Harnack Inequalities and ABP Estimates for Nonlinear Second-Order Elliptic Equations in Unbounded Domains

Abstract

We are concerned with fully nonlinear uniformly elliptic operators with a superlinear gradient term. We look for local estimates, such as weak Harnack inequality and local maximum principle, and their extension up to the boundary. As applications, we deduce ABP-type estimates and weak maximum principles in general unbounded domains, a strong maximum principle, and a Liouville-type theorem.

1. Introduction

The qualitative theory of second-order elliptic equations received a strong effort from Harnack inequalities. Here, we will make use of this powerful technique to study continuous viscosity solutions 𝑢 of fully nonlinear elliptic equations (𝐹=𝑓):𝐹𝑥,𝑢(𝑥),𝐷𝑢(𝑥),𝐷²𝑢(𝑥)=𝑓(𝑥),𝑥∈Ω, (1.1)in unbounded domains Ω of 𝑅^𝑛, where 𝐹 is a real function of 𝑥∈Ω, 𝑡∈𝑅, 𝑝∈𝑅^𝑛 and 𝑋 in the set 𝒮^𝑛 of 𝑛×𝑛 real symmetric matrices.

We recall that 𝐹 is (degenerate) elliptic if 𝐹 is nondecreasing in 𝑋 and uniformly elliptic if there exist (ellipticity) constants 𝜆 and Λ such that 0<𝜆≤Λ and𝜆tr(𝑌)≤𝐹(𝑥,𝑡,𝑝,𝑋+𝑌)−𝐹(𝑥,𝑡,𝑝,𝑋)≤Λtr(𝑌), (1.2)for (1.2)𝑌≥0, that is (1.2)𝑌 is semidefinite positive, where (1.2)tr(𝑌) denotes the trace of the matrix (1.2)𝑌.

In the class of uniformly elliptic operators, there are two extremal ones, well known as Pucci maximal and minimal operators, respectively:𝒫⁺_𝜆,Λ(𝑋)=Λtr𝑋⁺−𝜆tr𝑋⁻,𝒫⁻_𝜆,Λ(𝑋)=𝜆tr𝑋⁺−Λtr𝑋⁻, (1.2)where 𝑋^± are the positive and negative parts of 𝑋, which can be decomposed in a unique way as 𝑋=𝑋⁺−𝑋⁻ with 𝑋^±≥0 and 𝑋⁺𝑋⁻=0. Other examples of fully nonlinear uniformly elliptic operators can be found in [1–3].

Throughout this paper, we will consider elliptic operators with the structure conditions𝐹(𝑥,𝑡,𝑝,𝑋)≥𝒫⁻_𝜆,Λ(𝑋)−𝑏(𝑥)|𝑝|^𝑞,𝐹(𝑥,𝑡,𝑝,𝑋)≤𝒫⁺_𝜆,Λ(𝑋)+𝑏(𝑥)|𝑝|^𝑞, (1.4)(1.5)where 𝒫^± are the extremal Pucci operators, 𝑏(𝑥) is a continuous function and the exponent 𝑞∈[1,2], so that the gradient term can have a superlinear, at most quadratic growth.

Remark 1.1. The above structure conditions are exactly equivalent to the uniform ellipticity when 𝐹 is linear in the variable 𝑋∈𝒮^𝑛. In the nonlinear case they allow a slight generalization. Let us consider, for 0<𝜆<Λ and 𝑡≥0, the function1ℎ(𝑡)=Λ𝑡,if0≤𝑡≤Λ,11,ifΛ1<𝑡≤𝜆,1𝜆𝑡,if𝑡>𝜆, (1.6)then the operator 𝐹=ℎ(tr(𝑋⁺))−ℎ(tr(𝑋⁻)) is elliptic and satisfies both the conditions (1.4) and (1.5), even that it is not uniformly elliptic.
However, if (1.5) (resp., (1.4)) holds, then subsolutions (resp., supersolutions) of the equation 𝐹=𝑓 are subsolutions (resp., supersolutions) of uniformly elliptic equations, and this is needed to prove our results.

We will be concerned principally with the following topics in unbounded domains; see [4–6] for classical results.

