Generalized Picone identity for Finsler $p$-Laplacian and its applications
UDC 517.9 We prove a generalized Picone-type identity for Finsler $p$-Laplacian and use it to establish some qualitative results for some boundary-value problems involving Finsler $p$-Laplacian.  
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2021
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| author | Dwivedi, G. Dwivedi, G. |
| author_facet | Dwivedi, G. Dwivedi, G. |
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| description | UDC 517.9
We prove a generalized Picone-type identity for Finsler $p$-Laplacian and use it to establish some qualitative results for some boundary-value problems involving Finsler $p$-Laplacian.
 
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DOI: 10.37863/umzh.v73i11.1050
UDC 517.9
G. Dwivedi (Birla Inst. Technology and Sci. Pilani, Rajasthan, India)
GENERALIZED PICONE IDENTITY
FOR FINSLER \bfitp -LAPLACIAN AND ITS APPLICATIONS*
УЗАГАЛЬНЕНА ТОТОЖНIСТЬ ПIКОНЕ
ДЛЯ \bfitp -ЛАПЛАСIАНА ФIНСЛЕРА ТА ЇЇ ЗАСТОСУВАННЯ
We prove a generalized Picone-type identity for Finsler p-Laplacian and use it to establish some qualitative results for
some boundary-value problems involving Finsler p-Laplacian.
Доведено узагальнену тотожнiсть типу Пiконе для p-лапласiана Фiнслера, яку потiм використано для отримання
деяких якiсних результатiв для граничних задач, що включають p-лапласiан Фiнслера.
1. Introduction. In this paper, we establish a generalized Picone identity for the class of operators
\Delta H,pu := \mathrm{d}\mathrm{i}\mathrm{v}(H(\nabla u)p - 1\nabla \xi H(\nabla u)), (1.1)
where p > 1, H : \BbbR n \rightarrow [0,\infty ), n \geq 2, is a strictly convex, twice differentiable function which
is positively homogeneous of degree 1, \Delta and \Delta \xi denote the usual gradient operators with respect
to variable x and \xi , respectively. The operators of the form (1.1) are called Finsler p-Laplacian or
anisotropic p-Laplacian. A prototype function H is given by
H(\xi ) = \| \xi \| r =
\Biggl(
n\sum
i=1
| \xi i| r
\Biggr) 1/r
, r > 1.
For this choice of H, the operator (1.1) reduces to
\Delta H,pv = \mathrm{d}\mathrm{i}\mathrm{v}
\bigl(
\| \nabla v\| p - 2
r \nabla rv
\bigr)
, (1.2)
where \nabla rv =
\bigl(
| vx1 | r - 2vx1 , . . . , | vxn | r - 2vxn
\bigr)
. (1.2) reduces to p-Laplacian if r = 2 and p \in (1,\infty ),
while it reduces to pseudo p-Laplacian if r = p > 1. In case of r = p = 2, we get standard Laplace
operator from (1.2).
Finsler p-Laplacian has been studied by several authors. V. Ferone and B. Kawohl [18] proved
some properties of Finsler p-Laplacian such as existence of fundamental solution, maximum prin-
ciple, comparison principle, mean value property etc. Belloni et al. [6] obtained positivity and
simplicity of the first eigenvalue and Faber – Krahn inequality for Finsler p-Laplacian with Dirichlet
boundary conditions. They also established symmetry of positive solutions to
- \Delta H,nu = f(u) in \Omega ,
u = 0 on \partial \Omega ,
where \Omega is a smooth and bounded domain in \BbbR n. G. Wang and C. Xia [26] obtained a lower bound for
the first eigenvalue for Finsler p-Laplacian with Neumann boundary conditions. F. Della Pietra and
* This paper was supported by Science and Engineering Research Board, India (the grant MTR/2018/000233).
c\bigcirc G. DWIVEDI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11 1451
1452 G. DWIVEDI
N. Gavitone [9, 10] discussed existence and properties of the first eigenvalue of Finsler p-Laplacian
with Dirichlet and Robin boundary conditions. G. Wang and C. Xia [27] studied blow up analysis
for the problem - \Delta H,2u = V (x)eu in dimension 2. We refer to [8, 22, 24, 29] and reference cited
therein for some further existence and qualitative results involving Finsler p-Laplacian.
