Generalized superharmonic functions with strongly nonlinear operator

We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In particular, Harnack's Principle and Minimum Principle are provided for $\mathcal{A}$-superharmonic functions and boundary Harnack inequality is proven for $\mathcal{A}$-harmonic functions.


Introduction
The cornerstone of the classical potential theory is the Dirichlet problem for harmonic functions. The focus of the nonlinear potential theory is similar, however, harmonic functions are replaced by p-harmonic functions, that is, continuous solutions to the p-Laplace equation −∆ p u = −div(|Du| p−2 Du) = 0, 1 < p < ∞. There are known attempts to adapt the theory to the case when the exponent varies in space, that is p = p(x) for x ∈ Ω or the growth is non-polynomial. Inspired by the significant attention paid lately to problems with strongly nonstandard and nonuniformly elliptic growth e.g. [10,14,19,26,42,45] we aim at developing basics of potential theory for problems with essentially broader class of operators embracing in one theory as special cases Orlicz, variable exponent and double-phase generalizations of p-Laplacian. To cover whole the mentioned range of general growth problems we employ the framework described in the monograph [22]. Let us stress that unlike the classical studies [30,34] the operator we consider does not enjoy homogeneity of a form A(x, kξ) = |k| p−2 kA(x, ξ). Consequently, our class of solutions is not invariant with respect to scalar multiplication. Moreover, we allow for operators whose ellipticity is allowed to vary dramatically in the space variable. What is more, we do not need to assume in the definition of A-superharmonic function that it is integrable with some positive power, which is typically imposed in the variable exponent case, cf. e.g. [23,38].
We study fine properties of A-superharmonic functions defined by the Comparison Principle with respect to continuous solutions to −divA(x, Du) = 0. Here A is assumed to have generalized Orlicz growth expressed by the means of an inhomogeneous convex Φ-functions ϕ : Ω × [0, ∞) → [0, ∞) satisfying natural nondegeneracy and balance conditions, see Section 2 for details. The solutions belongs to the Musielak-Orlicz-Sobolev space W 1,ϕ(·) (Ω) described carefully in the monograph [22]. The assumptions on the operator are summarized below, and will be referred to as (A) throughout the paper. measurable and z → A(·, z) is continuous. Assume further that the following growth and coercivity assumptions hold true for almost all x ∈ Ω and all z ∈ R n \ {0}: with absolute constants c A 1 , c A 2 > 0 and some function ϕ : Ω× [0, ∞) → [0, ∞) being measurable with respect to the first variable, convex with respect to the second one and satisfying (A0), (A1), (aInc) p and (aDec) q with some 1 < p ≤ q < ∞. The precise statement of these conditions is given in Section 2. We collect all parameters of the problem as data = data(p, q, c A 1 , c A 2 ). Moreover, let A be monotone in the sense that for a.a. x ∈ Ω and any distinct z 1 , z 2 ∈ R n it holds that We shall consider weak solutions, A-supersolutions, A-superharmonic, and Aharmonic functions related to the problem (2) − div A(x, Du) = 0 in Ω.
For precise definitions see Section 3.
Special cases. Besides the p-Laplace operator case, corresponding to the choice of ϕ(x, s) = s p , 1 < p < ∞ we cover by one approach a wide range of more degenerate operators. When we take ϕ(x, s) = s p(x) , with p : Ω → R such that 1 < p − Ω ≤ p(x) ≤ p + Ω < ∞ and satisfying log-Hölder condition (a special case of (A1)), we render the so-called variable exponent, or the p(x)-Laplace equation 0 = −∆ p(x) u = −div(|Du| p(x)−2 Du).
