In this section, we study the solvability of the following linear sub-Laplace Dirichlet problem

$\left\{\begin{array}{cc}Lu-\lambda \left(\xi \right)u=f,\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ u=\phi ,\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \Omega ,\hfill \end{array}\right.$

(3.1)

where $\lambda \left(\xi \right)\in C\left(\stackrel{\u0304}{\Omega}\right)$ satisfies *λ*(*ξ*) > 0.

**Definition 3.1**.

*A bounded open set* Ω ⊂

*G is said to satisfy the outer sphere condition at ξ*_{0} ∈

*∂* Ω,

*if there exists a ball B*_{
G
}(

*η, r*)

*lying in G\* Ω

*such that*$\partial {B}_{G}\left(\eta ,r\right)\cap \partial \Omega =\left\{{\xi}_{0}\right\}.$

The definition in the case of general degenerate elliptic operator can be seen in [3]. Notice that in the H-type group case, every bounded convex subset accords with the condition of the outer sphere. In particular, the gauge balls in H-type group are convex domains (see [4]). From Theorem 2.12 in [13] and Theorem 5.2 in [2] respectively, one has the following two lemmas.

**Lemma 3.2**.

*(Maximum principle) Let* Ω

*be a connected open set in a Carnot group G. If u* ∈

*C*^{2}(Ω)

*satisfies*$Lu-\lambda \left(\xi \right)u\ge 0\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}\Omega ,$

*then u cannot achieve a nonnegative maximum at an interior point unless u ≡* constant *in* Ω.

**Lemma 3.3**.

*Let* Ω

*be a bounded domain in G. Then there exists a family of open subsets, denoted by* $\mathcal{F}=\left\{\omega :\stackrel{\u0304}{\omega}\subset \Omega \right\}$,

*which is a base for the topology of* Ω

*for which the Dirichlet problem*$\left\{\begin{array}{cc}Lu-\lambda \left(\xi \right)u=f,\hfill & in\phantom{\rule{2.77695pt}{0ex}}\omega ,\hfill \\ u=\phi ,\hfill & on\phantom{\rule{2.77695pt}{0ex}}\partial \omega \hfill \end{array}\right.$

(3.2)

*has a unique distributional solution* $u\in C\left(\stackrel{\u0304}{\omega}\right)$ for any $\omega \in \mathcal{F}$, $f\in C\left(\stackrel{\u0304}{\omega}\right)$ *and φ*∈ *C*(∂*ω*). *Furthermore, if f* ∈ *C*^{
∞
}(*ω*), *then u* ∈ *C*^{
∞
}(*ω*).

We give notions of subsolution and supersolution for the Dirichlet problem (3.1).

**Definition 3.4**.

*Let φ* ∈

*C*(∂Ω),

$f\in {C}^{\infty}\left(\stackrel{\u0304}{\Omega}\right)$.

*A function* $u\in C\left(\stackrel{\u0304}{\Omega}\right)$ *is called a subsolution of (3.1) if it fits the following properties:*- (i)

- (ii)
*for every* $\omega \in \mathcal{F}$ *and for every* $h\in {C}^{2}\left(\omega \right)\cap C\left(\stackrel{\u0304}{\omega}\right)$ *such that Lh - λ*(*ξ*)*h* = *f and u* ≤ *h on* ∂*ω, we also have u* ≤ *h in ω*.

The definition of supersolution is analogous.

**Lemma 3.5**. *Assume that u is a subsolution of (3.1) and v is a supersolution of (3.1), then either u* < *v in* Ω *or u ≡ v*.

