Existence results for a class of Kirchhoff-type systems with combined nonlinear effects
UDC 517.9 We study the existence of positive solutions for a nonlinear system $$M_1 \bigl(\int_{\Omega} |\nabla u|^p dx\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x|^{ap}| \nabla u|^{p-2}\nabla u) = \lambda | x| (a+1)p+c_1f(u, \upsilon ),\; x \in \Omega ,$$ $$M2 \bigl( \int_{ \Omega }| \nabla v| q d...
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| author | Afrouzi, G. A. Shakeri, S. Zahmatkesh, H. Афрузі, Г. А. Шакері, С. Захматкеш, Г. |
| author_facet | Afrouzi, G. A. Shakeri, S. Zahmatkesh, H. Афрузі, Г. А. Шакері, С. Захматкеш, Г. |
| author_sort | Afrouzi, G. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T10:12:29Z |
| description | UDC 517.9
We study the existence of positive solutions for a nonlinear system
$$M_1 \bigl(\int_{\Omega} |\nabla u|^p dx\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x|^{ap}| \nabla u|^{p-2}\nabla u) = \lambda | x| (a+1)p+c_1f(u, \upsilon ),\; x \in \Omega ,$$
$$M2 \bigl( \int_{ \Omega }| \nabla v| q dx
\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x| bq|\nabla \upsilon | q 2\nabla \upsilon ) = \lambda | x| (b+1)q+c_2g(u, \upsilon ),\; x \in \Omega ,$$
$$u = \upsilon = 0, x \in \partial \Omega ,$$
where $\Omega$ is a bounded smooth domain in $R^N$ with $0 \in \Omega,\; 1 < p, q < N, 0 \leq a < \cfrac{N-p}{p}, 0 \leq b < \cfrac{N-q}{p},$ а $c_1, c_2, \lambda$ are positive parameters. Here, $M_1,M_2, f$, and g satisfy certain conditions. We use the method of sub- and
supersolutions to establish our results. |
| first_indexed | 2026-03-24T02:05:45Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.9
G. A. Afrouzi (Dep. Math., Univ. Mazandaran, Babolsar, Iran),
S. Shakeri (Dep. Math., Ayatollah Amoli Branch, Islamic Azad Univ., Amol, Iran),
H. Zahmatkesh (Dep. Math., Univ. Mazandaran, Babolsar, Iran)
EXISTENCE RESULTS FOR A CLASS OF KIRCHHOFF-TYPE SYSTEMS
WITH COMBINED NONLINEAR EFFECTS
РЕЗУЛЬТАТИ ПРО IСНУВАННЯ РОЗВ’ЯЗКIВ ДЛЯ ОДНОГО КЛАСУ СИСТЕМ
ТИПУ КIРХГОФА З КОМБIНОВАНИМИ НЕЛIНIЙНИМИ ЕФЕКТАМИ
We study the existence of positive solutions for a nonlinear system
- M1
\left( \int
\Omega
| \nabla u| p dx
\right) \mathrm{d}\mathrm{i}\mathrm{v} (| x| - ap| \nabla u| p - 2\nabla u) = \lambda | x| - (a+1)p+c1f(u, \upsilon ), x \in \Omega ,
- M2
\left( \int
\Omega
| \nabla v| q dx
\right) \mathrm{d}\mathrm{i}\mathrm{v} (| x| - bq| \nabla \upsilon | q - 2\nabla \upsilon ) = \lambda | x| - (b+1)q+c2g(u, \upsilon ), x \in \Omega ,
u = \upsilon = 0, x \in \partial \Omega ,
where \Omega is a bounded smooth domain in \BbbR N with 0 \in \Omega , 1 < p, q < N, 0 \leq a <
N - p
p
, 0 \leq b <
N - q
q
, and
c1, c2, and \lambda are positive parameters. Here, M1,M2, f, and g satisfy certain conditions. We use the method of sub- and
supersolutions to establish our results.
Розглянуто проблему iснування додатних розв’язкiв нелiнiйної системи
- M1
\left( \int
\Omega
| \nabla u| p dx
\right) \mathrm{d}\mathrm{i}\mathrm{v} (| x| - ap| \nabla u| p - 2\nabla u) = \lambda | x| - (a+1)p+c1f(u, \upsilon ), x \in \Omega ,
- M2
\left( \int
\Omega
| \nabla v| q dx
\right) \mathrm{d}\mathrm{i}\mathrm{v} (| x| - bq| \nabla \upsilon | q - 2\nabla \upsilon ) = \lambda | x| - (b+1)q+c2g(u, \upsilon ), x \in \Omega ,
u = \upsilon = 0, x \in \partial \Omega ,
де \Omega — обмежена гладка область в \BbbR N з 0 \in \Omega , 1 < p, q < N, 0 \leq a <
N - p
p
, 0 \leq b <
N - q
q
, а c1, c2, \lambda —
додатнi параметри. Величини M1,M2, f та g задовольняють деякi умови. Нашi результати отримано за допомогою
методу суб- та суперрозв’язкiв.