[MP] maximum principle for u.s.c. subsolutions 𝑤 of 𝐹=0 in the viscosity sense (v.s.), in the form𝐹≥0inΩ,𝑤≤0on𝜕Ω,sup_Ω𝑤<+∞⟹𝑤≤0inΩ; (1.7)

[LT] Liouville theorem for continuous solutions of 𝐹=0 v.s., in the form𝐹=0inℝ^𝑛,sup_ℝ𝑛𝑤<+∞⟹𝑤=constantinℝ^𝑛. (1.8)Concerning MP, it is worth to note that the condition from above on the size of 𝑤 can be weakened in the framework of the Phragmén-Lindelöf theory (see, e.g., [7–9]) but not omitted at all, even for classical subsolutions (see, e.g., [4, 10]). It is also well known that MP fails to hold in general in exterior domains. In fact, due to the boundedness of the fundamental solution 𝑢(𝑥)=|𝑥|^2−𝑛 of the Laplace equation Δ𝑢=0, the function 𝑤=1−𝑢 provides a counterexample to MP in Ω=𝑅^𝑛⧵𝐵₁(0). Thus we introduce a local measure-geometric condition 𝐆_𝜎 in Ω at 𝑦∈𝑅^𝑛, which depends on the real parameter 𝜎∈(0,1) : there exists a ball 𝐵=𝐵(𝑦) such that𝑦∈𝐵,𝐵⧵Ω_𝑦≥𝜎|𝐵|, (1.9)where Ω_𝑦 is the connected component of 𝐵⧵𝜕Ω containing 𝑦.

If 𝐆_𝜎 is satisfied in Ω at all 𝑦∈Ω, we simply say that Ω is a 𝐰𝐆 domain (with parameter 𝜎). This is a generalization of condition 𝐆 of Cabré [10], which ultimately goes back to Berestycki et al. [11].

Let 𝑅(𝑦) denote the radius of the ball 𝐵=𝐵(𝑦) provided by condition 𝐰𝐆. We will call domains of cylindrical and conical type the 𝐰𝐆 domains such that 𝑅(𝑦)=𝑂(1) and 𝑅(𝑦)=𝑂(|𝑦|) as |𝑦|→+∞, respectively. Examples of the first kind are domains with finite measure, cylinders, slabs, complements of a periodic lattice of balls, whereas cones, and complements, in the plane, of logarithmic spirals, are examples of the second kind.

In [12], it is shown that MP holds true in a 𝐰𝐆 domain for strong solutions of a linear second-order uniformly elliptic operator 𝐹=tr𝐴(𝑥)𝑋; see also [13, 14] for earlier results and [15, 16] for viscosity solutions of a fully nonlinear operator with linear and quadratic growth in the gradient (i.e., in the case of the structure condition (1.5) with 𝑞=1 and 𝑞=2) provided that 𝑏(𝑥)=𝑂(1/|𝑥|) and 𝑏(𝑥)=𝑂(1) as |𝑥|→∞, respectively.

With the aim to find conditions on the coefficient 𝑏(𝑥) such that MP holds in 𝐰𝐆 domains when 1≤𝑞≤2, our result is the following.

Theorem 1.2 (MP). Let 0<𝜎<1 and 1≤𝑞≤2. Let Ω be a domain of 𝑅^𝑛 satisfying condition 𝐰𝐆 or alternatively such that, for a closed subset 𝐻 of 𝑅^𝑛, (i)𝑀𝑃 holds in each connected component of Ω⧵𝐻;(ii)condition 𝐆_𝜎 is satisfied in Ω at each 𝑦∈Ω∩𝐻. Suppose that 𝑤∈𝑈𝑆𝐶(Ω) is a viscosity solution of 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)≥0 and structure condition (1.5) holds with 𝑏∈𝐶(Ω), such that 𝑏(𝑥)=𝑂(1/|𝑥|^2−𝑞).
If 𝑤≤0 on 𝜕Ω and sup𝑤<+∞ in Ω, then 𝑤≤0 in Ω.

This yields indeed MP in a wider class of domains than 𝐰𝐆, for example, the cut plane and more generally the complement of continuous semi-infinite curves in 𝑅² and their generalizations to hypersurfaces in 𝑅^𝑛.

We also outline that the limit cases 𝑞=1 and 𝑞=2 of the above mentioned papers are obtained by continuity from the intermediate cases 1<𝑞<2, as it follows from Theorem 1.2. Nonetheless, there are technical improvements with respect to the previous works even in the limit cases.

Consider in particular a parabolic shaped domain Ω, satisfying condition 𝐰𝐆 with 𝑅(𝑦)=𝑂(|𝑦|^𝛼), 0<𝛼<1; the limit cases 𝛼=0 and 𝛼=1 correspond to domains of cylindrical and conical types, respectively.