*Proof*. Suppose that at some point

*η* ∈ Ω we have

*u*(

*η*) ≥

*v*(

*η*). Set

$M=\underset{\xi \in \Omega}{sup}\left(u-v\right)\left(\xi \right)\ge 0$. Take

*ξ*_{0} ∈ Ω such that (

*u - v*)(

*ξ*_{0}) =

*M*, and we can know that

*u - v ≡ M* in a neighborhood of

*ξ*_{0}. Otherwise there exists

$\omega \in \mathcal{F}$ such that

*ξ*_{0} ∈

*ω* but

*u - v* is not identically equal to

*M* on ∂

*ω*. Letting

$\mathit{\u016b}$ and

$\stackrel{\u0304}{v}$ denote the solutions of

*Lw - λ*(

*ξ*)

*w* =

*f* in

*ω*, equal to

*u* and

*v* on ∂

*ω* respectively. Since

*u* and

*v* are the subsolution and the supersolution respectively, we deduce from Definition 3.4 that

$\mathit{\u016b}\ge u$ and

$\stackrel{\u0304}{v}\le v$ in

*ω*. One sees that

$M=\underset{\xi \in \Omega}{sup}\left(u-v\right)\left(\xi \right)\ge \underset{\xi \in \partial \omega}{sup}\left(\mathit{\u016b}-\stackrel{\u0304}{v}\right)\left(\xi \right)\ge \left(\mathit{\u016b}-\stackrel{\u0304}{v}\right)\left({\xi}_{0}\right)\ge \left(u-v\right)\left({\xi}_{0}\right)=M,$

and hence all the equalities above hold. By Lemma 3.2 it follows that $\mathit{\u016b}-\stackrel{\u0304}{v}\equiv M$ in *ω* and hence *u - v ≡ M* on ∂*ω*, which contradicts the choice of *ω*.

The previous argument implies *u - v ≡ M* in Ω. Combining this with Definition 3.4-(i) we obtain *u ≡ v* in Ω. □

Let

$u\in C\left(\stackrel{\u0304}{\Omega}\right)$ be a subsolution of (3.1) and

$\omega \in \mathcal{F}$. Denote by

$\mathit{\u016b}$ the solution of the Dirichlet problem (see Lemma 3.3)

$\left\{\begin{array}{cc}L\mathit{\u016b}-\lambda \left(\xi \right)\mathit{\u016b}=f\left(\xi \right),\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\omega ,\hfill \\ \mathit{\u016b}=u,\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \omega ,\hfill \end{array}\right.$

and define in Ω the lifting of

*u* (in

*ω*) by

$U\left(\xi \right):=\left\{\begin{array}{c}\mathit{\u016b}\left(\xi \right),\phantom{\rule{2.77695pt}{0ex}}\xi \in \omega ,\hfill \\ u\left(\xi \right),\phantom{\rule{2.77695pt}{0ex}}\xi \in \Omega \backslash \omega .\hfill \end{array}\right.$

(3.3)

**Lemma 3.6**. *U*(*ξ*) *is a subsolution of (3.1)*.

*Proof*. Since *u*(*ξ*) is a subsolution of (3.1), it follows that *U*(*ξ*) = *u*(*ξ*) ≤ *φ*(*ξ*) on ∂Ω. Let ${\omega}^{\prime}\in \mathcal{F}$ and $h\in {C}^{2}\left({\omega}^{\prime}\right)\cap C\left(\overline{{\omega}^{\prime}}\right)$ such that *Lh - λ*(*ξ*)*h* = *f* and *U* ≤ *h* on ∂*ω*'. If *ω* ∩ *ω*' = *ϕ*, then *u* = *U* ≤ *h* on ∂*ω*'. It leads to *U* = *u* ≤ *h* in *ω*';

Suppose now *ω* ∩ *ω*' = *ϕ*. Since *u* ≤ *U*, we have *u* ≤ *h* on *∂ω*' and then *u* ≤ *h* in *ω*'. In particular, *u* ≤ *h* in *ω*'*\ω*, i.e. *U* ≤ *h* in *ω*'*\ω*. Thus, we have $\mathit{\u016b}\le h$ on ∂(*ω*' ∩ *ω*). As $L\left(\mathit{\u016b}-h\right)-\lambda \left(\xi \right)\left(\mathit{\u016b}-h\right)=0$ in *ω*' ∩ *ω* and $\mathit{\u016b}-h\le 0$ on ∂(*ω*' ∩ *ω*), it yields by Lemma 3.2 that $\mathit{\u016b}\le h$ in *ω*' ∩ *ω*, and therefore *U* ≤ *h* in *ω*' ∩ *ω*. □

The following result is a trivial consequence of Definition 3.4.

