kirchhoff problem with the critical Sobolev exponent
- (a \| u\| p + b)\Delta pu = up
\ast - 1 + \lambda g (x) in IRN ,
u \in W 1,p
\bigl(
IRN
\bigr)
,
(\scrP \lambda )
where N \geq 3, 1 < p < N, \Delta p is the p-Laplacian operator, \| .\| is the usual norm in W 1,p
\bigl(
IRN
\bigr)
given by
\| u\| p =
\int
IRN
| \nabla u| pdx,
p\ast = pN/ (N - p) is the critical Sobolev exponent of the embedding\bigl(
W 1,p(IRN ), \| .\|
\bigr)
\lhook \rightarrow
\Bigl(
Lq
\bigl(
IRN
\bigr)
, \| .\| q
\Bigr)
with q \in [p, p\ast ] and \| u\| qq =
\int
IRN
| u| qdx is the norm in Lq
\bigl(
IRN
\bigr)
, a and b are two positive constants,
\lambda is a positive parameter and g belongs to
\bigl(
W 1,p
\bigl(
IRN
\bigr) \bigr) \ast
such that
\int
IRN
gu\ast dx \not = 0, where u\ast is a
function defined below in (1),
\Bigl( \bigl(
W 1,p(IRN )
\bigr) \ast
is the dual of W 1,p(IRN )
\Bigr)
.
In recent years, the Kirchhoff-type problems in bounded or unbounded domaine have been studied
in many papers by using variational methods. Some interesting studies can be found in [1, 4 – 6, 8].
Since the Sobolev embedding
\bigl(
W 1,p(IRN ), \| .\|
\bigr)
\lhook \rightarrow
\Bigl(
Lq
\bigl(
IRN
\bigr)
, \| .\| q
\Bigr)
is not compact for all q \in [p,
p\ast ], many authors considered the following Kirchhoff-type problem without the critical Sobolev
exponent
- (a \| u\| p + b)\Delta pu+ V (x)u = h(x, u) in IRN , (\scrP V )
where V \in C
\bigl(
IRN , IR
\bigr)
and h \in C
\bigl(
IRN \times IR, IR
\bigr)
is subcritical, satisfies sufficiently conditions to
show the boundedness of any Palais Smale or Cerami sequence. They imposed some conditions on
c\bigcirc A. BENAISSA, A. MATALLAH, 2020
184 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
NONHOMOGENEOUS ELLIPTIC KIRCHHOFF EQUATIONS OF THE p-LAPLACIAN TYPE 185
the weight function V (x) for recovering the compactness of Sobolev embedding (see, for example,
[11]). We should mention here that, to the best of our knowledge, there is no result concerning
Kirchhoff equations of p-Laplacian type with the presence of nonlinear term of critical growth and
without potential term in higher dimension.
Main result of this paper is the following theorem.
Theorem 1.1. Assume that a > 0, b > 0, N = 3k, p = 2k, and k \in IN\ast . Then there exists
\Lambda \ast > 0 such that problem (\scrP \lambda ) has at least two nontrivial solutions for any \lambda \in (0,\Lambda \ast ) .
This paper is organized as follows. In Section 2, we give some technical results which allow us
to give a variational approach of our main result that we prove in Section 3.
2. Auxiliary results. In this paper we use the following notation: X = W 1,2k
\bigl(
IR3k
\bigr)
where
k \in IN\ast , | | .| | \ast denotes the norm in X\ast , B\rho is the ball centred at 0 and of radius \rho , \rightarrow (resp., \rightharpoonup )
denotes strong (resp., weak) convergence and \circ n (1) denotes \circ n (1) \rightarrow 0 as n \rightarrow \infty . S is the best
Sobolev constant defined by
S = \mathrm{i}\mathrm{n}\mathrm{f}
u\in W 1,p(IRN
)
\| u\| p
\| u\| pp\ast
, (1)
it is well known that S is attained in IRN by a function u\ast (see [10]).
