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Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups

Zixia Yuan1* and Guanxiu Yuan2

Author Affiliations

1 School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu 611731, China

2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China

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Journal of Inequalities and Applications 2012, 2012:136  doi:10.1186/1029-242X-2012-136


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/136


Received:7 December 2011
Accepted:12 June 2012
Published:12 June 2012

© 2012 Yuan and Yuan; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of this article, is to study the Dirichlet problems of the sub-Laplace equation Lu + f(ξ, u) = 0, where L is the sub-Laplacian on the Carnot group G and f is a smooth function. By extending the Perron method in the Euclidean space to the Carnot group and constructing barrier functions, we establish the existence and uniqueness of solutions for the linear Dirichlet problems under certain conditions on the domains. Furthermore, the solvability of semilinear Dirichlet problems is proved via the previous results and the monotone iteration scheme corresponding to the sub-Laplacian.

Mathematics Subject Classifications: 35J25, 35J70, 35J60.

Keywords:
Carnot group; sub-Laplace equation; Dirichlet problem; Perron method; monotone iteration scheme

1 Introduction

In this article we consider Dirichlet problems of the type

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M1">View MathML</a>

(1.1)

where Ω is a bounded domain in a Carnot group G and L is the sub-Laplacian. Some knowledge on G and L see next section. Hörmander's theorem permits us to judge the hypoellipticity of the operator L, i.e., if Lu Cthen u C(see [1]).

The investigation of the boundary value problems, concerning the operators in the form of the sum of squares of vector fields fulfilling Hörmander condition, has turned into the subject of several works, see [2-4]. The precursory work of Bony [2] proved a maximum principle and the solvability of the Dirichlet problem in the sense of Perron-Wiener. The Wiener type regularity of boundary points for the Dirichlet problem was considered in [3]. Thanks to the previous results, Capogna et al. [4] established the solvability of the Dirichlet problem when the boundary datum belongs to Lp, 1 < p , in the group of Heisenberg type.

The Perron method (see [5,6]) and the monotone iteration scheme (see [7,8]) are well-known constructive methods for solving linear and semilinear Dirichlet problems, respectively. Brandolini et al. [9] applied these methods to the Dirichlet problems for sub-Laplace equations on the gauge balls in the Heisenberg group which is the simplest Carnot group of step two. Let us notice that the balls possess of legible properties. However, we do not see the reseach to the problems on other domains using these methods. Concerning the construction of barrier function, Brandolini et al. [9] used the result given in [10], which holds in the setting of Heisenberg group.

Our work is motivated by [9]. We try to extend the existence of solutions for semilinear Dirichlet problems on the Heisenberg balls in [9] to general Carnot domains. To do so, the Perron method in the Carnot group is used in this article. Based on the work in [3], we construct a barrier function in a domain of the Carnot group (see Lemma 3.10) under the hypothesis of the outer sphere condition to discuss the boundary behaviour of the Perron solutions. The method to obtain a barrier function is essentially similar to the one in [9]. Then we prove the existence of solutions for linear sub-Laplace Dirichlet problems. In the discussion of semilinear Dirichlet problems, we will use monotone iteration scheme. The main difficulty we meet is that the sub-Laplacian L does not have explicit expression. To overcome it, we use the regularity of L in [1].

The article is organized as follows. In the next section, we recall some basic definitions and collect some known results on the Carnot group which will play a role in the following sections. Section 3 is devoted to the study of the Perron method for linear equations. By finding a barrier function related to the sub-Laplacian L, we prove that the Perron solutions for linear Dirichlet problems are continuous up to the boundary. The main results are Theorems 3.8, 3.11, and 3.13. In Section 4, using the results in Section 3 and the monotone iteration scheme, we provide the solutions of the semilinear Dirichlet problems in Carnot groups with some available supersolutions and subsolutions. Finally, we give an existence of solution to the sub-Laplace equation on the whole group of Heisenberg type (a specific Carnot group of step two). The main results in this section are Theorems 4.2 and 4.3.

2 Carnot groups

We will consider G = (ℝN, ·) as a Carnot group with a group operation · and a family of dilations, compatible with the Lie structure.