1. Introduction. In this paper we study the existence of positive solution for the nonlinear system
- M1
\left( \int
\Omega
| \nabla u| p dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - ap| \nabla u| p - 2\nabla u
\bigr)
= \lambda | x| - (a+1)p+c1f(u, \upsilon ), x \in \Omega ,
c\bigcirc G. A. AFROUZI, S. SHAKERI, H. ZAHMATKESH, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4 571
572 G. A. AFROUZI, S. SHAKERI, H. ZAHMATKESH
- M2
\left( \int
\Omega
| \nabla v| q dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - bq| \nabla \upsilon | q - 2\nabla \upsilon
\bigr)
= \lambda | x| - (b+1)q+c2g(u, \upsilon ), x \in \Omega , (1.1)
u = v = 0 on \partial \Omega ,
where \Omega is a bounded smooth domain of \BbbR N with 0 \in \Omega , 1 < p, q < N, 0 \leq a <
N - p
p
,
0 \leq b <
N - q
q
and c1, c2, and \lambda are positive parameters. Here M1, M2 satisfy the following
condition:
(H1) Mi : \BbbR +
0 \rightarrow \BbbR +, i = 1, 2, are two continuous and increasing functions and 0 < mi \leq
\leq Mi(t) \leq mi,\infty for all t \in \BbbR +
0 , where \BbbR +
0 : = [0,+\infty ).
Moreover f, g : [0,\infty )\times [0,\infty ) \rightarrow [0,\infty ) are nondecreasing continuous functions. System (1.1)
is related to the stationary problem of a model introduced by Kirchhoff [16]. More precisely, Kirch-
hoff proposed a model given by the equation
\rho
\partial 2u
\partial t2
-
\left( P0
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)
where \rho , P0, h, and E are all constants. This equation extends the classical D’Alembert wave
equation. A distinguishing feature of equation (1.2) is that the equations a nonlocal coefficient
P0
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; hence, the equation is no
longer a pointwise identity. Nonlocal problems can be used for modeling, for example, physical and
biological systems for which u describes a process which depends on the average of itself, such as
the population density. The elliptic problems involving more general operator, such as the degenerate
quasilinear elliptic operator given by - \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - ap| \nabla u| p - 2\nabla u
\bigr)
, were motivated by the following
Caaffarelli, Kohn and Nirenberg’s inequality (see [6, 24]). The study of this type of problem is
motivated by its various applications, for example, in fluid mechanics, in Newtonian fluids, in flow
through porous media and in glaciology (see [4, 8]). On the other hand, quasilinear elliptic systems
has an extensive practical background. It can be used to describe the multiplicate chemical reaction
catalyzed by the catalyst grains under constant or variant temperature, it can be used in the theory of
quasiregular and quasiconformal mappings in Riemannian manifolds with boundary (see [13, 23]),
and can be a simple model of tubular chemical reaction, more naturally, it can be a correspondence
of the stable station of dynamical system determined by the reaction-diffusion system, see Ladde and
Lakshmikantham et al. [17]. More naturally, it can be the populations of two competing species [10].
So, the study of positive solutions of elliptic systems has more practical meanings. We refer to
[5, 12, 14, 22] for additional results on elliptic problems. We are inspired by the ideas in the
interesting paper [21], in which the authors considered (1.1) in the case M1(t) = M2(t) \equiv 1.
Using the sub-supersolution method combining a comparison principle introduced in [3], the authors
established the existence of a positive solution for (1.1) when the parameter \lambda is large. The concepts
of sub- and supersolution were introduced by Nagumo [19] in 1937 who proved, using also the
shooting method, the existence of at least one solution for a class of nonlinear Sturm – Liouville
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
EXISTENCE RESULTS FOR A CLASS OF KIRCHHOFF-TYPE SYSTEMS . . . 573
problems. In fact, the premises of the sub- and supersolution method can be traced back to Picard.
