In this article we consider Dirichlet problems of the type

L u + f ( ξ , u ) = 0 , in Ω , u = φ , on ∂ Ω ,

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 ∈ C^∞ then 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 234. 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 L^p, 1 < p ≤ ∞, in the group of Heisenberg type.

The Perron method (see 56) and the monotone iteration scheme (see 78) 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.

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

Following 1112, a Carnot group G of step r ≥ 1 is a simple connected nilpotent Lie group whose Lie algebra g ̃ admits a stratification. That is, there exist linear subspaces V₁, . . ., V_r of g̃ such that

g ̃ = ⊕ j = 1 r V j , [ V 1 , V i ] = V i + 1 for i = 1 , 2 , … , r - 1 and [ V 1 , V r ] = { 0 } .

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

δ λ ( ξ ) = δ λ ( x ( 1 ) , x ( 2 ) , … , x ( r ) ) = ( λ x ( 1 ) , λ 2 x ( 2 ) , … , λ r x ( r ) ) .

Here ξ = (x⁽¹⁾, x⁽²⁾, . . ., x^(r)), x ( i ) ∈ ℝ N i for i = 1, . . ., r and N₁ + · · · + N_r = N. We denote by Q = ∑ j = 1 r j N j the homogeneous dimension of G attached to the dilations {δ_λ}_{λ > 0}. Let m = N₁ and X = {X₁, . . ., X_m} be the dimension and a basis of V₁, respectively. Let Xu = {X₁u, . . ., X_mu} denote the horizontal gradient for a function u. The sub-Laplacian associated with X on G is given by

L = ∑ j = 1 m X j 2 .

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

u * v ( ξ ) = ∫ G u ( η ) v ( η - 1 ⋅ ξ ) d G ( η ) = ∫ G u ( ξ ⋅ η - 1 ) v ( η ) d G ( η ) ,

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 ρ ( ξ ) ∈ C 0 ∞ G \ { e } ∩ C ( G ) 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:

B G ( ξ , R ) = { η ∈ G : ρ ( ξ , η ) = ρ ( ξ - 1 ⋅ η ) < R } .

Let D ′ denote the space of distributions on G. The non-isotropic Sobolev space S^{k, p} is defined by

S k , p = { f ∈ D ′ : D α f ∈ L p ( G ) , | α | ≤ k } ,

where α = (α₁, . . ., α_l) is a multi-index, D α = D α 1 D α 2 ⋯ D α l , and D α j ∈ { X 1 , … , X m } . In the space S^{k, p}, we shall adopt the norm

f S k , p = sup α ≤ k D α f L p .

For a domain Ω in G, we define S^{k, p}(Ω, loc) as the space of distributions f such that for every ψ ( ξ ) ∈ C 0 ∞ ( Ω ) we have fψ ∈ S^{k, p}. Let 0 < β < ∞, we employ the following non-isotropic Lipschitz spaces:

(i) for 0 < β < 1,

Γ β : = f ∈ L ∞ ∩ C 0 : sup ξ , η | f ( η ⋅ ξ ) - f ( η ) | ( ρ ( ξ , e ) ) β < ∞ ,

(ii) for β = 1,

Γ 1 : = f ∈ L ∞ ∩ C 0 : sup ξ , η | f ( η ⋅ ξ ) + f ( η ⋅ ξ - 1 ) - 2 f ( η ) | ( ρ ( ξ , e ) ) β < ∞ ,

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

Γ β : = f ∈ L ∞ ∩ C 0 : D α f ∈ Γ β ′ , | α | ≤ k .

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 f , g ∈ D ′ ( Ω ) satisfy Lf = g in Ω. If g ∈ S^{k, p}(Ω, loc) (1 < p < ∞, k ≥ 0) then f ∈ S^k+2,p(Ω, loc).

(ii) Suppose 1 < p < ∞ and β = k - Q p > 0 , then S^{k, p} ⊂ Γ^β.