Following [11,12], a Carnot group G of step r ≥ 1 is a simple connected nilpotent Lie group whose Lie algebra <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M2">View MathML</a> admits a stratification. That is, there exist linear subspaces V1, . . ., Vr of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M3">View MathML</a>

Via the exponential map, it is possible to induce on G a family of non-isotropic dilations defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M4">View MathML</a>

Here ξ = (x(1), x(2), . . ., x(r)), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M5">View MathML</a> for i = 1, . . ., r and N1 + · · · + Nr = N. We denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M6">View MathML</a> the homogeneous dimension of G attached to the dilations {δλ}λ > 0. Let m = N1 and X = {X1, . . ., Xm} be the dimension and a basis of V1, respectively. Let Xu = {X1u, . . ., Xmu} denote the horizontal gradient for a function u. The sub-Laplacian associated with X on G is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M7">View MathML</a>

If u and v are two measurable functions on G, their convolution is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M8">View MathML</a>

where dG(η) denotes a fixed Haar measure on G.

Let e be the identity on G. For ξ G, we denote by ξ-1 the inverse of ξ with respect to the group operation. By [1], there exists a norm function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M9">View MathML</a> satisfying

(1) ρ(ξ) ≥ 0; Moreover, ρ(ξ) = 0 if and only if ξ = e;

(2) ρ(ξ) = ρ(ξ-1).

The open ball of radius R centered at ξ is expressed as the set:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M10">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M11">View MathML</a> denote the space of distributions on G. The non-isotropic Sobolev space Sk, p is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M12">View MathML</a>

where α = (α1, . . ., αl) is a multi-index, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M13">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M14">View MathML</a>. In the space Sk, p, we shall adopt the norm

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M15">View MathML</a>

For a domain Ω in G, we define Sk, p(Ω, loc) as the space of distributions f such that for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M16">View MathML</a> we have Sk, p. Let 0 < β < , we employ the following non-isotropic Lipschitz spaces:

(i) for 0 < β < 1,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M17">View MathML</a>

(ii) for β = 1,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M18">View MathML</a>

(iii) for β = k + β' where k = 1, 2, 3, . . . and, 0 < β' ≤ 1,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M19">View MathML</a>

We refer the reader to [1] for more information on the above.

The following results are useful.

Proposition 2.1. (i) Suppose Ω ⊂ G is an open set, and suppose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M20">View MathML</a>satisfy Lf = g in Ω. If g Sk, p(Ω, loc) (1 < p < ∞, k ≥ 0) then f Sk+2,p(Ω, loc).

(ii) Suppose 1 < p < ∞ and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M21">View MathML</a>, then Sk, p ⊂ Γβ.

Part (i) and (ii) are contained, respectively, in Theorems 6.1 and 5.15 of [1].

3 The Perron method and barrier function for linear problem

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M22">View MathML</a>

(3.1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M23">View MathML</a> satisfies λ(ξ) > 0.

Definition 3.1. A bounded open set Ω ⊂ G is said to satisfy the outer sphere condition at ξ0 Ω, if there exists a ball BG(η, r) lying in G\such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M24">View MathML</a>

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 be a connected open set in a Carnot group G. If u C2(Ω) satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M25">View MathML</a>

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

Lemma 3.3. Let be a bounded domain in G. Then there exists a family of open subsets, denoted by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M26">View MathML</a>, which is a base for the topology of for which the Dirichlet problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M27">View MathML</a>

(3.2)

has a unique distributional solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M28">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M30">View MathML</a> 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), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M31">View MathML</a>. A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32">View MathML</a>is called a subsolution of (3.1) if it fits the following properties:

(i) u φ on ∂Ω;

(ii) for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29">View MathML</a>and for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M33">View MathML</a> 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 or u ≡ v.