He applied, in the early 1880s, the method of successive approximations to argue the existence of
solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear
problems. This is the starting point of the use of sub- and supersolutions in connection with monotone
methods. Picard’s techniques were applied later by Poincaré [20] in connection with problems arising
in astrophysics.
2. Preliminary results. In this paper, we denote W 1,r
0
\bigl(
\Omega , | x| - ar
\bigr)
, the completion of C\infty
0 (\Omega ),
with respect to the norm \| u\| =
\biggl( \int
\Omega
| x| - ar| \nabla u| rdx
\biggr) 1/r
with r = p, q. To precisely state our
existence result we consider the eigenvalue problem
- \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - sr| \nabla \phi | r - 2\nabla \phi
\bigr)
= \lambda | x| - (s+1)p+t| \phi | r - 2\phi , x \in \Omega , \phi = 0, x \in \partial \Omega . (2.1)
For r = p, s = a and t = c1, let \phi 1,p be the eigenfunction corresponding to the first eigenvalue \lambda 1,p
of (2.1) such that \phi 1,p(x) > 0 in \Omega and \| \phi 1,p\| \infty = 1, and for r = q, s = b and t = c2, let \phi 1,q
be the eigenfunction corresponding to the first eigenvalue \lambda 1;q of (2.1) such that \phi 1,q(x) > 0 in \Omega
and \| \phi 1,q\| \infty = 1 (see [18, 25]). It can be shown that
\partial \phi 1,r
\partial n
< 0 on \partial \Omega for r = p, q. Here n is the
outward normal. This result is well known and hence, depending on \Omega there exist positive constants
\epsilon , \delta , \sigma r such that
\lambda 1,r| x| - (s+1)r+t\phi r1,r - | x| - sr| \nabla \phi 1,r| r \leq - \epsilon , x \in \Omega \delta , (2.2)
\phi 1,r \geq \sigma r , x \in \Omega \setminus \Omega \delta , (2.3)
with r = p, q; s = a, b; t = c1, c2 and \Omega \delta =
\bigl\{
x \in \Omega | d(x, \partial \Omega ) \leq \delta
\bigr\}
(see [18]). We will also
consider the unique solution (\zeta p(x), \zeta q(x)) \in W 1,p
0 (\Omega , \| x\| - ap)\times W 1,q
0 (\Omega , \| x\| - bq) for the system
- \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - ap| \nabla \zeta p| p - 2\nabla \zeta p
\bigr)
= | x| - (a+1)p+c1 , x \in \Omega ,
- \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - bq| \nabla \zeta q| q - 2\nabla \zeta q
\bigr)
= | x| - (b+1)q+c2 , x \in \Omega ,
u = \upsilon = 0, x \in \partial \Omega ,
to discuss our existence result. It is known that \zeta r(x) > 0 in \Omega and
\partial \zeta r(x)
\partial n
< 0 on \partial \Omega for r = p, q
(see [18]).
3. Existence results. In this section, we shall establish our existence result via the method
of sub- and supersolutions. A pair of nonnegative functions (\psi 1, \psi 2), (z1, z2) are called a weak
subsolution and supersolution of (1.1) if they satisfy (\psi 1, \psi 2) = (0, 0) = (z1, z2) on \partial \Omega and
M1
\left( \int
\Omega
| \nabla \psi 1| p dx
\right) \int
\Omega
| x| - ap| \nabla \psi 1| p - 2\nabla \psi 1 \cdot \nabla w dx \leq \lambda
\int
\Omega
| x| - (a+1)p+c1f(\psi 1, \psi 2)w dx,
M2
\left( \int
\Omega
| \nabla \psi 2| q dx
\right) \int
\Omega
| x| - bq| \nabla \psi 2| q - 2\nabla \psi 2 \cdot \nabla w dx \leq \lambda
\int
\Omega
| x| - (b+1)q+c2g(\psi 1, \psi 2)w dx,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
574 G. A. AFROUZI, S. SHAKERI, H. ZAHMATKESH
and
M1
\left( \int
\Omega
| \nabla z1| p dx
\right) \int
\Omega
| x| - ap| \nabla z1| p - 2\nabla z1 \cdot \nabla w dx \geq \lambda
\int
\Omega
| x| - (a+1)p+c1f(z1, z2)w dx,
M2
\left( \int
\Omega
| \nabla z2| q dx
\right) \int
\Omega
| x| - bq| \nabla z2| q - 2\nabla z2 \cdot \nabla w dx \geq \lambda
\int
\Omega
| x| - (b+1)q+c2g(z1, z2)w dx,
for all w \in W =
\bigl\{
w \in C\infty
0 (\Omega )| w \geq 0 \in \Omega
\bigr\}
.