Since our approach is variational, we define the functional I\lambda by
I\lambda (u) =
a
4k
| | u| | 4k + b
2k
| | u| | 2k - 1
6k
\| u\| 6k6k - \lambda
\int
IR3k
gu dx
for all k \in IN\ast and u \in X. It is clear that I\lambda is well defined in X and belongs to C1 (X, IR) .
u \in X is said to be a weak solution of problem (\scrP \lambda ) if it satisfies\Bigl(
a| | u| | 2k + b
\Bigr) \int
IR3k
| \nabla u| 2k - 2\nabla u\nabla \varphi dx -
\int
IR3k
u6k - 1\varphi dx - \lambda
\int
IR3k
g\varphi dx = 0 for all \varphi \in X.
To prove our main result, we need following lemmas.
Lemma 2.1. Let (un) \subset X be a (PS)c sequence of I\lambda for some c \in IR. Then un \rightharpoonup u in X for
some u with I \prime \lambda (u) = 0.
Proof. We have
c+ \circ n (1) = I\lambda (un) and \circ n (1) =
\bigl\langle
I \prime \lambda (un) , un
\bigr\rangle
, (2)
then
c+ \circ n (1) = I\lambda (un) -
1
6k
\bigl\langle
I \prime \lambda (un) , un
\bigr\rangle
\geq
\geq a
12k
| | un| | 4k +
b
3k
| | un| | 2k - \lambda
6k - 1
6k
\| g\| \ast \| un\| .
Hence, (un) is bounded in X. Up to a subsequence if necessary, we obtain
un \rightharpoonup u in X,un \rightharpoonup u in L6k
\Bigl(
IR3k
\Bigr)
, un \rightarrow u a.e. in IR3k, and \nabla un \rightarrow \nabla u a.e. in IR3k.
Thus, \langle I \prime \lambda (un) , \varphi \rangle = 0 for all \varphi \in C\infty
0
\bigl(
IR3k
\bigr)
, which means that I \prime \lambda (u) = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
186 A. BENAISSA, A. MATALLAH
Lemma 2.2. There exist positive constants \Lambda 1, \rho 1 and \delta 1 such that for all \lambda \in (0,\Lambda 1) we have
I\lambda (u)| \partial B\rho 1
\geq \delta 1 and I\lambda (u)| B\rho 1
\geq - 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast .
Proof. Let u \in X\setminus \{ 0\} and \rho = \| u\| . Then by Sobolev and Hölder inequalities, we have
I\lambda (u) \geq
a
4k
\rho 4k +
b
2k
\rho 2k - S - 3
6k
\rho 6k - \lambda \| g\| \ast \rho .
By applying the inequality
\alpha \beta <
1
p1
\alpha p1 +
1
p2
\beta p2
for any \alpha , \beta , p1, p2 > 0 such that
1
p1
+
1
p2
= 1, we get
\lambda \| g\| \ast \rho =
\Bigl(
\lambda (4k - 1)/4k \| g\| \ast
\Bigr) \Bigl(
\lambda 1/4k\rho
\Bigr)
\leq
\leq 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast +
1
2k
\lambda 1/2\rho 2k.
Therefore, if \lambda \leq b2, then
I\lambda (u) \geq \Psi (\rho ) - 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast ,
where
\Psi (\rho ) =
a
4k
\rho 4k - S - 3
6k
\rho 6k and, hence, \mathrm{m}\mathrm{a}\mathrm{x}
\rho \geq 0
\Psi (\rho ) = \Psi
\Bigl( \bigl(
aS3
\bigr) 1/2k\Bigr)
.
Taking
\Lambda 1 = \mathrm{m}\mathrm{i}\mathrm{n}
\Biggl\{
b2,
\biggl[
k
2k - 1
\Psi
\Bigl( \bigl(
aS3
\bigr) 1/2k\Bigr) \biggr] 4k - 2
4k - 1
\| g\|
4k
1 - 4k
\ast
\Biggr\}
and \delta 1 =
1
2
\Psi
\Bigl( \bigl(
aS3
\bigr) 1/2k\Bigr)
.
