Abstract
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.
Similar content being viewed by others
1 Introduction and main results
In recent years, the anisotropic Laplacian
has been considerably concerned. Note that if \({p_{i}} = 2\) (\({i = 1, \ldots,n}\)), then (1.1) becomes the classical Laplacian; if \({p_{i}} = p = \mathrm{const}\), then (1.1) is the pseudo-p-Laplacian (see [1])
The anisotropic Laplacian has not only the widespread practical background in the natural science, but also the important theoretical value in the mathematics. For example, it reflects anisotropic physical properties of some reinforced materials (Lions [2] and Tang [3]), and describes the dynamics of fluids in the anisotropic media when the conductivities of the media are different in each direction [4, 5]. The equations associated with (1.1) are also deduced in the image processing [6]. Existence, integrability, boundedness, and continuity of solutions to anisotropic elliptic equations have received much attention; see [7–15] and the references therein. In this paper, we prove 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 given. Before giving the main results of this paper, we briefly recall the existing results for the isotropic case.
Picone [16] considered the homogeneous linear second order differential system
where u and v are differentiable functions in x, and proved the identity that, for the differentiable function \(v(x) \ne0\),
then a Sturmian comparison principle and the oscillation theory of solutions were obtained via (1.2). Picone [17] (see also Allegretto [18]) generalized (1.2) to a Laplacian that, for differentiable functions \(v > 0\) and \(u \geq0\),
Allegretto and Huang [19], Dunninger [20] independently extended (1.3) to a p-Laplacian, for differentiable functions \(v > 0\) and \(u \geq0\),
and applied (1.4) to derive a Sturmian comparison principle, Liouville’s theorem, the Hardy inequality, and some profound results for p-Laplace equations and systems. For other generalizations of the Picone identities and applications, see Bal [21], Dwivedi [22], Dwivedi and Tyagi [23], Niu, Zhang and Wang [24], Tyagi [25]. These results indicate that Picone identities are seemingly simple in form, but extremely useful in the study of partial differential equations, and they have become an important tool in the analysis.
Our main results are as follows.
Theorem 1.1
Anisotropic Picone identity
Let \(v > 0\) and \(u \geq0\) be two differentiable functions in the set \(\Omega \subset{R^{n}}\), and denote
where \({p_{i}} > 1\) (\({i = 1, \ldots,n}\)). Then
Moreover, we have
furthermore, \(L(u,v) = 0\) a.e. in Ω if and only if \(u = cv\) a.e. in Ω, c is a positive constant.
Remark 1.2
If \({p_{i}} = 2\) (\({i = 1, \ldots,n}\)) in (1.5) and (1.6), we have (1.3) from (1.7). If \({p_{i}} = p = \mathrm{const}\) (\({i = 1, \ldots,n}\)) in (1.5) and (1.6), the result in [26] follows. Moreover, the identity in Theorem 1.1 is different from the one in [26].
Theorem 1.3
Anisotropic Hardy type inequality
Let \(u \in C_{0}^{1} ( A )\), \(1 < {p_{i}} < n\), \(i = 1, \ldots,n\), \(A = \{ x \in{R^{n}}| {{x_{i}} \ne0,i = 1, \ldots,n} \}\). Then we have
This paper is organized as follows: The proofs of Theorem 1.1 and a Sturmian comparison principle to the anisotropic elliptic equation are given in Section 2; Section 3 is devoted to the proof of Theorem 1.3 in which a key ingredient is to choose a suitable auxiliary function (see (3.3) below) for the anisotropic case. Two corollaries are also furnished.
2 Proof of Theorem 1.1
Proof of Theorem 1.1
One derives easily that
which is (1.7). To check \(L(u,v) \geq0\), we rewrite \(L(u,v)\) by
where
Recall Young’s inequality: for \(a \geq0\) and \(b \geq0\),
where \({p_{i}} > 1\), \({q_{i}} > 1\) (\(i = 1, \ldots,n\)) and \(\frac{1}{ {{p_{i}}}} + \frac{1}{{{q_{i}}}} = 1\); the equality holds if and only if \({a^{{p_{i}}}} = {b^{{q_{i}}}}\), namely, \(a = {b^{\frac{1}{{{p_{i}} - 1}}}}\). We take \(a = \vert {\frac{{\partial u}}{{\partial {x_{i}}}}} \vert \) and \(b = { ( {\frac{u}{v}\vert {\frac{{\partial v}}{{\partial{x_{i}}}}} \vert } )^{{p_{i}} - 1}}\) in (2.2) to obtain
and so \(\mathit{I} \geq0\) from (2.3). Clearly, \(\mathit{II} \geq0\) in virtue of \(\vert {\frac{{\partial v}}{ {\partial{x_{i}}}}} \vert \vert {\frac{{\partial u}}{ {\partial{x_{i}}}}} \vert - \frac{{\partial v}}{ {\partial{x_{i}}}}\frac{{\partial u}}{ {\partial{x_{i}}}} \geq0\). Hence \(L(u,v) \geq0\) from (2.1).
