Quasilinear equations with source terms on Carnot groups

In this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of Lane--Emden type with measure data on a Carnot group $\mathbb G$ of arbitrary step. The quasilinear part involves operators of the $p$-Laplacian type $\Delta_{\mathbb G,\,p}\,$, $1<p<\infty$. These results are based on new a priori estimates of solutions in terms of nonlinear potentials of Th. Wolff's type. As a consequence, we characterize completely removable singularities, and prove a Liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub-Laplacian ($p=2$) on the Heisenberg group.


Introduction
In this paper we study the solvability problem and pointwise estimates of solutions for a class of quasilinear Lane-Emden type equations with measure data on Carnot groups of arbitrary step. A complete characterization of removable singularities for the corresponding homogeneous equations as well as a Liouville type theorem for supersolutions will also be obtained as a consequence.
The basic setting of our study is a given Carnot group G of step r ≥ 1, i.e., a connected and simply connected stratified nilpotent Lie group whose Lie algebra G admits a stratification G = V 1 ⊕ V 2 ⊕ · · · ⊕ V r and is generated via commutations by its first (horizontal) layer V 1 (see Sect. 2). Given a basic {X j } m j=1 of V 1 , the associated p-Laplacian operator ∆ G, p , 1 < p < ∞, is defined by where Xu = X 1 uX 1 + X 2 uX 2 + · · · + X m uX m , is the horizontal gradient of u with length |Xu| = m i=1 |X i u| 2 1/2 . We study the following Lane-Emden type equation on a bounded open set Ω ⊂ G: −∆ G, p u = u q + ω in Ω, u = 0 on ∂Ω, (1.1) where q > p − 1 > 0, and ω is a given nonnegative finite measure on Ω. Our objective is to obtain necessary and sufficient conditions on the measure ω for the existence of solutions to (1.1), and to give a complete characterization of removable singularities for the corresponding homogeneous equation: (1.2) − ∆ G, p u = u q in Ω.
Equations similar to (1.1) in the entire group G are also considered with applications to Liouville type theorems for the differential inequality Such problems have been studied in depth in our previous work [PV1], [PV2] in the standard Euclidean setting; see also earlier work in [BP], [AP], [BV1], and [BV2]. However, in the setting of Carnot groups, the failure of the Besicovitch covering lemma (see [SW], [KR]) and the lack of a perfect dyadic grid of cubes cause major difficulties. We observe that even in the setting of the Heisenberg group, the simplest model of a non-commutative Carnot group, Liouville type theorems for the differential inequality (1.3) are known only in the sub-Laplacian case, i.e., p = 2 (see [GL], [BCC], [PVe]).
A substantial part of our study of (1.1) is devoted to integral inequalities for both linear and nonlinear potential operators and their discrete analogues over "approximate" dyadic grids of cubes constructed in [SW] and [Chr] in the general setting of homogeneous spaces.
For each α > 0, the Bessel potential of a locally integrable function f in this setting is defined by where G α is the Bessel kernel of order α on G given by (1.4) G α (x) = 1 Γ(α/2)ˆ∞ 0 t α/2−1 e −t h(x, t)dt.
In (1.4) h(x, t) is the heat kernel associated with the sub-Laplacian ∆ G = ∆ G, 2 whose basic properties can be found in [Fol], [VSC]. We also write for each locally µ-integrable function f . When dealing with solutions on the entire group G and Liouville type theorems we need to use another linear potential, the Riesz potential. For each 0 < α < M and f ∈ L 1 loc (G), it is defined by where d cc is the Carnot-Carathéodory distance on G, and M is the homogeneous dimension of G (see Sect. 2). Associated with the kernel G α is the Bessel capacity C α, s (·), s > 1, defined by (see [AH], Sec. 2.6, in the Euclidean case) . These capacities will play an essential role in our characterizations of the existence of solutions and removable singularities, as well as Liouville type theorems for the Lane-Emden type equation. We will also need the following dual definition of these capacities (see [Lu,Theorem 2.10]; [AH,Theorem 2.2.7] in the Euclidean case): where M + (E) denotes the set of all nonnegative measures supported on E.
