Lipschitz Regularity for a Non-Homogeneous Highly Degenerate q-Laplace Equation in the Heisenberg Group

Abstract

In the Heisenberg group ${\mathbb{H}}^n$ and for exponents $2\le q\le 4$, the local Lipschitz ($C^{0,1}$) regularity is demonstrated for weak solutions to a non-homogeneous highly degenerate q-Laplace equation of the form $-{\sum }_{i=1}^{2n}{X}_i\left({\lambda }_i(Xu)X_{i}u\right)=g(x)$, where ${\lambda }_i(z)={(z_i^2+z_{i+n}^2)}^{{\displaystyle \frac{q-2}{2}}}$ for $i=1,\dots ,n$ and ${\lambda }_i(z)={(z_i^2+z_{i-n}^2)}^{{\displaystyle \frac{q-2}{2}}}$ for $i=n+1,\dots ,2n$. Our approach draws heavily on the work of X. Zhong. However, owing to technical limitations, the regularity of weak solutions for $1<q<2$ and $4<q<\infty$ remains unclear.

1. Introduction

Understanding the regularity of solutions to nonlinear elliptic equations has been a driving force in analysis for decades. A central model is the p-Laplacian ${\Delta }_{p}u=0$, whose weak solutions are known to be local $C^{1,\alpha }$ for every $1<p<\infty$. This optimal result emerged from a series of landmark contributions. Ural’ceva [1] and Uhlenbeck [2] treated the super-quadratic case $p>2$; DiBenedetto [3] and Tolksdorf [4] later covered the full range. Subsequently, Manfredi and Weitsman [5] employed the Cordes condition [6,7,8] to derive second-order Sobolev regularity $u\in W_{\mathrm{loc}}^{2,2}$ under the restriction $1<p<3+{\displaystyle \frac{2}{n-2}}$, a result later recovered and extended by Dong et al. [9] using different techniques.

Sub-Riemannian geometries introduce additional non-commutative features that profoundly affect regularity. In the Heisenberg group ${\mathbb{H}}^n$, a systematic study of the p-Laplacian began with [10,11,12,13,14,15], which established $C^{0,1}$ and $C^{1,\alpha }$ estimates for $2\le p<4$ (see also the monograph [16]). A decisive advance was made by Zhong [17], who proved the same regularity for all $1<p<\infty$; his results were later refined by Mukherjee and Zhong [18]. These achievements paved the way for second-order estimates. $W_{\mathrm{loc}}^{2,2}$ regularity was proven by Liu et al. [19] for $1<p\le 4$ (case $n=1$) and for $1<p<3+{\displaystyle \frac{1}{n-1}}$ (case $n>1$), building on the work of Domokos [20]. In SU(3), Domokos and Manfredi [21,22] achieved full $C^{0,1}$ for all p and $C^{1,\alpha }$ for $p\ge 2$. Yu [23] extended the second-order theory to $1<p<{\displaystyle \frac{7}{2}}$, while Citti and Mukherjee [24] unified the treatment of p-Laplacian equations under step-two Hörmander vector fields, building on the ideas of Zhong and Mukherjee. This trajectory highlights the deep interplay between geometric structure and nonlinear potential theory.

Parallel to these developments, a growing body of research has focused on orthotropic or widely degenerate elliptic equations, where the principal part vanishes on a large set. For such equations, the optimal regularity is typically Lipschitz continuity rather than $C^{1,\alpha }$. In the Euclidean setting, Bousquet, Brasco and their collaborators have systematically studied orthotropic functionals with nonstandard growth conditions, obtaining local Lipschitz bounds for the gradient under various exponent ranges [25,26]. These results are closely connected to models of congested optimal transport, as first explored by Brasco and Carlier [27,28]. Demengel [29] proved interior Lipschitz regularity for viscosity and weak solutions of the pseudo p-Laplacian equation, while Lindqvist and Ricciotti [30] obtained gradient regularity for anisotropic equations in the plane. In the singular regime $1<p<2$, Ricciotti [31] established regularity of derivatives for p-orthotropic functions. More generally, Duzaar and Mingione [32] developed a general approach to local Lipschitz regularity for degenerate elliptic systems, which has influenced many subsequent works.

Most recently, Circelli et al. [33] studied the homogeneous version of the orthotropic q-Laplacian-type equation in ${\mathbb{H}}^n$ (i.e., Equation (1) with $g\equiv 0$). They proved local Lipschitz regularity for weak solutions when $2<q<\infty$, following an approach inspired by Zhong [17]. The present paper extends their analysis to the non-homogeneous case $g\not\equiv 0$. Although our overall strategy shares some ingredients with [33] (such as the use of mixed Caccioppoli inequalities and Moser iteration), the presence of the source term introduces substantial new difficulties. In particular, the derivation of uniform Caccioppoli estimates (Lemmas 1, 2 and 4) and the treatment of the vertical derivative term $Tu$ (Lemma 3) require proofs that are fundamentally different from those in [33]. Lemmas 1–4 in this paper rely on novel estimates that have no direct analogue in the homogeneous setting. Thus the present work can be viewed as a non-homogeneous counterpart of [33], complementing their result and offering a more complete picture of the regularity theory for orthotropic q-Laplacian-type equations in the Heisenberg group. In addition, several recent preprints on Riemannian approximation techniques for sub-Riemannian problems [34,35] provide useful methodological background, though we do not directly employ those approximations here.

For $2\le q<\infty$, we study the local Lipschitz regularity of weak solutions to a non-homogeneous highly degenerate q-Laplace equation of the form

$$-{\displaystyle \sum _{i=1}^{2n}}{X}_i\left({\lambda }_i(Xu)X_{i}u\right)=g(x)\mathrm{in}\Omega ,$$

(1)

where the domain $\Omega$ lies in the n-th Heisenberg group ${\mathbb{H}}^n$. The functions ${\lambda }_i:{\mathbb{R}}^{2n}\to \mathbb{R}$ are prescribed by

$${\lambda }_i(z)=\left\{\begin{array}{cc}{(\delta +z_i^2+z_{i+n}^2)}^{{\displaystyle \frac{q-2}{2}}},\hfill & i=1,\dots ,n,\hfill \\ {(\delta +z_i^2+z_{i-n}^2)}^{{\displaystyle \frac{q-2}{2}}},\hfill & i=n+1,\dots ,2n,\hfill \end{array}\right.$$

(2)

with $\delta \in [0,1]$; in (1), each ${\lambda }_i$ is taken at $z=Xu$. A function $u\in H{W}_{\mathrm{loc}}^{1,q}(\Omega )$ is said to be a weak solution of (1) if the integral identity

$${\displaystyle \sum _{i=1}^{2n}}{\int }_{\Omega }{\lambda }_i(Xu)X_{i}u{X}_i\varphi \mathrm{d}x={\int }_{\Omega }g(x)\varphi \mathrm{d}x$$

(3)

holds for all test functions $\varphi \in C_0^{\infty }(\Omega )$. Here $H{W}_{\mathrm{loc}}^{1,q}(\Omega )$ denotes the standard local horizontal Sobolev space; we refer the reader to Section 3 for details. When $\delta >0$, the solution u is smooth, i.e., $u\in C^{\infty }(\Omega )$, by the elliptic regularity theory for the non-degenerate equation (see also [16,17,33]). All a priori estimates in this paper are first derived for such smooth solutions and are subsequently extended to weak solutions by a standard mollification argument. Since the estimates are uniform with respect to the regularization parameter, they pass to the limit $\delta \to 0$ and thus remain valid for weak solutions.