Next, let us recall some historical developments in Picone identity. The classical Picone iden-
tity [23] 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\nabla v = | \nabla u| 2 - \nabla
\biggl(
u2
v
\biggr)
\nabla v \geq 0. (1.3)
(1.3) has an enormous applications to second-order elliptic equations and systems (see, for
instance, [1, 2, 21] and the references therein). In order to apply (1.3) to p-Laplace equations,
W. Allegretto and Y. X. Huang [3] extended (1.3) as follows.
Theorem 1.1 [3]. Let v > 0 and u \geq 0 be differentiable functions in a domain \Omega of \BbbR n. Denote
L(u, v) = | \nabla u| p + (p - 1)
up
vp
| \nabla v| p - p
up - 1
vp - 1
| \nabla v| p - 2\nabla u\nabla v,
R(u, v) = | \nabla u| p - \nabla
\biggl(
up
vp - 1
\biggr)
| \nabla v| p - 2\nabla v.
Then L(u, v) = R(u, v). Moreover, L(u, v) \geq 0 and L(u, v) = 0 a.e. in \Omega if and only if \nabla
\Bigl( u
v
\Bigr)
= 0
a.e. in \Omega .
A nonlinear analogue of Theorem 1.1 was proved by J. Tyagi [25] in case of p = 2 and by K. Bal
[4] in general case. The results of K. Bal [4] was further generalized by T. Feng [16] as follows.
Theorem 1.2 [16]. Let v > 0 and u \geq 0 be differentiable functions in a domain \Omega \subseteq \BbbR n,
n \geq 3. Assume that differentiable functions g(u) and f(v) satisfy that for p > 1, q > 1,
1
p
+
1
q
= 1,
g(u)f \prime (v)| \nabla v| p
[f(v)]2
\geq p
q
\biggl[
g\prime (u)| \nabla v| p - 1
pf(v)
\biggr] q
,
where g(u), g\prime (u) > 0 for u > 0; g(u), g\prime (u) = 0 for u = 0, and f(v), f \prime (v) > 0. Denote
L(u, v) = | \nabla u| p - g\prime (u)| \nabla v| p - 2\nabla v\nabla u
f(v)
+
g(u)f \prime (v)| \nabla v| p
[f(v)]2
,
R(u, v) = | \nabla u| p - \nabla
\biggl(
g(u)
f(v)
\biggr)
| \nabla v| p - 2\nabla v.
Then L(u, v) = R(u, v) \geq 0. Moreover, L(u, v) = 0 a.e. in \Omega if and only if
\nabla
\Bigl( u
v
\Bigr)
= 0, | \nabla u| p =
\biggl[
g\prime (u)| \nabla v| p - 1
pf(v)
\biggr] q
,
p
q
\biggl[
g\prime (u)| \nabla v| p - 1
pf(v)
\biggr] q
=
g(u)f \prime (v)| \nabla v| p
[f(v)]2
a.e. in \Omega .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
GENERALIZED PICONE IDENTITY FOR FINSLER p-LAPLACIAN AND ITS APPLICATIONS 1453
There are several other interesting articles dealing with Picone identity in different contexts. For
instance, for a Picone-type identity to higher order half linear differentiable operators, we refer to [20]
and the references therein, for Picone identities to half-linear elliptic operators with p(x)-Laplacians,
we refer to [28] for Picone-type identity to pseudo p-Laplacian with variable power, we refer to [7]
and for Picone identity for biharmonic operators and applications, we refer to [11 – 14, 17]. J. Jaroš
[19] proved a Picone identity for the class of operators (1.1). Their main result is follows.