Then the conclusion holds.
Lemma 2.3. Let (un) \subset X be a (PS)c sequence of I\lambda for some c \in IR such that un \rightharpoonup u in
X. Then
either un \rightarrow u or c \geq I\lambda (u) + Ca,b,k,S ,
where Ca,b,k,S =
\Bigl[
a+
\bigl(
a2 + 4bS3
\bigr) 1/2\Bigr] \biggl[ aS6
48k
\Bigl[
a+
\bigl(
a2 + 4bS3
\bigr) 1/2\Bigr]
+
bS3
6k
\biggr]
.
Proof. By the proof of Lemma 2.1, we obtain (un) is a bounded sequence in X. Furthermore,
if vn = un - u, we derive vn \rightharpoonup 0 in X. Then, by using Brezis – Lieb lemma [3], we have
\| un\| 2k = \| vn\| 2k + \| u\| 2k + on(1) and \| un\| 6k6k = \| vn\| 6k6k + \| u\| 6k6k + on(1). (3)
Putting together (2) and (3), we get
c+ on(1) = I\lambda (u) +
a
4k
\| vn\| 4k +
b
2k
\| vn\| 2k +
a
2k
\| vn\| 2k \| u\| 2k -
1
6k
\| vn\| 6k6k
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
NONHOMOGENEOUS ELLIPTIC KIRCHHOFF EQUATIONS OF THE p-LAPLACIAN TYPE 187
and
on(1) = a \| vn\| 4k + b \| vn\| 2k + 2a \| vn\| 2k \| u\| 2k - \| vn\| 6k6k . (4)
Therefore,
c+ on(1) = I\lambda (un) -
1
6k
\bigl\langle
I \prime \lambda (un) , un
\bigr\rangle
=
= I\lambda (u) +
a
12k
\| vn\| 4k +
b
3k
\| vn\| 2k +
a
6k
\| vn\| 2k \| u\| 2k . (5)
Assume that \| vn\| \rightarrow l > 0, then by (5) and the Sobolev inequality we obtain
S - 3l6k \geq al4 + bl2k,
this implies that
l2k \geq a
2
S3 +
1
2
S3
\bigl(
a2 + 4bS - 3
\bigr) 1/2
.
From the above inequality and (5), we conclude
c \geq I\lambda (u) +
a
12k
l4k +
b
3k
l2k \geq
\geq I\lambda (u) +
aS6
48k
\Bigl[
a+
\bigl(
a2 + 4bS3
\bigr) 1/2\Bigr] 2
+
bS3
6k
\Bigl[
a+
\bigl(
a2 + 4bS3
\bigr) 1/2\Bigr]
=
= I\lambda (u) + Ca,b,k,S .
Lemma 2.3 is proved.
3. Proof of the Theorem 1.1. The proof is given in two parts.
3.1. Existence of a local minimizer. By Lemma 2.2, we define
c1 = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
I\lambda (u) ;u \in \=B\rho 1
\bigr\}
.
Since g \not \equiv 0, we can choose \Phi \in C\infty
0
\bigl(
IR3k\setminus \{ 0\}
\bigr)
such that
\int
IR3k
g\Phi dx > 0. Hence, there exists
t0 > 0 small enough such that \| t0\Phi \| < \rho 1 and
I\lambda (t0\Phi ) =
a
4k
t4k0 \| \Phi \| 4k + b
2k
t2k0 \| \Phi \| 2k - t6k0
6k
\| \Phi \| 6k6k - \lambda t0
\int
IR3k
g\Phi dx < 0,
which implies that c1 < 0 = I\lambda (0). Using the Ekeland’s variational principle [7], for the complete
metric space \=B\rho 1 with respect to the norm of X, we obtain the result that there exists a (PS)c1
sequence (un) \subset \=B\rho 1 such that un \rightharpoonup u1 in X for some u1 with \| u1\| \leq \rho 1. Assume un \not \rightarrow u1 in
X, then it follows from Lemma 2.3 that
c1 \geq I\lambda (u) + Ca,b,k,S > c1,
which is a contradiction. Thus u1 is a nontrivial solution of (\scrP \lambda ) with negative energy.