Within the framework studied in [19] solutions to double phase version of the p-Laplacian 0 = −div A(x, Du) = −div ω(x) |Du| p−2 + a(x)|Du| q−2 Du are analysed with 1 < p ≤ q < ∞, possibly vanishing weight 0 ≤ a ∈ C 0,α (Ω) and q/p ≤ 1 + α/n (a special case of (A1); sharp for density of regular functions) and with a bounded, measurable, separated from zero weight ω. We embrace also the borderline case between the double phase space and the variable exponent one, cf. [5]. Namely, we consider solutions to with 1 < p < ∞, log-Hölder continuous a and a bounded, measurable, separated from zero weight ω. Having an N -function B ∈ ∆ 2 ∩ ∇ 2 , we can allow for problems with the leading part of the operator with growth driven by ϕ(x, s) = B(s) with an example of 0 = −div A(x, Du) = −div ω(x) B(|Du|) |Du| 2 Du with a bounded, measurable, and separated from zero weight ω. To give more new examples one can consider problems stated in weighted Orlicz (if ϕ(x, s) = a(x)B(s)), variable exponent double phase (if ϕ(x, s) = s p(x) + a(x)s q(x) ), or multi phase Orlicz cases (if ϕ(x, s) = i a i (x)B i (s)), as long as ϕ(x, s) is comparable to a function doubling with respect to the second variable and it satisfies the nondegeneracy and continuity assumptions (A0)-(A1), see Section 2.
State of art. The key references for already classical nonlinear potential theory are [2,30,34], but its fundaments date back further to [29,31]. A complete overview of the theory for equations with p-growth is presented in [37]. The first generalization of potential theory towards nonstandard growth is done in the weighted case [43,47]. So far significant attention was put on the variable exponent case, see e.g. [4,23,24,28,38], and analysis of related problems over metric spaces [7], but to our best knowledge the Orlicz case is not yet covered by any comprehensive study stemming from [39,41].
Let us mention the recent advances within the theory. Supersolutions to (2) are in fact solutions to measure data problems with nonnegative measure, that enjoy lately the separate interest, cf. [1, 11-13, 16, 18, 35, 37, 44] concentrating on their existence and gradient estimates. The generalization of studies on removable sets for Hölder continuous solutions provided by [36] to the case of strongly non-uniformly elliptic operators has been carried out lately in [14,15]. There are available various regularity results for related quasiminimizers having Orlicz or generalized Orlicz growth [21,25,26,28,32,33,41]. For other recent developments in the understanding of the functional setting we refer also to [3,17,20,27].
Results and organization. Section 2 is devoted to notation and basic information on the setting. In Section 3 we define weak solutions, A-supersolutions, Aharmonic and A-superharmonic functions and provide proofs of their fundamental properties including the Harnack inequality for A-harmonic functions (Theorem 1). Further analysis of A-superharmonic functions is carried out in Section 4. We prove there Harnack's Principle (Theorem 2), fundamental properties of Poisson's modification (Theorem 3), and Strong Minimum Principle (Theorem 4) together with their important consequence of the boundary Harnack inequality (Theorem 5) for A-harmonic functions.

Preliminaries
2.1. Notation. In the following we shall adopt the customary convention of denoting by c a constant that may vary from line to line. Sometimes to skip rewriting a constant, we use . By a ≃ b, we mean a b and b a. By B R we shall denote a ball usually skipping prescribing its center, when it is not important. Then by cB R = B cR we mean a ball with the same center as B R , but with rescaled radius cR. With U ⊂ R n being a measurable set with finite and positive n-dimensional Lebesgue measure |U | > 0, and with f : we mean the integral average of f over U . We make use of symmetric truncation on level k > 0, T k : R → R, defined as follows

Generalized Orlicz functions.
To handle in one framework with log-Hölder continuous variable exponent power functions, as well as doubling Orlicz ones and double phase within a natural range of parameters, we employ the formalism introduced in the monograph [22].
In fact, within our framework there exist some constants depending only on ϕ for which we have (3) ϕ (x, ϕ(x, s)/s) ∼ ϕ(x, s) for a.e. x ∈ Ω and all s > 0.
The Musielak-Orlicz space is defined as the set Moreover, the following Hölder inequality holds true We define the Musielak-Orlicz-Sobolev space W 1,ϕ(·) (Ω) as follows where D stands for distributional derivative. The space is considered endowed with the norm (Ω) we denote a closure of C ∞ 0 (Ω) under the above norm. Because of the growth conditions W 1,ϕ(·) (Ω) is a separable and reflexive space. Moreover, smooth functions are dense there.
then strong (norm) topology of W 1,ϕ(·) (Ω) coincides with the sequensional modular topology. Moreover, smooth functions are dense in this space in both topologies.