We make the following assumptions:
(H2) f, g : [0,\infty )\times [0,\infty ) \rightarrow [0,\infty ) are C1 functions such that fu, f\upsilon , gu, g\upsilon \geq 0 and
\mathrm{l}\mathrm{i}\mathrm{m}
u,\upsilon \rightarrow \infty
f(u, \upsilon ) = \mathrm{l}\mathrm{i}\mathrm{m}
u,\upsilon \rightarrow \infty
g(u, \upsilon ) = \infty ;
(H3) for every A > 0,
\mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow \infty
f
\bigl(
x,A[g(x, x)]1/q - 1
\bigr)
xp - 1
= 0;
(H4) \mathrm{l}\mathrm{i}\mathrm{m}x\rightarrow \infty
g(x, x)
xq - 1
= 0.
A key role in our arguments will be played by the following auxiliary result. Its proof is similar
to those presented in [9], the reader can consult further the papers [1, 2, 15].
Lemma 3.1. Assume that M : \BbbR +
0 \rightarrow \BbbR + is continuous and increasing, and there exists m0 > 0
such that M(t) \geq m0 for all t \in \BbbR +
0 . If the functions u, v \in W 1,p
0 (\Omega , | x| - ap) satisfy
M
\left( \int
\Omega
| \nabla u| p dx
\right) \int
\Omega
| x| - ap| \nabla u| p - 2\nabla u \cdot \nabla \varphi dx \leq
\leq M
\left( \int
\Omega
| \nabla v| p dx
\right) \int
\Omega
| x| - ap| \nabla v| p - 2\nabla v \cdot \nabla \varphi dx (3.1)
for all \varphi \in W 1,p
0 (\Omega , | x| - ap), \varphi \geq 0, then u \leq v in \Omega .
From Lemma 3.1 we can establish the basic principle of the sub- and supersolutions method for
nonlocal systems. Indeed, we consider the nonlocal system
- M1
\left( \int
\Omega
| \nabla u| p dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - ap| \nabla u| p - 2\nabla u
\bigr)
= | x| - (a+1)p+c1h(x, u, v) in \Omega ,
- M2
\left( \int
\Omega
| \nabla v| q dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - bq| \nabla v| q - 2\nabla v
\bigr)
= | x| - (b+1)q+c2k(x, u, v) in \Omega , (3.2)
u = v = 0 on x \in \partial \Omega ,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
EXISTENCE RESULTS FOR A CLASS OF KIRCHHOFF-TYPE SYSTEMS . . . 575
where \Omega is a bounded smooth domain of \BbbR N and h, k : \Omega \times \BbbR \times \BbbR \rightarrow \BbbR satisfy the following
conditions:
(HK1) h(x, s, t) and k(x, s, t) are Carathéodory functions and they are bounded if s, t belong to
bounded sets;
(HK2) there exists a function g : \BbbR \rightarrow \BbbR being continuous, nondecreasing, with g(0) = 0,
0 \leq g(s) \leq C
\bigl(
1 + | s| min\{ p,q\} - 1
\bigr)
for some C > 0, and applications s \mapsto \rightarrow h(x, s, t) + g(s) and
t \mapsto \rightarrow k(x, s, t) + g(t) are nondecreasing, for a.e. x \in \Omega .
If u, v \in L\infty (\Omega ), with u(x) \leq v(x) for a.e. x \in \Omega , we denote by [u, v] the set \{ w \in L\infty (\Omega ) :
u(x) \leq w(x) \leq v(x) for a.e. x \in \Omega \} . Using Lemma 3.1 and the method as in the proof of Theo-
rem 2.4 of [18] (see also Section 4 of [7]), we can establish a version of the abstract lower and upper
solution method for our class of the operators as follows.
Proposition 3.1. Let M1,M2 : \BbbR +
0 \rightarrow \BbbR + be two functions satisfying the condition (H1). As-
sume that the functions h, k satisfy the conditions (HK1) and (HK2). Assume that (u, v), (u, v), are
respectively, a weak subsolution and a weak supersolution of system (3.2) with u(x) \leq u(x) and
v(x) \leq v(x) for a.e. x \in \Omega . Then there exists a minimal (u\ast , v\ast )
\bigl(
and, respectively, a maximal
(u\ast , v\ast )
\bigr)
weak solution for system (3.2) in the set [u, u] \times [v, v]. In particular, every weak solution
(u, v) \in [u, u] \times [v, v] of system (3.2) satisfies u\ast (x) \leq u(x) \leq u\ast (x) and v\ast (x) \leq v(x) \leq v\ast (x)
for a.e. x \in \Omega .