The main result of this paper (Theorem 1, stated in Section 2) asserts that for $2<q\le 4$, every weak solution u of Equation (1) is intrinsically locally Lipschitz, i.e., $Xu\in L_{\mathrm{loc}}^{\infty }(\Omega )$, and satisfies a quantitative gradient estimate involving the source term g.

When $q=2$, the coefficient functions in (2) satisfy ${\lambda }_i(z)\equiv 1$ so that Equation (1) reduces to the non-homogeneous sub-Laplacian

$$-{\displaystyle \sum _{i=1}^{2n}}{X}_i^{2}u=g(x)\mathrm{in}\Omega ,$$

i.e., $-{\Delta }_{{\mathbb{H}}^n}u=g(x)$, where ${\Delta }_{{\mathbb{H}}^n}={\sum }_{i=1}^{2n}{X}_i^2$ is the canonical sub-Laplacian on the Heisenberg group. By the classical hypoellipticity theorem of Hörmander [36], the sub-Laplacian ${\Delta }_{{\mathbb{H}}^n}$ is hypoelliptic since the vector fields $\{X_1,\dots ,X_{2n}\}$ generate the full Lie algebra and satisfy Hörmander’s condition of step two. Consequently, if $g\in C^{\infty }(\Omega )$, every solution u of $-{\Delta }_{{\mathbb{H}}^n}u=g$ belongs to $C^{\infty }(\Omega )$; for less regular g, the solution inherits corresponding Sobolev or Hölder regularity. This smoothness result for the Heisenberg sub-Laplacian was further developed in the foundational works of Folland [37] and Folland-Stein [38] and is treated systematically in the monograph by Bonfiglioli, Lanconelli and Uguzzoni [39].

The restriction $2<q\le 4$ is inherent to our method, while the remaining exponent ranges $1<q<2$ and $4<q<\infty$ are left open.

The paper is organized as follows. Section 2 states the main theorem and outlines the overall proof strategy. Section 3 collects the necessary preliminaries regarding the Heisenberg group and the horizontal Sobolev spaces. In Section 4, we establish uniform Caccioppoli-type estimates (Lemmas 1–4). Section 5 is devoted to a Moser iteration argument, which yields an $L^{\infty }$ bound for the horizontal gradient; subsequently, we pass to the limit to recover the original solution and thereby prove the desired Lipschitz regularity.

2. Main Theorem and Proof Method

Let u be a weak solution to Equation (1). Our aim is to establish its local Lipschitz regularity, i.e., $u\in C_{\mathrm{loc}}^{0,1}$. For the range $2<q\le 4$, we demonstrate that $Xu\in L_{\mathrm{loc}}^{\infty }$. The precise result is stated below.

Theorem 1.

Assume $u\in H{W}_{\mathrm{loc}}^{1,q}(\Omega )$ is a weak solution of the non-homogeneous highly degenerate q-Laplace Equation (1) in ${\mathbb{H}}^n$, with the non-homogeneous term $g(x)$ satisfying $Xg,Tg\in L_{\mathrm{loc}}^{\infty }(\Omega )$. Then, when $2<q\le 4$, u is intrinsically local Lipschitz continuous, namely $u\in C_{\mathrm{loc}}^{0,1}$, and satisfies the gradient estimate: there exists a constant $C=C(n,q,r_0)>0$ depending only on $n,q,r_0$ such that for every ball $B(2r)\subset B(2r_0)\subset \Omega$,

$${\parallel Xu\parallel }_{L^{\infty }B(r)}\le Cmax\left\{{\left({⨍}_{B(2r)}{|Xu|}^q\mathrm{d}x\right)}^{{\displaystyle \frac{1}{q}}},{\left({\parallel Tg\parallel }_{L^{\infty }(B(2r_0))}+{\parallel Xg\parallel }_{L^{\infty }(B(2r_0))}\right)}^{{\displaystyle \frac{1}{q-1}}}\right\}.$$

(4)

This follows directly from the bound $Xu\in L_{\mathrm{loc}}^{\infty }$ and the fact that u is absolutely continuous along horizontal curves.

Corollary 1.

Under the assumptions of Theorem 1, u is locally Lipschitz with respect to the Carnot–Carathéodory distance: for every compact set $U⋐\Omega$, there exists a constant $C_U>0$ such that

$$|u(x)-u(y)|\le C_U{d}_{CC}(x,y)\forall x,y\in U.$$

The proof of Theorem 1 employs Moser’s iteration scheme and hinges on the derivation of suitable Caccioppoli-type estimates for horizontal gradients. First, we establish two pivotal Caccioppoli-type inequalities for the quantities ${\sum }_{i=1}^n{\lambda }_i(Xu)\left({(X_j{X}_{i}u)}^2+{(X_j{X}_{i+n}u)}^2\right)$ and ${\sum }_{i=1}^n{\lambda }_i(Xu)\left({(X_{i}Tu)}^2+{(X_{i+n}Tu)}^2\right)$, recorded in Lemmas 1 and 2 respectively; these provide control over second-order horizontal derivatives. Next, using the Lie bracket relation $T=X_1{X}_{n+1}-X_{n+1}{X}_1$ together with Lemma 2, we derive a $L^{q+\beta }$ norm estimate for the vertical derivative $Tu$ (Lemma 3), which is valid for $2<q\le 4$ and $\beta \ge 0$. With Lemma 3 at hand, we revisit the integral bounds from Lemma 1 and carefully estimate the terms involving $Tu$, thereby eliminating undesired contributions and obtaining a refined Caccioppoli-type inequality that involves only horizontal derivatives; this key estimate is presented in Lemma 4 (see inequality (17)) and holds for $2<q\le 4$. Finally, combining this refined inequality with a standard Moser’s iteration argument yields a uniform $L^{\infty }$ bound for the horizontal gradient $Xu$ on interior balls, which in turn implies the intrinsic local Lipschitz regularity $u\in C_{\mathrm{loc}}^{0,1}(\Omega )$. The quantitative gradient estimate stated in Theorem 1 follows directly from the iteration scheme.

3. Preliminaries

3.1. Heisenberg Group and Horizontal Sobolev Space

The Heisenberg group ${\mathbb{H}}^n$ is realized as ${\mathbb{R}}^{2n+1}$ equipped with a non-commutative product law

$$x\circ y:=\left(x_1+y_1,\dots ,x_{2n}+y_{2n},x_{2n+1}+y_{2n+1}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}(x_i{y}_{n+i}-x_{n+i}{y}_i)\right).$$