Theorem 1.3. Let \Omega \subseteq \BbbR n be a domain and H be an arbitrary norm in \BbbR n which is of class
C1 for x \not = 0. Assume that u, v \in W 1,p
loc (\Omega ) \cap C(\Omega ) with v(x) \not = 0 in \Omega and denote
\Phi (u, v) := H(\nabla u)p + (p - 1)
| u| p
| v| p
H(\nabla v)p - p
| u| p - 2u
| v| p - 2v
\langle \nabla u,H(\nabla v)p - 1\nabla \xi H(\nabla v)\rangle .
Then
H(\nabla u)p -
\biggl\langle
\nabla
\biggl(
| u| p
| v| p - 2v
\biggr)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
= \Phi (u, v)
and \Phi (u, v) \geq 0 a.e. in \Omega . If , in addition, H(\xi )p is strictly convex in \BbbR n, then \Phi (u, v) = 0 a.e. in
\Omega if and only if u is a constant multiple of v in \Omega .
Bal et al. [5] generalized Theorem 1.3 as follows.
Theorem 1.4. Let \Omega \subseteq \BbbR n be a domain. For u, v \in W 1,p
loc (\Omega ) \cap C(\Omega ) with u \geq 0 and v > 0,
define
A(u, v) = H(\nabla u)p - p
up - 1
f(v)
\bigl\langle
\nabla u,H(\nabla v)p - 1\nabla \xi (\nabla v)
\bigr\rangle
+
upf \prime (v)
(f(v))2
H(\nabla v)p =
= H(\nabla u)p -
\biggl\langle
\nabla
\biggl(
up
f(v)
\biggr)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
\geq 0
for f \in \BbbM :=
\Bigl\{
f : (0,\infty ) \rightarrow (0,\infty )f : f \prime (y) \geq (p - 1)f(y)
p - 1
p - 2
\Bigr\}
\subset C1((0,\infty )). Moreover,
A(u, v) = 0 a.e. in \Omega if and only if u = cv a.e. in \Omega , where c is a constant.
In this paper, we prove a new nonlinear Picone-type identity, which is a generalization of Theo-
rem 1.4. The main result of this paper is follows.
Theorem 1.5. Let H be an arbitrary norm in \BbbR n which is of class C1(\BbbR n\setminus \{ 0\} ). Assume that
u, v \in W 1,p
loc (\Omega ) \cap C(\Omega ) with u \geq 0 and v > 0, where \Omega \subseteq \BbbR n is a domain. Assume that g and f
are twice differentiable functions satisfying
g(u)f \prime (v)H(\nabla v)p
(f(v))2
\geq p
q
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
,
where g(u), g\prime (u) > 0 for u > 0, g(u), g\prime (u) = 0 if u = 0, f(v), f \prime (v) > 0. Denote
L(u, v) = H(\nabla u)p +
g(u)f \prime (v)
(f(v))2
H(\nabla v)p -
\biggl\langle
g\prime (u)
f(v)
\nabla u,H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
,
R(u, v) = H(\nabla u)p -
\biggl\langle
\nabla
\biggl(
g(u)
f(v)
\biggr)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
.
Then (i) L(u, v) = R(u, v) \geq 0; (ii) L(u, v) = 0 a.e. in \Omega if and only if
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1454 G. DWIVEDI
\nabla u =
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v,
g(u)f \prime (v)H(\nabla v)p
(f(v))2
\geq p
q
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
, (1.4)
H(\nabla u)p =
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
. (1.5)
Remark 1.1. 1. If we choose g(u) = up in Theorem 1.5, then we obtain Picone identity of Bal
et al. [5].
2. If we choose g(u) = up and f(v) = vp - 1 in Theorem 1.5, then we obtain Picone identity of
J. Jaroš [19].
3. If we choose H(\xi ) =
\Bigl( \sum n
i=1
| \xi i| p
\Bigr) 1/p
in Theorem 1.5, then we obtain Picone identity of
T. Feng [16].
This paper is organized as follows. In Section 2, we state some elementary properties of an
arbitrary norm. In Section 3, we prove Theorem 1.5 and Section 4 deals with some applications of
Theorem 1.5.