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

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

L u - λ ( ξ ) u = f , in Ω , u = φ , on ∂ Ω ,

where λ ( ξ ) ∈ C ( Ω ̄ ) 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 B_G(η, r) lying in G\Ω such that

∂ B G ( η , r ) ∩ ∂ Ω = { ξ 0 } .

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 ∈ C²(Ω) satisfies

L u - λ ( ξ ) u ≥ 0 i n Ω ,

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 F = { ω : ω ̄ ⊂ Ω } , which is a base for the topology of Ω for which the Dirichlet problem

L u - λ ( ξ ) u = f , i n ω , u = φ , o n ∂ ω

has a unique distributional solution u ∈ C ( ω ̄ ) for any ω ∈ F , f ∈ C ( ω ̄ ) 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 ∈ C ∞ ( Ω ̄ ) . A function u ∈ C ( Ω ̄ ) is called a subsolution of (3.1) if it fits the following properties:

(i) u ≤ φ on ∂Ω;

(ii) for every ω∈F and for every h ∈ C 2 ( ω ) ∩ C ( ω ̄ ) 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 = sup ξ ∈ Ω ( u - v ) ( ξ ) ≥ 0 . 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 ω∈F such that ξ₀ ∈ ω but u - v is not identically equal to M on ∂ω. Letting ū   and 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 ū ≥ u and v ̄ ≤ v in ω. One sees that

M = sup ξ ∈ Ω ( u - v ) ( ξ ) ≥ sup ξ ∈ ∂ ω ( ū - v ̄ ) ( ξ ) ≥ ( ū - v ̄ ) ( ξ 0 ) ≥ ( u - v ) ( ξ 0 ) = M ,

and hence all the equalities above hold. By Lemma 3.2 it follows that ū - v ̄ ≡ 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∈C(Ω̄) be a subsolution of (3.1) and ω∈F. Denote by ū  the solution of the Dirichlet problem (see Lemma 3.3)

L ū - λ ( ξ ) ū = f ( ξ ) , in ω , ū = u , on ∂ ω ,

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

U ( ξ ) : = ū ( ξ ) , ξ ∈ ω , u ( ξ ) , ξ ∈ Ω \ ω .

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 ω ′ ∈ F and h ∈ C 2 ( ω ′ ) ∩ C ( ω ′ ¯ ) 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 ū ≤ h on ∂(ω' ∩ ω). As L ( ū - h ) - λ ( ξ ) ( ū - h ) = 0 in ω' ∩ ω and ū - h ≤ 0 on ∂(ω' ∩ ω), it yields by Lemma 3.2 that ū≤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 u₁, u₂, . . ., u_l be subsolutions of (3.1). Then the function

v = max { u 1 , u 2 , … , u l }

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 u ( ξ ) : = sup v ∈ S v ( ξ ) 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 {v_n}_n∈ℕ such that v_n(η) → u(η). By replacing v_n with max {v₁, . . ., v_n}, we may assume that v₁ ≤ v₂ ≤ · · · ≤ v_n ≤ · · ·. Let ω∈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 ( ξ ) : = lim n → ∞ V n ( ξ ) . 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 ū ∈ S such that V ( ζ ) < ū ( ζ ) . Define the increasing sequence w n = max { ū , V n } and then the corresponding liftings W_n. Set W ( ξ ) : = lim n → ∞ W n ( ξ ) . 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 ( ζ ) < ū ( ζ ) ≤ W ( ζ ) . 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 ( ξ ) ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) 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 Ω ̄ \ { ζ } , 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

L w = - 1 , i n Ω , w ( ξ ) = ρ ( ξ , ζ ) , o n ∂ Ω

has a unique solution w ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) fulfilling w(ξ) > 0 on Ω̄\{ζ} and w(ζ) = 0.

