Harnack inequality for subelliptic p-Laplacian equations of Schrödinger type
Journal of Inequalities and Applications volume 2013, Article number: 160 (2013)
In this paper, we establish the Harnack inequality for weak solutions of nonlinear subelliptic p-Laplacian equations of Schrödinger type
when the singular potential V is in the Kato-Stummel type class with respect to the Carnot-Carathéodory metric.
Let Ω be a bounded open subset of for . Given a family of vector fields , we assume that each component is locally Lipschitz continuous for . We identify the vector field with if , and when , it is understood in the distributional sense that
In this paper, we are interested in nonlinear equations involving the subelliptic p-Laplace operator and the singular potential V belonging to the Kato-Stummel type class. We consider the nonlinear subelliptic p-Laplacian equations of Schrödinger type in Ω
where , denotes the adjoint of and each entry of the bounded measurable coefficient matrix satisfies and . We suppose that the operator satisfies the following X-ellipticity condition:
where λ is a positive constant and denotes the standard inner product in . The notion of X-ellipticity was implicity introduced in  by Franchi and Lanconelli in 1982, after which it was intensively studied in a series of works [2–5], etc. In 2000, it was explicitly developed in  by Lanconelli and Kogoj.
For reader’s convenience, let us recall the notion of control distance (or the Carnot-Carathéodory distance) associated to the family X. An absolutely continuous path is said to be an X-subunit if , with , for almost every . Assuming that is X-connected, i.e., for every , there exists at least one X-subunit path connecting x and y, we define
The mapping is a metric on . It can be proved that implies , where is the Euclidean norm. Hereafter, all the distances mentioned in this context are designated with respect to the metric d unless we indicate it specifically. In particular, denotes the ball with the control metric, and denotes the Lebesgue measure of the set . Following the literature , we define the Sobolev spaces for as follows:
equipped with the norm
Before stating our main results, we present several assumptions of control distance with respect to the vector fields X in and the singular potential V which we will use in the next sections.
(H1) (Metric equivalence) If , then we have that .
(H2) (Homogeneous dimension) There are positive constants , , and such that for all and , the following relation is valid:
where the number Q is chosen as the least integer such that the above inequality holds, which is called the homogeneous dimension of X in Ω.
From the assumptions of control distance , the following Sobolev embedding inequality is valid : If , then for . Furthermore, there exists a constant , then for all ,
According to , we have the Poincaré inequality, that is, there is a positive constant C such that the following inequality holds:
For the subelliptic p-Laplace operator in , the Green function is given by , that is, when , we have the following decay estimates:
where is the smallest metric ball such that . Also, the Green function estimates for the divergence form of subelliptic operators with nonsmooth coefficients were obtained in  for .
The assumption on V is that , the Kato-Stummel type class, which means that V is a local integrable function such that , where
Let , , where
We remark that if and , then is in fact the Morrey space for some appropriate μ. Moreover, the inclusion is trivial.
This article is devoted to presenting the elementary proof of the local estimate and the Harnack inequality for the solutions of Equation (1.1), which is the extension of the related results in [5, 11], where or , and the Harnack inequality for p-subLaplacian in .
Our main results are as follows.
Theorem 1.1 Let be a weak solution to Equation (1.1), and let with and . Assume that with . Then
with the constant depending only on Q, p, λ and .
Theorem 1.2 Let be a non-negative weak solution to Equation (1.1) in Ω and , and let . Then there exists a constant depending only on Q, p, λ and such that
Corollary 1.1 Under the assumptions of Theorem 1.2, the non-negative weak solution of Equation (1.1) is locally Hölder continuous in Ω.
We say is a weak solution of Equation (1.1) if u satisfies
for any test function .
Throughout this paper, unless otherwise indicated, C is used to denote a positive constant that is not necessarily the same at each occurrence, which depends at most on Q, p, λ, V and Ω.
2 Local boundedness of solutions
Lemma 2.1 (Embedding lemma)
Let Ω be an open, bounded and connected set in , and suppose and . Then there exists a positive constant such that for any and , we have
Proof Let be the Green function for in . We set
This is well defined, and since . Moreover, is the weak solution to the equation in , and also on .