Note that as a consequence of [6, Lemma 2.1] for every function u, such that T k (u) ∈ W 1,ϕ(·) (Ω) for every k > 0 (with T k given by (2.1)) there exists a (unique) measurable function Z u : Ω → R n such that (6) DT k (u) = χ {|u|<k} Z u for a.e. in Ω and for every k > 0.
With an abuse of notation, we denote Z u simply by Du and call it a generalized gradient.

Various types of solutions and the notion of A-harmonicity
All the problems are considered under Assumption (A).

Definitions and basic remarks.
Existence and uniqueness of A-harmonic functions is proven in [15]. (Ω).
Moreover, u is locally bounded and for every E ⋐ Ω we have We call a function u ∈ W By density of smooth functions we can use actually test functions from W 1,ϕ(·) 0 (Ω).
The classes of A-superharmonic and A-subharmonic are defined by the Comparison Principle.

We say that an upper semicontinuous function u is
The above definitions have the following direct consequences.
We have the following estimate for A-supersolutions.
It is well known that solutions, subsolutions, and supersolutions can be described by the theory of quasiminimizers. Since many of the results on quasiminizers from [21] apply to our A-harmonic functions we shall recall the definition.
Among all functions having the same 'boundary datum' w ∈ W 1,ϕ(·) (Ω) the function u ∈ W 1,ϕ(·) is a quasiminimizer if it has the least energy up to a factor C, (Ω) and holds true with an absolute constant C > 0 for every v ∈ W 1,ϕ(·) 0 (Ω). We call a function u superquasiminimizer (subquasiminimizer) if (9) holds for all v as above that are additionally nonnegative (nonpositive). Proof. Let us take an arbitrary v ∈ W 1,ϕ(·) 0 (Ω). We may write (Ω), and upon testing the equation (2) with v we obtain Then by coercivity of A, Young's inequality, growth of A and doubling growth, for every ε > 0 we have Let us choose ε > 0 small enough for the first term on the right-hand side can be absorbed on the left-hand side. By rearranging terms, and using the fact that u + v = w +ṽ we get that Hence we get the claim.
By the same calculations as in the above proof we have the following corollary.
For more properties of solutions to related obstacle problems see also [8,9,14,23,32,46]. In particular, in [32] several basic properties of quasiminimizers to related variational obstacle problem are proven. (1) if u is a solution to the K ψ,w (Ω)-obstacle problem (11), then there exists c = c(data, n), such that (2) if u is a A-supersolution to (8) in Ω, then there exists c = c(data, n), such that Note that in fact in [32,Proposition 4.3] only (1) is proven in detail, but (2) follows by the same arguments.  Proof. When we fix x ∈ Ω and ε > 0, by the assumption we can find a regular open set D ⋐ Ω, such that v < u + ε on ∂D. Pick a decreasing sequence {φ k } ⊂ C ∞ (Ω) converging to v pointwise in D. Since ∂D is compact by lower semicontinuity of (u + ε) we infer that φ k ≤ u + ε on ∂D for some k. We take a function h being Aharmonic in D coinciding with φ k . By definition it is continuous up to a boundary of D. Therefore, v ≤ h ≤ u + ε on ∂D and so v ≤ h ≤ u + ε in D as well. We get the claim by letting ε → 0. (i) If a 1 , a 2 ∈ R, a 1 ≥ 0, and u is A-superharmonic in Ω, then so is a 1 u + a 2 .  Since w is lower semicontinuous, for every x ∈ ∂D ∩ E it holds that Consequently, for every x ∈ ∂(D ∩ E) one has  Proof. Since u is continuous and finite a.e. (because of the fact that it belongs to W
Let G ⋐ Ω be an open set, and let h be a continuous, A-harmonic function in G, such that h ≤ u on ∂G. Fix ǫ > 0 and choose and open set E ⋐ G such that u + ǫ ≥ h in G \ E. Since the function min{u + ǫ − h, 0} has compact support, it belongs to W 1,ϕ(·) (E). Hence Lemma 3.5 implies u + ǫ ≥ h in E, and therefore a.e. in G. Since the function is continuous, the inequality is true in each point of G. The ǫ was chosen arbitrary, so the claim follows.