Its Hausdorff (homogeneous) dimension equals $2n+2$. The Lie algebra ${\mathfrak{h}}^n$ splits into $V_1\oplus V_2$ with $V_1=\mathrm{span}\{X_1,\dots ,X_{2n}\}$ and $V_2=\mathrm{span}\left\{T\right\}$. The left-invariant vector fields generating $V_1$ are

$$X_i:={\partial }_{x_i}-{\displaystyle \frac{x_{n+i}}{2}}{\partial }_{x_{2n+1}}(1\le i\le n),X_{n+i}:={\partial }_{x_{n+i}}+{\displaystyle \frac{x_i}{2}}{\partial }_{x_{2n+1}}(1\le i\le n),$$

and satisfy Hörmander’s condition at step two. The vertical direction arises from the commutator

$$T:={\partial }_{x_{2n+1}}=[X_i,X_{n+i}](i=1,\dots ,n).$$

Dilations ${\delta }_{\kappa }(x)=(\kappa x_1,\dots ,\kappa x_{2n},{\kappa }^2{x}_{2n+1})$ are group automorphisms, and the Carnot–Carathéodory distance is induced by horizontal curves. The only non-trivial Lie bracket relations are

$$[X_i,X_{i\pm n}]=\mp T,[X_i,X_j]=0(j\ne i\pm n),[X_i,T]=0.$$

For a differentiable function u, the horizontal gradient is $Xu=(X_{1}u,\dots ,X_{2n}u)$. Now take an open connected domain $\Omega \subset {\mathbb{H}}^n$. For a $C^1$ function $\mu$ on $\Omega$, the horizontal gradient is the vector $X\mu =(X_1\mu ,\dots ,X_{2n}\mu )\in {\mathbb{R}}^{2n}$. If $\mu$ is $C^2$, its second-order horizontal derivative is the matrix $XX\mu ={(X_i{X}_j\mu )}_{1\le i,j\le 2n}$. Their standard Euclidean norms are

$$|X\mu |={\left({\displaystyle \sum _{i=1}^{2n}}{|X_i\mu |}^2\right)}^{1/2},|XX\mu |={\left({\displaystyle \sum _{i,j=1}^{2n}}{|X_i{X}_j\mu |}^2\right)}^{1/2}.$$

For $1\le p<\infty$, the first-order horizontal Sobolev space is

$$H{W}^{1,p}(\Omega ):=\left\{\mu \in L^p(\Omega ):X\mu \in L^p(\Omega ,{\mathbb{R}}^{2n})\right\},$$

equipped with the norm ${\parallel \mu \parallel }_{H{W}^{1,p}(\Omega )}={\parallel \mu \parallel }_{L^p(\Omega )}+{\parallel X\mu \parallel }_{L^p(\Omega )}$. Its local counterpart $H{W}_{\mathrm{loc}}^{1,p}(\Omega )$ comprises those functions belonging to $H{W}^{1,p}(U)$ for every compactly contained subdomain $U⋐\Omega$; this serves as the appropriate framework for local regularity studies.

3.2. Notation

Throughout the article we employ the following conventional symbols.

• $C^{k,\alpha }(\Omega )$: Hölder space with integer $k\ge 0$ and exponent $\alpha \in [0,1]$.
• $C_0^{\infty }(\Omega )$: smooth functions compactly supported in $\Omega$.
• ${\parallel \mu \parallel }_{L^p(\Omega )}:=({\int }_{\Omega }{|\mu |}^p{\mathrm{d}x)}^{1/p}$ for $1\le p<\infty$.
• ${\parallel \mu \parallel }_{L^{\infty }(\Omega )}:={ess\; sup}_{\Omega }|\mu |$ (or ${max}_{\Omega }|\mu |$ when $\mu$ is continuous).
• ${⨍}_U\mu \mathrm{d}x:={\displaystyle \frac{1}{|U|}}{\int }_U\mu \mathrm{d}x$.

To simplify writing and proof, we define the following functions.

• $F_j(u):={\sum }_{i=1}^n{\lambda }_i(Xu)\left({(X_j{X}_{i}u)}^2+{(X_j{X}_{i+n}u)}^2\right)$.
• $F(u):={\sum }_{j=1}^{2n}{F}_j(u)$.
• $E_j(u):={\sum }_{i=1}^n{\lambda }_i(Xu)\left({(X_i{X}_{j}u)}^2+{(X_{i+n}{X}_{j}u)}^2\right)$.
• $E_T(u):={\sum }_{i=1}^n{\lambda }_i(Xu)\left({(X_{i}Tu)}^2+{(X_{i+n}Tu)}^2\right)$.
• $a(u):={\sum }_{i=1}^n{\lambda }_i(Xu)$.
• $b(u):={\left(\delta ++{|Xu|}^2\right)}^{{\displaystyle \frac{1}{2}}}$.
• $f(z):={\displaystyle \frac{1}{q}}{\sum }_{i=1}^n{\left(z_i^2+z_{i+n}^2\right)}^{{\displaystyle \frac{q}{2}}}$, so that $D_{i}f(z)={\lambda }_i(z)z_i$.

Here, when $2\le q<\infty$, for any $z,\xi \in {\mathbb{R}}^{2n}$,

$${\displaystyle \sum _{i=1}^n}{\lambda }_i(z)({\xi }_i^2+{\xi }_{i+n}^2)\le \langle D^{2}f(z)\xi ,\xi \rangle \le (q-1){\displaystyle \sum _{i=1}^n}{\lambda }_i(z)({\xi }_i^2+{\xi }_{i+n}^2),$$

and

$$|Df(z)|\le {(\delta +|z|}^2{)}^{{\displaystyle \frac{q-2}{2}}}|z|.$$

4. Main Lemmas

In this section we collect the key estimates needed for the proof of Theorem 1. Throughout the paper, we assume that the non-homogeneous term $g(x)$ satisfies $Xg,Tg\in L_{\mathrm{loc}}^{\infty }(\Omega )$. Lemma 1 provides a fundamental Caccioppoli-type inequality that controls second-order horizontal derivatives at the expense of a term involving the vertical derivative $Tu$ and the source term $Xg$.

Lemma 1.

Given a weak solution u to (1), there exists a constant $C=C(n,q)>0$ for which the following holds for any non-negative test function $\eta \in C_0^{\infty }(\Omega )$ and any $\beta \ge 0$:

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x\le & C(\beta +1){\int }_{\Omega }\left({|X\eta |}^2+\eta |T\eta |\right)b{(u)}^{\beta +2}a(u)\mathrm{d}x\hfill \\ \hfill & +C{(\beta +1)}^2{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x+C{\int }_{\Omega }|Xg|{\eta }^{2}b{(u)}^{\beta +1}\mathrm{d}x.\hfill \end{array}$$

(5)

Proof. 

For $\beta \ge 0$ and $l=1,\dots ,2n$, applying $\varphi =X_l({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)$ to test Equation (1), then by integration by parts, one gets

$${\int }_{\Omega }{\displaystyle \sum _{i=1}^{2n}}{D}_{i}f(Xu)X_i(X_l({\eta }^{2}b{(u)}^{\beta }{X}_{l}u))\mathrm{d}x=-{\int }_{\Omega }{X}_{l}g{\eta }^{2}b{(u)}^{\beta }{X}_{l}u\mathrm{d}x.$$

Assuming that $l=1,\dots ,n$ and applying the Lie bracket relations $X_{l+n}{X}_l=-T+X_l{X}_{l+n}$ and $X_i{X}_l=X_l{X}_i(i\ne l+n)$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^{2n}}{\int }_{\Omega }{X}_l(D_{i}f(Xu))X_i({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)\mathrm{d}x+{\int }_{\Omega }{D}_{l+n}f(Xu)T({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)\mathrm{d}x\hfill \\ \hfill & ={\int }_{\Omega }{X}_{l}g{\eta }^{2}b{(u)}^{\beta }{X}_{l}u\mathrm{d}x.\hfill \end{array}$$

From this, applying the Lie bracket relations $X_{l+n}{X}_l=-T+X_l{X}_{l+n}$ and $X_i{X}_l=X_l{X}_i$$(i\ne l+n)$ to the first term, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i,j=1}^{2n}}{\int }_{\Omega }{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_l{X}_{i}u{\eta }^{2}b{(u)}^{\beta }\mathrm{d}x-{\displaystyle \sum _{j=1}^{2n}}{\int }_{\Omega }{D}_{l+n,j}^{2}f(Xu)X_l{X}_{j}uTu{\eta }^{2}b{(u)}^{\beta }\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \sum _{i,j=1}^{2n}}{\int }_{\Omega }{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_i({\eta }^{2}b{(u)}^{\beta })X_{l}u\mathrm{d}x+{\int }_{\Omega }{D}_{l+n}f(Xu)T({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)\mathrm{d}x\hfill \\ \hfill & ={\int }_{\Omega }{X}_{l}g{\eta }^{2}b{(u)}^{\beta }{X}_{l}u\mathrm{d}x.\hfill \end{array}$$