2. Preliminaries. In this section, we recall some elementary properties of an arbitrary norm
on \BbbR n. For further details, we refer to [19] and references therein. Let H : \BbbR n \rightarrow [0,\infty ) be any
arbitrary norm in \BbbR n, i.e., a strictly convex, twice differentiable function such that:
(i) H(\xi ) > 0 for any \xi \not = 0,
(ii) H(t\xi ) = | t| H(\xi ) for all \xi \in \BbbR n and t \in \BbbR ,
(iii) if H is C1(\BbbR n\setminus \{ 0\} ), then \nabla \xi H(t\xi ) = sgn t\nabla \xi H(\xi ) for all \xi \not = 0 and t \not = 0,
(iv) \langle \xi ,\nabla \xi H(\xi )\rangle = H(\xi ) for all \xi \in \BbbR n, where the left-hand side is zero for \xi = 0,
(v) there exist constant 0 < c1 \leq c2 such that c1| x| \leq H(x) \leq c2| x| .
Next, we define the dual norm H0 of H by
H0(x) = \mathrm{s}\mathrm{u}\mathrm{p}
\xi \not =0
\langle x, \xi \rangle
H(\xi )
,
where \langle \cdot , \cdot \rangle is the usual inner product in \BbbR n.
Any norm H of class C1 for \xi \not = 0 and its dual H0 satisfy the following properties:
(i) H0(\nabla H(\xi )) = 1 for \xi \in \BbbR n\setminus \{ 0\} ,
(ii) H(\nabla H0(x)) = 1 for x \in \BbbR n\setminus \{ 0\} ,
(iii) H[H0(x)\nabla H0(x)]\nabla \xi [H0(x)\nabla H0(x)] = x,
(iv) H0[H(\xi )\nabla \xi H(\xi )]\nabla H0[H(\xi )\nabla \xi H(\xi )] = \xi ,
where (ii), (iii) hold for all x, \xi \in \BbbR n, H(0)\nabla \xi H(0) and H0(0)\nabla H0(0) are defined to be 0.
The Hölder-type inequality for norm H as follows:
H(\xi )H0(x) \geq \langle x, \xi \rangle , (2.1)
and equality holds if and only if H0(x) = H(\eta ).
Next, we state an elementary lemma. For a proof we refer to [19].
Lemma 2.1. Let H be a norm in \BbbR n such that H \in C1(\BbbR n\setminus \{ 0\} ) and Hp, 1 < p < \infty , is
strictly convex. If
H(\xi )p + (p - 1)H(\eta )p - p\langle \xi ,H(\eta )p - 1\nabla H(\eta )\rangle = 0
for some \xi , \eta \in \BbbR n, \eta \not = 0, and H(\xi ) = H(\eta ), then \xi = \eta .
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
GENERALIZED PICONE IDENTITY FOR FINSLER p-LAPLACIAN AND ITS APPLICATIONS 1455
Lemma 2.2 (Young’s inequality). If a and b are two nonnegative real numbers and p and q are
such that
1
p
+
1
q
= 1, then equality
ab \leq ap
p
+
bq
q
(2.2)
holds if and only if ap = bq.
Proof. For a proof, we refer to [15].