Based on an argument of [12], eventually passing to a smaller 𝑟_𝑦≤𝑅(𝑦), we can suppose that condition 𝐆_𝜎 is satisfied with |𝐵⧵Ω_𝑦|=𝜎|𝐵| exactly. We get the new following variant of ABP estimate.

Theorem 1.3 (ABP). Let 0<𝜎,𝜏<1, 𝜏^′>1, 𝑅₀,𝛽≥0, 1≤𝑞≤2, and 𝑁>0. Let Ω be a 𝐰𝐆 domain, such that condition 𝐆_𝜎 in Ω is fulfilled at each 𝑦∈Ω with 𝑅(𝑦)≤𝑅₀+𝛽|𝑦|^𝛼, 0≤𝛼≤1. Assume that 𝐹 satisfies the structure condition (1.5), with 𝑏,𝑓∈𝐶(Ω) and 𝑏₀∶=sup_Ω|𝑏(𝑥)|(1+|𝑥|^{𝛼(2−𝑞)})<+∞.
If 𝑤∈𝑈𝑆𝐶(Ω) is a viscosity solution of 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)≥𝑓 such that 𝑤≤𝑁 in Ω and 𝑤≤0 on 𝜕Ω, thensup_Ω𝑤≤𝐶lim_𝜀→0+sup_{𝑦∈Ω;|𝑦|≥𝜀𝑟𝑦}𝑟_𝑦𝑓⁻_{𝐿𝑛(Ω∩𝐵𝑦′𝑟𝑦𝜏𝜀𝑟,𝜏)}, (1.10)where 𝐶 is a positive constant depending on 𝑛, 𝑞, 𝜆, Λ, 𝑏₀𝑁^𝑞−1, 𝜎, 𝜏, 𝜏^′, 𝑅₀, 𝛽.

Note that in the case of a domain of cylindrical type (𝛼=0), it is sufficient to have 𝑏(𝑥)=𝑂(1), for all 𝑞∈[1,2], as well as in the case of a quadratic growth in the gradient variable (𝑞=2), for all 𝛼∈[0,1].

This result extends the previous ones contained in [10, 14] for the linear case, and [8, 16], dealing with fully nonlinear equations, in the limit situations of cylindrical/conical domains and linear/quadratic gradient terms.

Remark 1.4. In general, unless 𝑞=1, the above ABP type estimate is different from the so-called ABP maximum principles since 𝐶 depends on the upper bound 𝑁 of 𝑤 if 𝑏₀>0 and 𝑞>1. For ABP-type estimates of this kind in bounded domains we refer to [17]. Counterexamples to the ABP maximum principle can been found in [17–19].

Consider now Ω=𝑅^𝑛. The classical Liouville theorem says that harmonic functions in the entire 𝑅^𝑛, which are bounded either above or below, are constant. This result continues to hold for strong solutions of quasilinear uniformly elliptic equations; see [20]. For viscosity solutions of fully nonlinear uniformly elliptic equations with an additive gradient term having linear growth, we refer to [21, 22]. Our result is the following.

Theorem 1.5 (Liouville theorem). Let 𝑤∈𝐶(𝑅^𝑛) be such that 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)=0 in the viscosity sense, and assume structure conditions (1.4) and (1.5), with 𝑏∈𝐶(𝑅^𝑛) such that 𝑏(𝑥)=𝑂(1/|𝑥|^2−𝑞) as |𝑥|→+∞. If 𝑤 is bounded either above or below, then 𝑤 is constant.

Remark 1.6. Under some additional assumptions, Liouville-type results also hold in un-bounded domains of 𝑅^𝑛 containing balls of arbitrary large radius; see [23].

Our main tools are Krylov-Safonov Harnack inequalities and local MP; see [20] for strong solutions of quasilinear uniformly elliptic equations. For viscosity solutions and 𝐹 satisfying the structure condition (1.4), they can be found in [3] if 𝑏=0 and in [24] if 𝑞=2; see also [25]. In the case of linear or superlinear, almost quadratic, growth in the gradient (1≤𝑞<2), weak Harnack (wH) inequality and local MP can be deduced using arguments of [17], in which a (full) Harnack inequality has been established for 𝐿^𝑝 viscosity solutions; see also [26].

Nevertheless, for convenience of the reader we believed that it is worth to report systematically on this kind of inequalities in Section 3.