If \(u = cv\), c is a positive constant, then clearly \(L(u,v) = 0\). Now let us conclude that \(L(u,v) = 0\) implies \(u = cv\). In fact, if \(L(u,v)({x_{0}}) = 0\), \({x_{0}} \in\Omega\), then we consider the two cases \(u({x_{0}}) \ne0\) and \(u({x_{0}}) = 0\), respectively.
(a) If \(u({x_{0}}) \ne0\), then \(\mathit{I} = 0\) and \(\mathit{II} = 0\). One shows by \(\mathit{I} = 0\) that
Using \(\mathit{II} = 0\), it implies
Putting (2.5) into (2.4) yields \(u = cv\).
(b) If \(u({x_{0}}) = 0\), then we denote \(S = \{x \in\Omega | {u(x) = 0} \}\) and \(\frac{{\partial u}}{{\partial {x_{i}}}} = 0\) a.e. in S. Thus
which shows \(u = cv\). The proof of Theorem 1.1 is completed. □
Let us address anisotropic Sobolev spaces; see Adams [27], Lu [28], Troisi [29] etc. Given a domain \(\Omega \subset {R^{n}}\), \({p_{i}} > 1\), \(i = 1,2, \ldots,n\). We define two anisotropic Sobolev spaces by
and
with the norms
and
respectively. Note that \(W_{0}^{1, ( {{p_{i}}} )}(\Omega)\) is the closure of \(C_{0}^{\infty}(\Omega)\) in \({W^{1, ( {{p_{i}}} )}}(\Omega)\). It is well known that \({W^{1, ( {{p_{i}}} )}}(\Omega)\) and \(W_{0}^{1, ( {{p_{i}}} )}(\Omega)\) are both separable and reflexive Banach spaces.
We will show a Sturmian comparison principle to the anisotropic elliptic equation by Theorem 1.1.
Proposition 2.1
Let \({f_{1}}(x)\) and \({f_{2}}(x)\) be two continuous functions with \({f_{1}}(x) < {f_{2}}(x)\) in the bounded domain Ω. Assume that there exists a positive function \(u \in W_{0}^{1, ( {{p_{i}}} )}(\Omega)\) satisfying
Then any nontrivial solution v to the following anisotropic elliptic equation:
must change sign.
Proof
Suppose that v to (2.7) does not change sign, without loss of generality, let \(v > 0\) in Ω. By (2.6), (2.7), and (1.7), we observe
which is a contradiction. This completes the proof. □
3 Proof of Theorem 1.3
To prove Theorem 1.3, we need a lemma from Theorem 1.1.
Lemma 3.1
If there exist a constant \({k_{i}} > 0\) and a function \({h_{i}}(x)\), \(i = 1, \ldots,n\), such that a differentiable function \(v>0\) in the set Ω satisfies
then, for any \(0 \leq u \in C_{0}^{1}(\Omega)\), we have
Proof
which implies (3.2). □
Proof of Theorem 1.3
Without loss of generality, we let \(0 \leq u \in C_{0}^{\infty}\). To use Lemma 3.1, we introduce the auxiliary function
where \({\beta_{j}} = \frac{{{p_{j}} - 1}}{{{p_{j}}}}\) and \({\overline{v} _{i}} = \prod_{j = 1,j \ne i}^{n} {{{\vert {{x_{j}}} \vert }^{{\beta_{j}}}}} \), hence
and
Taking \({k_{i}} = { ( {\frac{{{p_{i}} - 1}}{{{p_{i}}}}} )^{{p_{i}}}}\) and \({h_{i}}(x) = \frac{1}{{{{\vert {{x_{i}}} \vert }^{{p_{i}}}}}}\), and using Lemma 3.1, we obtain (1.8). □
Corollary 3.2
For \(u \in C_{0}^{1} ( A )\), it follows that
Proof
Letting \({p_{i}} = 2\) (\(i = 1, \ldots,n\)) in (1.8) and noting the elementary inequality
we have by taking \({a_{i}} = {\vert {{x_{i}}} \vert ^{2}}\),
□
Corollary 3.3
If \(p > 2\), then, for \(u \in C_{0}^{1} ( {A} )\), it follows that
Proof
Let \({p_{i}} = p > 2\) (\(i = 1, \ldots,n\)) in (1.8). Recall the inequality
which gives
Taking \({a_{i}} = \frac{1}{{\vert {{x_{i}}} \vert }}\) in (3.8), it implies by (3.6) that
Putting (3.9) into the right-hand side of (1.8),
On the other hand,
References
Belloni, M, Kawohl, B: The pseudo-p-Laplace eigenvalue problem and viscosity solutions as \(p \to\infty\). ESAIM Control Optim. Calc. Var. 10(1), 28-52 (2004)
Lions, JL: Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969)
Tang, Q: Regularity of minimizer of non-isotropic integrals of the calculus of variations. Ann. Mat. Pura Appl. 164(1), 77-87 (1993)