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

ũ : = - Γ * χ Ω ,

where χ_Ω denotes the indicator function. Since Γ ( ξ ) ∈ L l o c p for 1 ≤ p < Q Q - 2 , it yields ũ ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) .

According to Corollary 10 in 3, the problem

L v = 0 , in Ω , v ( ξ ) = ρ ( ξ , ζ ) - ũ ( ξ ) , on ∂ Ω

has a unique solution v ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) . Since L ũ = - χ Ω (see Corollary 2.8 in 1), it follows that w : = v + ũ is the desired solution of (3.4). □

Theorem 3.11. Let Ω be as in Lemma 3.10. Suppose φ ∈ C(∂Ω) and f ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) . Then the Dirichlet problem (3.1) possesses a unique solution u ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) .

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

| φ ( ξ ) - φ ( ζ ) | < ε .

Let w(ξ) be the barrier function related to L at ζ constructed in Lemma 3.10. Set M = sup ξ ∈ ∂ Ω | φ ( ξ ) | and choose k₁ > 0 such that k₁w(ξ) ≥ 2M if ρ(ξ, ζ) ≥ δ. Set k 2 = [ | φ ( ζ ) | + ε ] max ξ ∈ Ω λ ( ξ ) + sup ξ ∈ Ω | f ( ξ ) | , 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,

L w 1 - λ ( ξ ) w 1 = - k - λ ( ξ ) φ ( ζ ) - λ ( ξ ) ε - k λ ( ξ ) w ( ξ ) ≤ f in Ω .

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

w 2 ( ξ ) ≤ u ( ξ ) ≤ w 1 ( ξ )

and then

| u ( ξ ) - φ ( ζ ) | ≤ ε + k w ( ξ ) .

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

Remark 3.12. Let f∈C∞(Ω)∩C(Ω̄) and u be the solution of

L u - λ ( ξ ) u = f , in Ω , u = 0 , on ∂ Ω .

Elementary calculations show that - 1 min ξ ∈ Ω λ ( ξ ) | | f | | L ∞ ( Ω ) and 1 min ξ ∈ Ω λ ( ξ ) | | f | | L ∞ ( Ω ) are a subsolution and a supersolution of (3.5) respectively. Thus, | | u | | L ∞ ( Ω ) ≤ 1 min ξ ∈ Ω λ ( ξ ) | | f | | L ∞ ( Ω ) . It provides a L^∞ estimate for the solution of (3.5).

Theorem 3.13. Set φ ∈ C(∂Ω) and f ∈ C ( Ω ̄ ) . Then there exists a unique solution u∈C(Ω̄) to (3.1) in the sense of distribution.

Proof. Take a sequence f n ( ξ ) ∈ C ∞ ( Ω ) ∩ C ( Ω ̄ ) , 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

L v - λ ( ξ ) v = f n ( ξ ) , in Ω , u = φ , on ∂ Ω .

We obtain, in view of Remark 3.12,

| | u n - u m | | L ∞ ( Ω ) ≤ 1 min ξ ∈ Ω λ ( ξ ) | | f n - f m | | L ∞ ( Ω ) .

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

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 μ ∈ C ( Ω ̄ ) is called a supersolution of (1.1) if it satisfies

L μ + f ( ξ , μ ) ≤ 0 , in Ω , μ ( ξ ) ≥ φ ( ξ ) , on ∂ Ω .

Analogously, a function ν ∈ C ( Ω ̄ ) is called a subsolution of (1.1) if it satisfies

L ν + f ( ξ , ν ) ≥ 0 , in Ω , ν ( ξ ) ≤ φ ( ξ ) , on ∂ Ω .

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 Tψ ≥ 0. In the following we are ready to construct a smooth solution of (1.1) commencing with a subsolution and a supersolution in S^1,2(Ω, loc) by the monotone iteration scheme. We first prove a maximum principle.

