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An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. (English) Zbl 0802.35024

From the introduction: Let ${X}_{1},\cdots ,{X}_{m}$ be ${C}^{\infty }$ vector fields in ${ℝ}^{n}$ satisfying Hörmander’s condition for hypoellipticity: rank Lie $\left[{X}_{1},\cdots ,{X}_{m}\right]=n$ at every point $x\in {ℝ}^{n}$. Denote by ${X}_{j}^{*}$ the formal adjoint of ${X}_{j}$. The linear operator $ℒ={\sum }_{j=1}^{m}{X}_{j}^{*}{X}_{j}$ is the subelliptic Laplacian associated to the vector fields ${X}_{1},\cdots ,{X}_{m}$.

Given an open set $U\subset {ℝ}^{n}$, and a function $u\in {C}^{1}\left(U\right)$, denote by ${D}_{ℒ}u=\left({X}_{1}u,\cdots ,{X}_{m}u\right)$ the subelliptic gradient of $u$. For $1 we consider the functional

${J}_{p}\left(u\right)={\int }_{U}{|{D}_{ℒ}u|}^{p}\phantom{\rule{4pt}{0ex}}dx={\int }_{U}{\left[\sum _{j=1}^{m}{\left({X}_{j}u\right)}^{2}\right]}^{p/2}dx,$

and define ${S}^{1,p}\left(U\right)$ to be the completion of ${C}_{0}^{1}\left(U\right)$ in the norm generated by ${J}_{p}$. The Euler equation of ${J}_{p}$ is

$\sum _{j=1}^{m}{X}_{j}^{*}\left(|{D}_{ℒ}{u|}^{p-2}{X}_{j}u\right)=0·\phantom{\rule{2.em}{0ex}}\left(1\right)$

We call the operator in (1) the subelliptic $p$-Laplacian. Critical points of ${J}_{p}$ are (weak) solutions of (1), and vice-versa.

In this paper, we propose to study a general class of nonlinear subelliptic equations, whose prototype is constituted by (1) above. Our objectives are: a) To establish an optimal embedding result of Sobolev type for the subelliptic spaces ${S}^{1,p}$; b) To prove a Harnack type inequality for nonnegative solutions. From b) the Hölder continuity of solutions with respect to the $\left({X}_{1},\cdots ,{X}_{m}\right)$-control distance will follow.

##### MSC:
 65H10 Systems of nonlinear equations (numerical methods) 35B45 A priori estimates for solutions of PDE 31C45 Nonlinear potential theory, etc.