We shall prove that A-superharmonic functions can be approximated from below by A-supersolutions. Proof. Since u is lower semicontinuous in K, it is bounded from below and there exists a nondecreasing sequence {φ j } of Lipschitz functions on K such that u = lim j→∞ φ j in K. For nonnegative u, obviously φ j , j ∈ N can be chosen nonnegative as well. Let u j be the solution of the K φj ,φj (K)-obstacle problem which by Proposition 3.9 is continuous and Moreover, u j is A-harmonic in A j . By Comparison Principle from Proposition 4.1 we infer that the sequence {u j } is nondecreasing. Since u is A-superharmonic, we have u j ≤ u in A j . Then consequently φ j ≤ u j ≤ u in K. Passing to the limit with j → ∞ we get that u = lim j→∞ u j , what completes the proof.
Lemma 4.6. If u is A-superharmonic in Ω and locally bounded from above, then u ∈ W
Proof. Fix open sets E ⋐ G ⋐ Ω. By Proposition 4.5 there exists a nondecreasing sequence of continuous A-supersolutions {u j } in G such that u = lim j→∞ u j pointwise in G. Since u is locally bounded we may assume u j ≤ u < 0 in G. It follows from Proposition 3.10 that the sequence {|Du j |} is locally bounded in L ϕ(·) (Ω). Since u j → u a.e., it follows that u ∈ W 1,ϕ(·) (G), and Du j ⇀ Du weakly in L ϕ(·) (G).
We need to show now that u is an A-supersolution in Ω. To this end we first prove that (up to a subsequence) gradients {Du j } converge a.e. in G. We start with proving that Choose η ∈ C ∞ 0 (G) such that 0 ≤ η ≤ 1, and η = 1 in E. Using ψ = η(u − u j ) as a test function for the A-supersolution u j and applying the Hölder inequality, the doubling property of ϕ, and the Lebesgue dominated monotone convergence theorem we obtain
Then, since η A(x, Du) − A(x, Du j ) · Du − Du j ≥ 0 a.e. in G, we conclude with (12). Since the integrand in I j is nonnegative, we may pick up a subsequence (still denoted u j ) such that (13) A Fix x ∈ E such that (13) is valid, and that |Du(x)| < ∞. Upon choosing further subsequence we may assume that Since we have (13) is true, it must follow that |ξ| < ∞.

Since the mapping ζ → A(x, ζ) is continuous, we have
and it follows that ξ = Du(x), and for a.e. x ∈ E, and A(·, Du j ) ⇀ A(·, Du) weakly in L ϕ(·) . Therefore that u is an A-supersolution of (8).
Since E was arbitrary this concludes the proof.

4.2.
Harnack's inequalities. In order to get strong Harnack's inequality for Aharmonic function and weak Harnack's inequality for A-superharmonic functions we need related estimates proved for A-subsolutions and A-supersolutions. Having Lemma 3.7 we can specify results derived for quasiminizers in [21] to our case. (Ω) being A-subsolution in Ω there exist constants R 0 = R 0 (n) > 0 and C = C(data, n, R 0 , ess sup BR 0 u) > 0, such that for all R ∈ (0, R 0 ], s > 0 and k ∈ R. (Ω) A-supersolution in Ω there exist constants R 0 = R(n) > 0, s 0 = s 0 (data, n) > 0 and C = C(data, n) > 0, such that Let us comment on the above result. For the application in [21] dependency of s 0 on other parameters is not important and so -not studied with attention. Actually, this theorem is not proven in detail in [21], but refers to standard arguments presented in [26,28]. Their re-verification enables to find s 0 = s 0 (data, n).
Since A-harmonic function is an A-subsolution and and A-supersolution at the same time (Lemma 3.4), by Propositions 4.7 and 4.8 we infer the full Harnack inequality.