Relocating the first and third terms to the left-hand side gives

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_l{X}_{i}u\mathrm{d}x+{\int }_{\Omega }{X}_{l}u{\eta }^2{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_i(b{(u)}^{\beta })\mathrm{d}x\hfill \\ \hfill & =-2{\int }_{\Omega }\eta X_{l}ub{(u)}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_i\eta \mathrm{d}x+{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{j=1}^{2n}}{D}_{l+n,j}^{2}f(Xu)X_l{X}_{j}uTu\mathrm{d}x\hfill \\ \hfill & -{\int }_{\Omega }{D}_{l+n}f(Xu)T({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)\mathrm{d}x+{\int }_{\Omega }{X}_{l}g{\eta }^{2}b{(u)}^{\beta }{X}_{l}u\mathrm{d}x.\hfill \end{array}$$

(6)

Note that

$$\begin{array}{cc}\hfill X_i(b{(u)}^{\beta })& =\beta b{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}{X}_i{X}_{k}u{X}_{k}u\hfill \\ \hfill & =\beta b{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}{X}_k{X}_{i}u{X}_{k}u+\beta b{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}[X_i,X_k]u{X}_{k}u.\hfill \end{array}$$

Hence, Equation (6) becomes

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_l{X}_{i}u\mathrm{d}x\hfill \\ \hfill & +{\int }_{\Omega }{X}_{l}u{\eta }^2{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u\beta b{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}{X}_k{X}_{i}u{X}_{k}u\mathrm{d}x\hfill \\ \hfill & =-2{\int }_{\Omega }\eta X_{l}ub{(u)}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_i\eta \mathrm{d}x+{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{j=1}^{2n}}{D}_{l+n,j}^{2}f(Xu)X_l{X}_{j}uTu\mathrm{d}x\hfill \\ \hfill & -{\int }_{\Omega }{D}_{l+n}f(Xu)T({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)\mathrm{d}x\hfill \\ \hfill & -{\int }_{\Omega }{X}_{l}u{\eta }^2{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u\beta b{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}[X_i,X_k]u{X}_{k}u\mathrm{d}x+{\int }_{\Omega }{X}_{l}g{\eta }^{2}b{(u)}^{\beta }{X}_{l}u\mathrm{d}x\hfill \\ \hfill & =:R_1^l+R_2^l+R_3^l+R_4^l+R_5^l.\hfill \end{array}$$

(7)

For $l=n+1,\dots ,2n$, by using the same method, an estimate similar to Equation (7) can be obtained.

Summing over l from 1 to $2n$, we obtain

$$\begin{array}{cc}\hfill L_1+L_2:=& {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i,j,l=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_l{X}_{j}u{X}_l{X}_{i}u\mathrm{d}x\hfill \\ \hfill & +\beta {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta -2}{\displaystyle \sum _{i,j,l=1}^{2n}}{D}_{i,j}^{2}f(Xu)\langle X{X}_{j}u,Xu\rangle \langle X{X}_{i}u,Xu\rangle \mathrm{d}x\hfill \\ \hfill & ={\displaystyle \sum _{l=1}^{2n}}\left(R_1^l+R_2^l+R_3^l+R_4^l+R_5^l\right).\hfill \end{array}$$

(8)

The definition of $f(Xu)$ and $\beta \ge 0$ yield

$$L_1\ge {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x\mathrm{and}{L}_2\ge 0,$$

which, together with Equation (8), yield

$${\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x\le {\displaystyle \sum _{l=1}^{2n}}\left(|R_1^l|+|R_2^l|+|R_3^l|+|R_4^l|+|R_5^l|\right).$$

(9)

For $R_1^l$, the definition of $f(Xu)$ and Young’s inequality yield

$$\begin{array}{cc}\hfill |R_1^l|\le & \kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i=1}^n}{\lambda }_i(Xu)\left(|X{X}_i{u|}^2+{|X{X}_{i+n}u|}^2\right)\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{C}{\kappa }}{\int }_{\Omega }b{(u)}^{\beta +2}{\displaystyle \sum _{i=1}^n}{\lambda }_i(Xu){|X\eta |}^2\mathrm{d}x\hfill \\ \hfill & =\kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x+{\displaystyle \frac{C}{\kappa }}{\int }_{\Omega }{|X\eta |}^{2}b{(u)}^{\beta +2}a(u)\mathrm{d}x,\hfill \end{array}$$

(10)

where $\kappa$ is positive and small enough.

For $R_2^l$, the definition of $f(Xu)$ and Young’s inequality yield

$$\begin{array}{cc}\hfill |R_2^l|\le & \kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i=1}^n}{\lambda }_i(Xu)\left(|X{X}_i{u|}^2+{|X{X}_{i+n}u|}^2\right)\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{C}{\kappa }}{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{\displaystyle \sum _{i=1}^n}{\lambda }_i(Xu){|Tu|}^2\mathrm{d}x\hfill \\ \hfill & =\kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x+{\displaystyle \frac{C}{\kappa }}{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }a(u){|Tu|}^2\mathrm{d}x.\hfill \end{array}$$

(11)

For $R_3^l$, the Lie bracket relation $T{X}_i=X_{i}T$ yields

$$T({\eta }^{2}b{(u)}^{\beta }{X}_{l}u)=2\eta T\eta b{(u)}^{\beta }{X}_{l}u+\beta {\eta }^2{X}_{l}ub{(u)}^{\beta -2}{\displaystyle \sum _{k=1}^{2n}}{X}_{k}u{X}_{k}Tu+{\eta }^{2}b{(u)}^{\beta }{X}_{l}Tu,$$

which, together with integration by parts, yields

$$\begin{array}{cc}\hfill R_3^l=& -{\int }_{\Omega }{D}_{l+n}f(Xu)2\eta T\eta B{(u)}^{\beta }{X}_{l}u\mathrm{d}x+{\int }_{\Omega }{X}_k(D_{l+n}f(Xu)\beta {\eta }^2{X}_{l}ub{(u)}^{\beta -2}{X}_{k}u)Tu\mathrm{d}x\hfill \\ \hfill & +{\int }_{\Omega }{X}_l(D_{l+n}f(Xu){\eta }^{2}b{(u)}^{\beta })Tu\mathrm{d}x,\hfill \end{array}$$

where $l=1,\dots ,n$. From this, the definition of $f(Xu)$ and Young’s inequality yield

$$\begin{array}{cc}\hfill |R_3^l|\le & \kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x+C(\beta +1){\int }_{\Omega }\left({|X\eta |}^2+\eta |T\eta |\right)b{(u)}^{\beta +2}a(u)\mathrm{d}x\hfill \\ \hfill & +{\displaystyle \frac{C{(\beta +1)}^2}{\kappa }}{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x.\hfill \end{array}$$

(12)

For $l=n+1,\dots ,2n$, we can obtain the same estimate as Inequality (12) using the same method.