For and , it is easy to see from the Young inequality that
From the uniform X-ellipticity assumption (1.2), a direct calculation shows
when φ and . Assume that , then by the inequality (2.1), we get that
for any . In addition, we note that
This and the inequality (2.2) with yield that
From the inequalities (2.3) and (2.4), and the estimation of ψ, we obtain that
To complete the proof of the lemma, we employ the method of the proof of Lemma 3.3 in . Given , let be a finite partition of unity of such that with and . Set and further restrict . Therefore,
We now choose ε such that
which implies that
By summing in j, it follows that
with the constant . It is clear that the inequality (2.6) implies the desired conclusion. □
Lemma 2.2 Let Ω be an open, bounded and connected set in , and suppose with and . Then, for any , there exists a constant such that for any , we have
Proof Repeating the proof of Lemma 2.1 and noting , we can deduce from the inequality (2.6) that the lemma holds with and as long as we choose
We are ready to show the local maximal estimates, i.e., Theorem 1.1.
Proof of Theorem 1.1 For and , where h is a positive number which will be determined later, we define the function by
Note that is non-decreasing, non-negative and bounded for each fixed M, and for any . Let be the weak solution to Equation (1.1), and let . We consider the function . It is easy to see that
for any . Moreover, .
Thus, for any non-negative function , we can choose as a testing function in Equation (1.1), which yields
Recalling that and so , using the estimates (2.7) and the fact that , we can then obtain that
Using the inequality (2.2) with , we have
where . We set and assume , then we rewrite the inequality above as follows:
Now we take , and so . By Lemma 2.2 we have, for any ,
with the positive constants C and τ independent of f, u and ϕ. From this and the inequality (2.8) it follows that
where C is a positive constant depending on Q, p and λ.
By the Sobolev embedding inequality, we get from (2.9) that
with absolute constant C, where .
Let and be such that , taking in such a way that in , in and , and recalling and , we thus obtain
Letting , then
with the constant C independent of , , q and ν.
By the standard iteration argument , we set and for . Hence the previous inequality (2.10) becomes
with the constant C independent of ν. Recalling and that , we get the conclusion. This completes the proof of the theorem. □
3 Harnack inequality of solutions
In this section we give the proof of Theorem 1.2, i.e., the Harnack inequality.
Proof of Theorem 1.2 Proceeding as in the proof of Theorem 1.1, it is sufficient to prove the theorem by assuming . The general case follows by dilations. We set with and , and take as a test function in the inequality (1.3), where is a non-negative smooth function such that and . We obtain
Then it follows from the Cauchy inequality that
Then, from the above inequality, in the case , we see that
where and . Simultaneously, in the case , we can see that
To our destination, we exploit the inequality (3.2) and use Lemma 2.1 with to get
where C is a positive constant depending only on Q, p, λ and . Choosing in such a way that in , and , where is an arbitrary open ball contained in , we then obtain that
Thus, by the assumption of (H3) Poincaré inequality, we get that U is a BMO function, and so the John-Nirenberg lemma for BMO function yields that there exist positive constants and C such that
for any real number and . Then from the previous inequality, recalling that , we have
Now we turn our attention to (3.1). By Lemma 2.2 we obtain
with the constant depending only Q and p. We choose ε such that , and so we deduce that
Let and be such that , choosing in such a way that in , in and . From (3.4) and the Sobolev embedding inequality, we obtain
with and the constant depending only on Q, p and τ.
Put and take the q th root of each side of (3.5) for positive q and negative q, respectively. Then by the iteration arguments used in the proof of Theorem 1.1, we choose the initial index with some , if , such that lies midway between two consecutive iterates of for , which certifies that , and we choose the initial index if ,
Therefore, from (3.3), (3.6), (3.7) and the Hölder inequality, we have arrived at
which implies the desired conclusion of Theorem 1.2. □
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The second author is supported by the NSF of China (No. 11161044).
The authors declare that they have no competing interests.
The authors did not provide this information.
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Guo, Y., Jiang, Y. Harnack inequality for subelliptic p-Laplacian equations of Schrödinger type. J Inequal Appl 2013, 160 (2013). https://doi.org/10.1186/1029-242X-2013-160
- Harnack inequality
- subelliptic p-Laplacian
- control distance