The nonlinear potential we use below is the (truncated) Wolff's potential W R α, p originally introduced in [HW]. In our setting, for α > 0, p > 1, and 0 < R ≤ ∞, it is defined for each nonnegative measure µ on G by where B t (x) is the Carnot-Carathéodory ball centered at x of radius t (see Sect. 2). For our purpose we introduce the following notion of solutions for p-Laplace equations with general measure as data. This will serve as an efficient substitution for the notion of renormalized solutions introduced in [DMOP] in the Euclidean setting.
Definition 1.1. For a nonnegative finite measure µ on Ω, we say that u is a solution to −∆ G, p u = µ in Ω, u = 0 on ∂Ω, (1.7) in the potential theoretic sense if u is p-superharmonic in Ω, min{u, k} ∈ S 1, p 0 (Ω) for every k > 0, u satisfies a pointwise bound and for every ϕ ∈ C ∞ 0 (Ω) one haŝ From this definition we see right away that potential theoretic solutions to (1.7) are also distributional solutions. However, the converse is not necessarily true as easily seen by a simple example (see [Kil]). The existence of potential theoretic solutions to (1.7) will be obtained in Corollary 4.2, whereas their uniqueness is unknown even in the Euclidean setting.
In Definition 1.1 the notation S 1, p 0 (Ω) stands for the completion of C ∞ 0 (Ω) under the norm of the horizontal Sobolev space S 1, p (Ω) (see Sect. 3), and in (1.8), A is a universal constant independent of x, u, µ, and Ω. For the notion of p-superharmonic functions on Carnot groups see Sect. 3. We are now ready to state the first result of the paper.
Conversely, there exists a constant C 0 = C 0 (M, p, q) > 0 such that if any one of the statements (i)-(v) holds with C ≤ C 0 then equation (1.9) has a nonnegative potential theoretic solution u ∈ L q (Ω) for any nonnegative finite measure ω. Moreover, u satisfies the following pointwise estimate u ≤ κ W 2R 1, p ω. Our second result is about removable singularities of solutions to homogeneous equations, which is in fact a consequence of Theorem 1.2. Theorem 1.3. Let q > p − 1 > 0 and let E be a compact subset of Ω. Then any solution u to the problem (1.14) is also a solution to a similar problem with Ω in place of Ω \ E if and only if The proof of Theorems 1.2 and 1.3 will be given at the end of Sect. 4. In case the bounded domain Ω in Theorem 1.2 is replaced by the whole group G, then Riesz potentials and the corresponding Riesz capacity must be used, and we have the following result.
Theorem 1.4. Let 1 < p < M , q > p − 1 and let ω be a nonnegative locally finite measure on G. If the equation has a nonnegative p-superharmonic distributional solution u ∈ L q loc (G), then there exists a constant C > 0 such that statements (i)-(vi) below hold true.
(i) For every compact set E ⊂ G, holds for all g ∈ L q p−1 (dω), g ≥ 0.
(v) The inequalityˆB holds for all Carnot-Carathéodory balls B ⊂ G.
. Conversely, there exists a constant C 0 = C 0 (M, p, q) > 0 such that if any one of the statements (ii)-(vi) holds with C ≤ C 0 then equation (1.15) has a nonnegative p-superharmonic solution u ∈ L q loc (G). Moreover, u satisfies the following pointwise two-sided estimate Theorem 1.4 yields the following Liouville type theorem. We observe that for p = 2 this Liouville type theorem is new even in the Heisenberg group. For p = 2, as mentioned earlier, such a result was obtained in [GL], [BCC], and [PVe] in the setting of the Heisenberg group. However, the approach of using test functions and integration by parts in these papers does not seem to work in the general setting of Carnot groups of arbitrary step.