As the previous ones, our approach follows the lines of [3], based on the methods of [27, 28] and on the ABP maximum principle for viscosity solutions in bounded domains, due to Caffarelli [29].

Remark 1.7. In deriving wH inequality and local MP, we only need the Alexandroff-Bakelman-Pucci (ABP) estimate with 𝑞=1 and 𝑓 continuous, so [30, Proposition 2.12] and also [17, Theorem 4.1] in the case of linear growth in the gradient term, are sufficient to our purpose. But we notice that new ABP-type estimates have been established for 𝐿^𝑝-viscosity solutions of equations with discontinuous coefficients by Koike and Święch [19, 27] for 𝑞∈[1,2] and 𝑓∈𝐿^𝑝.

Remark 1.8. In the case of a superlinear first-order term, wH inequality and local MP are obtained by interpolation between the linear and quadratic cases, eliminating the square gradient term by means of an exponential transformation used before by Trudinger [24], see Lemmas 3.1 and 3.2 below. This kind of ideas have been also considered by Sirakov in [31].

What we definitely need are, for MP, the scaled boundary wH inequality (3.16), derived in Section 3 by means of typical viscosity methods, and, for technical reasons, its version in annular regions (3.24), and, for LT, the scaled Harnack inequality (3.11). Moreover, using the interior wH inequality (3.7) and assuming the structure condition (1.5), we also state a strong MP theorem, according to which a subsolution 𝑢 of equation 𝐹=0 cannot achieve a positive maximum inside any domain (open connected set) of 𝑅^𝑛 unless it is constant; see Theorem 5.1 below. For a different approach, based on Hopf lemma, and more general versions see [32].

The paper is organized as follows. In Section 2, we recall some basic results of elliptic theory for viscosity solutions of second-order fully nonlinear equations with a linear gradient term; in Section 3, we extend local maximum principle and weak Harnack inequality, even up to the boundary, to the case of a superlinear gradient term; these results are applied in Section 4 to get Alexandroff-Bakelman-Pucci-type estimates and maximum principles, with the proof of Theorems 1.2 and 1.3; finally, a strong maximum principle is derived and the proof of Liouville theorem (Theorem 1.5) is given in Section 5. In the appendix, for the sake of completeness, we show the basic weak Harnack inequality and local MP for a uniformly elliptic operator with an additive first-order term having linear growth in the gradient.

2. Basic Estimates (Linear Gradient Term)

Let Ω be a domain of 𝑅^𝑛, and denote by 𝑈𝑆𝐶(Ω) and 𝐿𝑆𝐶(Ω), respectively, the sets of the upper and lower semicontinuous functions in Ω. The function 𝑢∈𝑈𝑆𝐶(Ω) is said to be a viscosity subsolution of 𝐹=𝑓 if𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)≥𝑓(𝑥) (2.1)at any point 𝑥∈Ω and for all 𝜑∈𝐶²(Ω) such that 𝜑−𝑢 has a local minimum in 𝑥. Similarly, a viscosity supersolution 𝑢∈𝐿𝑆𝐶(Ω) of 𝐹=𝑓 satisfies𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)≤𝑓(𝑥) (2.2)at any point 𝑥∈Ω and for all 𝜑∈𝐶²(Ω) such that 𝑢−𝜑 has a local minimum in 𝑥.

We may also assume that 𝜑(𝑥)=𝑢(𝑥) in the above definition, that is the graph of the test function 𝜑 touches that one of 𝑢 from above for subsolutions and from below for supersolutions [3]. Moreover, if 𝐹 is continuous in the matrix-variable, as for uniformly elliptic operators, then we may assume that 𝜑(𝑥) is a paraboloid, that is a quadratic polynomial.

We will make use of the following version of the ABP estimate, in which Γ⁺_𝑢 denotes the upper contact setΓ⁺_𝑢=𝑥∈Ω/∃𝑝∈ℝ^𝑛𝑠.𝑡.𝑢(𝑦)≤𝑢(𝑥)+𝑝⋅(𝑦−𝑥)for𝑥∈Ω (2.3)of the graph of the function 𝑢. Using [30, Proposition 2.12] or [17, Theorem 4.1], we have the following.

