Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications
UDC 517.9We prove a nonlinear analogue of Picone's identity for $\Delta_{\gamma}$-Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system. &...
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| author | Luyen, D. T. Luyen, D. T. |
| author_facet | Luyen, D. T. Luyen, D. T. |
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| description | UDC 517.9We prove a nonlinear analogue of Picone's identity for $\Delta_{\gamma}$-Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system.
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| doi_str_mv | 10.37863/umzh.v73i4.639 |
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DOI: 10.37863/umzh.v73i4.639
UDC 517.9
D. T. Luyen (Hoa Lu Univ., Ninh Nhat, Vietnam)
PICONE’S IDENTITY FOR \bfDelta \bfitgamma -LAPLACE OPERATOR
AND ITS APPLICATIONS*
ТОТОЖНIСТЬ ПIКОНЕ ДЛЯ \bfDelta \bfitgamma -ОПЕРАТОРА ЛАПЛАСА
ТА ЇЇ ЗАСТОСУВАННЯ
We prove a nonlinear analogue of Picone’s identity for \Delta \gamma -Laplace operator. As an application, we give a Hardy type
inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and
degenerate elliptic system.
Доведено нелiнiйний аналог тотожностi Пiконе для \Delta \gamma -оператора Лапласа. Як застосування наведено нерiвнiсть
типу Гардi та принцип порiвняння Штурма. Також доведено строгу монотоннiсть власного значення принципу та
виродженої елiптичної системи.
1. Introduction. It is a well-known fact that in the qualitative theory of elliptic PDEs, Picone’s
identity plays an important role. The classical Picone’s identity says that if u and v are differentiable
functions such that v > 0 and u \geq 0, then
| \nabla u| 2 + u2
v2
| \nabla v| 2 - 2
u
v
\nabla u \cdot \nabla v = | \nabla u| 2 - \nabla
\biggl(
u2
v
\biggr)
\cdot \nabla v \geq 0. (1.1)
(1.1) has an enormous applications to second-order elliptic equations and systems (see, for instance,
[1 – 3, 22] and the references therein). Nonlinear analogue of (1.1) is established by J. Tyagi [29]. In
order to apply (1.1) to p-Laplace equations, (1.1) is extended by W. Allegretto and Y. X. Huang [4].
Nonlinear analogue of Picone’s type identity for p-Laplace equations is established by K. Bal [6].
In this article we establish the nonlinear analogue of generalized Picone’s identity for \Delta \gamma -Laplace
operator and its applications.
This paper is organized as follows. In Section 2, we recall the definition of the \Delta \gamma -Laplace
operator and the associated functional setting. We further give examples for the class of \Delta \gamma -Laplace
operator. Section 3 deals with nonlinear analogue of Picone’s identity. In Section 4, we give several
application of Picone’s identity to \Delta \gamma -Laplace equations.
2. The \bfDelta \bfitgamma -Laplace operator. The \Delta \gamma -operator was considered by B. Franchi and E. Lanconelli
in [7, 8], and recently reconsidered in [10] under the additional assumption that the operator is
homogeneous of degree two with respect to a group dilation in \BbbR N . We consider the operators of the
form
\Delta \gamma :=
N\sum
j=1
\partial xj
\bigl(
\gamma 2j \partial xj
\bigr)
, \partial xj =
\partial
\partial xj
, j = 1, 2, . . . , N.
Here, the functions \gamma j : \BbbR N - \rightarrow \BbbR are assumed to be continuous, different from zero and of class
C1 in \BbbR N\setminus \Pi , where
* This research was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED)
(grant no. 101.02-2020.13).
c\bigcirc D. T. LUYEN, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4 515
516 D. T. LUYEN
\Pi :=
\left\{ x = (x1, x2, . . . , xN ) \in \BbbR N :
N\prod
j=1
xj = 0
\right\} .