Harnack's Principle for A-superharmonic functions.
We are going to characterize the limit of nondecreasing sequence of A-superharmonic functions and their gradients. Proof. The proof is presented in three steps. We start with motivating that the limit function is either A-superharmonic or u ≡ ∞, then we concentrate on gradients initially proving the claim for a priori globally bounded sequence {u i } and conclude by passing to the limit with the bound.
Step 1. Since u i are lower semicontinuous, so is u. The following fact holds: Given a compact set K ⋐ Ω, if h ∈ C(K), ǫ > 0 is a small fixed number, and u > h − ǫ on K, then, for i sufficiently large, u i > h − ǫ. Indeed, let's argue by contradiction. Assume that for every i there exists x i ∈ K, such that Since K is compact, we can assume that is in the contradiction with the fact that u > h − ǫ on K.
Using this fact we can prove that the limit function u = lim i→∞ u i is Asuperharmonic unless u ≡ ∞. Choose an open Ω ′ ⋐ Ω and h ∈ C(Ω ′ ) an Aharmonic function. Assume the inequality u ≥ h holds on ∂Ω ′ . It follows that for every ǫ > 0 on ∂Ω ′ we have u > h − ǫ and, from the aforementioned fact, it follows that u i > h−ǫ on ∂Ω ′ . Since all u i are A-superharmonic, Proposition 4.1 yields that u i ≥ h − ǫ on Ω ′ . Therefore u ≥ h − ǫ on Ω ′ . Since ǫ is arbitrary, we have u ≥ h on Ω ′ . Therefore the Comparison Principle from definition of A-superharmonic holds unless u ≡ ∞ in Ω. Finally, u = lim i→∞ u i is A-superharmonic unless u ≡ ∞.
Step 2. Assume 0 ≤ u i ≤ k for all i with k > 1 and choose open sets E ⋐ G ⋐ Ω. By Lemma 3.6 we get that ̺ ϕ(·),G (Du i ) ≤ ck q with c = c(data, n) > 0 uniform with respect to i. Then, by doubling properties of ϕ, we infer that (14) Du i L ϕ(·) (G) ≤ c(data, n, k).
Further, it has a non-relabelled subsequence converging a.e. in G to u ∈ W 1,ϕ(·) (G). Let us show that (15) Du j → Du a.e. in E.
We fix arbitrary ε ∈ (0, 1), denote and estimate its measure. We have Let η ∈ C ∞ 0 (G) be such that 1 E ≤ η ≤ 1 G . We define Then w i 1 η and w i 2 η are nonnegative functions from W 1,ϕ(·) 0 (G) and can be used as test functions. Since u and u i , i = 1, 2, . . ., are A-supersolutions we already know that u i → u weakly in W 1,ϕ(·) (E ′ ). By growth condition we can estimate like in (7) and by (14) we have with c > 0 independent of i and ε. Analogously Summing up the above observations we have The left-hand side is nonnegative by the monotonicity of the operator, so due to (16) we have with c > 0 independent of i and ε. By letting ε → 0 we get that |E j | → 0. Because of the strict monotonicity of the operator, we infer (15). We can conclude the proof of this step by choosing a diagonal subsequence.
Step 3. Now we concentrate on the general case. For every k = 1, 2, . . . we select subsequences {u in Ω. We note that v k increases to a function, which is A-harmonic or equivalently infinite. Additionally, v k = T k (u). The diagonally chosen subsequence {u

Poisson modification. The Poisson modification of an A-superharmonic in a regular set E carries the idea of local smoothing of A-superharmonic functions.
A point is called regular if at this point the boundary value of any Musielak-Orlicz-Sobolev function is attained not only in the Sobolev sense but also pointwise. A set is called regular if all of its regular points are regular. See [21] for the result that if the complement of Ω is locally fat at x 0 ∈ ∂Ω in the capacity sense, then x 0 is regular. Thereby of course polyhedra and balls are regular.