Lemma 2.1 (ABP estimate). Let 𝑢∈𝐿𝑆𝐶(𝐵) be a viscosity supersolution of the equation𝒫⁻_𝜆,Λ𝐷²𝑢−𝑏₀|𝐷𝑢|=𝑓 (2.4)in a ball 𝐵 of unit radius, such that 𝑢≥0 on 𝜕𝐵, where 𝑓∈𝐿^𝑛(𝐵)∩𝐶(𝐵), for some constant 𝑏₀≥0. Thensup_𝐵𝑢⁻≤𝐶𝑓⁺_{𝐿𝑛(Γ+𝑢−)}, (2.5)for a positive constant 𝐶=𝐶(𝑛,𝜆,Λ,𝑏₀). Similarly, if 𝑢∈𝑈𝑆𝐶(𝐵) is a viscosity subsolution of the equation𝒫⁺_𝜆,Λ𝐷²𝑢+𝑏₀|𝐷𝑢|=𝑓 (2.6)such that 𝑢≤0 on 𝜕𝐵, thensup_𝐵𝑢⁺≤𝐶𝑓⁻_{𝐿𝑛(Γ+𝑢+)}. (2.7)

From Lemma 2.1, we obtain the following results, see the appendix, which extend [3, Theorem 4.8, (1) and (2)]; see also [15].

Here we denote by 𝐵_𝑟 a ball centered at 𝑥₀∈𝑅^𝑛 of radius 𝑟>0.

Lemma 2.2 (wH inequality). Let 𝑏₀≥0 and 0<𝜏<1. Suppose that 𝑢∈𝐿𝑆𝐶(𝐵_1/𝜏) is a viscosity supersolution of (2.4), with 𝑓∈𝐶(𝐵_1/𝜏), and 𝑢≥0 in 𝐵_1/𝜏. Then1|𝐵₁|_𝐵1𝑢^𝑝01/𝑝0≤𝐶inf_𝐵1𝑢+𝑓⁺_{𝐿𝑛(𝐵1/𝜏)}, (2.8)where 𝐶 and 𝑝₀ are positive numbers, depending on 𝑛,𝜆,Λ,𝑏₀, and 𝜏.

Lemma 2.3 (local MP). Let 𝑏₀≥0 and 0<𝜏<1. Suppose that 𝑢∈𝑈𝑆𝐶(𝐵₁) is a viscosity subsolution of (2.6) with 𝑓∈𝐶(𝐵₁). Then for all 𝑝>0,sup_𝐵𝜏1𝑢≤𝐶|𝐵₁|_𝐵1𝑢^+𝑝1/𝑝+𝑓⁻_{𝐿𝑛(𝐵1)}, (2.9)where 𝐶 is a positive constant, depending on 𝑛,𝜆,Λ,𝑏₀,𝜏, and 𝑝.

3. Interior and Boundary Harnack Estimates and Local MP (Superlinear Gradient Term)

Firstly, we extend interior estimates (2.8) and (2.9) to fully nonlinear operators 𝐹 with a superlinear first-order term, such that, respectively, (1.4) and (1.5) hold.

Lemma 3.1 (wH inequality). Let 𝑏₀≥0, 0<𝜏<1, and 1≤𝑞≤2. Suppose that 𝑢∈𝐿𝑆𝐶(𝐵_1/𝜏) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤𝑓, under structure condition (1.4) with 𝑏≤𝑏₀, 𝑓∈𝐶(𝐵_1/𝜏), and 0≤𝑢≤1 in 𝐵_1/𝜏. Then (2.8) holds with positive constants 𝐶 and 𝑝₀, depending on 𝑛,𝜆,Λ,𝑏₀,𝜏, and 𝑞.

Lemma 3.2 (local MP). Let 𝑏₀≥0, 0<𝜏<1, and 1≤𝑞≤2. Suppose that 𝑢∈𝑈𝑆𝐶(𝐵₁) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≥𝑓, under structure condition (1.5), with 𝑏≤𝑏₀, 𝑓∈𝐶(𝐵₁), and 𝑢≤1. Then (2.9) holds for all 𝑝>0 with a positive constant 𝐶, depending on 𝑛,𝜆,Λ,𝑏₀,𝑞,𝜏, and 𝑝.