Moreover, we assume the following properties:
i) There exists a group of dilations \{ \delta t\} t>0 such that
\delta t : \BbbR N - \rightarrow \BbbR , \delta t (x1, . . . , xN ) = (t\varepsilon 1x1, . . . , t
\varepsilon NxN ) , 1 = \varepsilon 1 \leq \varepsilon 2 \leq . . . \leq \varepsilon N ,
such that \gamma j is \delta t-homogeneous of degree \varepsilon j - 1, i.e.,
\gamma j (\delta t (x)) = t\varepsilon j - 1\gamma j (x) \forall x \in \BbbR N \forall t > 0, j = 1, . . . , N.
The number \widetilde N :=
N\sum
j=1
\varepsilon j
is called the homogeneous dimension of \BbbR N with respect to \{ \delta t\} t>0.
ii) \gamma 1 = 1, \gamma j (x) = \gamma j (x1, x2, . . . , xj - 1) , j = 2, . . . , N.
iii) There exists a constant \rho \geq 0 such that
0 \leq xk\partial xk
\gamma j (x) \leq \rho \gamma j (x) \forall k \in \{ 1, 2, . . . , j - 1\} \forall j = 2, . . . , N,
and for every x \in \BbbR N
+ :=
\bigl\{
(x1, . . . , xN ) \in \BbbR N : xj \geq 0 \forall j = 1, 2, . . . , N
\bigr\}
.
iv) Equalities \gamma j (x) = \gamma j (x
\ast ) , j = 1, 2, . . . , N, are satisfied for every x \in \BbbR N , where
x\ast = (| x1| , . . . , | xN | ) if x = (x1, x2, . . . , xN ).
Many aspects of the theory of degenerate elliptic differential operators are presented in monographs
[27, 28] (see also some recent results in [5, 10 – 20, 23 – 26] and the references therein).
Definition 2.1. By Sp
\gamma (\Omega ), 1 \leq p < +\infty , we will denote the set of all functions u \in Lp(\Omega )
such that \gamma j\partial xju \in Lp(\Omega ) for all j = 1, . . . , N. We define the norm in this space as follows:
\| u\| Sp
\gamma (\Omega ) =
\left\{
\int
\Omega
\left( | u| p +
N\sum
j=1
\bigm| \bigm| \gamma j\partial xju
\bigm| \bigm| p\right) dx
\right\}
1
p
.
If p = 2 we can also define the scalar product in S2
\gamma (\Omega ) as follows:
(u, v)S2
\gamma (\Omega ) = (u, v)L2(\Omega ) +
N\sum
j=1
(\gamma j\partial xju, \gamma j\partial xjv)L2(\Omega ).
The space Sp
\gamma ,0(\Omega ) is defined as the closure of C1
0 (\Omega ) in the space Sp
\gamma (\Omega ).
Set
\nabla \gamma u := (\gamma 1\partial x1u, \gamma 2\partial x2u, . . . , \gamma N\partial xNu) , | \nabla \gamma u| :=
\left( N\sum
j=1
\bigm| \bigm| \gamma j\partial xju
\bigm| \bigm| 2\right) 1
2
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
PICONE’S IDENTITY FOR \Delta \gamma -LAPLACE OPERATOR AND ITS APPLICATIONS 517
We now give some examples of the \Delta \gamma -Laplace operator. We use the following notations: we
split \BbbR N into
\BbbR N = \BbbR N1 \times \BbbR N2 \times \BbbR N3 ,
and write
x =
\Bigl(
x(1), x(2), x(3)
\Bigr)
, x(i) =
\Bigl(
x
(i)
1 , x
(i)
2 , . . . , x
(i)
Ni
\Bigr)
\in \BbbR Ni ,
| x(i)| 2 =
Ni\sum
j=1
| x(i)j | 2, i = 1, 2, 3.
We denote the classical Laplace operator in \BbbR Ni by
\Delta x(i) =
Ni\sum
j=1
\partial 2
x
(i)
j
.