**Lemma 3.7**.

*Let u*_{1},

*u*_{2}, . . .,

*u*_{
l
} *be subsolutions of (3.1). Then the function*$v=max\left\{{u}_{1},{u}_{2},\phantom{\rule{2.77695pt}{0ex}}\dots ,\phantom{\rule{2.77695pt}{0ex}}{u}_{l}\right\}$

*is also a subsolution of (3.1)*.

Let *S* denote the set of all subsolutions of (3.1). Notice that *S* is not empty, since -*k*^{2} ∈ *S* for *k* large enough. The basic result via the Perron method is contained in the following theorem.

**Theorem 3.8**. *The function* $u\left(\xi \right):=\underset{v\in S}{sup}v\left(\xi \right)$ *satisfies Lu - λ*(*ξ*)*u* = *f in* Ω.

*Proof*. Notice that *k*^{2}, for *k* large enough, is a supersolution of (3.1). By Lemma 3.5, we deduce *v* ≤ *k*^{2} for any *v* ∈ *S*, so *u* is well defined. Let *η* be an arbitrary fixed point of Ω. By the definition of *u*, there exists a sequence {*v*_{
n
}}_{n∈ℕ}such that *v*_{
n
}(*η*) *→ u*(*η*). By replacing *v*_{
n
} with max {*v*_{1}, . . ., *v*_{
n
}}, we may assume that *v*_{1} ≤ *v*_{2} ≤ *· · ·* ≤ *v*_{
n
} ≤ *· · ·*. Let $\omega \in \mathcal{F}$ be such that *η* ∈ *ω* and define *V*_{
n
}(*η*) to be the lifting of *v*_{
n
} in *ω* according to (3.3). From Lemma 3.2, *V*_{
n
} is also increasing and, since *V*_{
n
} ∈ *S* (see Lemma 3.6) and *V*_{
n
} ≥ *v*_{
n
}, it gets *V*_{
n
}(*η*) *→ u*(*η*). Set $V\left(\xi \right):=\underset{n\to \infty}{lim}{V}_{n}\left(\xi \right)$. Obviously, we have that *V* ≤ *u* in Ω and *V* (*η*) = *u*(*η*). Noting that every *V*_{
n
} satisfies *LV*_{
n
} *- λ*(*ξ*)*V*_{
n
} = *f* in *ω*, we have, by the dominated convergence theorem that the function *V* satisfies *LV - λ*(*ξ*)*V* = *f* in the distributional sense in *ω*. Since *f* ∈ *C*^{
∞
}(*ω*), we have *V*(*ξ*) ∈ *C*^{
∞
}(*ω*) in view of the hypoellipticity of the operator *L - λ*(*ξ*).

We conclude that *V ≡ u* in *ω*. In fact, suppose *V*(*ζ*) < *u*(*ζ*) for some *ζ* ∈ *ω*, then there exists a function $\mathit{\u016b}\in S$ such that $V\left(\zeta \right)<\mathit{\u016b}\left(\zeta \right)$. Define the increasing sequence ${w}_{n}=max\left\{\mathit{\u016b},{V}_{n}\right\}$ and then the corresponding liftings *W*_{
n
}. Set $W\left(\xi \right):=\phantom{\rule{2.77695pt}{0ex}}\underset{n\to \infty}{lim}\phantom{\rule{2.77695pt}{0ex}}{W}_{n}\left(\xi \right)$. Analogously to *V, W* satisfies *LW - λ*(*ξ*)*W* = *f*. Since *V*_{
n
} ≤ *w*_{
n
} ≤ *W*_{
n
}, we obtain *V* ≤ *W*. The equalities *V*(*η*) = *u*(*η*) = *W*(*η*) and Lemma 3.2 imply that *V ≡ W* in Ω. This is in contradiction with $V\left(\zeta \right)<\mathit{\u016b}\left(\zeta \right)\le W\left(\zeta \right)$. Consequently, *V ≡ u* in *ω* and *u* satisfies *Lu - λ*(*ξ*)*u* = *f* in the classical sense. The arbitrariness of *η* leads to the desired result. □

**Definition 3.9**.

*Let ζ* ∈ ∂Ω.