3.2. Existence of Mountain Pass type solution. The existence of a Mountain Pass type solution
follows immediately from the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
188 A. BENAISSA, A. MATALLAH
Lemma 3.1. Let \Lambda 2 > 0 such that
- 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast + Ca,b,k,S > 0 \forall \lambda \in (0,\Lambda 2).
Then there exist z\ast \in X and 0 < \Lambda \ast \leq \Lambda 2 such that
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
I\lambda (tz\ast ) < c1 + Ca,b,k,S \forall \lambda \in (0,\Lambda \ast ).
Proof. Since
\int
IR3k
gu\ast dx \not = 0, we can choose z\ast (x) = u\ast (x) or z\ast (x) = - u\ast (x) such
that
\int
IR3k
gz\ast dx > 0.
We consider functions
\Phi 1(t) =
at4k
4k
\| z\ast \| 4k +
bt2k
2k
\| z\ast \| 2k -
t6k
6k
\| z\ast \| 6k6k
and
\Phi 2(t) = \Phi 1(t) - \lambda t
\int
IR3k
gz\ast dx.
So, for all \lambda \in (0,\Lambda 2), we have
\Phi 2(0) = 0 < - 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast + Ca,b,k,S .
Hence, by the continuity of \Phi 2(t), there exists t1 > 0 small enough such that
\Phi 2(t) < - 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast + Ca,b,k,S \forall t \in (0, t1) .
On the other hand, the function \Phi 1(t) attains its maximum at
t2k\ast =
a \| z\ast \| 4k +
\Bigl(
a2 \| z\ast \| 8k + 4b \| z\ast \| 2k \| z\ast \| 6k6k
\Bigr) 1/2
2 \| z\ast \| 6k6k
.
From the definition of S, we have
at4k\ast
4k
\| z\ast \| 4k =
a
4k
\| z\ast \| 4k
\left[ a \| z\ast \| 4k +
\Bigl(
a2 \| z\ast \| 8k + 4b \| z\ast \| 2k \| z\ast \| 6k6k
\Bigr) 1/2
2 \| z\ast \| 66
\right]
2
=
=
a
16k
\left[ a \| z\ast \| 6k
\| z\ast \| 6k6k
+
\Biggl[
a2 \| z\ast \| 12k + 4b \| z\ast \| 6k \| z\ast \| 6k6k
\| z\ast \| 12k6k
\Biggr] 1/2
\right] 2
=
=
a
16k
\Bigl[
aS3 +
\bigl[
a2S6 + 4bS3
\bigr] 1/2\Bigr] 2
.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
NONHOMOGENEOUS ELLIPTIC KIRCHHOFF EQUATIONS OF THE p-LAPLACIAN TYPE 189
Similarly, we obtain
bt2k\ast
2k
\| z\ast \| 2k =
b
4k
\Bigl[
aS3 +
\bigl(
a2S6 + 4bS3
\bigr) 1/2\Bigr]
and
t6k\ast
6k
\| z\ast \| 6k6k =
S - 3
48k
\Bigl[
aS3 +
\bigl(
a2S6 + 4bS3
\bigr) 1/2\Bigr] 3
.
By the above estimates, we deduce that \mathrm{s}\mathrm{u}\mathrm{p}t\geq 0\Phi 1(t) \leq Ca,b,k,S .
On the other hand, by using Lemma 2.2, we see that
c1 \geq - 2k - 1
2k
\lambda (4k - 1)/(4k - 2) \| g\| 2k/(2k - 1)
\ast \forall \lambda \in (0,\Lambda 1),
furthermore, if
\lambda <
\left( 2kt1
2k - 1
\| g\|
- 2k
2k - 1
\ast
\int
IR3k
gz\ast dx
\right)
4k - 2
,
we get
c1 > - t1\lambda
\int
IR3k
gz\ast dx.