M −p , then the inequality −∆ G, p u ≥ u q admits no nontrivial nonnegative p-superharmonic distributional solutions in G.
The proofs of Theorem 1.4 and Corollary 1.5 will be given in Sect. 5.

Preliminaries on Carnot groups
Let G be a Lie group, i.e., a differentiable manifold endowed with a group structure such that the map G × G → G defined by (x, y) → xy −1 is C ∞ . Here y −1 is the inverse of y and xy −1 denotes the group multiplication of x by y −1 . We will denote by respectively, the left-and right-translations on G. A vector field X on G is called left-invariant if for each x 0 ∈ G, Under the Lie bracket operation on vector fields, the set of left-invariant vector fields on G forms a Lie algebra called the Lie algebra of G and is denoted by G. Note that we can identify G with the tangent space G e to G at the identity e ∈ G via the isomorphism α : G → G e defined by α(X) = X(e) and thus dim G = dim G = N , the topological dimension of G.
A Carnot group G of step r is a connected and simply connected Lie group whose Lie algebra G admits a nilpotent stratification of step r, i.e., Let {X j } m j=1 be a basis for the first layer V 1 (also called the horizontal layer) of G. Then for 2 ≤ i ≤ r, we can choose a basis {X ij }, 1 ≤ j ≤ dim(V i ), for V i consisting of commutators of length i. In particular, X 1j = X j for j = 1, . . . , m, and m = dim(V 1 ). We then define an inner product < · , · > on G by declaring the X ij 's to be orthonormal. Since G is connected and simply connected, the exponential map exp is a global diffeomorphism from G onto G (see [VSC], [Va]). Thus for each x ∈ G, there is a uniquê Thus the maps φ ij : form a system of global coordinates on G which are called the exponential coordinates. Henceforth we will always use these coordinates and simply write Such an identification of G with its Lie algebra is justified by the Baker-Cambell-Hausdorff formula (see, e.g., [Va]) where H(X, Y ) = X + Y + 1 2 [X, Y ] + · · · with the dots indicating a finite linear combination of terms containing commutators of order two and higher. If we define a group law * on G by then the group G can be identified with (G, * ) via the exponential coordinates. Note that from the Baker-Cambell-Hausdorff formula we have where P ij (x 0 , x) depends only on the coordinates φ kl (x 0 ) and φ kl (x) with k < i. Thus the determinant of dL x 0 is equal to 1, and the same properties hold for the right translation R x 0 and its differential dR x 0 as well. It follows that Lebesgue measure on G is lifted via the exponential mapping exp to a bi-invariant Haar measure on G, which we will denote by dx.
For a given function f : G → R, the action of X ∈ G on f is specified by the equation For t > 0, we define the dilation δ t : G → G by which obviously satisfies |δ t (x)| = t|x| and |x −1 | = |x|. This homogeneous norm generates a quasi-metric ρ(x, y) = |x −1 y| equivalent to the Carnot-Carathéodory metric d cc on G (see [NSW], [VSC]). Here where the infimum is taken over all curves γ : Such a curve is called a horizontal curve connecting x, y ∈ G. By Chow-Rashevsky's accessibility theorem (see [Cho], [Ra]), any two points x, y ∈ G can be joined by a horizontal curve of finite length and hence d cc is a left-invariant metric on G. We will denote by the Carnot-Carathéodory metric ball centered at x with radius R. Note that there is c = c(G) such that where for a Borel set E ⊂ G we write |E| for´E dx. Moreover, by homogeneity and left-invariance we have and for x, x ′ , y ∈ G,

p-superharmonic functions on Carnot groups
Let p > 1 and let Ω be an open set in G. Recall from the previous section that X = (X 1 , X 2 , . . . , X m ) = (X 11 , X 12 , . . . , X 1m ) is an orthonormal basic for the first layer V 1 of G. The horizontal Sobolev space S 1, p (Ω) associated with the system X is defined by where X i u is understood in the sense of distributions, i.e., The corresponding local Sobolev space S 1, p loc (Ω) is defined similarly, with L p loc (Ω) in place of L p (Ω). We will denote by S 1, p 0 (Ω) the completion of Recall that for a smooth function u on G, the p-Laplacian of u is defined by It is known that every weak solution to (3.1) has a continuous representative (see [TW], [HKM]), and such continuous solutions are called p-harmonic functions on Ω. On the other hand, if u ∈ S 1, p loc (Ω) and The following fundamental connection between supersolutions to (3.1) and p-superharmonic functions can be found in [TW].