Proof of Lemmas 3.1 and 3.2. . We only show the proof of Lemma 3.2, since that one of Lemma 3.1 is similar. By the structure condition (1.5), we have𝒫⁺_𝜆,Λ𝐷²𝑢+𝑏₀|𝐷𝑢|^𝑞≥𝑓(𝑥) (3.1)and also, in the viscosity sense,𝒫⁺_𝜆,Λ𝐷²𝑢⁺+𝑏₀𝐷𝑢^+𝑞≥−𝑓⁻(𝑥). (3.2)From this, by Young's inequality, it follows that𝒫⁺_𝜆,Λ𝐷²𝑢⁺+𝑏₁𝐷𝑢⁺+𝑏₂𝐷𝑢⁺²≥−𝑓⁻(𝑥) (3.3)with𝑏₁=(2−𝑞)𝑏₀^1/𝑞,𝑏₂=(𝑞−1)𝑏₀^2/𝑞. (3.4)Using the transformation 𝑢⁺=(𝜆/𝑏₂)log(1+(𝑏₂/𝜆)𝑣), then the USC function 𝑣=(𝜆/𝑏₂)(exp((𝑏₂/𝜆)𝑢⁺)−1) satisfies the differential inequality𝒫⁺_𝜆,Λ𝐷²𝑣+𝑏₁|𝐷𝑣|≥−𝑓⁻𝑏(𝑥)1+²𝜆𝑣(𝑥) (3.5)in 𝐵_1/𝜏. Therefore, we can apply Lemma 2.3 to the subsolution 𝑣. To conclude the proof of Lemma 3.2, it is sufficient to observe that𝑢⁺𝜆≤𝑣≤𝑏₂𝑏exp²𝜆−1𝑢⁺. (3.6)

Rescaling variables and functions, we highlight the dependence on geometric parameters.

Theorem 3.3 (scaled wH inequality). Let 𝑏₀≥0, 0<𝜏<1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝐿𝑆𝐶(𝐵_𝑅/𝜏) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤𝑓, under structure condition (1.4), with 𝑏≤𝑏₀, 𝑓∈𝐶(𝐵_𝑅/𝜏), and 0≤𝑢≤𝑁 in 𝐵_𝑅/𝜏. Then1|𝐵_𝑅|_𝐵𝑅𝑢^𝑝01/𝑝0≤𝐶inf_𝐵𝑅𝑢+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏)}, (3.7)with positive constants 𝐶 and 𝑝₀, depending on 𝑛,𝜆,Λ,𝑞,𝜏, and 𝑏₀𝑁^𝑞−1𝑅^2−𝑞.

Proof. Considering, for 𝑦∈𝐵_1/𝜏, the function 𝑣(𝑦), defined by 𝑢(𝑥)=𝑁𝑣(𝑥/𝑅), we have𝒫⁻_𝜆,Λ𝐷²𝑣−𝑏₀𝑁^𝑞−1𝑅^2−𝑞|𝐷𝑣|^𝑞≤𝑅²𝑁⁻¹𝑓⁺(𝑅𝑦). (3.8)Thus, applying Lemma 3.1, we get1|𝐵₁|_𝐵1𝑣^𝑝01/𝑝0≤𝐶inf_𝐵1𝑣+𝑅²𝑁⁻¹𝑓⁺(𝑅𝑦)_{𝐿𝑛𝑦(𝐵1/𝜏)}, (3.9)with 𝐶=𝐶(𝑛,𝜆,Λ,𝑞,𝜏,𝑏₀𝑁^𝑞−1𝑅^2−𝑞), from which the assert follows.

Note that constants 𝑝₀ and 𝐶 of the above wH inequality depend in general on the upper bound 𝑁 for the supersolution and on the radius 𝑅 of the ball, but in the case 𝑞=1 there is no dependence on 𝑁 and in the case 𝑞=2 no dependence on 𝑅.

In the same manner as in Theorem 3.3 for wH inequality, we make the dependence on the geometric constants explicit in the following local MP.

Theorem 3.4 (scaled local MP). Let 𝑏₀≥0, 0<𝜏<1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝑈𝑆𝐶(𝐵_𝑅) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≥𝑓, under structure condition (1.5), with 𝑏≤𝑏₀, 𝑓∈𝐶(𝐵_𝑅), and 𝑢≤𝑁. Then for all 𝑝>0,sup_𝐵𝜏𝑅1𝑢≤𝐶|𝐵_𝑅|_𝐵𝑅𝑢^+𝑝1/𝑝+𝑅𝑓⁻_{𝐿𝑛(𝐵𝑅)}, (3.10)with a positive constant 𝐶, depending on 𝑛,𝜆,Λ,𝑞,𝜏,𝑏₀𝑁^𝑞−1𝑅^2−𝑞, and 𝑝.

Combining Theorems 3.3 and 3.4, we get the full Harnack inequality for solutions.