Antontsev, SN, Díaz, JI, Shmarev, S: Energy Methods for Free Boundary Problems: Applications to Nonlinear PDEs and Fluid Mechanics. Springer, Berlin (2012)
Bear, J: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)
Weickert, J: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
Alves, CO, El Hamidi, A: Existence of solution for a anisotropic equation with critical exponent. Differ. Integral Equ. 21(1-2), 25-40 (2008)
Cianchi, A: Symmetrization in anisotropic elliptic problems. Commun. Partial Differ. Equ. 32(5), 693-717 (2007)
Cîrstea, FC, Vétois, J: Fundamental solutions for anisotropic elliptic equations: existence and a priori estimates. Commun. Partial Differ. Equ. 40(4), 727-765 (2015)
Di Castro, A, Montefusco, E: Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations. Nonlinear Anal. 70(11), 4093-4105 (2009)
Fragalà, I, Gazzola, F, Kawohl, B: Existence and nonexistence results for anisotropic quasilinear elliptic equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21(5), 715-734 (2004)
Innamorati, A, Leonetti, F: Global integrability for weak solutions to some anisotropic elliptic equations. Nonlinear Anal. 113(5), 430-434 (2015)
Lieberman, GM: Gradient estimates for anisotropic elliptic equations. Adv. Differ. Equ. 10(7), 767-812 (2005)
Liskevich, V, Skrypnik, II: Hölder continuity of solutions to an anisotropic elliptic equation. Nonlinear Anal. 71(5-6), 1699-1708 (2009)
Tersenov, AS, Tersenov, AS: The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations. J. Differ. Equ. 235(2), 376-396 (2007)
Picone, M: Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 11, 1-144 (1910)
Picone, M: Un teorema sulle soluzioni delle equazioni lineari ellittiche autoaggiunte alle derivate parziali del secondo-ordine. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 20, 213-219 (1911)
Allegretto, W: Sturmianian theorems for second order systems. Proc. Am. Math. Soc. 94(2), 291-296 (1985)
Allegretto, W, Huang, Y: A Picone’s identity for the p-Laplacian and applications. Nonlinear Anal. 32(7), 819-830 (1998)
Dunninger, DR: A Sturm comparison theorem for some degenerate quasilinear elliptic operators. Boll. Unione Mat. Ital., A 9, 117-121 (1995)
Bal, K: Generalized Picone’s identity and its applications. Electron. J. Differ. Equ. 2013, 243 (2013)
Dwivedi, G, Tyagi, J: Remarks on the qualitative questions for biharmonic operators. Taiwan. J. Math. 19(6), 1743-1758 (2015)
Dwivedi, G, Tyagi, J: Picone’s identity for biharmonic operators on Heisenberg group and its applications. NoDEA Nonlinear Differ. Equ. Appl. 23(2), 1-26 (2016)
Niu, P, Zhang, H, Wang, Y: Hardy type and Rellich type inequalities on the Heisenberg group. Proc. Am. Math. Soc. 129(129), 3623-3630 (2001)
Tyagi, J: A nonlinear Picone’s identity and its applications. Appl. Math. Lett. 26(6), 624-626 (2013)
Jaroš, J: Caccioppoli estimates through an anisotropic Picone’s identity. Proc. Am. Math. Soc. 143(3), 1137-1144 (2015)
Adams, RA: Sobolev Spaces. Academic Press, New York (1975)
Lu, W: On embedding theorem of spaces of functions with partial derivatives summable with different powers. Vestn. Leningr. State Univ. 7, 23-37 (1961)
Troisi, M: Teoremi di inclusione per spazi di Sobolev non isotropi. Ric. Mat. 18(1), 3-24 (1969)
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Feng, T., Cui, X. Anisotropic Picone identities and anisotropic Hardy inequalities. J Inequal Appl 2017, 16 (2017). https://doi.org/10.1186/s13660-017-1292-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-017-1292-4