*Then a function* $w\left(\xi \right)\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ *is called a barrier function related to the sub-Laplacian L at ζ if the following two conditions hold:*- (i)

- (ii)
*w*(*ξ*) > 0 *on* $\stackrel{\u0304}{\Omega}\backslash \left\{\zeta \right\}$, *w*(*ζ*) = 0.

**Lemma 3.10**.

*Let* Ω ⊂

*G be a bounded open domain which satisfies the outer sphere condition at every point of the boundary* ∂Ω.

*Then for every ζ* ∈ ∂Ω,

*the Dirichlet problem*$\left\{\begin{array}{cc}Lw=-1,\hfill & in\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ w\left(\xi \right)=\rho \left(\xi ,\phantom{\rule{2.77695pt}{0ex}}\zeta \right),\hfill & on\phantom{\rule{2.77695pt}{0ex}}\partial \Omega \hfill \end{array}\right.$

(3.4)

*has a unique solution* $w\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ *fulfilling w*(*ξ*) > 0 *on* $\stackrel{\u0304}{\Omega}\backslash \left\{\zeta \right\}$ *and w*(*ζ*) = 0.

*Proof*. From [

1], let Γ(

*ξ*) =

*C*_{
Q
}*ρ*(

*ξ, e*)

^{-(Q-2)}be the fundamental solution of the sub-Laplacian

*L*. Define the convolution

$\mathit{\u0169}\phantom{\rule{2.77695pt}{0ex}}:=-\Gamma *{\chi}_{\Omega},$

where χ_{Ω} denotes the indicator function. Since $\Gamma \left(\xi \right)\in {L}_{loc}^{p}$ for $1\le p<\frac{Q}{Q-2}$, it yields $\mathit{\u0169}\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$.

According to Corollary 10 in [

3], the problem

$\left\{\begin{array}{cc}Lv=0,\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ v\left(\xi \right)=\rho \left(\xi ,\phantom{\rule{2.77695pt}{0ex}}\zeta \right)-\mathit{\u0169}\left(\xi \right),\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \Omega \hfill \end{array}\right.$

has a unique solution $v\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$. Since $L\mathit{\u0169}=-{\chi}_{\Omega}$ (see Corollary 2.8 in [1]), it follows that $w:=v+\mathit{\u0169}$ is the desired solution of (3.4). □

**Theorem 3.11**. *Let* Ω *be as in Lemma 3.10. Suppose φ* ∈ *C*(∂Ω) *and* $f\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$. *Then the Dirichlet problem (3.1) possesses a unique solution* $u\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$.

*Proof*. Uniqueness is a direct consequence of Lemma 3.2. Theorem 3.8 provides the existence of the solution *u* ∈ *C*^{
∞
}(Ω). To complete the proof of the theorem, it needs only to examine that *u* is continuous up to the boundary of Ω.

Let

*ζ* ∈

*∂* Ω. Since

*φ* ∈

*C*(∂Ω), it follows that for any

*ε* > 0 there exists some

*δ* > 0 such that for every

*ξ* ∈ ∂Ω with

*ρ*(

*ξ, ζ*) <

*δ*, we have

$|\phi \left(\xi \right)-\phi \left(\zeta \right)|\phantom{\rule{2.77695pt}{0ex}}<\epsilon .$

Let

*w*(

*ξ*) be the barrier function related to

*L* at

*ζ* constructed in Lemma 3.10. Set

$M=\underset{\xi \in \partial \Omega}{sup}\left|\phi \left(\xi \right)\right|$ and choose

*k*_{1} > 0 such that

*k*_{1}*w*(

*ξ*) ≥ 2

*M* if

*ρ*(

*ξ, ζ*) ≥

*δ*. Set

${k}_{2}=\left[\left|\phi \left(\zeta \right)\right|+\epsilon \right]\underset{\xi \in \Omega}{max}\lambda \left(\xi \right)+\underset{\xi \in \Omega}{sup}\left|f\left(\xi \right)\right|$, and