Proof. Suppose that at some point η ∈ Ω we have u(η) ≥ v(η). Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M34">View MathML</a>. Take ξ0 ∈ Ω such that (u - v)(ξ0) = M, and we can know that u - v ≡ M in a neighborhood of ξ0. Otherwise there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29">View MathML</a> such that ξ0 ω but u - v is not identically equal to M on ∂ω. Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M35">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M36">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M37">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M38">View MathML</a> in ω. One sees that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M39">View MathML</a>

and hence all the equalities above hold. By Lemma 3.2 it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M40">View MathML</a> 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32">View MathML</a> be a subsolution of (3.1) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29">View MathML</a>. Denote by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M35">View MathML</a> the solution of the Dirichlet problem (see Lemma 3.3)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M41">View MathML</a>

and define in Ω the lifting of u (in ω) by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M42">View MathML</a>

(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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M43">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M44">View MathML</a> 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', i.e. U h in ω'. Thus, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M45">View MathML</a> on ∂(ω' ∩ ω). As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M46">View MathML</a> in ω' ∩ ω and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M47">View MathML</a> on ∂(ω' ∩ ω), it yields by Lemma 3.2 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M45">View MathML</a> 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 u1, u2, . . ., ul be subsolutions of (3.1). Then the function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M48">View MathML</a>

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 -k2 S for k large enough. The basic result via the Perron method is contained in the following theorem.

Theorem 3.8. The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M49">View MathML</a>satisfies Lu - λ(ξ)u = f in Ω.

Proof. Notice that k2, for k large enough, is a supersolution of (3.1). By Lemma 3.5, we deduce v k2 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 {vn}n∈ℕ such that vn(η) → u(η). By replacing vn with max {v1, . . ., vn}, we may assume that v1 v2 · · · vn · · ·. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M29">View MathML</a> be such that η ω and define Vn(η) to be the lifting of vn in ω according to (3.3). From Lemma 3.2, Vn is also increasing and, since Vn S (see Lemma 3.6) and Vn vn, it gets Vn(η) → u(η). Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M50">View MathML</a>. Obviously, we have that V u in Ω and V (η) = u(η). Noting that every Vn satisfies LVn - λ(ξ)Vn = 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M51">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M52">View MathML</a>. Define the increasing sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M53">View MathML</a> and then the corresponding liftings Wn. Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M54">View MathML</a>. Analogously to V, W satisfies LW - λ(ξ)W = f. Since Vn wn Wn, 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M55">View MathML</a>. 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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M56">View MathML</a>is called a barrier function related to the sub-Laplacian L at ζ if the following two conditions hold:

(i) Lw(ξ) ≤ -1 in Ω;

(ii) w(ξ) > 0 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M57">View MathML</a>, 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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M58">View MathML</a>

(3.4)

has a unique solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M59">View MathML</a> fulfilling w(ξ) > 0 on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M57">View MathML</a>and w(ζ) = 0.

Proof. From [1], let Γ(ξ) = CQρ(ξ, e)-(Q-2) be the fundamental solution of the sub-Laplacian L. Define the convolution

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M60">View MathML</a>

where χdenotes the indicator function. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M61">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M62">View MathML</a>, it yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M63">View MathML</a>.

According to Corollary 10 in [3], the problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M64">View MathML</a>

has a unique solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M65">View MathML</a>. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M66">View MathML</a> (see Corollary 2.8 in [1]), it follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M67">View MathML</a> is the desired solution of (3.4). □

Theorem 3.11. Let be as in Lemma 3.10. Suppose φ C(∂Ω) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M68">View MathML</a>. Then the Dirichlet problem (3.1) possesses a unique solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69">View MathML</a>.

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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M70">View MathML</a>

Let w(ξ) be the barrier function related to L at ζ constructed in Lemma 3.10. Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M71">View MathML</a> and choose k1 > 0 such that k1w(ξ) ≥ 2M if ρ(ξ, ζ) ≥ δ. Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M72">View MathML</a>, and k = max{k1, k2}. Define that w1(ξ): = φ(ζ) + ε + kw(ξ) and w2(ξ): = φ(ζ) - ε - kw(ξ). Then we see in view of Lemma 3.10,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M73">View MathML</a>

On the one hand, w1(ξ) = φ(ζ) + ε + kw(ξ) ≥ φ(ζ) + ε > φ(ξ) when ρ(ξ, ζ) < δ; On the other hand, w1(ξ) ≥ φ(ζ) + ε + 2M > φ(ξ) when ρ(ξ, ζ) ≥ δ. Combining these with Lemma 3.2 we can conclude that w1(ξ) is a supersolution of (3.1). Analogously, w2(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

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M74">View MathML</a>

and then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M75">View MathML</a>

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

Remark 3.12. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M68">View MathML</a> and u be the solution of

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M76">View MathML</a>

(3.5)

Elementary calculations show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M77">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M78">View MathML</a> are a subsolution and a supersolution of (3.5) respectively. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M79">View MathML</a>. It provides a Lestimate for the solution of (3.5).