3. Proof of Theorem 1.5. It is easy to see that
\nabla
\biggl(
g(u)
f(v)
\biggr)
=
f(v)g\prime (u)\nabla u - g(u)f \prime (v)\nabla v
(f(v))2
,
R(u, v) = H(\nabla u)p -
\biggl\langle
\nabla
\biggl(
g(u)
f(v)
\biggr)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
=
= H(\nabla u)p -
\biggl\langle
g\prime (u)\nabla u
f(v)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
+
\biggl\langle
g(u)f \prime (v)\nabla v
(f(v))2
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
=
= H(\nabla u)p -
\biggl\langle
g\prime (u)\nabla u
f(v)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
+
g(u)f \prime (v)
(f(v))2
H(\nabla v)p =
= H(\nabla u)p +
g(u)f \prime (v)
(f(v))2
H(\nabla v)p - g\prime (u)H(\nabla v)p - 1H(\nabla u)
f(v)
+
g\prime (u)H(\nabla v)p - 1H(\nabla u)
f(v)
-
-
\biggl\langle
g\prime (u)\nabla u
f(v)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
=
= p
\biggl[
H(\nabla u)p
p
+
1
q
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q\biggr]
- g\prime (u)H(\nabla v)p - 1H(\nabla u)
f(v)\underbrace{} \underbrace{}
(I)
+
+
g(u)f \prime (v)H(\nabla v)p
(f(v))2
- p
q
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
\underbrace{} \underbrace{}
(II)
+
+
g\prime (u)H(\nabla v)p - 1H(\nabla u)
f(v)
-
\biggl\langle
g\prime (u)\nabla u
f(v)
, H(\nabla v)p - 1\nabla \xi H(\nabla v)
\biggr\rangle
\underbrace{} \underbrace{}
(III)
.
Now, we plan to show that (I), (II), (III) are nonnegative. Take a = H(\nabla u) and b =
=
g\prime (u)H(\nabla v)p - 1
pf(v)
in (2.2), we get
g\prime (u)H(\nabla v)p - 1H(\nabla u)
pf(v)
\leq H(\nabla u)p
p
+
1
q
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
,
which implies (I) \geq 0. By using (1.4), we obtain (II) \geq 0. To show (III) \geq 0, let us rewrite
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1456 G. DWIVEDI
(III) = H(\nabla u)H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
-
-
\Biggl\langle
\nabla u, H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
\nabla \xi H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) \Biggr\rangle
.
Take \xi = \nabla u, x = H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
\nabla \xi H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr)
in (2.1), we obtain
\Biggl\langle
H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
\nabla \xi H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr)
,\nabla u
\Biggr\rangle
\leq
\leq H(\nabla u)H0
\left( H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
\nabla \xi H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) \right) =
= H(\nabla u)H
\Biggl( \biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
=
=
g\prime (u)
f(v)
H(\nabla u)H(\nabla v)p - 1 \geq 0.
The equality in (I) holds if and only if
H(\nabla u)p =
\biggl(
g\prime (u)H(\nabla v)p - 1
pf(v)
\biggr) q
=
\biggl(
g\prime (u)
pf(v)
\biggr) q
H(\nabla v)p,
H(\nabla u) = H
\Biggl( \biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v
\Biggr)
a.e. in \Omega .
If
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v \not = 0 for some x0 \in S := \{ x \in \Omega : R(u, v) = 0\} , then
(I) = H(\nabla u)p + (p - 1)
\biggl(
g\prime (u)
pf(v)
\biggr) q
H(\nabla v)p - g\prime (u)
f(v)
H(\nabla v)p - 1H(\nabla u) =
= H(\nabla u)p + (p - 1)H
\Biggl( \biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p
-
- p
\Biggl\langle
\nabla u,H
\Biggl( \biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v
\Biggr) p - 1
\nabla \xi H
\Biggl( \biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v
\Biggr) \Biggr\rangle
.
When (I) = 0, by Lemma 2.1,
\nabla u =
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
GENERALIZED PICONE IDENTITY FOR FINSLER p-LAPLACIAN AND ITS APPLICATIONS 1457
If
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v = 0 for some subset S0 of S, then \nabla u = 0 a.e. in S0 which implies \nabla u =
=
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v.
Theorem 1.5 is proved.
4. Applications of Theorem 1.5. In this section, we use Theorem 1.5 to prove some qualitative
result. We assume that \Omega is a smooth and bounded domain in \BbbR n and the functions f and g satisfy
assumption of Theorem 1.5. First, we prove a Hardy-type inequality.