Theorem 3.5 (Harnack inequality). Let 𝑏₀≥0, 0<𝜏<1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝐶(𝐵_𝑅/𝜏) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 in 𝐵_𝑅/𝜏, under structure conditions (1.4) and (1.5), with 𝑏≤𝑏₀, 𝑓∈𝐶(𝐵_𝑅/𝜏), and 0≤𝑢≤𝑁. Thensup_𝐵𝑅𝑢≤𝐶inf_𝐵𝑅𝑢+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏)}, (3.11)with a positive constant 𝐶=𝐶(𝑛,𝜆,Λ,𝑞,𝜏,𝑏₀𝑁^𝑞−1𝑅^2−𝑞).

We wish to extend the above estimates up to the boundary, that is, to balls intersecting the boundary of the domain 𝐴⊂𝑅^𝑛, where the solutions are defined. For this purpose we will need suitable extensions of such solutions outside 𝐴. Precisely, take concentric balls 𝐵_𝜏𝑅⊂𝐵_𝑅⊂𝐵_𝑅/𝜏 such that 𝐵_𝜏𝑅∩𝐴≠∅ and 𝐵_𝑅/𝜏⧵𝐴≠∅. For a nonnegative viscosity supersolution 𝑢∈𝐿𝑆𝐶(𝐴) of equation 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 in 𝐴, we put𝑚=inf_{𝐵𝑅/𝜏∩𝜕𝐴}𝑢;𝑢⁻_𝑚(𝑥)=min𝑢(𝑥),𝑚,if𝑥∈𝐴,𝑚,if𝑥∉𝐴, (3.12)where 0<𝜏<1. Similarly, for a viscosity subsolution 𝑢∈USC(𝐴), we put𝑀=sup_{𝐵𝑅∩𝜕𝐴}𝑢⁺;𝑢⁺_𝑀𝑢(𝑥)=max⁺(𝑥),𝑀,if𝑥∈𝐴,𝑀,if𝑥∉𝐴. (3.13)Denote also by 𝑓⁺₀ and 𝑓⁻₀ the continuations of 𝑓⁺ and 𝑓⁻ vanishing outside 𝐴, respectively. Following [3, Proposition 2.8] and using the structure conditions (1.4) and (1.5), we have𝒫⁻_𝜆,Λ𝐷²𝑢⁻_𝑚−𝑏₀𝐷𝑢⁻_𝑚^𝑞≤𝑓⁺₀ (3.14)in 𝐵_𝑅/𝜏 for a viscosity supersolution 𝑢∈LSC(𝐴), and𝒫⁺_𝜆,Λ𝐷²𝑢⁺_𝜏+𝑏₀𝐷𝑢⁺_𝜏^𝑞≥−𝑓⁻₀ (3.15)in 𝐵_𝑅 for a viscosity subsolution 𝑢∈𝑈𝑆𝐶(𝐴).

Observe that, if 𝑓⁺=0 on 𝜕𝐴, then 𝑓⁺₀ is continuous, and then we can apply Theorem 3.3 to get a boundary wH inequality. Similarly, if 𝑓⁻=0 on 𝜕𝐴, we can use Theorem 3.4 to deduce a boundary local MP.

Nevertheless, even in the general case, when 𝑓⁺₀ and 𝑓⁻₀ are not necessarily continuous, we can get boundary estimates by means of an approximation process, as shown here below, where we use the notations defined just above.

Theorem 3.6 (boundary wH inequality). Let 𝑏₀≥0, 0<𝜏<1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝐿𝑆𝐶(𝐴) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤𝑓, under structure condition (1.4), with 𝑏(𝑥)≤𝑏₀, 𝑓∈𝐶(𝐴), and 0≤𝑢≤𝑁 in 𝐴. Then1|𝐵_𝑅|_𝐵𝑅𝑢⁻_𝑚^𝑝01/𝑝0≤𝐶inf_{𝐵𝑅∩𝐴}𝑢+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏∩𝐴)}, (3.16)with positive constants 𝐶 and 𝑝₀, depending on 𝑛,𝜆,Λ,𝑞,𝜏, and 𝑏₀𝑁^𝑞−1𝑅^2−𝑞.