For $R_4^l$, the Lie bracket relations $[X_i,X_{i\pm n}]=\mp T$ and $[X_i,X_k]=0(k\ne i\pm n)$, the definition of $f(Xu)$ and Young’s inequality yield

$$|R_4^l|\le \kappa {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x+{\displaystyle \frac{C{\beta }^2}{\kappa }}{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x.$$

(13)

For $R_5^l$, we obtain

$$|R_5^l|\le C{\int }_{\Omega }|Xg|{\eta }^{2}b{(u)}^{\beta +1}\mathrm{d}x.$$

(14)

Combining (9)–(14), then selecting a sufficiently small $\kappa$, we obtain (5). □

Lemma 2 deals with the vertical derivative $Tu$ itself. By choosing a suitable test function and exploiting the structure of the equation, we derive a Caccioppoli-type inequality for $Tu$ that involves the source term $Tg$.

Lemma 2.

Given a weak solution u to (1), there exists a constant $C=C(n,q)>0$ for which the following holds for any non-negative test function $\eta \in C_0^{\infty }(\Omega )$ and any $\beta \ge 0$:

$${\int }_{\Omega }{\eta }^2{|Tu|}^{\beta }{E}_T(u)\mathrm{d}x\le {\displaystyle \frac{C}{{(\beta +1)}^2}}{\int }_{\Omega }{|X\eta |}^2{|Tu|}^{\beta +2}a(u)\mathrm{d}x+{\displaystyle \frac{C}{\beta +1}}{\int }_{\Omega }|Tg|{\eta }^2{|Tu|}^{\beta +1}\mathrm{d}x.$$

(15)

Proof. 

For $\beta \ge 0$, applying $\varphi =T({\eta }^2|Tu{|}^{\beta }Tu)$ to test Equation (1), then by integration by parts, one gets

$${\int }_{\Omega }{\displaystyle \sum _{i=1}^{2n}}{D}_{i}f(Xu)X_{i}T({\eta }^2{|Tu|}^{\beta }Tu)\mathrm{d}x=-{\int }_{\Omega }Tg{\eta }^2{|Tu|}^{\beta }Tu\mathrm{d}x.$$

Applying the Lie bracket relation $T{X}_i=X_{i}T$ and integration by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\eta }^2{|Tu|}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_{j}Tu{X}_{i}Tu\mathrm{d}x\hfill \\ \hfill & =-{\int }_{\Omega }Tu{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_{j}Tu{X}_i({\eta }^2{|Tu|}^{\beta })\mathrm{d}x+{\int }_{\Omega }Tg{\eta }^2{|Tu|}^{\beta }Tu\mathrm{d}x.\hfill \end{array}$$

Noting that

$$X_i({\eta }^2{|Tu|}^{\beta })=2\eta X_i{\eta |Tu|}^{\beta }+\beta {\eta }^2{|Tu|}^{\beta -2}Tu{X}_{i}Tu,$$

we obtain

$$\begin{array}{cc}\hfill L:=& {\int }_{\Omega }{\eta }^2{|Tu|}^{\beta }{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_{j}Tu{X}_{i}Tu\mathrm{d}x\hfill \\ \hfill =& -{\displaystyle \frac{1}{\beta +1}}{\int }_{\Omega }Tu{\displaystyle \sum _{i,j=1}^{2n}}{D}_{i,j}^{2}f(Xu)X_{j}Tu{X}_{i}2\eta X_i{\eta |Tu|}^{\beta }\mathrm{d}x+{\displaystyle \frac{1}{\beta +1}}{\int }_{\Omega }Tg{\eta }^2{|Tu|}^{\beta }Tu\mathrm{d}x\hfill \\ \hfill =& :R_1+R_2.\hfill \end{array}$$

The definition of $f(Xu)$ yields

$$L\ge {\int }_{\Omega }{\eta }^2{|Tu|}^{\beta }{E}_T(u)\mathrm{d}x.$$

For $R_1$, then by Young’s inequality, we obtain

$$|R_1|\le \kappa {\int }_{\Omega }{\eta }^2{|Tu|}^{\beta }{E}_T(u)\mathrm{d}x+{\displaystyle \frac{C}{{(\beta +1)}^2\kappa }}{\int }_{\Omega }{|X\eta |}^2{|Tu|}^{\beta +2}a(u)\mathrm{d}x,$$

where $\kappa$ is positive and small enough.

For $R_2$, we obtain

$$|R_2|\le {\displaystyle \frac{1}{\beta +1}}{\int }_{\Omega }|Tg|{\eta }^2{|Tu|}^{\beta +1}\mathrm{d}x.$$

Combining above estimates, then selecting a sufficiently small $\kappa$, we obtain (15). □

To eliminate the undesirable ${\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x$ term appearing in Lemma 1, we require higher integrability of $Tu$. Lemma 3, valid for $2\le q\le 4$, provides an estimate of the $L^{q+\beta }$ norm of $Tu$ in terms of the source term $Tg$, which is derived from Lemma 2.

Lemma 3.

Given $2<q\le 4$ and a weak solution u to (1), there exists a constant $C=C(n,q)>0$ for which the following holds for any non-negative test function $\eta \in C_0^{\infty }(\Omega )$ and any $\beta \ge 0$:

$$\begin{array}{cc}\hfill & {\left({\int }_{\Omega }{|Tu|}^{q+\beta }{\eta }^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{1}{q+\beta }}}\hfill \\ \hfill & \le {C(q+\beta )\parallel X\eta \parallel }_{L^{\infty }}{\left({\int }_{\mathrm{supp}(\eta )}a{(u)}^{{\displaystyle \frac{q+\beta }{q-2}}}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{q+\beta }}}\hfill \\ \hfill & +C{(q+\beta )}^{{\displaystyle \frac{2}{3}}}{\left({\int }_{\mathrm{supp}(\eta )}a{(u)}^{{\displaystyle \frac{q+\beta }{q-2}}}\mathrm{d}x\right)}^{{\displaystyle \frac{4-q}{3(q+\beta )}}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{3(q+\beta )}}}.\hfill \end{array}$$

(16)

Proof. 

Denote

$$L:={\int }_{\Omega }{|Tu|}^{q+\beta }{\eta }^{q+\beta }\mathrm{d}x.$$

The Lie bracket relation $T=X_1{X}_{n+1}-X_{n+1}{X}_1$ yields

$${|Tu|}^{q+\beta }={|Tu|}^{q-2+\beta }Tu(X_1{X}_{n+1}u-X_{n+1}{X}_{1}u),$$

which, together with integration by parts, yields

$$\begin{array}{cc}\hfill L=& {\int }_{\Omega }{|Tu|}^{q-2+\beta }Tu(X_1{X}_{n+1}u-X_{n+1}{X}_{1}u){\eta }^{q+\beta }\mathrm{d}x\hfill \\ \hfill =& -(q-1+\beta ){\int }_{\Omega }{|Tu|}^{q-2+\beta }(X_{1}Tu{X}_{n+1}u-X_{n+1}Tu{X}_{1}u){\eta }^{q+\beta }\mathrm{d}x\hfill \\ \hfill & -(q+\beta ){\int }_{\Omega }{|Tu|}^{q-2+\beta }Tu(X_{n+1}u{X}_1\eta -X_{1}u{X}_{n+1}\eta ){\eta }^{q-1+\beta }\mathrm{d}x\hfill \\ \hfill \le & (q+\beta ){\int }_{\Omega }{|Tu|}^{q-2+\beta }(|X_{1}Tu||X_{n+1}u|+|X_{n+1}Tu||X_{1}u|){\eta }^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +(q+\beta ){\int }_{\Omega }{|Tu|}^{q-1+\beta }(|X_{n+1}u|+|X_{1}u|)|X\eta |{\eta }^{q-1+\beta }\mathrm{d}x=:R_1+R_2.\hfill \end{array}$$

Here we use the identity $T=X_k{X}_{n+k}-X_{n+k}{X}_k$, valid for each $k=1,\dots n$. Since the estimate for $|Tu|$ does not depend on k, it is enough to fix one particular value, e.g., $k=1$.