Example 2.1 (see [11, 17]). Let \alpha be a real positive number. The operator
\Delta \gamma = \Delta x(1) + | x(1)| 2\alpha (\Delta x(2) +\Delta x(3)),
where
\gamma =
\bigl(
1, 1, . . . , 1\underbrace{} \underbrace{}
N1 - times
, | x(1)| \alpha , . . . , | x(1)| \alpha \underbrace{} \underbrace{}
(N2+N3) - times
\bigr)
,
is called the Grushin operator (see [9]).
Example 2.2 (see [11, 17]). Let \alpha , \beta be nonnegative real numbers. The operator
\Delta \gamma = \Delta x(1) +\Delta x(2) + | x(1)| 2\alpha | x(2)| 2\beta \Delta x(3) ,
where
\gamma =
\bigl(
1, 1, . . . , 1\underbrace{} \underbrace{}
(N1+N2) - times
, | x(1)| \alpha | x(2)| \beta , . . . , | x(1)| \alpha | x(2)| \beta \underbrace{} \underbrace{}
N3 - times
\bigr)
,
is called the strongly degenerate elliptic operators (see [24, 28]).
3. Generalized Picone’s inequality.
Theorem 3.1. Let v > 0 and u \geq 0 be be two non-constant differentiable functions in \Omega . Also
assume that f \in C1(\BbbR , (0,\infty )) satisfies f \prime (y) \geq 1 for all y \in (0,\infty ). Define
L(u, v) = | \nabla \gamma u| 2 -
2u\nabla \gamma u \cdot \nabla \gamma v
f(v)
+
u2f \prime (v) | \nabla \gamma v| 2
f2(v)
,
R(u, v) = | \nabla \gamma u| 2 - \nabla \gamma
\biggl(
u2
f(v)
\biggr)
\cdot \nabla \gamma v.
Then L(u, v) = R(u, v) \geq 0. Moreover, L(u, v) = 0 a.e. in \Omega if and only if \nabla \gamma
\Bigl( u
v
\Bigr)
= 0 a.e. in \Omega ,
i.e., u = kv for some constant k in each component of \Omega .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
518 D. T. LUYEN
Proof. Expanding R(u, v) one easily sees that L(u, v) = R(u, v). To show L(u, v) \geq 0 we
proceed as follows:
L(u, v) = | \nabla \gamma u| 2 -
2u\nabla \gamma u \cdot \nabla \gamma v
f(v)
+
u2f \prime (v) | \nabla \gamma v| 2
f2(v)
=
= | \nabla \gamma u| 2 +
u2f \prime (v) | \nabla \gamma v| 2
f2(v)
- 2u | \nabla \gamma u| | \nabla \gamma v|
f(v)
+
+
2u
f(v)
(| \nabla \gamma u| | \nabla \gamma v| - \nabla \gamma u \cdot \nabla \gamma v) =
=
\Biggl(
| \nabla \gamma u| 2 +
u2 | \nabla \gamma v| 2
f2(v)
\Biggr)
- u2 | \nabla \gamma v| 2
f2(v)
- 2u | \nabla \gamma u| | \nabla \gamma v|
f(v)
+
+
u2f \prime (v) | \nabla \gamma v| 2
f2(v)
+
2u
f(v)
(| \nabla \gamma u| | \nabla \gamma v| - \nabla \gamma u \cdot \nabla \gamma v) .
By using Cauchy’s inequality, we get
| \nabla \gamma u| 2 +
u2 | \nabla \gamma v| 2
f2(v)
\geq 2u | \nabla \gamma u| | \nabla \gamma v|
f(v)
. (3.1)
Which is possible since both u and f are non negative. Equality holds when
| \nabla \gamma u| =
u
f(v)
| \nabla \gamma v| . (3.2)
Again using the fact that f \prime (y) \geq 1, we have
u2f \prime (v) | \nabla \gamma v| 2
f2(v)
\geq u2 | \nabla \gamma v| 2
f2(v)
. (3.3)
Equality holds when
f \prime (v) = 1. (3.4)
Combining (3.1) and (3.3), we obtain L(u, v) \geq 0. Equality holds when (3.2) and (3.4) together with
| \nabla \gamma u| | \nabla \gamma v| = \nabla \gamma u \cdot \nabla \gamma v holds simultaneously.