Let us consider a function u, which is A-superharmonic and finite a.e. in Ω and a regular open set E ⋐ Ω. We define for each x ∈ ∂E} and the Poisson modification of u in E by it follows that P (u, E) ≤ h in E. On the other hand, by the Comparison Principle (Proposition 4.1) we get that h i ≤ P (u, E) in E for every i. Therefore P (u, E)| E = h is A-harmonic in E. This reasoning also shows that P (u, E) is lower semicontinuous and, by Lemma 4.3, it is also A-superharmonic in Ω.

Minimum and Maximum
Principles. Before we prove the principles, we need to prove the following lemmas. Proof. It is enough to show that u = 0 in a given ball B ⋐ Ω. By lower semicontinuity of u infer that it is nonpositive. By Lemma 4.6, we get that u ∈ W 1,ϕ(·) (Ω).
Let v = P (u, B) be the Poisson modification of u in B. By Theorem 3 we have that v is continuous in B and v ≤ u ≤ 0. Therefore v is an A-supersolution in Ω and (Ω). Moreover, where the last equality holds because Du = 0 a.e. in Ω. But then, we directly get that Dv = 0 and v = 0 a.e. in Ω. By continuity of v in B we get that v = 0 everywhere in B. In the view of v ≤ u ≤ 0, we get that also u ≡ 0 in Ω.
Lemma 4.11. If u is A-superharmonic and finite a.e. in Ω, then for every x ∈ Ω it holds that u(x) = lim inf y→x u(y) = ess lim inf y→x u(y).
Proof. We fix arbitrary x ∈ Ω and by lower semicontinuity u(x) ≤ lim inf y→x u(y) ≤ ess lim inf y→x u(y) =: a. Let ε ∈ (0, a) and B = B(x, r) ⊂ Ω be such that u(y) > a − ε for a.e. y ∈ B. By Corollary 4.2 function v = min{u − a + ε, 0} is Asuperharmonic in Ω and v = 0 a.e. in B. By Lemma 4.10 v ≡ 0 in Ω, but then u(x) ≥ a − ε. Letting ε → 0 we obtain that u(x) = a and the claim is proven.
Note that within our regime s → ψ(·, s) is strictly increasing, but not necessarily convex. Although in general ψ does not generate the Musielak-Orlicz space, we still can define ̺ ψ(·),Ω by (4) useful in quantifying the uniform estimates for trucations in the following lemma. Proof. The result is classical when p = q, [30]. Therefore, we present the proof only for p < q. We start with observing that |{x ∈ B : ϕ(x, |Du|) > s}| ≤ |{x ∈ B : |u| > k}| + |{x ∈ B : ϕ(x, |Du|) > s, |u| ≤ k}| Let us first estimate the volume of superlevel sets of u using Tchebyszev inequality, Poincaré inequality, assumptions on the growth of ϕ, and (18). For all sufficiently large k we have Similarly by Tchebyszev inequality and (18) we can estimate also Altogether for all sufficiently large s (i.e. s > k p 0 ) we have that Recall that due to (3) there exists C > 0 uniform in x such that ψ(x, s) ≥ C ϕ −1 (x, ϕ(x, s)), so for some c > 0 independent of x. Since the case q = p is trivial for these estimates, it suffices to consider q > p. Then − q ′ p ′ < −1 and we get the uniform integrability of {ψ(x, |DT k u|)} k , thus the claim follows.
Let us sum up the information on integrability of gradients of truncations of A-superharmonic functions.  in Ω there exist constants R A 0 = R A 0 (n) > 0, s 0 = s 0 (data, n) > 0 as in the weak Harnack inequality (Proposition 4.8), and C = C(data, n) > 0, such that for every k > 1 we have Proof. The proof is based on Remark 4.13 and Propoposition 4.8 that provides weak Harnack inequality for an A-supersolution v holding with constant C = C(data, n) and for balls with radius R < R 0 (n) and so small that ̺ ϕ(·),B3R 0 (Dv) ≤ 1.