Then u is p-superharmonic and u = u a.e.
From this proposition it follows that we may assume all supersolutions to be lower semicontinuous. Therefore a function u is a supersolution to (3.1) if and only if u is p-superharmonic and belongs to S 1, p loc (Ω). Note that a p-superharmonic function u does not necessarily belong to S 1, p loc (Ω), but its truncation min{u, k} does for every integer k. Using this we set defined a.e. If either u ∈ L ∞ (Ω) or u ∈ S 1, 1 loc (Ω), then Xu coincides with the regular distributional horizontal gradient of u. In general we have the following gradient estimate [TW] (see also [HKM]).
is a nonnegative distribution in Ω for a p-superharmonic function u. It follows that there is a positive (not necessarily finite) Radon measure denoted by µ[u] such that in Ω.
The close relation between p-superharmonic functions and measures generated by them is established in the weak continuity theorem due to Trudinger and Wang [TW]. for all ϕ ∈ C ∞ 0 (Ω). The following pointwise estimates by means of Wolff's potentials were also proved in [TW] which extend earlier results due to Kilpeläinen and Malý [KM2] to the subelliptic setting. They will play an essential role in this paper.
Theorem 3.4 ( [TW]). Suppose u ≥ 0 is a p-superharmonic function in B 3r (x). If µ = −∆ G, p u, then where C 1 , C 2 and C 3 are positive constants depending only on M and p. Consequently, if −∆ G, p u = µ on G and inf G u = 0 then

Lane-Emden type equations and related inequalities
In this section we fix a standard mollifier ζ on G, i.e., a function ζ ∈ C ∞ 0 (G) which is radially decreasing and is supported in {x ∈ G : |x| ≤ 1} such that´ζdx = 1. Also, for n ≥ 1 we denote by ζ n the function defined by ζ n (x) = 1 n ζ( x n ). The following theorem gives an existence result and global pointwise estimates for a quasilinear equation with measure data.
Theorem 4.1. Suppose that Ω is bounded and µ is a nonnegative finite measure on Ω. Let u n be the unique solution in S 1, p 0 (Ω) of (4.1) − ∆ G, p u n = ζ n * µ in Ω.
Then there is a subsequence {u n k } of {u n } and a p-superharmonic function u on Ω such that u = lim k→∞ u n k a.e.
Moreover, u solves the equation in the sense of Definition 1.1.
Consequently, by Sobolev's embedding theorem we obtain min{u n , k} This gives Now arguing as in [KM1] we can find subsequences {u n k }, {v n k } and psuperharmonic functions u, v on Ω such that u n k → u, v n k → v a.e. Hence from (4.5) and Theorem 3.3 we see that u is a distributional solution of (4.2). Similarly, v also solves (4.2) in the distributional sense with B in place of Ω, and (4.5), (4.6) hold for v n with B in place of Ω as well. In particular, this implies Thus in view of (4.4) and Theorem 3.4 we get where x ∈ Ω and d(x) = dist(x, ∂B). Note that we have used the fact that d(x) ≥ R in the last inequality. Finally, from this and (4.7) we obtain the pointwise estimate for all x ∈ Ω. Thus u solves (4.2) in the potential theoretic sense and the proof is complete.
