\(L^{p}\)-Estimates for quasilinear subelliptic equations with VMO coefficients under the controllable growth

Before imposing some structural assumptions on \(A_{ij}(x,u)\) and \(b(x,u,Xu)\); and stating our main result, let us first recall a few of notations and basic facts involving the Carnot-Carathéodory metric induced by the smooth vector fields X on \(\mathbb{R}^{n}\).

Let us now review some recent studies on the subelliptic topic. Recent tremendous studies on subelliptic PDEs arising from non-commuting vector fields have been well developed; for details see [3, 6, 7, 12–19] and references therein. The regularity of subelliptic operators was first introduced by Hörmander in [20], which stimulated people’s interest in subelliptic problems to a large degree. Since then, many important results about the fundamental solution to subelliptic operators and harmonic analysis theory on stratified nilpotent Lie groups have been obtained by Folland [21], Rothschild and Stein [13], and Nagel,Stein and Wainger [3]. These results laid a solid foundation for further investigation of subelliptic Partial Differential Equations theory. Up to the 1990s, the function theory and harmonic analysis tools on Carnot groups, such as the Sobolev embedding inequality of X-gradient and the isoperimetric inequality, became increasingly mature; for more details see [1, 3, 5, 13, 16, 21–24] and references therein. From this, there was a large amount of literature as regards the problems of subelliptic PDEs on the Carnot-Carathéodory metric space. For instance, Capogna, Danielli and Garofalo [1] in 1993 studied an embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Meanwhile, the Harnack inequality for solutions to quasilinear subelliptic differential equations concerning (1.1), a class of degenerate subelliptic equations and a class of strongly degenerate Schrödinger operators were already established by Franchi, Lu and Wheeden in [14, 22, 25], respectively. In addition, Lu [15] also obtained the existence and size estimates for the Green’s functions of differential operators constructed from degenerate vector fields. Lately, Xu and Zuily in [6] obtained Schauder estimates of quasilinear subelliptic equations with smooth coefficients under natural growth, and further proved that their weak solutions are smooth if all given datum are smooth. Recently, Bramanti, Brandolini and Fanciullo [9, 26–28] studied \(L^{p}\)-estimates for nonvariational hypoelliptic operators with VMO coefficients, Schauder estimates for parabolic nondivergence operators, \(C^{k,\alpha }\)-regularity to quasilinear equations, and BMO estimates for nonvariational operators with discontinuous coefficients structured on Hörmander’s vector fields, respectively. Closely related to this article, we would like to mention that Di Fazio and Fanciullo established \(L^{p}\)-theory of nonlinear operators with coefficients under the Cordes conditions in Heisenberg group [29] and Carnot groups [30]. As for the problems of geometric subelliptic equations, Jost and Xu [31] and Wang [8] got partial regularities of subelliptic harmonic mappings, respectively. Moreover, a similar result for subelliptic p-harmonic mappings was obtained by Hajłasz and Strzelecki in [32]. Zheng and Feng [18] very recently gave various estimates of the Green’s function of quasilinear subelliptic equations and their applications to regularity problems.

On the other hand, there has been tremendous work on the \(L^{p}\)-theory of elliptic equations with discontinuous coefficients. It was remarkable and unpredictable that Chiarenza et al. [33] first derived results as regards \(W^{2,p}\)-estimates for elliptic equations with VMO coefficients based on the Calderón-Zygmund theorem and estimates of commutators. Later, \(L^{p}\)-estimates were extended to divergence form elliptic equations with discontinuous coefficients by Di Fazio in [34]. In recent years, two different approaches to elliptic and parabolic equations with VMO coefficients were developed in Byun and Wang [35] and Dong, Kim, and Krylov [36], respectively. Zhu, Bramanti and Niu [37] recently attained interior \(L^{p}\)-estimates for divergence degenerate elliptic systems in Carnot groups by way of rather geometric arguments from Byun and Wang’s work on series, which include the Hardy-Littlewood maximal functions and modified versions of the Vitali covering lemma. In general, when quasilinear or nonlinear equations are considered, the regularity of weak solutions has been investigated under various growth conditions on lower order terms. For the quasilinear setting, Zheng and Feng [38] recently obtained an optimal interior Hölder continuity by way of an interior reverse Hölder’s inequality for quasilinear elliptic equations with the controlled growth conditions under the assumption that the leading coefficients are in the class of VMO functions with respect to x variables. Later, Dong and Kim [2] used a unified approach to get global Hölder continuity by the interior and boundary reverse Hölder’s inequalities and bootstrap argument for quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. Yu and Zheng [39] also derived the same Hölder continuity to quasilinear elliptic equations with the controlled growth based on a modified A-harmonic approximation and the Caccioppoli inequality, also see [40, 41].