*k* = max{

*k*_{1},

*k*_{2}}. Define that

*w*_{1}(

*ξ*): =

*φ*(

*ζ*) +

*ε* +

*kw*(

*ξ*) and

*w*_{2}(

*ξ*): =

*φ*(

*ζ*) -

*ε* -

*kw*(

*ξ*). Then we see in view of Lemma 3.10,

$L{w}_{1}-\lambda \left(\xi \right){w}_{1}=-k-\lambda \left(\xi \right)\phi \left(\zeta \right)-\lambda \left(\xi \right)\epsilon -k\lambda \left(\xi \right)w\left(\xi \right)\le f\phantom{\rule{1em}{0ex}}\mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega .$

On the one hand,

*w*_{1}(

*ξ*) =

*φ*(

*ζ*) +

*ε* +

*kw*(

*ξ*) ≥

*φ*(

*ζ*) +

*ε* >

*φ*(

*ξ*) when

*ρ*(

*ξ, ζ*) <

*δ*; On the other hand,

*w*_{1}(

*ξ*) ≥

*φ*(

*ζ*) +

*ε* + 2

*M* >

*φ*(

*ξ*) when

*ρ*(

*ξ, ζ*) ≥

*δ*. Combining these with Lemma 3.2 we can conclude that

*w*_{1}(

*ξ*) is a supersolution of (3.1). Analogously,

*w*_{2}(

*ξ*) is a subsolution of (3.1). Hence from the choice of

*u* and the fact that every supersolution dominates every subsolution, we have in Ω that

${w}_{2}\left(\xi \right)\le u\left(\xi \right)\le {w}_{1}\left(\xi \right)$

and then

$|u\left(\xi \right)-\phi \left(\zeta \right)|\phantom{\rule{2.77695pt}{0ex}}\le \epsilon \phantom{\rule{2.77695pt}{0ex}}+kw\left(\xi \right).$

Since *w*(*ξ*) *→* 0 as *ξ → ζ*, we obtain *u*(*ξ*) *→ φ*(*ζ*) as *ξ → ζ*. □

*Remark* 3.12. Let

$f\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ and

*u* be the solution of

$\left\{\begin{array}{cc}Lu-\lambda \left(\xi \right)u=f,\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ u=0,\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \Omega .\hfill \end{array}\right.$

(3.5)

Elementary calculations show that $-\frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}\left|\right|f|{|}_{{L}^{\infty}\left(\Omega \right)}$ and $\frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}\left|\right|f|{|}_{{L}^{\infty}\left(\Omega \right)}$ are a subsolution and a supersolution of (3.5) respectively. Thus, $\left|\right|u|{|}_{{L}^{\infty}\left(\Omega \right)}\le \frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}|\left|f\right|{|}_{{L}^{\infty}\left(\Omega \right)}$. It provides a *L*^{
∞
} estimate for the solution of (3.5).

**Theorem 3.13**. *Set φ* ∈ *C*(∂Ω) *and* $f\in C\left(\stackrel{\u0304}{\Omega}\right)$. *Then there exists a unique solution* $u\in C\left(\stackrel{\u0304}{\Omega}\right)$ *to (3.1) in the sense of distribution*.

*Proof*. Take a sequence

${f}_{n}\left(\xi \right)\in {C}^{\infty}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$,

*n* = 1, 2, . . ., so that {

*f*_{
n
}(

*ξ*)} converges uniformly to

*f* in Ω. Denote by

*u*_{
n
} the corresponding solution of the Dirichlet problem

$\left\{\begin{array}{cc}Lv-\lambda \left(\xi \right)v={f}_{n}\left(\xi \right),\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega ,\hfill \\ u=\phi ,\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \Omega .\hfill \end{array}\right.$

We obtain, in view of Remark 3.12,

$\left|\right|{u}_{n}-{u}_{m}|{|}_{{L}^{\infty}\left(\Omega \right)}\le \frac{1}{\underset{\xi \in \Omega}{min}\lambda \left(\xi \right)}||{f}_{n}-{f}_{m}|{|}_{{L}^{\infty}\left(\Omega \right)}.$

In conclusion, {*u*_{
n
}} converges uniformly to a continuous function *u* which is the required solution. □