Lemma 4.1. Assume that u ∈ S 1 , 2 ( Ω ) ∩ C ( Ω ̄ ) satisfies

L u - λ ( ξ ) u ≥ 0 ,

where λ(ξ)∈C(Ω̄) and λ(ξ) > 0. If u ≤ 0 on ∂Ω, then sup ξ ∈ Ω u ( ξ ) ≤ 0 .

Proof. Suppose that the conclusion fails. Since u is continuous on Ω ̄ , there exists a point ξ₀ ∈ Ω such that u(ξ₀) > 0. Fix ε > 0 so small that u(ξ₀) - ε > 0. Consequently, the function u_ε : = 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

∫ Ω X u ⋅ X u ε d G = ∫ Ω - u ε L u d G ≤ ∫ Ω - λ ( ξ ) u u ε d G ≤ 0 .

When u_ε > 0, it follows Xu_ε = 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 Ω 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, ν ∈ S 1 , 2 ( Ω , l o c ) ∩ C ( Ω ̄ ) , ν ≤ μ, and a < min ν < max μ < b. Then there exists a solution u∈C∞(Ω)∩C(Ω̄) of (1.1) satisfying ν ≤ u ≤ μ.

Proof. Take K > 0 such that

∂ f ∂ u + K 2 > 0

on Ω ̄ × [ min ν , max μ ] . Let v = Tu denote the unique solution in C ( Ω ̄ ) of the Dirichlet problem (see Theorem 3.11)

( L - K 2 ) v = - [ f ( ξ , u ) + K 2 u ] , in Ω , v ( ξ ) = φ ( ξ ) , on ∂ Ω .

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

( L - K 2 ) T u 1 = - [ f ( ξ , u 1 ) + K 2 u 1 ] , ( L - K 2 ) T u 2 = - [ f ( ξ , u 2 ) + K 2 u 2 ] ,

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

( L - K 2 ) w = - [ f ( ξ , u 1 ) - f ( ξ , u 2 ) + K 2 ( u 1 - u 2 ) ]

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

L w = K 2 w - [ f ( ξ , u 1 ) - f ( ξ , u 2 ) + K 2 ( u 1 - u 2 ) ] ,

we get w S^2,2(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 = Tμ. As

( L - K 2 ) u 1 = - [ f ( ξ , μ ) + K 2 μ ] ,

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

( L - K 2 ) ( u 1 - μ ) = - [ L μ + f ( ξ , μ ) ] ≥ 0

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

Define u_n+1 = Tu_n. The monotoneity of T yields

μ ≥ u 1 ≥ u 2 ≥ ⋯ .

Analogously, starting from ν, we obtain a nondecreasing sequence

ν ≤ v 1 ≤ v 2 ≤ ⋯ ,

where v₁ = Tν, v_n+1 = Tv_n. Moreover, ν ≤ μ implies v₁ = Tν ≤ Tμ = u₁ and, therefore, v_n ≤ u_n for each n ∈ ℕ. Thus

ν ≤ v 1 ≤ v 2 ≤ ⋯ ≤ v n ≤ u n ≤ ⋯ ≤ u 2 ≤ u 1 ≤ μ ,

so that the limit u = lim n → ∞ u n is well defined in Ω̄. Recall that

( L - K 2 ) u n + 1 = - [ f ( ξ , u n ) + K 2 u n ] .

The dominated convergence theorem shows that

L u + f ( ξ , u ) = 0

in the distributional sense. According to Proposition 2.1-(i) and the fact that f(ξ, u) ∈ L^p(Ω) for 1 < p < +∞ one has u ∈ S^2,p(Ω, loc). Iterating the process, we get u ∈ S^{k, p}(Ω, loc) for k ≥ 0. Let ψ ∈ C 0 ∞ ( Ω ) . The definition in Section 2 gives ψu ∈ S^{k, p}. Furthermore, we obtain u ∈ C^∞(Ω) in view of Proposition 2.1-(ii). Combining this with (4.3) we have u∈C∞(Ω)∩C(Ω̄) 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 41113 and the references therein.