Theorem 4.1 (Hardy-type inequality). Let \Omega be a bounded domain in \BbbR n and v \in C\infty
c (\Omega ) be
such that
- \Delta H,pv \geq \lambda k(x)f(v), v > 0 in \Omega
for some \lambda > 0 and nonnegative function k \in L\infty (\Omega ). Then, for any u \in C\infty
c (\Omega ), u \geq 0 and
g(u) \in C\infty
c (\Omega ), \int
\Omega
H(\nabla u)pdx \geq \lambda
\int
\Omega
k(x)g(u)dx.
Proof. Take \phi \in C\infty
c (\Omega ), \phi > 0. By Theorem 1.5, we have
0 \leq
\int
\Omega
L(\phi , v) dx =
=
\int
\Omega
R(\phi , v) dx =
\int
\Omega
H(\nabla \phi )pdx - \Delta
\biggl(
g(\phi )
f(v)
\biggr)
H(\nabla v)p - 1\nabla \xi H(\nabla v)dx =
=
\int
\Omega
H(\nabla \phi )pdx+
\int
\Omega
g(\phi )
f(v)
\Delta H,pv dx \leq
\leq
\int
\Omega
H(\nabla \phi )pdx - \lambda
\int
\Omega
g(\phi )k(x) dx,
and letting \phi \rightarrow u, we get \int
\Omega
H(\nabla u)pdx \geq \lambda
\int
\Omega
k(x)g(u)dx.
Theorem 4.1 is proved.
Next, we prove a comparison result.
Theorem 4.2. Let k1(x) and k2(x) be two continuous functions such that k1(x) < k2(x) on a
bounded domain \Omega \subset \BbbR n. If there exists a function u \in C2(\Omega ) satisfying
- \Delta H,pu =
k1(x)g(u)
u
in \Omega ,
u > 0, g(u) > 0 in \Omega ,
u = 0 = g(u) on \partial \Omega ,
(4.1)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1458 G. DWIVEDI
then any nontrivial solution v to the equation
- \Delta H,pv = k2(x)f(v) in \Omega (4.2)
must change sign.
Proof. Assume that v does not change sign and \varepsilon > 0. Since g(u) = 0 on \partial \Omega ,
g(u)
f(v) + \varepsilon
\in
\in W 1,p
0 (\Omega ). By Theorem 1.5,
0 \leq
\int
\Omega
L(u, v)dx =
\int
\Omega
R(u, v)dx =
=
\int
\Omega
H(\nabla u)pdx -
\int
\Omega
\biggl\langle
\nabla
\biggl(
g(u)
f(v) + \varepsilon
\biggr)
, H(\nabla v)p - 1\nabla \xi (\nabla v)
\biggr\rangle
dx =
=
\int
\Omega
H(\nabla u)pdx+
\int
\Omega
g(u)
f(v) + \varepsilon
\Delta H,pvdx.
As \varepsilon \rightarrow 0, we obtain \int
\Omega
H(\nabla u)pdx+
\int
\Omega
g(u)
f(v)
\Delta H,pvdx \geq 0. (4.3)
On using (4.1) and (4.2) in (4.3), we have\int
\Omega
(k1(x) - k2(x))g(u)dx \geq 0,
which is a contradiction because k1(x) < k2(x) and g(u) > 0.
Theorem 4.2 is proved.
Finally, we establish a qualitative result concerning a system of equations involving Finsler p-
Laplacian.
Theorem 4.3. Let \Omega be a bounded domain in \BbbR n and (u, v) \in C2(\Omega ) \times C2(\Omega ) be a positive
solution to the elliptic system
- \Delta H,pu = f(v) in \Omega ,
- \Delta H,pv =
(f(v))2u
g(u)
in \Omega ,
u > 0, v > 0, g(u), f(v) > 0 in \Omega ,
u = 0 = g(u) on \partial \Omega .
Then \nabla u =
\biggl(
g\prime (u)
pf(v)
\biggr) 1/p - 1
\nabla v.
