Harnack inequality for subelliptic p-Laplacian equations of Schrödinger type
© Guo and Jiang; licensee Springer. 2013
Received: 14 May 2012
Accepted: 14 March 2013
Published: 8 April 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.
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.
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 .
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.
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 .
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.
with the constant depending only on Q, p, λ and .
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.
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)
This is well defined, and since . Moreover, is the weak solution to the equation in , and also on .
with the constant . It is clear that the inequality (2.6) implies the desired conclusion. □
We are ready to show the local maximal estimates, i.e., Theorem 1.1.
for any . Moreover, .
where C is a positive constant depending on Q, p and λ.
with absolute constant C, where .
with the constant C independent of , , q and ν.
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.
with and the constant depending only on Q, p and τ.
which implies the desired conclusion of Theorem 1.2. □
The second author is supported by the NSF of China (No. 11161044).
- Franchi B, Lanconelli E: Une métrique associée à une class d’opérateurs elliptiques dégénérés. Conference on Linear Partial and Pseudo-Differential Operators 1984, 105–114. (in French). Special issue on Rend. Sem. Mat. Univ. Politec. Torino 1983, Torino 1982Google Scholar
- Capogna L, Danielli D, Garofalo N: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Commun. Partial Differ. Equ. 1993, 18(9–10):1765–1794.MathSciNetView ArticleGoogle Scholar
- Fazio GD, Zamboni P: Unique continuation of non negative solutions to quasilinear subelliptic equations in Carnot Carathéodory spaces. Commun. Appl. Nonlinear Anal. 2003, 10(2):97–105.Google Scholar
- Mazzoni G: Green function for X -elliptic operators. Manuscr. Math. 2004, 115(2):207–238. 10.1007/s00229-004-0494-5MathSciNetView ArticleGoogle Scholar
- Zheng S, Feng Z: Green functions for a class of nonlinear degenerate operators with X -ellipticity. Trans. Am. Math. Soc. 2012, 364(7):3627–3655. 10.1090/S0002-9947-2012-05617-0MathSciNetView ArticleGoogle Scholar
- Lanconelli E, Kogoj AE: X -elliptic operators and X -control distances, contributions in honor of the memory of Ennio De Giorgi. Ric. Mat. 2000, 49: 223–243. (suppl)MathSciNetGoogle Scholar
- Lu G: Weighted Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications. Rev. Mat. Iberoam. 1992, 8(3):367–439.View ArticleGoogle Scholar
- Jerison D: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 1986, 53(2):503–523. 10.1215/S0012-7094-86-05329-9MathSciNetView ArticleGoogle Scholar
- Lu G: The sharp Poincaré inequality for free vector fields: an endpoint result. Rev. Mat. Iberoam. 1994, 10(2):453–466.View ArticleGoogle Scholar
- Lu G: Existence and size estimates for the Green’s functions of differential operators constructed from degenerate vector fields. Commun. Partial Differ. Equ. 1992, 17(7–8):1213–1251.Google Scholar
- Lu G: On Harnack’s inequality for a class of strongly degenerate Schrödinger operators formed by vector fields. Differ. Integral Equ. 1994, 7(1):73–100.Google Scholar
- Lu G: Embedding theorems into Lipschitz and BMO spaces and applications to quasilinear subelliptic differential equations. Publ. Mat. 1996, 40(2):301–329.MathSciNetView ArticleGoogle Scholar
- Serrin J: Local behavior of solutions of quasilinear equations. Acta Math. 1964, 111(1):247–302. 10.1007/BF02391014MathSciNetView ArticleGoogle Scholar
- Zamboni P: Hölder continuity for solutions of linear degenerate elliptic equations under minimal assumptions. J. Differ. Equ. 2002, 182(1):121–140. 10.1006/jdeq.2001.4094MathSciNetView ArticleGoogle Scholar
- Grutierrez C: Harnack’s inequality for degenerate Schrödinger operators. Trans. Am. Math. Soc. 1989, 312(1):403–419.Google Scholar
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