For $R_2$, by Hölder’s inequality, we obtain

$$R_2\le (q+\beta ){\parallel X\eta \parallel }_{L^{\infty }}{S}^{{\displaystyle \frac{1}{q+\beta }}}{L}^{{\displaystyle \frac{q-1+\beta }{q+\beta }}},$$

where $S:={\int }_{\mathrm{supp}(\eta )}(|X_1{u|}^{q+\beta }+|X_{n+1}u{|}^{q+\beta })\mathrm{d}x$.

For $R_1$, when $2<q\le 4$, by Hölder’s inequality, we obtain

$$R_1\le (q+\beta )M^{{\displaystyle \frac{1}{2}}}{S}^{{\displaystyle \frac{4-q}{2(q+\beta )}}}{L}^{{\displaystyle \frac{2q-4+\beta }{2(q+\beta )}}},$$

where $M:={\int }_{\Omega }{|Tu|}^{\beta }(|X_{n+1}{u|}^{q-2}|X_1{Tu|}^2+|X_1{u|}^{q-2}|X_{n+1}Tu{|}^2){\eta }^{4+\beta }\mathrm{d}x$. The definition of ${\lambda }_i(z)$ yields

$$S\le {\int }_{\mathrm{supp}(\eta )}{\lambda }_1{(Xu)}^{{\displaystyle \frac{q+\beta }{q-2}}}\mathrm{d}x={\int }_{\mathrm{supp}(\eta )}(\delta +|X_1{u|}^2+|X_{1+n}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{2}}}\mathrm{d}x,$$

and

$$M\le {\int }_{\Omega }{|Tu|}^{\beta }{\lambda }_1(Xu)(|X_1{Tu|}^2+|X_{n+1}Tu{|}^2){\eta }^{4+\beta }\mathrm{d}x.$$

Applying Lemma 2 with $\eta \to {\eta }^{2+\beta /2}$, we obtain

$$\begin{array}{cc}\hfill M\le & C{\int }_{\Omega }{\eta }^{\beta +2}{|X\eta |}^2{|Tu|}^{\beta +2}a(u)\mathrm{d}x+C{\int }_{\Omega }|Tg|{\eta }^{\beta +4}{|Tu|}^{\beta +1}\mathrm{d}x\hfill \\ \hfill \le & {C\parallel X\eta \parallel }_{L^{\infty }}^2{A}^{{\displaystyle \frac{q-2}{q+\beta }}}{L}^{{\displaystyle \frac{\beta +2}{q+\beta }}}+C{L}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{q+\beta }}},\hfill \end{array}$$

where $A:={\int }_{\mathrm{supp}(\eta )}a{(u)}^{{\displaystyle \frac{q+\beta }{q-2}}}\mathrm{d}x$. Thus

$$\begin{array}{c}\hfill R_1\le C(q+\beta ){\parallel X\eta \parallel }_{L^{\infty }}^2{A}^{{\displaystyle \frac{1}{q+\beta }}}{L}^{{\displaystyle \frac{q-1+\beta }{q+\beta }}}+C(q+\beta )A^{{\displaystyle \frac{4-q}{2(q+\beta )}}}{L}^{{\displaystyle \frac{2q-3+2\beta }{2(q+\beta )}}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{2(q+\beta )}}}.\end{array}$$

The restriction $q\le 4$ is critical here: when $q>4$, the exponent $(4-q)/(2(q+\beta ))$ becomes negative, and the term involving A cannot be absorbed. This is the precise reason why our method does not extend to the super-quadratic regime $q>4$. When $q=4$, the exponent $(4-q)/(2(q+\beta ))$ vanishes, and the corresponding term is simply absorbed into the constant. Thus the estimate remains valid at the endpoint.

Noting that ${\lambda }_1(Xu)\le a(u)$ and combining above estimates, then by Young’s inequality, one gets (16). □

Finally, we combine Lemma 3 with Lemma 1 to absorb the term containing ${|Tu|}^2$. This yields a refined Caccioppoli inequality that involves only horizontal derivatives and no longer contains the vertical derivative $Tu$. Lemma 4 is the crucial tool for the subsequent Moser’s iteration.

Lemma 4.

Given $2<q\le 4$ and a weak solution u to (1), there exists a constant $C=C(n,q)>0$ for which the following holds for any non-negative test function $\eta \in C_0^{\infty }(\Omega )$ and any $\beta \ge 0$:

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x\hfill \\ \hfill & \le C{(q+\beta )}^4\left({\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta T\eta \parallel }_{L^{\infty }}\right){\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +C{(q+\beta )}^{2+{\displaystyle \frac{4}{3}}}{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q+2+3\beta }{3(q+\beta )}}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{2(q-1)}{3(q+\beta )}}}\hfill \\ \hfill & +C{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\left({\int }_{\Omega }{|Xg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{2(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{q+\beta }}}.\hfill \end{array}$$

(17)

Proof. 

Noting that ${\lambda }_i(Xu)\le b{(u)}^{q-2}$, then we use Hölder’s inequality to re-estimate the second term on the right side of Inequality (5) in Lemma 1 and obtain

$$\begin{array}{cc}\hfill {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x\le & C{\int }_{\Omega }{\eta }^{2}b{(u)}^{q-2+\beta }{|Tu|}^2\mathrm{d}x\hfill \\ \hfill \le & {\left({\int }_{\Omega }{|Tu|}^{q+\beta }{\eta }^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{2}{q+\beta }}}{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q-2+\beta }{q+\beta }}},\hfill \end{array}$$

which, together with Lemma 3, yields

$$\begin{array}{cc}\hfill & {\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }{|Tu|}^{2}a(u)\mathrm{d}x\hfill \\ \hfill & \le C{\int }_{\Omega }{\eta }^{2}b{(u)}^{q-2+\beta }{|Tu|}^2\mathrm{d}x\hfill \\ \hfill & \le C{(q+\beta )}^2{\parallel X\eta \parallel }_{L^{\infty }}^2{\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +C{(q+\beta )}^{{\displaystyle \frac{4}{3}}}{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q+2+3\beta }{3(q+\beta )}}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{2(q-1)}{3(q+\beta )}}},\hfill \end{array}$$

Re-estimating the second term on the right side of Inequality (5) in Lemma 1, one gets

$$\begin{array}{c}\hfill {\int }_{\Omega }\left({|X\eta |}^2+\eta |T\eta |\right)b{(u)}^{\beta +2}a(u)\mathrm{d}x\le {C(\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta T\eta \parallel }_{L^{\infty }}){\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x.\end{array}$$

Using Hölder’s inequality to re-estimate the last term on the right side of Inequality (5), we obtain

$$\begin{array}{c}\hfill {\int }_{\Omega }|Xg|{\eta }^{2}b{(u)}^{\beta +1}\mathrm{d}x\le C{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\left({\int }_{\Omega }{|Xg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{2(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{q+\beta }}}.\end{array}$$

Combining above estimates and Lemma 1, we obtain (17). □

5. Proof of Theorem 1

We now proceed to the proof of Theorem 1. The argument is based on Moser’s iteration scheme applied directly to the refined Caccioppoli-type estimate (Lemma 4).

The proof of Theorem 1.