Corollary 4.2. For any nonnegative finite measure µ on Ω, there exists a potential theoretic solution to equation (1.7).
We now construct a solution to a nonlinear equation with a power source term under a certain iterated Wolff's potential condition. This condition turns out to be sharp as we will see later.
Theorem 4.3. Let ω be a nonnegative finite measure on Ω. Let p > 1 and q > p − 1. Suppose that R = diam(Ω), and and A is the constant in Definition 1.1. Then there is a solution u ∈ L q (Ω) to the equation −∆ G, p u = u q + ω in Ω, u = 0 on ∂Ω. n we see from (4.12) and (4.13) that u (1) ≤ u (2) a.e. and hence everywhere since they are p-superharmonic. Thus by induction we can find an increasing sequence {u (k) } such that u (1) satisfies (4.11) and for k ≥ 2, in the sense of Definition 1.1. Note that we have In view of these estimates and the condition (4.8) we get where c 1 = A and c 2 = A max{1, 2 p ′ −2 }(c q(p ′ −1) 1 C + 1). By induction we can find a sequence {c k } k≥1 of positive numbers such that for all k ≥ 2. It is then easy to see that c k ≤ A max{1, 2 p ′ −2 }q q−p+1 for all k ≥ 1 as long as C satisfies (4.9). Thus Therefore, {u (k) } converges pointwise increasingly to a nonnegative function u for which u ≤ κ W 2R 1, p ω. Finally, in view of (4.14) and Theorem 3.3 we see that u solves (4.10) in the sense of Definition 1.1. This completes the proof of the theorem.
In the general context of homogeneous spaces, it was proved in [SW] and [Chr] that for λ = 8, and for any (large negative) integer m, there are points {x k j } ⊂ G and a family of sets D m = {E k j }, k = m, m+1, . . . and j = 1, 2, . . .
We shall say that the family D = ∞ m=−∞ D m is a dyadic cube decomposition of G, and call sets in D dyadic cubes and denote them by Q. Note that the cubes in D m 1 may have no relation to those in D m 2 if m 1 and m 2 are different. If Q = E k j ∈ D m for some m, we say Q is centered at x k j and define the side length of Q to be ℓ(Q) = λ k . We also denote by Q * the containing ball B λ k+1 (x k j ) of Q and by Q * * the ball B 2λ k+2 (x k j ).
This implies that the ball Q * * = B 2λ k+2 (x k j 1 ) is contained in the union of at most d dyadic cubes of side length λ k for some constant d = d(G).
For an integer m, let Λ = {λ Q } Q∈Dm , λ Q ≥ 0, and let σ be a positive locally finite Borel measure on G such that λ Q = 0 whenever σ(Q) = 0. We will follow the convention that 0 · ∞ = 0. For 1 < s < +∞, we define The proof of following proposition will be omitted as it is similar to the one given in [COV] in the case G = R N and D m is the set of all standard dyadic cubes in R N .
We next consider the following quantities. For each integer m, a dyadic cube P ∈ D m , and a nonnegative Borel measure µ on G we define Here α > 0, p > 1, q > p − 1, and the sum is taken over all dyadic cubes Q ∈ D m such that Q ⊂ P .
Proposition 4.6. There exist constants C i > 0, i = 1, 2, 3, independent of m, P , and µ such that if Q ⊂ P and λ Q = 0 otherwise. Applying Proposition 4.5 with dσ = χ P dx and s = q p−1 > 1 we obtain Furthermore, since q p−1 > 1, where we have used Proposition 4.5 in the second inequality. We next observe that for p ≤ 2, B m 2 (P, µ) ≤ B m 3 (P, µ). Thus it remains to show that for p > 2, B m 2 (P, µ) ≤ C B m 3 (P, µ). Since q > p − 1 > 1, by Proposition 4.5 we have On the other hand, by Hölder's inequality the sum in the above square brackets can be estimated by Hence, combining the preceding inequalities, we obtain This completes the proof of the proposition.