Now we are ready to state our existence result.
Theorem 3.1. Assume (H1) – (H4) hold. Then the system (1.1) admits a positive solution when
\lambda is large enough.
Proof. Since f, g are continuous and nondecreasing, we have f(x, y), g(x, y) \geq - a0 for all
x, y \geq 0 and for some a0 > 0. Choose \eta > 0 such that
\eta \leq \mathrm{M}\mathrm{i}\mathrm{n}
\bigl\{
| x| - (a+1)p+c1 , | x| - (b+1)q+c2
\bigr\}
in \Omega \delta . We shall verify that
(\psi 1,\lambda , \psi 2,\lambda ) =
\Biggl( \biggl[
\lambda a0\eta
\epsilon m1,\infty
\biggr] 1
p - 1
\biggl(
p - 1
p
\biggr)
\phi
p
p - 1
1,p ,
\biggl[
\lambda a0\eta
\epsilon m2,\infty
\biggr] 1
q - 1
\biggl(
q - 1
q
\biggr)
\phi
q
q - 1
1,q
\Biggr)
is a subsolution of (1.1). Let w \in W. Then a calculation shows that
M1
\left( \int
\Omega
| \nabla \psi 1,\lambda | p dx
\right) \int
\Omega
| x| - ap| \nabla \psi 1,\lambda | p - 2\nabla \psi 1,\lambda \cdot \nabla w dx =
=M1
\left( \int
\Omega
| \nabla \psi 1,\lambda | p dx
\right) \biggl( \lambda a0\eta
\epsilon m1,\infty
\biggr) \int
\Omega
| x| - ap\phi 1,p| \nabla \phi 1,p| p - 2\nabla \phi 1,p \cdot \nabla w dx =
=M1
\left( \int
\Omega
| \nabla \psi 1,\lambda | p dx
\right) \biggl( \lambda a0\eta
\epsilon m1,\infty
\biggr) \int
\Omega
| x| - ap| \nabla \phi 1,p| p - 2\nabla \phi 1,p
\bigl[
\nabla (\phi 1,pw) - | \nabla \phi 1,p| pw
\bigr]
dx \leq
\leq
\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega
\bigl[
\lambda 1,p| x| - (a+1)p+c1\phi p1,p - | x| - ap| \nabla \phi 1,p| p
\bigr]
w dx.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
576 G. A. AFROUZI, S. SHAKERI, H. ZAHMATKESH
Similarly
M2
\left( \int
\Omega
| \nabla \psi 2,\lambda | q dx
\right) \int
\Omega
| x| - bq| \nabla \psi 2,\lambda | q - 2\nabla \psi 2,\lambda \cdot \nabla w dx \leq
\leq
\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega
\bigl[
\lambda 1,q| x| - (b+1)q+c2\phi q1,q - | x| - bq| \nabla \phi 1,q| q
\bigr]
w dx.
First we consider the case x \in \Omega \delta . We have
\lambda 1,p| x| - (a+1)p+c1\phi p1,p - | x| - ap| \nabla \phi 1,p| p \leq - \epsilon
on \Omega \delta . Since \psi 1,\lambda (x), \psi 2,\lambda (x) \geq 0 in \Omega it follows that
- a0\eta \leq \mathrm{M}\mathrm{i}\mathrm{n}
\bigl\{
| x| - (a+1)p+c1f(\psi 1,\lambda , \psi 2,\lambda ), | x| - (b+1)q+c2g(\psi 1,\lambda , \psi 2,\lambda )
\bigr\}
in \Omega \delta . Hence, we have\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega \delta
\bigl[
\lambda 1,p| x| - (a+1)p+c1\phi p1,p - | x| - ap| \nabla \phi 1,p| p
\bigr]
w dx \leq
\leq - \lambda a0\eta
\int
\Omega \delta
w dx \leq \lambda
\int
\Omega \delta
| x| - (a+1)p+c1f(\psi 1,\lambda , \psi 2,\lambda )w dx.
A similar argument shows that\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega \delta
\bigl[
\lambda 1,p| x| - (b+1)q+c2\phi q1,q - | x| - bq| \nabla \phi 1,q| q
\bigr]
w dx \leq
\leq \lambda
\int
\Omega \delta
| x| - (b+1)q+c2g(\psi 1,\lambda , \psi 2,\lambda )w dx.