Theorem 3.13. Set φ C(∂Ω) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M80">View MathML</a>. Then there exists a unique solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M32">View MathML</a>to (3.1) in the sense of distribution.

Proof. Take a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M81">View MathML</a>, n = 1, 2, . . ., so that {fn(ξ)} converges uniformly to f in Ω. Denote by un the corresponding solution of the Dirichlet problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M82">View MathML</a>

We obtain, in view of Remark 3.12,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M83">View MathML</a>

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

4 The monotone iteration scheme for semilinear equation

Let Ω be a bounded open domain in a Carnot group G. Consider Dirichlet problem (1.1), where f(ξ, u) is a smooth function of ξ and u, φ C(∂Ω). A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M84">View MathML</a> is called a supersolution of (1.1) if it satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M85">View MathML</a>

Analogously, a function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M86">View MathML</a> is called a subsolution of (1.1) if it satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M87">View MathML</a>

The above inequalities are both in the sense of distribution. Here, a function T ≥ 0 means that for any positive test function ψ, we have ≥ 0. In the following we are ready to construct a smooth solution of (1.1) commencing with a subsolution and a supersolution in S1,2(Ω, loc) by the monotone iteration scheme. We first prove a maximum principle.

Lemma 4.1. Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M88">View MathML</a>satisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M89">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M23">View MathML</a>and λ(ξ) > 0. If u ≤ 0 on ∂Ω, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M90">View MathML</a>.

Proof. Suppose that the conclusion fails. Since u is continuous on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M91">View MathML</a>, there exists a point ξ0 ∈ Ω such that u(ξ0) > 0. Fix ε > 0 so small that u(ξ0) - ε > 0. Consequently, the function uε : = max{u - ε, 0} is non-negative and has compact support in Ω as u ≤ 0 on ∂Ω. By the distribution meaning of solutions, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M92">View MathML</a>

(4.1)

When uε > 0, it follows Xuε = Xu and Xu is not identically zero. In fact, if Xu ≡ 0, then u ≡ u(ξ0) > 0 in Ω which contradicts the assumption that u ≤ 0 on ∂Ω. Consequently the left hand side of (4.1) is positive, a contradiction. This completes the proof of the lemma. □

Theorem 4.2. Let be as in Lemma 3.10. Let f C(G × (a, b)) and φ C(∂Ω). Suppose that μ and ν are, respectively, a supersolution and a subsolution of (1.1) with μ, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M93">View MathML</a>, ν μ, and a < min ν < max μ < b. Then there exists a solution <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69">View MathML</a>of (1.1) satisfying ν u μ.

Proof. Take K > 0 such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M94">View MathML</a>

(4.2)

on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M95">View MathML</a>. Let v = Tu denote the unique solution in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M96">View MathML</a> of the Dirichlet problem (see Theorem 3.11)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M97">View MathML</a>

We claim that the nonlinear transformation T is monotone. To establish this we set u1 < u2 and notice that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M98">View MathML</a>

and Tu1 = Tu2 = φ on ∂Ω. Letting w = Tu1 - Tu2, we can obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M99">View MathML</a>

and w = 0 on ∂Ω. As f(ξ, u) + K2u is increasing in u by (4.2), it yields (L - K2) w ≥ 0. From

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M100">View MathML</a>

we get w S2,2(Ω, loc) by Lw L2(Ω) and Proposition 2.1-(i). It follows that w ≤ 0 in Ω by applying Lemma 4.1, therefore, Tu1 Tu2 and T is monotone. We now begin the iteration scheme.

Let u1 = . As

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M101">View MathML</a>

and u1 = φ on ∂Ω, we get by a trivial calculation that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M102">View MathML</a>

and u1 - μ ≤ 0 on ∂Ω. Arguing as in the previous gives u1 μ in Ω.