Solving for (3.4) one obtains f(v) = v. So, if L(u, v)(x0) = 0 and u(x0) \not = 0, then (3.1) together
with f(v) = v and | \nabla \gamma u| | \nabla \gamma v| = \nabla \gamma u \cdot \nabla \gamma v yields, i.e., \nabla \gamma u = (u/v)\nabla \gamma v or \nabla \gamma (u/v)(x0) = 0.
On the other hand, if \Lambda = \{ x \in \Omega , u(x) = 0\} , then \nabla \gamma u = 0 a.e. in \Lambda (see [17]), and thus
\nabla \gamma (u/v) = 0 a.e. in \Omega . We conclude that \nabla \gamma (u/v) = 0 a.e. in \Omega and consequently u = kv for
some constant k.
Remark 3.1. If \gamma = (1, 1, . . . , 1\underbrace{} \underbrace{}
N - times
) and f(y) = y, we get the classical Picone’s identity (1.1)
for Laplacian operator.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
PICONE’S IDENTITY FOR \Delta \gamma -LAPLACE OPERATOR AND ITS APPLICATIONS 519
4. Applications. In this section, we will give some applications of nonlinear Picone’s identity
following the spirit of [4].
4.1. Hardy type result. We start with establishing a Hardy type inequality for \Delta \gamma -Laplace
operator.
Theorem 4.1. Assume that there is a v \in C1(\Omega ) satisfying
- \Delta \gamma v \geq \lambda gf(v), v > 0 in \Omega ,
for some \lambda > 0 and nonnegative continuous function g. Then, for any u \in C\infty
0 (\Omega ), u \geq 0, it holds
that \int
\Omega
| \nabla \gamma u| 2 dx \geq \lambda
\int
\Omega
gu2dx, (4.1)
where f \in C1(\BbbR , (0,\infty )) satisfies f \prime (y) \geq 1 for all y \in (0,\infty ).
Proof. Take \phi \in C\infty
0 (\Omega ), \phi > 0. By Theorem 3.1, we have
0 \leq
\int
\Omega
L(\phi , v)dx =
=
\int
\Omega
R(\phi , v)dx =
\int
\Omega
\biggl(
| \nabla \gamma \phi | 2 - \nabla \gamma
\biggl(
\phi 2
f(v)
\biggr)
\cdot \nabla \gamma v
\biggr)
dx =
=
\int
\Omega
\biggl(
| \nabla \gamma \phi | 2 +
\phi 2
f(v)
\Delta \gamma v
\biggr)
dx \leq
\leq
\int
\Omega
\Bigl(
| \nabla \gamma \phi | 2 - \lambda \phi 2g
\Bigr)
dx.
Letting \phi \rightarrow u, we get (4.1).
4.2. Strumium comparison principle. Comparison principles play vital role in study of partial
differential equations. Here, we establish nonlinear version of Sturmian comparison principle for
\Delta \gamma -Laplace operator.
Theorem 4.2. Let f1 and f2 are two weight functions such that f1(\xi ) < f2(\xi ) for all \xi \in \Omega
and f \in C1(\BbbR , (0,\infty )) satisfies f \prime (y) \geq 1 for all y \in (0,\infty ). If there is a positive solution u
satisfying
- \Delta \gamma u = f1(x)u in \Omega , u = 0 on \partial \Omega ,
then any nontrivial solution v of
- \Delta \gamma v = f2(x)f(v) in \Omega , v = 0 on \partial \Omega , (4.2)
must change sign.