Remark 4.7. From Remark 4.4 we see that for any β > 0. Thus the chain of inequalities in (4.15) still holds if µ(Q) is replaced by µ(Q * * ) in the definition of B m i (P, µ), i = 1, 2, 3.
Lemma 4.8. Let α > 0 and p > 1. Then for any integer m, and (4.17)ˆr In (4.17) [log λ r] stands for the integral part of the real number log λ r.
Proof. To prove (4.16) we may assume that λ m ≤ λ −3 r since dyadic cubes in D m have side length not smaller than λ m . Then [log λ r]− 3 ≥ m. Observe that where the last inequality follows from the fact that for k ≥ 0 and x ∈ E Thus we obtain (4.16). Similarly, to prove (4.17) we may assume that m < 0 and we havê Here E −k−1+[log λ r] j ∈ D m+[log λ r] , and the last inequality follows since for This gives (4.17) and completes the proof of the lemma.
The result obtained in the following theorem may be considered as an analogue of Wolff's inequality (see [HW], [PV1]) which is crucial in our approach to quasilinear Lane-Emden type equations later on.
Theorem 4.9. Let α > 0, p > 1 and q > p − 1. Then for any 0 < r < ∞ and any nonnegative Borel measure µ on G, where the constants of equivalence are independent of r and µ.
Proof. Let k ∈ Z be such that r λ < λ k ≤ r. For any interger m ≤ 0, by Lemma 4.8 we havê Thus by Proposition 4.6 and Remark 4.7 we obtain where the last inequality follows from Remark 4.4. This giveŝ Analogously, from (4.16) in Lemma 4.8 we obtain To prove (4.18), we choose an interger k so that λ k+1 < r 4 ≤ λ k+2 and as in Remark 4.4, it can be seen that for x ∈ E k j 1 ⊂ D k for some j 1 ≥ 1 the ball B λ 3 r (x) is contained in the union of at most d cubes in {E k j } j≥1 ⊂ D k for some constant d = d(G). That is,

Thus we obtain
which gives (4.18) and completes the proof of the theorem.
We also have a continuous version of Wolff's inequality which is known in the standard Euclidean setting [PV1].
Theorem 4.10. Let α > 0, p > 1 and q > p − 1. Then for any 0 < r < ∞ and any nonnegative Borel measure µ on G, where the constants in these equivalences are independent µ.
Proof. By arguing as in the proof of Theorem 4.9 we also find that Thus by Theorem 4.9, On the other hand, by Wolff's inequality (see [CLL], and [AH], [Tu] in the Euclidean setting), , which gives the theorem.
Similarly, in the case r = ∞ we have the following Wolff type inequality.
Theorem 4.11. Let α > 0, 1 < p < M/α and q > p − 1. Then for any nonnegative Borel measure µ on G, where the constants in these equivalences are independent of µ.
We are now in a position to prove the first main result of the paper.
Proof of Theorem 1.2. It is known that (i) ⇔ (ii) at least in the elliptic case, i.e., on R N (see, e.g., [AH]), and the proof given in [AH] works also on Carnot groups. Next, by duality and Theorem 4.10 we have (i) ⇔ (iii). Also, observe that (iii) ⇒ (iv) by letting g = χ B in (1.11). Moreover, we have the implication (iv) ⇒ (v) by following the proof given in [PV1,Theorem 2.10] in the elliptic case. Thus from Theorem 4.3 we obtain the last conclusion of the theorem.
Therefore, it is left to show that the existence of a solution u to (1.9) implies (i). To this end, we let µ = u q + ω and δ(x) = dist(x, ∂Ω). From the lower Wolff's potential estimate in Theorem 3.4 we have for all g ∈ L q p−1 (dµ), g ≥ 0, and supp(g) ⊂ Ω ′ . Inequality (4.21), duality, and the facts that ω ≤ µ and supp(ω) ⊂ Ω ′ finally yield This completes the proof of the theorem.