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GENERALIZED PICONE IDENTITY FOR FINSLER p-LAPLACIAN AND ITS APPLICATIONS 1459
Proof. For any \phi 1, \phi 2 \in W 1,p
0 (\Omega ), we get\int
\Omega
H(\nabla u)p - 1\nabla \xi H(\nabla u)\nabla \phi 1dx =
\int
\Omega
f(v)\phi 1dx,
\int
\Omega
H(\nabla v)p - 1\nabla \xi H(\nabla v)\nabla \phi 2dx =
\int
\Omega
(f(v))2u
g(u)
\phi 2dx.
Let \varepsilon > 0. Since g(u) = 0 on \partial \Omega ,
g(u)
f(v) + \varepsilon
\in W 1,p
0 (\Omega ). On choosing \phi 1 = u, \phi 2 =
g(u)
f(v) + \varepsilon
, we
obtain \int
\Omega
H(\nabla u)p - 1\nabla \xi H(\nabla u)\nabla udx =
\int
\Omega
f(v)udx, (4.4)
\int
\Omega
H(\nabla v)p - 1\nabla \xi H(\nabla v)\nabla
\biggl(
g(u)
f(v) + \varepsilon
\biggr)
dx =
\int
\Omega
uf(v)dx. (4.5)
On using (4.4) and (4.5), we have\int
\Omega
H(\nabla v)p - 1\nabla \xi H(\nabla v)\nabla
\biggl(
g(u)
f(v) + \varepsilon
\biggr)
dx =
\int
\Omega
uf(v)dx =
=
\int
\Omega
H(\nabla u)p - 1\nabla \xi H(\nabla u)\nabla udx =
=
\int
\Omega
H(\nabla u)pdx.
As \varepsilon \rightarrow 0, we obtain \int
\Omega
L(u, v)dx =
\int
\Omega
R(u, v)dx = 0
and, by Theorem 1.5, \nabla u =
\biggl(
g\prime (u)
f(v)
\biggr) 1/p - 1
a.e. in \Omega .
Theorem 4.3 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
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| id | umjimathkievua-article-1050 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:29Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/46/2346aff8511c25bcdbc8aa7dbb2ca046.pdf |
| spelling | umjimathkievua-article-10502025-03-31T08:46:33Z Generalized Picone identity for Finsler $p$-Laplacian and its applications Generalized Picone identity for Finsler $p$-Laplacian and its applications Dwivedi, G. Dwivedi, G. Picone identity Finsler p-Laplacian Picone identity Finsler p-Laplacian UDC 517.9 We prove a generalized Picone-type identity for Finsler $p$-Laplacian and use it to establish some qualitative results for some boundary-value problems involving Finsler $p$-Laplacian. &nbsp; УДК 517.9 Узагальнена тотожнiсть Пiконе для $p$ -лапласiана Фiнслера та її застосування Доведено узагальнену тотожність типу Піконе для $p$-лапласіана Фінслера, яку потім використано для отримання деяких якісних результатів для граничних задач, що включають $p$-лапласіан Фінслера. Institute of Mathematics, NAS of Ukraine 2021-11-23 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1050 10.37863/umzh.v73i11.1050 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 11 (2021); 1451 - 1460 Український математичний журнал; Том 73 № 11 (2021); 1451 - 1460 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1050/9155 Copyright (c) 2021 Gaurav Dwivedi |
| spellingShingle | Dwivedi, G. Dwivedi, G. Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_alt | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_full | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_fullStr | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_full_unstemmed | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_short | Generalized Picone identity for Finsler $p$-Laplacian and its applications |
| title_sort | generalized picone identity for finsler $p$-laplacian and its applications |
| topic_facet | Picone identity Finsler p-Laplacian Picone identity Finsler p-Laplacian |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1050 |
| work_keys_str_mv | AT dwivedig generalizedpiconeidentityforfinslerplaplaciananditsapplications AT dwivedig generalizedpiconeidentityforfinslerplaplaciananditsapplications |