On the other hand, on \Omega \setminus \Omega \delta we have \phi 1,p \geq \sigma p and \phi 1,q \geq \sigma q for some 0 < \sigma p, \sigma q < 1. Therefore,
\psi 1,\lambda \geq
\biggl(
\lambda a0\eta
m1,\infty \epsilon
\biggr) 1
p - 1
\biggl(
p - 1
p
\biggr)
\sigma
p
p - 1
p \rightarrow \infty , (3.3)
\psi 2,\lambda \geq
\biggl(
\lambda a0\eta
m2,\infty \epsilon
\biggr) 1
q - 1
\biggl(
q - 1
q
\biggr)
\sigma
q
q - 1
q \rightarrow \infty (3.4)
as \lambda \rightarrow \infty , uniformly in \Omega \setminus \Omega \delta . By (3.3), (3.4) and (H2) we can find \lambda \ast sufficiently large such that
f(\psi 1,\lambda , \psi 2,\lambda ), g(\psi 1,\lambda , \psi 2,\lambda ) \geq
a0\eta
\epsilon
\mathrm{M}\mathrm{a}\mathrm{x}
\bigl\{
\lambda 1,p, \lambda 1,q
\bigr\}
for all x \in \Omega \setminus \Omega \delta and for all \lambda \geq \lambda \ast . Hence,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
EXISTENCE RESULTS FOR A CLASS OF KIRCHHOFF-TYPE SYSTEMS . . . 577\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega \setminus \Omega \delta
\bigl[
\lambda 1,p| x| - (a+1)p+c1\phi p1,p - | x| - ap| \nabla \phi 1,p| p
\bigr]
w dx \leq
\leq
\biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega \setminus \Omega \delta
| x| - (a+1)p+c1\lambda 1,pw dx \leq
\leq \lambda
\int
\Omega \setminus \Omega \delta
| x| - (a+1)p+c1f(\psi 1,\lambda , \psi 2,\lambda )w dx.
Similarly \biggl(
\lambda a0\eta
\epsilon
\biggr) \int
\Omega \setminus \Omega \delta
\bigl[
\lambda 1,p| x| - (b+1)q+c2\phi q1,q - | x| - bq| \nabla \phi 1,q| q
\bigr]
w dx \leq
\leq
\int
\Omega \setminus \Omega \delta
| x| - (b+1)q+c2g(\psi 1,\lambda , \psi 2,\lambda )w dx.
Hence,
M1
\left( \int
\Omega
| \nabla \psi 1,\lambda | p dx
\right) \int
\Omega
| x| - ap| \nabla \psi 1,\lambda | p - 2| \nabla \psi 1,\lambda | \cdot \nabla w dx \leq
\leq
\int
\Omega
| x| - (a+1)p+c1f(\psi 1,\lambda , \psi 2,\lambda )w dx,
M2
\left( \int
\Omega
| \nabla \psi 2,\lambda | q dx
\right) \int
\Omega
| x| - bq| \nabla \psi 2,\lambda | q - 2| \nabla \psi 2,\lambda | \cdot \nabla w dx \leq
\leq
\int
\Omega
| x| - (b+1)q+c2g(\psi 1,\lambda , \psi 2,\lambda )w dx,
i.e., (\psi 1,\lambda , \psi 2,\lambda ) is a subsolution of (1.1).
Now, we will prove there exists a M large enough so that
(z1, z2) =
=
\left( M\theta - 1
p
\biggl(
\lambda
m0
\biggr) 1
p - 1
\zeta p(x),
\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1\biggl(
\lambda
m0
\biggr) 1
q - 1
\zeta q(x)
\right)
is a supersolution of (1.1); where \theta r = \| \zeta r\| \infty ; r = p, q, \lambda \geq \lambda \ast and m0 = \mathrm{M}\mathrm{i}\mathrm{n}\{ m1,m2\} .
A calculation shows that
M1
\left( \int
\Omega
| \nabla z1| p dx
\right) \int
\Omega
| x| - ap| \nabla z1| p - 2\nabla z1 \cdot \nabla w dx =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
578 G. A. AFROUZI, S. SHAKERI, H. ZAHMATKESH
=M1
\left( \int
\Omega
| \nabla z1| p dx
\right) \biggl( \lambda
m0
\biggr) \bigl(
M\theta - 1
p
\bigr) p - 1
\int
\Omega
| x| - ap| \nabla \zeta p| p - 2\nabla \zeta p\nabla w dx =
=M1
\left( \int
\Omega
| \nabla z1| p dx
\right) \theta 1 - p
p
\Biggl[
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr] p - 1 \int
\Omega
| x| - (a+1)p+c1w dx \geq
\geq m1\theta
1 - p
p
\Biggl[
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr] p - 1 \int
\Omega
| x| - (a+1)p+c1w dx.