The only explanation is required whenever |Dv| ≥ 1 a.e. in the considered ball. Then for every k > 1 there exists R 1 (k) such that we get (19) for T k v over balls such that R < min{R 1 (k), R 0 (n)} and ̺ ϕ(·),B 3R 1 (k) (DT k v) ≤ 1. Of course, then there exists R A 0 (k) ∈ (0, R 1 (k)), such that we have (19) for R < min{R 1 (k), R 0 (n)} and ̺ ψ(·), Note that it is Remark 4.13 that allows us to choose R A 0 independently of k.
We are in a position to prove that an A-harmonic function cannot attain its minimum nor maximum in a domain.
Theorem 4 (Strong Minimum Principle for A-superharmonic functions). Suppose u is A-superharmonic and finite a.e. in connected set Ω. If u attains its minimum inside Ω, then u is a constant function.
Proof. We consider v = (u − inf Ω u), which by Corollary 4.2 is A-superharmonic. Let E = {x ∈ Ω : v(x) = 0}, which by lower semicontinuity of v (Lemma 4.11) is nonempty and relatively closed in Ω. Having in hand Remark 4.13 we can choose B = B(x, R) ⊂ 3B ⋐ Ω with radius smaller than R A 0 from Lemma 4.14 and such that ̺ ψ(·),B3R (Du) ≤ 1 where ψ is as in (17). Therefore, in the rest of the proof we restrict ourselves to a ball B. By Corollary 4.2 functions v and T k v are Asuperharmonic in 3B. Moreover, by Lemma 4.6 we infer that {T k v} is a sequence of A-supersolutions integrable uniformly in the sense of Lemma 4.12. We take any y ∈ B -a Lebesgue's point of T k v for every k and choose B ′ = B ′ (y, R ′ ) ⋐ B. Let us also fix arbitrary k > 0. We have the weak Harnack inequality from Lemma 4.14 for T k v on B ′ yielding with s 0 , C > 0 independent of k. Letting R ′ → 0 we get that T k v(y) = 0. Lebesgue's points of T k v for every k are dense in B, we get that T k v ≡ 0 a.e. in B. By arguments as in Lemma 4.10 we get that T k v ≡ 0 in B, but then B ⊂ E and E has to be an open set. Since Ω is connected, E is the only nonempty and relatively closed open set in Ω, that is E = Ω. Therefore T k v ≡ 0 in Ω. As k > 0 was arbitrary v = u − inf Ω u ≡ 0 in Ω as well.
The classical consequence of Strong Minimum Principle, we get its weaker form. By the very definition of an A-subharmonic function one gets the following direct consequence of the above fact. Having Theorem 4 and Corollary 4.16, we infer that if u is A-harmonic in Ω, then it attains its minimum and maximum on ∂Ω. In other words A-harmonic functions have the following Liouville-type property. Theorem 5 (Boundary Harnack inequality for A-harmonic functions). For a nonnegative function u which is A-harmonic in a connected set Ω there exist R 0 = R(n) > 0 and C = C(data, n, R 0 , ess sup BR 0 u) > 0, such that for all R ∈ (0, R 0 ] provided B 3R ⋐ Ω and ̺ ψ(·),B3R (Du) ≤ 1, where ψ is given by (17).
Proof. It suffices to note that by Lemma 3.4 we can use Minimum Principle of Corollary 4.15 and Maximum Principle of Corollary 4.16. Then by Harnack inequality of Theorem 1 the proof is complete.
Corollary 4.18. Suppose u is A-harmonic in B 3 2 R \ B R , with R < R 0 from Theorem 5, then exists C = C(data, n, R 0 , ess sup BR 0 u) > 0, such that Proof. Fix ε > 0 small enough for B R ⋐ B 4 3 R−ε ⊂ B 4 3 R+ε ⋐ B 3 2 R . Of course, then u is A-harmonic in B 4 3 R+ε \ B 4 3 R−ε . We cover the annulus with finite number of balls of equal radius as prescribed in the theorem and such that ̺ ψ(·),B (Du) ≤ 1, which is possible due to Remark 4.13. Let us observe that due to the Harnack's inequality from Theorem 5 we have Since u is continuous in B 3 2 R \ B R , passing with ε → 0 we get the claim.