Let G be a Carnot group of step two whose Lie algebra g ̃ = V 1 ⊕ V 2 . Consider the map J : V₂ → End(V₁) defined by

⟨ J ( ξ 2 ) ξ 1 ′ , ξ 1 ″ ⟩ = ⟨ ξ 2 , [ ξ 1 ′ , ξ 1 ″ ] ⟩ , for ξ 1 ′ , ξ 1 ″ ∈ V 1 and ξ 2 ∈ V 2 .

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 B_G(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) S^1,2(G, loc) ∩ C(G) be respectively a subsolution and a supersolution of the problem

L u + f ( ξ , u ) = 0 ,

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

u - ≤ u ≤ u +

in G.

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

L v + f ( ξ , v ) = 0 , in B G ( e , m ) , v ( ξ ) = u + ( ξ ) , on ∂ B G ( e , m )

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

Define

u m ( ξ ) = v m ( ξ ) , ξ ∈ B G ( e , m ) , u + ( ξ ) , ξ ∉ B G ( e , m ) .

Obviously, u_- ≤ u_m ≤ u_m-1. We need to prove that u_m is a supersolution of (4.4). To see this, take a positive test function ψ ( ξ ) ∈ C 0 ∞ ( G ) . From the divergence theorem, we obtain

∫ B G ( e , m ) v m L ψ d G = ∫ B G ( e , m ) ψ L v m d G + ∫ ∂ B G ( e , m ) v m ⟨ A ∇ ψ , n → ⟩ d S - ∫ ∂ B G ( e , m ) ψ ⟨ A ∇ v m , n → ⟩ d S

and

∫ G \ B G ( e , m ) u + L ψ d G = ∫ G \ B G ( e , m ) ψ L u + d G + ∫ ∂ B G ( e , m ) ψ ⟨ A ∇ u + , n → ⟩ d S - ∫ ∂ B G ( e , m ) u + ⟨ A ∇ ψ , n → ⟩ d S .

The above two identities give

∫ G u m L ψ d G = ∫ B G ( e , m ) ψ L v m d G + ∫ G \ B G ( e , m ) ψ L u + d G + ∫ ∂ B G ( e , m ) ψ ⟨ A ∇ ( u + - v m ) , n → ⟩ d S ,

where n → denotes the outerward normal to ∂B_G(e, m), and A is a fixed positive semi-definite matrix (see 413). Therefore, we may restrict ourselves to the case in which ⟨ A ∇ ( u + - v m ) , n → ⟩ represents the derivative of u₊ - v_m in an outward direction with respect to ∂B_G(e, m). Moreover, since u₊ - v_m ≥ 0 in B_G(e, m) and u₊ - v_m = 0 on ∂B_G(e, m), it follows

ψ ⟨ A ∇ ( u + - v m ) , n → ⟩ ≤ 0 for ξ ∈ ∂ B G ( e , m ) .

Substitution in (4.5) gives

∫ G u m L ψ d G ≤ - ∫ B G ( e , m ) ψ f ( ξ , v m ) d G - ∫ G \ B G ( e , m ) ψ f ( ξ , u + ) d G = - ∫ G ψ f ( ξ , u m ) d G .

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

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

(i) {u_m} is nonincreasing, and u_- ≤ u_m ≤ u₊;

(ii) Every u_m satisfies Lu_m + f(ξ, u_m) = 0 in B_G(e, m).

Set u ( ξ ) = lim m → ∞ u m ( ξ ) . We observe that {u_m} is a sequence of solutions of (4.4) on any B_G(e, k) for m ≥ k. It follows that u is a solution on B_G(e, k). Arguing as in Theorem 4.2 we know u ∈ C^∞ (B_G(e, k)). The arbitrariness of k implies u ∈ C^∞(G). Therefore, it holds that u is the required solution of (4.4). □

The authors declare that they have no competing interests.

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