Anisotropic Picone identities and anisotropic Hardy inequalities
Feng and Cui Journal of Inequalities and Applications
Anisotropic Picone identities and anisotropic Hardy inequalities
Tingfu Feng 0
Xuewei Cui 0
0 Department of Applied Mathematics, Northwestern Polytechnical University , Xi'an, Shaanxi 710072 , P.R. China
In this paper, we derive an anisotropic Picone identity for the anisotropic Laplacian, which contains some known Picone identities. As applications, a Sturmian comparison principle to the anisotropic elliptic equation and an anisotropic Hardy type inequality are shown. MSC: Primary 26D10; secondary 26D15

∂u pi– ∂u
∂xi
∂xi
∂u p– ∂u
∂xi
∂xi
Allegretto and Huang [], Dunninger [] independently extended (.) to a pLaplacian,
for differentiable functions v > and u ≥ ,
u u
= ∇u + v ∇v – v ∇v · ∇u
= ∇u – ∇
∇v.
= ∇up – ∇
(a(x)u ) + b(x)u = ,
(a(x)v ) + b(x)v = ,
where u and v are differentiable functions in x, and proved the identity that, for the
differentiable function v(x) = ,
Theorem . (Anisotropic Picone identity) Let v > and u ≥ be two differentiable
functions in the set ⊂ Rn, and denote
R(u, v) =
L(u, v) =
∂u pi
∂xi
∂u pi
∂xi
upi– ∂v pi– ∂v ∂u
pi vpi– ∂xi ∂xi ∂xi
∂v pi– ∂v
∂xi
∂xi
∂u pi
dx ≥
2 Proof of Theorem 1.1
Proof of Theorem . One derives easily that
R(u, v) = L(u, v).
Moreover, we have
L(u, v) ≥ ;
∂u pi
∂xi
∂u pi
∂xi
∂u pi
∂xi
= L(u, v),
∂u pi
∂xi
R(u, v) =
L(u, v) =
n ∂ upi ∂v pi– ∂v
i= ∂xi vpi– ∂xi ∂xi
n piupi– ∂∂xui vpi– – upi (pi – )vpi– ∂∂xvi ∂v pi– ∂v
∂xi
∂xi
upi– ∂v pi– ∂v ∂u
pi vpi– ∂xi ∂xi ∂xi
upi– ∂v pi– ∂u
pi vpi– ∂xi ∂xi
I =
II =
i=
n
∂u pi + pi –
pi pi ∂xi pi
upi– ∂v pi– ∂u
pi vpi– ∂xi ∂xi
∂∂xvi ∂∂xui – ∂∂xvi ∂∂xui .
∂u pi + pi –
≤ pi pi ∂xi pi
∂∂xui = uv ∂∂xvi .
Using II = , it implies
∂∂xui = c ∂∂xvi .
Putting (.) into (.) yields u = cv.
(b) If u(x) = , then we denote S = {x ∈ u(x) = } and ∂∂xui = a.e. in S. Thus
with the norms
u W,(pi)( ) =
u dx +
∂u pi
∂xi
u W,(pi)( ) =
∂u pi
∂xi
in= f(x)upi–, x ∈ ,
upi ∂
i= vpi– ∂xi
∂v pi– ∂v
∂xi
∂xi
f(x)upi , x ∈ ,
Proof Suppose that v to (.) does not change sign, without loss of generality, let v >
in . By (.), (.), and (.), we observe
must change sign.
≤
L(u, v) dx =
R(u, v) dx
∂u pi
∂xi
∂v pi– ∂v
∂xi
∂xi
which is a contradiction. This completes the proof.
3 Proof of Theorem 1.3
To prove Theorem ., we need a lemma from Theorem ..
v =
∂ v pi– ∂ v
∂ xi
∂ xi
≥ kihi(x)vpi–,
then, for any ≤ u ∈ C( ), we have
∂ u pi
∂ xi
dx ≥
hi(x)upi dx.
Proof By (.) and (.), we see
≤
L(u, v) dx =
R(u, v) dx
where βj = ppj–
j and vi =
∂ u pi
∂ xi
∂ u pi
∂ xi
∂ u pi
∂ xi
∂ v pi– ∂ v
∂ xi
∂ xi
∂ v pi– ∂ v
∂ xi
∂ xi
hi(x)upi dx,
which implies (.).
Proof of Theorem . Without loss of generality, we let ≤ u ∈ C∞. To use Lemma ., we
introduce the auxiliary function
∂ xi
∂ xi
∂ v pi–
∂ v pi– ∂ v
∂ xi
= βipi– v¯ipi–xiβipi–βi–pi+,
= βipi–vipi–xiβipi–βi–pi xi,
∂ v pi– ∂ v
∂ xi
∂ xi
i= ai
we have by taking ai = xi,
∇u dx =
for ai ≥ , i = , . . . , n,
∂ u
A ∂ xi
i= xi
∇up dx ≥
which gives
ai ≤
aip ≥ n– p–
for ai ≥ , i = , . . . , n,
i= xi
≥ n– p–
Putting (.) into the righthand side of (.),
On the other hand,
A i=
∂ u p
∂ xi
dx ≤
∂ xi
dx =
∇up dx.
Hence (.) is proved via (.) and (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 11271299), and the Natural
Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JM1203).
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