We next prove Theorem 1.3.
Proof of Theorem 1.3. We first suppose that C p, q q−p+1 (E) = 0. Since pq (q−p+1) > p by Theorems 4.1 and 4.9 in [Lu] we find C 1, p (E) = 0. On the other hand, by a result in [Fol] we have the identification S 1, p (G) = G 1 (L p (G)) with u S 1, p (G) ∼ = f L p (G) for any u with u = G 1 (f ). Thus we also have (4.22) cap 1, p (E, Ω) = 0, where cap 1, p (·, Ω) is a relative capacity adapted to Ω (see [TW], [HKM]) defined by Let u be a solution of (1.14). Using (4.22) and adapting the argument in [HKM] to this setting we see that the function is a p-superharmonic extension of u to the whole Ω. We now let ϕ be an arbitrary nonnegative function in C ∞ 0 (Ω). As in [BP,Lemme 2.2], we can construct a sequence {ϕ n } of nonnegative functions in C ∞ 0 (Ω \ E) such that (4.24) 0 ≤ ϕ n ≤ ϕ; ϕ n → ϕ almost everywhere.
Let K be an arbitrary compact subset of E and denote by µ K the restriction of µ E to K. We havê where we used (4.26) in the last inequality. On the other hand, it follows from the dual definition of capacity, see (1.5), that (4.28) µ K (K) ≤ C p, q q−p+1 (K) q−p+1 q G p (µ K ) q p−1 (G) . Thus we obtain from (4.27) and (4.28) that µ E (K) = µ K (K) ≤ C p, q q−p+1 (K).
Since this holds for all compact sets K by Theorem 1.2 we see that the equation (4.25) is solvable as long as ǫ > 0 is chosen small enough. But this would give us a contradiction and hence the proof is complete.

Global solutions and Liouville type theorems
In this section we sketch the proof of Theorem 1.4 and Corollary 1.5. To prove Theorem 1.4 one can proceed as in the proof of Theorem 1.2 but using Theorem 4.11 instead of Theorem 4.10, and the following global version of Theorem 4.3. The latest in turn can be proved as in [PV1,Theorem 5.3] by approximations and using pointwise estimates for potential theoretic solutions over arbitrarily large balls.
Theorem 5.1. Let ω ∈ M + (G), 1 < p < M , and q > p − 1. Suppose that and A is the constant used in Definition 1.1. Then there exists a p-superharmonic function u ∈ L q loc (R n ) such that −∆ G, p u = u q + ω, inf G u = 0, and for every x ∈ G, , where the constants c 1 , c 2 depend only p, q, and M .
We remark that in order to show that the existence of a solution u to (1.15) implies (i) and (ii) in Theorem 1.4 we need the following analogue of (4.21):ˆG [I p (gdµ)] q p−1 dx ≤ CˆG g q p−1 dµ, where µ = u q + ω. This can be shown to hold for all g ∈ L q p−1 (dµ), g ≥ 0, with no restriction on the support of g by using the lower bound in (3.3).
Finally, we give a proof of Corollary 1.5.
Proof of Corollary 1.5. Corollary 1.5 follows from Theorem 1.4 and the fact that for αs ≥ M the Riesz capacityĊ α, s (E) = 0 for every compact set E ⊂ G. To see the later note that for any nonnegative measure µ supported in a ball B R (e), R > 0, we have where |x| is the homogeneous norm of x (see Sect. 2). Thus using the condition αs ≥ M and [Fol, Corollary 1.6] we get I α * µ L s s−1 (G) = ∞ unless µ is identically zero. Therefore, in view of the dual definition of capacity, see (1.6), we obtainĊ α, s (E) = 0 for every compact set E ⊂ G.