Motivated by these recent papers, we are here devoted to the study of the \(L^{p}\)-theory of weak solutions to quasilinear divergence form subelliptic equations, which originates from these papers [2, 38, 39] with a controllable growth in the case of usual Euclidean metric with standard gradient. Another reason is partially inspired by interior \(L^{p}\)-estimates of divergence degenerate elliptic equations in Carnot groups [37]. Therefore, our aim of this paper is to attain the \(L^{p}\)-theory for quasilinear subelliptic equations, whose conclusions and approach are different from the Hölder estimates of weak solutions derived in [2, 38, 39]. Now, let us summarize our main result as follows.

The main difficulty of our proof is that the bootstrap argument in [2] cannot be applied directly to weak solutions under the controlled growth. To this end, we first establish an interior reverse Hölder’s inequality. Additionally, the relaxation of the regularity assumptions on the leading coefficients from uniform continuity to VMO relies on the \(L^{p}\)-theory of linear equations with VMO coefficients. So, it is another challenging role how to make use of \(L^{p}\)-theory of linear subelliptic equations with VMO coefficients, and here we do it by De Giorgi’s iteration argument. In the context, we do not consider boundary-value problems for the reasons that it also makes the presentation clearer.

The rest of the paper is organized as follows. In Section 2, we will recall some basic facts and improve an integrable index of equation (1.1) to one larger than 2 by the reverse Hölder inequality. In Section 3, we prove our main theorem, Theorem 1.4. Our argument is first to establish the boundedness of the weak solutions of equation (1.1) by using De Giorgi’s iteration, and then improve gradually the integrable index of X-gradient via the \(L^{p}\)-theory of linear subelliptic equations, perturbation argument and a bootstrap argument.

In the context, we adopt the usual convention of denoting by C a general constant, which may vary from line to line in the same chain of inequalities. This section is devoted to establishing the reverse Hölder inequality to equation (1.1) and introducing some useful lemmas. Now let us first recall the Sobolev embedding inequality with respect to the horizontal vector fields (cf. [1, 4, 5]). Indeed, one of our main techniques to do the Moser iteration is to use the Poincaré and Sobolev embedding theorems for vector fields satisfying Hörmander condition.

We would like to remark that the Poincaré and Sobolev embedding theorems for vector fields satisfying Hörmander condition are very important to study subelliptic PDE, which was first established due to Jerison’s work [42]. Later, the optimal result for \(q = \frac {pQ}{Q-p}\) and \(p > 1\) was first proved by Lu in his two very earlier papers [23, 25]. As their applications, Lu also established the Harnack inequality for a class of degenerate subelliptic equations. In the case \(q = \frac{pQ}{Q-p}\) and \(p \ge1\) (including \(p = 1\)), the optimal results above and isoperimetric inequalities were proved by Franchi, Lu and Wheeden in [22].

Next, we recall the following reverse Hölder inequality from Theorem 2.3 of Chapter 5 in [11].

On the basis of the reverse Hölder inequality above, we can obtain a self-improving integrability of X-gradient to equation (1.1) by a standard approach. For our papers to be self-contained we give its complete proof as follows.

For the convenience of studying quasilinear subelliptic equations (1.1), we here give the following conclusion, which can be found a similar conclusion from [43] in the case of Euclidean metric and usual gradient.

In order to get the boundedness of weak solutions to equation (1.1) under the controllable growth, we will use so-called De Giorgi’s iteration argument (cf. Lemma 5.1 in Chapter 5 of [44]). We denote by \(A_{k}=\{x\in\Omega: u(x)>k\} \) the distributional function of u on Ω, and by \(|A_{k}|\) denote the measure of \(A_{k}\) with the C.-C. metric.

Based on Lemma 3.4 above and Lemma 2.1 (Sobolev inequality of X-gradient), we obtain the following useful conclusion (cf. Lemma 5.2 in Chapter 2 of [44]).

Hence, in order to establish the boundedness of weak solutions to equation (1.1) we only need to prove that the weak solutions of equation (1.1) satisfy inequality (3.10).