To estimate $X\left(\eta {\left(\delta +|X_i{u|}^2\right)}^{{\displaystyle \frac{q+\beta }{4}}}\right)(i=1,\dots ,2n)$, we directly calculate and obtain

$$\begin{array}{cc}\hfill {\left|X\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)\right|}^2\le & {2|X\eta |}^2(\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{2}}}\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{(q+\beta )}^2{\eta }^2(\delta +|X_i{u|}^2{)}^{{\displaystyle \frac{q+\beta -2}{2}}}{|X{X}_{i}u|}^2,\hfill \end{array}$$

which, together with $\delta +|X_i{u|}^2\le {\lambda }_i{(Xu)}^{{\displaystyle \frac{2}{q-2}}}\le b{(u)}^2$, yields

$${\left|X\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)\right|}^2\le 2{|X\eta |}^{2}b{(u)}^{q+\beta }+{\displaystyle \frac{1}{2}}{(q+\beta )}^2{\eta }^{2}b{(u)}^{\beta }F(u).$$

Thus

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^{2n}}{\int }_{\Omega }{\left|X\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)\right|}^2\mathrm{d}x\hfill \\ \hfill & \le {C\parallel X\eta \parallel }_{\infty }^2{\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x+C{(q+\beta )}^2{\int }_{\Omega }{\eta }^{2}b{(u)}^{\beta }F(u)\mathrm{d}x.\hfill \end{array}$$

When $2<q\le 4$, using Lemma 4 to estimate the second term on the right-hand side of the above inequality, one gets

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i=1}^{2n}}{\int }_{\Omega }{\left|X\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)\right|}^2\mathrm{d}x\hfill \\ \hfill & \le C{(q+\beta )}^{6}N{\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +C{(q+\beta )}^{4+{\displaystyle \frac{4}{3}}}{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q+2+3\beta }{3(q+\beta )}}}{\left({\int }_{\Omega }{|Tg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{3(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{2(q-1)}{3(q+\beta )}}}\hfill \\ \hfill & +C{(q+\beta )}^2{\left({\int }_{\mathrm{supp}(\eta )}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\left({\int }_{\Omega }{|Xg|}^{{\displaystyle \frac{q+\beta }{q-1}}}{\eta }^{{\displaystyle \frac{2(q+\beta )}{q-1}}}\mathrm{d}x\right)}^{{\displaystyle \frac{q-1}{q+\beta }}},\hfill \end{array}$$

(18)

where $N:={\parallel X\eta \parallel }_{L^{\infty }}^2+{\parallel \eta T\eta \parallel }_{L^{\infty }}$.

Take concentric balls $B(\kappa r)\subset B(r)\subset B(r_0)\subset \Omega$ with $0<\kappa <1$. Set $r_j=\kappa r+(r-\kappa r)2^{-j}$ such that $r_j\searrow \kappa r$ and choose cut-off functions ${\eta }_j\in C^{\infty }(B(r_j))$ such that ${\eta }_j=1$ on $B(r_{j+1})$ and $\parallel X{\eta }_j{\parallel }_{L^{\infty }}\le C/(r_j-r_{j+1})$. From this, given the fact that ${|Xu|}^2={\sum }_{i=1}^{2n}{(X_{i}u)}^2$, the Sobolev inequality and (18) yield

$$\begin{array}{cc}\hfill {\left({⨍}_{B(r_j)}{\left(\eta b{(u)}^{{\displaystyle \frac{q+\beta }{2}}}\right)}^{{\displaystyle \frac{2n+2}{n}}}\mathrm{d}x\right)}^{{\displaystyle \frac{n}{n+1}}}& \le C{\displaystyle \sum _{i=1}^{2n}}{\left({⨍}_{B(r_j)}{\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)}^{{\displaystyle \frac{2n+2}{n}}}\mathrm{d}x\right)}^{{\displaystyle \frac{n}{n+1}}}\hfill \\ \hfill & \le C{r}_j^2{\displaystyle \sum _{i=1}^{2n}}{⨍}_{B(r_j)}{\left|X\left(\eta (\delta +|X_{i}u{{|}^2)}^{{\displaystyle \frac{q+\beta }{4}}}\right)\right|}^2\mathrm{d}x\hfill \\ \hfill & \le C{r}_j^2{(q+\beta )}^{6}N{⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +C{r}_j^2{(q+\beta )}^6{\left({⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q+2+3\beta }{3(q+\beta )}}}{\parallel Tg\parallel }_{L^{\infty }}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +C{r}_j^2{(q+\beta )}^2{\left({⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\parallel Xg\parallel }_{L^{\infty }},\hfill \end{array}$$

where $C=C(n,q)>0$. From this, applying Hölder’s inequality and the properties of cut-off functions ${\eta }_j$, we obtain

$$\begin{array}{cc}\hfill {\left({⨍}_{B(r_{j+1})}b{(u)}^{{\displaystyle \frac{(q+\beta )(n+1)}{n}}}\mathrm{d}x\right)}^{{\displaystyle \frac{n}{n+1}}}& \le C{\displaystyle \frac{{(q+\beta )}^6}{{(1-\kappa )}^2}}{⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\hfill \\ \hfill & +C{r}^2{(q+\beta )}^6{\left({⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{q+2+3\beta }{3(q+\beta )}}}{\parallel Tg\parallel }_{L^{\infty }}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +C{r}^2{(q+\beta )}^2{\left({⨍}_{B(r_j)}b{(u)}^{q+\beta }\mathrm{d}x\right)}^{{\displaystyle \frac{\beta +1}{q+\beta }}}{\parallel Xg\parallel }_{L^{\infty }}.\hfill \end{array}$$

(19)

Let ${\beta }_j=q{\left({\displaystyle \frac{n+1}{n}}\right)}^j-q\ge 0$. Denote ${\alpha }_j:=q+{\beta }_j=q{\left({\displaystyle \frac{n+1}{n}}\right)}^j\ge 2$ and

$$M_j:={\left({⨍}_{B(r_j)}b{(u)}^{{\alpha }_j}\mathrm{d}x\right)}^{{\displaystyle \frac{1}{{\alpha }_j}}}.$$

By $r\le r_0$, from (19), we have

$$M_{j+1}^{{\alpha }_{j+1}{\displaystyle \frac{n}{n+1}}}\le {\displaystyle \frac{C{\alpha }_j^6}{{(1-\kappa )}^2}}\left[M_j^{{\alpha }_j}+r_0^2{\parallel Tg\parallel }_{L^{\infty }}^2{M}_j^{{\alpha }_j-{\displaystyle \frac{2q-4+\beta }{3}}}+r_0^2{\parallel Xg\parallel }_{L^{\infty }}{M}_j^{{\alpha }_j-q+1}\right].$$

By Young’s inequality with exponents ${\displaystyle \frac{3(q+\beta )}{2q-4+\beta }}$ and ${\displaystyle \frac{3(q+\beta )}{q+2+3\beta }}$, the second term is controlled by

$$r_0^2{\parallel Tg\parallel }_{L^{\infty }}^2{M}_j^{{\alpha }_j-{\displaystyle \frac{2q-4+\beta }{3}}}\le C\left(M_j^{{\alpha }_j}+r_0^2{\parallel Tg\parallel }_{L^{\infty }}^2{M}_j^{{\alpha }_j}\right).$$