By monotonicity condition on f and (H3) we can choose M large enough so that
m1\theta
1 - p
p
\Biggl[
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr] p - 1
\geq
\geq \lambda f
\left( M\biggl( \lambda
m0
\biggr) 1
p - 1
,
\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1\biggl(
\lambda
m0
\biggr) 1
q - 1
\theta q
\right) \geq
\geq \lambda f
\left( M\theta - 1
p
\biggl(
\lambda
m0
\biggr) 1
p - 1
\zeta p(x),
\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1\biggl(
\lambda
m0
\biggr) 1
q - 1
\zeta q(x)
\right) =
= \lambda f(z1, z2).
Hence,
M1
\left( \int
\Omega
| \nabla z1| p dx
\right) \int
\Omega
| x| - ap| \nabla z1| p - 2\nabla z1 \cdot \nabla w dx \geq \lambda
\int
\Omega
| x| - (a+1)p+c1f(z1, z2)w dx.
Next, by (H4) for M large enough we have\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\leq
\biggl(
\lambda
m0
\biggr) 1
1 - q
\theta - 1
q .
Hence,
M2
\left( \int
\Omega
| \nabla z2| q dx
\right) \int
\Omega
| x| - bq| \nabla z2| q - 2\nabla z2 \cdot \nabla w dx =
=M2
\left( \int
\Omega
| \nabla z2| q dx
\right) \biggl( \lambda
m0
\biggr) \Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr]
\times
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
EXISTENCE RESULTS FOR A CLASS OF KIRCHHOFF-TYPE SYSTEMS . . . 579
\times
\int
\Omega
| x| - bq| \nabla \zeta p| q - 2\nabla \zeta p \cdot \nabla w dx =
=M2
\left( \int
\Omega
| \nabla z2| q dx
\right) \biggl( \lambda
m0
\biggr) \Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] \int
\Omega
| x| - (b+1)q+c2w dx \geq
\geq \lambda
\int
\Omega
| x| - (b+1)q+c2g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,
\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1 \biggl(
\lambda
m0
\biggr) 1
q - 1
\theta q
\Biggr)
w dx \geq
\geq \lambda
\int
\Omega
| x| - (b+1)q+c2g
\Biggl(
M\theta - 1
p
\biggl(
\lambda
m0
\biggr) 1
p - 1
\zeta p(x),
\Biggl[
g
\Biggl(
M
\biggl(
\lambda
m0
\biggr) 1
p - 1
,M
\biggl(
\lambda
m0
\biggr) 1
p - 1
\Biggr) \Biggr] 1
q - 1 \biggl(
\lambda
m0
\biggr) 1
q - 1
\zeta q(x)
\Biggr)
w dx =
= \lambda
\int
\Omega
| x| - (b+1)q+c2g(z1, z2)w dx,
i.e., (z1, z2) is a supersolution of (1.1) with zi \geq \psi i,\lambda , i = 1, 2, for a M large enough. Thus,
by Proposition 3.1 there exists a positive solution (u, \upsilon ) of (1.1) such that (\psi 1,\lambda , \psi 2,\lambda ) \leq (u, \upsilon ) \leq
\leq (z1, z2).
Theorem 3.1 is proved.
Example 3.1. Consider the problem
- M1
\left( \int
\Omega
| \nabla u| p dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - ap| \nabla u| p - 2\nabla u
\bigr)
= \lambda | x| - (a+1)p+c1(\upsilon \alpha + (u\upsilon )\beta - 1) in \Omega ,
- M2
\left( \int
\Omega
| \nabla v| q dx
\right) \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
| x| - bq| \nabla \upsilon | q - 2\nabla \upsilon
\bigr)
= \lambda | x| - (b+1)q+c2(u\sigma + (u\upsilon )
\gamma
2 - 1) \mathrm{i}\mathrm{n} \Omega , (3.5)
u = \upsilon = 0 in \partial \Omega ,
where \alpha , \beta , \sigma , and \gamma are positive parameters. Then it is easy to see that (3.5) satisfies the hypotheses
of Theorem 3.1 if \mathrm{m}\mathrm{a}\mathrm{x}\{ \sigma , \gamma \} \alpha
q - 1
< p - 1,
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}\{ \sigma , \gamma \} 1
q - 1
+1
\Bigr)
\beta < p - 1 and \mathrm{m}\mathrm{a}\mathrm{x}\{ \sigma , \gamma \} < q - 1.