Proof. Let us assume that there exists a solution v > 0 of (4.2) in \Omega . Then by Picone’s identity,
we have
0 \leq
\int
\Omega
L(u, v)dx =
\int
\Omega
R(u, v)dx =
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
520 D. T. LUYEN
=
\int
\Omega
\biggl(
| \nabla \gamma u| 2 - \nabla \gamma
\biggl(
u2
f(v)
\biggr)
\cdot \nabla \gamma v
\biggr)
dx =
=
\int
\Omega
\bigl(
f1(x)u
2 - f2(x)u
2
\bigr)
dx =
\int
\Omega
(f1(x) - f2(x))u
2dx < 0,
which is a contradiction. Hence, v changes sign in \Omega .
4.3. Strict monotonicity of principle eigenvalue in domain. Consider the indefinite eigenvalue
problem
- \Delta \gamma u = \lambda g(x)u in \Omega , u = 0 on \partial \Omega , (4.3)
where g(x) is indefinite weight function.
Theorem 4.3. Let \lambda +
1 (\Omega ) > 0 be the principle eigenvalue of (4.3), then suppose \Omega 1 \subset \Omega 2 and
\Omega 1 \not = \Omega 2. Then \lambda +
1 (\Omega 1) > \lambda +
1 (\Omega 2), if both exist.
Proof. Let ui be a positive eigenfunction associated with \lambda +
1 (\Omega i), i = 1, 2. Evidently, for
\phi \in C\infty
0 (\Omega 1), we obtain
0 \leq
\int
\Omega 1
L(\phi , u2)dx =
\int
\Omega
R(\phi , u2)dx =
=
\int
\Omega 1
\biggl(
| \nabla \gamma \phi | 2 - \nabla \gamma
\biggl(
\phi 2
f(u2)
\biggr)
\cdot \nabla \gamma u2
\biggr)
dx =
=
\int
\Omega 1
| \nabla \gamma \phi | 2 dx+
\int
\Omega 1
\phi 2
f(u2)
\Delta \gamma u2dx =
=
\int
\Omega 1
| \nabla \gamma \phi | 2 dx - \lambda +
1 (\Omega 2)
\int
\Omega 1
\phi 2
f(u2)
g(x)u2dx.
Letting \phi \rightarrow u1 and f(y) = y, we get
0 \leq
\int
\Omega 1
L(u1, u2)dx =
\bigl(
\lambda +
1 (\Omega 1) - \lambda +
1 (\Omega 2)
\bigr) \int
\Omega 1
g(x)u21dx.
This gives \lambda +
1 (\Omega 1) > \lambda +
1 (\Omega 2), as if \lambda +
1 (\Omega 1) = \lambda +
1 (\Omega 2). We conclude that u1 = ku2 which is not
possible as \Omega 1 \subset \Omega 2 and \Omega 1 \not = \Omega 2.
Remark 4.1. When g(x) = 1, we have \lambda 1(\Omega 1) > \lambda 1(\Omega 2) if \Omega 1 \subset \Omega 2 and \Omega 1 \not = \Omega 2.
4.4. Quasilinear system with singular nonlinearity. We will use Picone’s identity to establish a
linear relationship between solutions of a quasilinear system with singular nonlinearity. Consider the
singular degenerate elliptic system equations
- \Delta \gamma u = f(v) in \Omega ,
- \Delta \gamma v =
f2(v)
u
in \Omega ,
u > 0, v > 0 in \Omega ,
u = 0, v = 0 on \partial \Omega ,
(4.4)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
PICONE’S IDENTITY FOR \Delta \gamma -LAPLACE OPERATOR AND ITS APPLICATIONS 521
where f \in C1(\BbbR , (0,\infty )) satisfies f \prime (y) \geq 1 for all y \in (0,\infty ). We have the following result.
Theorem 4.4. Let (u, v) be a weak solution of (4.4). Then u = kv, where k is a constant.