Define un+1 = Tun. The monotoneity of T yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M103">View MathML</a>

Analogously, starting from ν, we obtain a nondecreasing sequence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M104">View MathML</a>

where v1 = Tν, vn+1 = Tvn. Moreover, ν μ implies v1 = = u1 and, therefore, vn un for each n ∈ ℕ. Thus

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M105">View MathML</a>

(4.3)

so that the limit <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M106">View MathML</a> is well defined in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M91">View MathML</a>. Recall that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M107">View MathML</a>

The dominated convergence theorem shows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M108">View MathML</a>

in the distributional sense. According to Proposition 2.1-(i) and the fact that f(ξ, u) ∈ Lp(Ω) for 1 < p < +one has u S2,p(Ω, loc). Iterating the process, we get u Sk, p(Ω, loc) for k ≥ 0. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M109">View MathML</a>. The definition in Section 2 gives ψu Sk, p. Furthermore, we obtain u C(Ω) in view of Proposition 2.1-(ii). Combining this with (4.3) we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M69">View MathML</a> which is the desired solution. □

We assume henceforth that G is of Heisenberg type. Such group was introduced by Kaplan [14] and has been subsequently studied by several authors, see [4,11,13] and the references therein.

Let G be a Carnot group of step two whose Lie algebra <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M110">View MathML</a>. Consider the map J : V2 → End(V1) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M111">View MathML</a>

G is said of Heisenberg type if for every ξ2 V2, with |ξ2| = 1, the map J (ξ2): V1 → V1 is orthogonal.

In the case of the Heisenberg type groups, the gauge balls coincide with the level sets of the fundamental solution (that is a radial function in this class of groups, see [14]), and the balls BG(e, R) invade G as R tends to +since the vector fields on G satisfy the Hörmander rank condition. Thus, we get the following existence theorem in the whole space G by making use of Theorem 4.2 and the result in [4] that the gauge balls in H-type group satisfy the outer sphere condition.

Theorem 4.3. Let G be a group of Heisenberg type. Let u-(ξ), u+(ξ) S1,2(G, loc) ∩ C(G) be respectively a subsolution and a supersolution of the problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M112">View MathML</a>

(4.4)

where f C(G × (a, b)) and a < u-(ξ) ≤ u+(ξ) < b. Then there exists a solution u C(G) of (4.4) satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M113">View MathML</a>

in G.

Proof. Let u0 = u+, set BG(e, m) be the gauge ball of radius m centered at identity e. We construct um inductively in the following manner. Let vm be the solution of the Dirichlet problem

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M114">View MathML</a>

obtained by means of Theorem 4.2 using u- and um-1, respectively, as a subsolution and a supersolution.

Define

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M115">View MathML</a>

Obviously, u- um um-1. We need to prove that um is a supersolution of (4.4). To see this, take a positive test function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M116">View MathML</a>. From the divergence theorem, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M117">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M118">View MathML</a>

The above two identities give

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M119">View MathML</a>

(4.5)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M120">View MathML</a> denotes the outerward normal to ∂BG(e, m), and A is a fixed positive semi-definite matrix (see [4,13]). Therefore, we may restrict ourselves to the case in which <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M121">View MathML</a> represents the derivative of u+ - vm in an outward direction with respect to ∂BG(e, m). Moreover, since u+ - vm ≥ 0 in BG(e, m) and u+ - vm = 0 on ∂BG(e, m), it follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M122">View MathML</a>

(4.6)

Substitution in (4.5) gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M123">View MathML</a>

This implies that um is a supersolution, and we can restart the monotone iteration scheme on BG(e, m+1).

In this way we obtain iteratively a sequence of supersolutions {um} satisfying the following properties:

(i) {um} is nonincreasing, and u- um u+;

(ii) Every um satisfies Lum + f(ξ, um) = 0 in BG(e, m).

Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/136/mathml/M124">View MathML</a>. We observe that {um} is a sequence of solutions of (4.4) on any BG(e, k) for m k. It follows that u is a solution on BG(e, k). Arguing as in Theorem 4.2 we know u C(BG(e, k)). The arbitrariness of k implies u C(G). Therefore, it holds that u is the required solution of (4.4). □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

We would like to thank Pengcheng Niu for research assistance and the two anonymous referees for very constructive comments. Zixia Yuan thanks the Mathematical Tianyuan Youth Foundation of China (No. 11026082) for financial support.

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