Taking
\Lambda \ast = \mathrm{m}\mathrm{i}\mathrm{n}
\left\{ \Lambda 1,\Lambda 2,
\left( 2kt1
2k - 1
\| g\|
- 2k
2k - 1
\ast
\int
IR3k
gz\ast dx
\right)
4k - 2\right\} .
Then we deduce that
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
I\lambda (tz\ast ) < c1 + Ca,b,k,S \forall \lambda \in (0,\Lambda \ast ) .
Note that I\lambda (0) = 0 and I\lambda (t2z\ast ) < 0 for t2 large enough, also from Lemma 2.2, we know that
I\lambda (u)| \partial B\rho 1
\geq \delta 1 > 0 \forall \lambda \in (0,\Lambda 1).
Then, by the Mountain Pass theorem [2], there exists a (PS)c2 sequence, where
c2 = \mathrm{i}\mathrm{n}\mathrm{f}
\gamma \in \Gamma
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,1]
I\lambda (\gamma (t)) ,
with
\Gamma = \{ \gamma \in C([0, 1] , X), \gamma (0) = 0 and \gamma (1) = t2z\ast \} .
By using Lemma 2.1, we have (un) has a subsequence, still denoted by (un) , such that un \rightharpoonup u2 in
X for some u2. Furthermore, we know, by Lemma 3.1, that
\mathrm{s}\mathrm{u}\mathrm{p}
t\geq 0
I\lambda (tz\ast ) < Ca,b,k,S + c1 \forall \lambda \in (0,\Lambda \ast ),
then, from Lemma 2.3, we deduce that un \rightarrow u2 in X. Thus we obtain a critical point u2 of I\lambda
satisfying I\lambda (u2) > 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
190 A. BENAISSA, A. MATALLAH
References
1. C. O. Alves, F. J. S. A. Correa, T. F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff-type,
Comput. Math. Appl., 49, 85 – 93 (2005).
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14, 349 – 381 (1973).
3. H. Brézis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc.
Amer. Math. Soc., 88, 486 – 490 (1983).
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5. S. J. Chena, L. Li, Multiple solutions for the nonhomogeneous Kirchhoff equation on IRN , Nonlinear Anal.: Real
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Received 13.02.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 2
|
| id | umjimathkievua-article-2359 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:24Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/48/d87990827527995e13b33eaa089e7348.pdf |
| spelling | umjimathkievua-article-23592020-04-07T12:11:44Z Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type Неоднорідні еліптичні рівняння Кірхгофа типу $p$-Лапласіана Benaissa, A. Matallah, A. Benaissa, A. Matallah, A. рівняння Кірхгофа критичний показник Соболева Kirchhoff equations laplacian type UDC 517.9 We use variational methods to study the existence and multiplicity of solutions for an nonhomogeneous $p$-Kirchhoff equation involving the critical Sobolev exponent. УДК 517.9 Варіаційні методи застосовуються для вивчення існування та кратності розв'язків неоднорідного еліптичного $p$-рівняння Кірхгоффа з критичним показником Соболева. Institute of Mathematics, NAS of Ukraine 2020-02-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2359 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 2 (2020); 184-190 Український математичний журнал; Том 72 № 2 (2020); 184-190 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2359/1558 Copyright (c) 2020 A. Benaissa,A. Benaissa |
| spellingShingle | Benaissa, A. Matallah, A. Benaissa, A. Matallah, A. Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title | Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title_alt | Неоднорідні еліптичні рівняння Кірхгофа типу $p$-Лапласіана |
| title_full | Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title_fullStr | Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title_full_unstemmed | Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title_short | Nonhomogeneous elliptic Kirchhoff equations of $p$-Laplacian type |
| title_sort | nonhomogeneous elliptic kirchhoff equations of $p$-laplacian type |
| topic_facet | рівняння Кірхгофа критичний показник Соболева Kirchhoff equations laplacian type |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2359 |
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