Proof. Let (u, v) be the weak solution of (4.4). Now for any \phi 1 and \phi 2 in S2
\gamma ,0(\Omega ), we have\int
\Omega
\nabla \gamma u \cdot \nabla \gamma \phi 1dx =
\int
\Omega
f(v)\phi 1dx, (4.5)
\int
\Omega
\nabla \gamma v \cdot \nabla \gamma \phi 2dx =
\int
\Omega
f2(v)
u
\phi 2dx. (4.6)
Choosing \phi 1 = u and \phi 2 = u2/f(v) in (4.5) and (4.6), we obtain\int
\Omega
| \nabla \gamma u| 2 dx =
\int
\Omega
f(v)udx =
\int
\Omega
\nabla \gamma v \cdot \nabla \gamma
\biggl(
u2
f(v)
\biggr)
dx.
Hence, we get \int
\Omega
R(u, v)dx =
\int
\Omega
\biggl(
| \nabla \gamma u| 2 - \nabla \gamma v \cdot \nabla \gamma
\biggl(
u2
f(v)
\biggr) \biggr)
dx = 0,
this gives R(u, v) = 0, which in turn implies that u = kv.
Acknowledgments. This paper was done while the author was staying at the Vietnam Institute
of Advanced Study in Mathematics (VIASM) as a research fellow. He would like to thank VIASM
for its hospitality and support.
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Received 13.03.17,
after revision — 29.09.20
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 4
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| id | umjimathkievua-article-639 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:26Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e1/eb2a1f4a969cb46623a412cb80fd6ce1.pdf |
| spelling | umjimathkievua-article-6392025-03-31T08:48:15Z Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications Picone’s identity for $\Delta_{\gamma}$ -Laplace operator and its applications Luyen, D. T. Luyen, D. T. $\Delta_{\gamma}$-Laplace operator Picone’s identit Sturmian comparison theore Monotonicity of the eigenvalu Hardy’s inequalit $\Delta_{\gamma}$-Laplace operator Picone’s identit Sturmian comparison theore Monotonicity of the eigenvalu Hardy’s inequalit UDC 517.9We prove a nonlinear analogue of Picone's identity for $\Delta_{\gamma}$-Laplace operator. As an application, we give a Hardy type inequality and Sturmian comparison principle.We also show the strict monotonicity of the principle eigenvalue and degenerate elliptic system. &nbsp; УДК517.9 Тотожнiсть Пiконе для $\Delta_{\gamma}$ -оператора Лапласа та її застосування Доведено нелінійний аналог тотожності Піконе для $\Delta_\gamma$-оператора Лапласа. Як застосування наведено нерівність типу Гарді та принцип порівняння Штурма.Також доведено строгу монотонність власного значення принципу та виродженої еліптичної системи. Institute of Mathematics, NAS of Ukraine 2021-04-21 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/639 10.37863/umzh.v73i4.639 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 4 (2021); 515 - 522 Український математичний журнал; Том 73 № 4 (2021); 515 - 522 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/639/9005 |
| spellingShingle | Luyen, D. T. Luyen, D. T. Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title | Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title_alt | Picone’s identity for $\Delta_{\gamma}$ -Laplace operator and its applications |
| title_full | Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title_fullStr | Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title_full_unstemmed | Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title_short | Picone’s identity for $\Delta_{\gamma}$-Laplace operator and its applications |
| title_sort | picone’s identity for $\delta_{\gamma}$-laplace operator and its applications |
| topic_facet | $\Delta_{\gamma}$-Laplace operator Picone’s identit Sturmian comparison theore Monotonicity of the eigenvalu Hardy’s inequalit $\Delta_{\gamma}$-Laplace operator Picone’s identit Sturmian comparison theore Monotonicity of the eigenvalu Hardy’s inequalit |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/639 |
| work_keys_str_mv | AT luyendt piconesidentityfordeltagammalaplaceoperatoranditsapplications AT luyendt piconesidentityfordeltagammalaplaceoperatoranditsapplications |