Similarly, the third term is controlled by Young’s inequality. We also corrected the exponents in the final estimate, with the constant $q-1$ now appearing consistently. Thus

$$\begin{array}{cc}\hfill M_{j+1}^{{\alpha }_{j+1}{\displaystyle \frac{n}{n+1}}}\le & {\displaystyle \frac{C{\alpha }_j^6}{{(1-\kappa )}^2}}\left[M_j^{q-1}+r_0^2{\parallel Tg\parallel }_{L^{\infty }}^{{\displaystyle \frac{2}{3}}}{M}_j^{{\displaystyle \frac{q-1}{3}}}+r_0^2{\parallel Xg\parallel }_{L^{\infty }}\right]M_j^{{\alpha }_j-q+1}\hfill \\ \hfill \le & C{\alpha }_j^6{\displaystyle \frac{1+r_0^2}{{(1-\kappa )}^2}}\left[{\parallel Tg\parallel }_{L^{\infty }}+{\parallel Xg\parallel }_{L^{\infty }}+M_j^{q-1}\right]M_j^{{\alpha }_j-q+1}.\hfill \end{array}$$

(20)

Denote

$${\overline{M}}_j:=max\left\{M_j,{\left({\parallel Tg\parallel }_{L^{\infty }}+{\parallel Xg\parallel }_{L^{\infty }}\right)}^{{\displaystyle \frac{1}{q-1}}}\right\}.$$

Then Inequality (20) reduces to

$$\begin{array}{c}\hfill {\overline{M}}_{j+1}^{{\alpha }_{j+1}{\displaystyle \frac{n}{n+1}}}\le C{\alpha }_j^6{\displaystyle \frac{1+r_0^2}{{(1-\kappa )}^2}}{\overline{M}}_j^{{\alpha }_j}.\end{array}$$

(21)

Iterating (21), we obtain

$${\overline{M}}_{j+1}\le \left({\displaystyle \prod _{i=0}^j}\left({\left({\displaystyle \frac{C(1+r_0^2)}{{(1-\kappa )}^2}}\right)}^{{\displaystyle \frac{n^i}{q{(n+1)}^i}}}{\left({\displaystyle \frac{q{(n+1)}^i}{n^i}}\right)}^{{\displaystyle \frac{6n^i}{q{(n+1)}^i}}}\right)\right){\overline{M}}_0,$$

where $C=C(n,q)>0$.

We now verify the convergence of the product of iteration constants arising from the Moser scheme. A direct computation yields

$$\begin{array}{cc}\hfill & {\displaystyle \prod _{i=0}^{\infty }}\left({\left({\displaystyle \frac{C(1+r_0^2)}{{(1-\kappa )}^2}}\right)}^{{\displaystyle \frac{n^i}{q{(n+1)}^i}}}{\left({\displaystyle \frac{q{(n+1)}^i}{n^i}}\right)}^{{\displaystyle \frac{6n^i}{q{(n+1)}^i}}}\right)\hfill \\ \hfill & ={\left({\displaystyle \frac{C(1+r_0^2)}{{(1-\kappa )}^2}}\right)}^{{\sum }_{i=0}^{\infty }{\displaystyle \frac{1}{q}}{\left({\displaystyle \frac{n}{n+1}}\right)}^i}{q}^{{\displaystyle \frac{6}{q}}{\sum }_{i=0}^{\infty }{\left({\displaystyle \frac{n}{n+1}}\right)}^i}{\left({\displaystyle \frac{n+1}{n}}\right)}^{{\displaystyle \frac{6}{q}}{\sum }_{i=0}^{\infty }i{\left({\displaystyle \frac{n}{n+1}}\right)}^i}\hfill \\ \hfill & ={\left({\displaystyle \frac{C(1+r_0^2)}{{(1-\kappa )}^2}}\right)}^{{\displaystyle \frac{n+1}{q}}}{q}^{{\displaystyle \frac{6(n+1)}{q}}}{\left({\displaystyle \frac{n+1}{n}}\right)}^{{\displaystyle \frac{6n(n+1)}{q}}}.\hfill \end{array}$$

Since the geometric series ${\sum }_{i=0}^{\infty }{(n/(n+1))}^i=n+1$ and ${\sum }_{i=0}^{\infty }i{(n/(n+1))}^i=n(n+1)$ both converge, the product is finite and defines a constant depending only on n, q, and $r_0$. Moreover, the factor ${(1-\kappa )}^{-2}$ reflects the standard dependence on the distance to the boundary in the iteration, which is ultimately absorbed into the geometric constant.

Consequently,

$$\underset{B(\kappa r)}{sup}b(u)=\underset{j\to \infty }{lim\; sup}{\overline{M}}_j\le C{\left({\displaystyle \frac{1+r_0^2}{{(1-\kappa )}^2}}\right)}^{{\displaystyle \frac{n+1}{q}}}{\overline{M}}_0,$$

where $C=C(n,q)>0$.

From this, setting $\delta \to 0$, one derives (4). Here we observe that the final estimate does not contain an explicit radius factor in the source term. This is consistent with the Heisenberg dilation

$${\delta }_{\rho }(x_1,\dots ,x_{2n},x_{2n+1})=(\rho x_1,\dots ,\rho x_{2n},{\rho }^2{x}_{2n+1}),$$

under which the horizontal gradient scales as $Xu\mapsto \rho Xu$, while the source terms scale as

$$Xg\mapsto \rho Xg,Tg\mapsto {\rho }^{2}Tg.$$

Consequently, the quantity

$${\left({\parallel Tg\parallel }_{L^{\infty }}+{\parallel Xg\parallel }_{L^{\infty }}\right)}^{{\displaystyle \frac{1}{q-1}}}$$

scales as $\rho$, matching the scaling of $|Xu|$. The absence of an explicit radius factor is therefore a consequence of the intrinsic homogeneity of the Heisenberg group. □

6. Conclusions

Weak solutions of the non-homogeneous highly degenerate q-Laplace equation

$$-{\displaystyle \sum _{i=1}^{2n}}{X}_i({\lambda }_i(Xu)X_{i}u)=g(x)\mathrm{in}{\mathbb{H}}^{\mathrm{n}}$$

are proved to be local Lipschitz, where the coefficients ${\lambda }_i$ are defined as in (2). For the exponent range $2<q\le 4$, we prove that weak solutions u are intrinsically locally Lipschitz, i.e., $u\in C_{\mathrm{loc}}^{0,1}(\Omega )$, and we provide a quantitative gradient estimate involving the source terms $Xg$ and $Tg$ (Theorem 1). This result complements the homogeneous case studied in [33] and extends the regularity theory to non-homogeneous orthotropic q-Laplacian-type equations in the Heisenberg group.

Compared with the homogeneous result of Circelli–Citti–Clop [33], our contribution consists of three main novelties: (i) Lemmas 1 and 2 contain new terms involving $Xg$ and $Tg$ that require additional estimates; (ii) Lemma 3 includes a new source term contribution that fundamentally changes the higher integrability argument; (iii) Lemma 4 and the final gradient estimate contain the source terms $Xg$ and $Tg$, which are absent in the homogeneous case. The proofs of Lemmas 1–4 are therefore substantially different from those in [33].

The restriction $q\in (2,4]$ plays a crucial role in Lemmas 3 and 4, where Hölder’s inequality and specific exponent relations are invoked. Currently, our method fails to cover the singular regime $1<q<2$ and the super-quadratic regime $q>4$. Consequently, our future efforts will be directed toward addressing these remaining ranges.

Two open problems are hereby proposed for subsequent research:

(i)

Show that weak solutions are local Lipschitz for $1<q<2$ and $4<q<\infty$ by refining the Caccioppoli estimates, possibly using different interpolation techniques or weighted inequalities.

(ii)

Generalize the aforementioned regularity theory to broader classes of sub-Riemannian manifolds, such as higher-step Carnot groups, where the non-commutativity becomes more intricate and the orthotropic structure may have no direct analogue.