References
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ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
|
| id | umjimathkievua-article-1458 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:45Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/20/a6f01e79b917930f644f83f74302d120.pdf |
| spelling | umjimathkievua-article-14582019-12-05T10:12:29Z Existence results for a class of Kirchhoff-type systems with combined nonlinear effects Результати про iснування розв’язкiв для одного класу систем типу Кiрхгофа з комбiнованими нелiнiйними ефектами Afrouzi, G. A. Shakeri, S. Zahmatkesh, H. Афрузі, Г. А. Шакері, С. Захматкеш, Г. UDC 517.9 We study the existence of positive solutions for a nonlinear system $$M_1 \bigl(\int_{\Omega} |\nabla u|^p dx\bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x|^{ap}| \nabla u|^{p-2}\nabla u) = \lambda | x| (a+1)p+c_1f(u, \upsilon ),\; x \in \Omega ,$$ $$M2 \bigl( \int_{ \Omega }| \nabla v| q dx \bigr)\mathrm{d}\mathrm{i}\mathrm{v} (| x| bq|\nabla \upsilon | q 2\nabla \upsilon ) = \lambda | x| (b+1)q+c_2g(u, \upsilon ),\; x \in \Omega ,$$ $$u = \upsilon = 0, x \in \partial \Omega ,$$ where $\Omega$ is a bounded smooth domain in $R^N$ with $0 \in \Omega,\; 1 < p, q < N, 0 \leq a < \cfrac{N-p}{p}, 0 \leq b < \cfrac{N-q}{p},$ а $c_1, c_2, \lambda$ are positive parameters. Here, $M_1,M_2, f$, and g satisfy certain conditions. We use the method of sub- and supersolutions to establish our results. УДК 517.9 Розглянуто проблему iснування додатних розв’язкiв нелiнiйної системи \begin{align*} &-M_1\left(\displaystyle\int\limits_\Omega |\nabla u|^{p}\,dx\right) {\rm div}\,(|x|^{-ap}|\nabla u|^{p-2}\nabla u)=\lambda |x|^{-(a+1)p+c_{1}}f(u,\upsilon),\quad x\in \Omega, \\ &-M_2\left(\displaystyle\int\limits_\Omega |\nabla v|^{q}\,dx\right) {\rm div}\,(|x|^{-bq}|\nabla \upsilon|^{q-2}\nabla \upsilon)=\lambda |x|^{-(b+1)q+c_{2}}g(u,\upsilon),\quad x\in \Omega, \\ &\,\,\,u=\upsilon =0,\quad x\in \partial \Omega, \end{align*} де $\Omega$ — обмежена гладка область в $R^N$ з $0 \in \Omega,\; 1 < p, q < N, 0 \leq a < \cfrac{N-p}{p}, 0 \leq b < \cfrac{N-q}{p},$ а $c_1, c_2, \lambda$ — додатнi параметри. Величини $M_1,M_2, f$ та $g$ задовольняють деякi умови. Нашi результати отримано за допомогою методу суб- та суперрозв’язкiв. Institute of Mathematics, NAS of Ukraine 2019-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1458 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 4 (2019); 571-580 Український математичний журнал; Том 71 № 4 (2019); 571-580 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1458/442 Copyright (c) 2019 Afrouzi G. A.; Shakeri S.; Zahmatkesh H. |
| spellingShingle | Afrouzi, G. A. Shakeri, S. Zahmatkesh, H. Афрузі, Г. А. Шакері, С. Захматкеш, Г. Existence results for a class of Kirchhoff-type systems with combined nonlinear effects |
| title | Existence results for a class of Kirchhoff-type
systems with combined nonlinear effects |
| title_alt | Результати про iснування розв’язкiв для одного класу систем типу Кiрхгофа з комбiнованими нелiнiйними ефектами |
| title_full | Existence results for a class of Kirchhoff-type
systems with combined nonlinear effects |
| title_fullStr | Existence results for a class of Kirchhoff-type
systems with combined nonlinear effects |
| title_full_unstemmed | Existence results for a class of Kirchhoff-type
systems with combined nonlinear effects |
| title_short | Existence results for a class of Kirchhoff-type
systems with combined nonlinear effects |
| title_sort | existence results for a class of kirchhoff-type
systems with combined nonlinear effects |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1458 |
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