Anisotropic Picone identities and anisotropic Hardy inequalities

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.


Introduction and main results
In recent years, the anisotropic Laplacian . Existence, integrability, boundedness, and continuity of solutions to anisotropic elliptic equations have received much attention; see [-] 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 [] considered the homogeneous linear second order differential system where u and v are differentiable functions in x, and proved the identity that, for the differ- then a Sturmian comparison principle and the oscillation theory of solutions were obtained via (.). Picone [] (see also Allegretto []) generalized (.) to a Laplacian that, for differentiable functions v >  and u ≥ ,  . 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 . (Anisotropic Picone identity)
Let v >  and u ≥  be two differentiable functions in the set ⊂ R n , and denote where p i >  (i = , . . . , n). Then .
This paper is organized as follows: The proofs of Theorem . and a Sturmian comparison principle to the anisotropic elliptic equation are given in Section ; Section  is devoted to the proof of Theorem . in which a key ingredient is to choose a suitable auxiliary function (see (.) below) for the anisotropic case. Two corollaries are also furnished.

Proof of Theorem 1.1
Proof of Theorem . One derives easily that Recall Young's inequality: for a ≥  and b ≥ , where p i > , q i >  (i = , . . . , n) and  p i +  q i = ; the equality holds if and only if which shows u = cv. The proof of Theorem . is completed.

Let us address anisotropic Sobolev spaces; see Adams [], Lu [], Troisi []
etc. Given a domain ⊂ R n , p i > , i = , , . . . , n. We define two anisotropic Sobolev spaces by with the norms Then any nontrivial solution v to the following anisotropic elliptic equation: Proof Suppose that v to (.) does not change sign, without loss of generality, let v >  in . By (.), (.), and (.), we observe which is a contradiction. This completes the proof.

Proof of Theorem 1.3
To prove Theorem ., we need a lemma from Theorem ..
Lemma . If there exist a constant k i >  and a function h i (x), i = , . . . , n, such that a differentiable function v >  in the set satisfies Proof By (.) and (.), we see which implies (.).
Proof of Theorem . Without loss of generality, we let  ≤ u ∈ C ∞  . To use Lemma ., we introduce the auxiliary function where β j = p j - p j and v i = n j=,j =i |x j | β j , hence Taking k i = ( p i - p i ) p i and h i (x) =  |x i | p i , and using Lemma ., we obtain (.).
Corollary . For u ∈ C   (A), it follows that we have by taking a i = |x i |  , Corollary . If p > , then, for u ∈ C   (A), it follows that