Proof. For 𝜀>0, set𝑚_𝜀=inf_{𝐼𝜀(𝜕𝐴)}𝑢,𝐼_𝜀(𝜕𝐴)=𝑥∈𝐵_𝑅/𝜏∩𝐴∶dist(𝑥,𝜕𝐴)≤𝜀, (3.17)and, for 𝑥∈𝐵_𝑅/𝜏,𝑢⁻_𝑚𝜀(𝑥)=min𝑢(𝑥),𝑚_𝜀𝑚,if𝑥∈𝐴,_𝜀𝑓,if𝑥∉𝐴,_𝜀(𝑥)=𝑓⁺(𝑥)𝜌dist(𝑥,ℝ^𝑛⧵𝐴)𝜀, (3.18)where𝜌(𝑡)=𝑡,if0≤𝑡<1,1,if𝑡≥1. (3.19)It is easy to check that 𝑢⁻_𝑚𝜀∈𝐿𝑆𝐶(𝐵_𝑅/𝜏), 0≤𝑢⁻_𝑚𝜀≤𝑁, 𝑓_𝜀∈𝐶(𝐵_𝑅/𝜏), and𝒫⁻_𝜆,Λ𝐷²𝑢⁻_𝑚𝜀−𝑏₀𝐷𝑢⁻_𝑚𝜀^𝑞≤𝑓_𝜀(𝑥) (3.20)in 𝐵_𝑅/𝜏. Therefore, we can apply Theorem 3.3 with 𝑢⁻_𝑚𝜀 instead of 𝑢 and 𝑓_𝜀 instead of 𝑓 to get1|𝐵_𝑅|_𝐵𝑅𝑢⁻_𝑚𝜀^𝑝01/𝑝0≤𝐶inf_𝐵𝑅𝑢⁻_𝑚𝜀+𝑅𝑓_{𝜀𝐿𝑛(𝐵𝑅/𝜏)}. (3.21)Note that inf_𝐵𝑅𝑢⁻_𝑚𝜀≤inf_{𝐵𝑅∩𝐴}𝑢 and 0≤𝑓_𝜀≤𝑓⁺ in 𝐴, 𝑓_𝜀=0 outside 𝐴. Also observing that, by lower semicontinuity,𝑚≤liminf_𝜀→0𝑚_𝜀 (3.22)and therefore, by Fatou's lemma,_𝐵𝑅𝑢⁻_𝑚^𝑝0≤liminf_{𝜀→0𝐵𝑅}𝑢⁻_𝑚𝜀^𝑝0, (3.23)from inequality (3.21) we get the assert.

In the sequel, we will make also use of a version of boundary wH inequality for annular regions 𝐵_{𝜀𝑅,𝑅}=𝐵_𝑅⧵𝐵_𝜀𝑅(0), 0<𝜀<1, which can be deduced by Theorem 3.6 reasoning as in [10, Theorem 3.1].

In this case 𝑚=inf_{𝜕𝐴∩𝐵′𝑅𝜀𝜏𝑅,𝜏}𝑢, where 0<𝜀≤1/2, 0<𝜏<1, 𝜏^′>1.

Corollary 3.7 (boundary wH inequality). Let 0<𝜏<1, 𝜏^′>1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝐿𝑆𝐶(𝐴) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤𝑓, under structure condition (1.4), with 𝑓∈𝐶(𝐴), and 0≤𝑢≤𝑁 in 𝐴. Then1|𝐵_{𝜀𝑅,𝑅}|_{𝐵𝜀𝑅,𝑅}𝑢⁻_𝑚^𝑝01/𝑝0≤𝐶inf_{𝐴∩𝐵𝜀𝑅,𝑅}𝑢+𝑅‖𝑓‖_𝐿^𝑛_{(𝐴∩𝐵′𝑅𝜀𝜏𝑅,𝜏)}, (3.24)with positive constants 𝐶 and 𝑝₀, depending on 𝑛,𝜆,Λ,𝑞,𝜏,𝜏^′, and 𝑁^𝑞−1𝑅^2−𝑞‖𝑏‖_𝐿^∞_{(𝐴∩𝐵𝜀𝜏𝑅,𝑅/𝜏)}.

In a similar manner, we extend the local MP up to the boundary.

Theorem 3.8 (boundary local MP). Let 𝑏₀≥0, 0<𝜏<1, 𝑁>0, and 1≤𝑞≤2. Suppose that 𝑢∈𝑈𝑆𝐶(𝐴) is a viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≥𝑓, under structure condition (1.5), with 𝑓∈𝐶(𝐴) and 𝑢≤𝑁 in 𝐴. Then for all 𝑝>0,sup_{𝐵𝜏𝑅∩𝐴}1𝑢≤𝐶|𝐵_𝑅|_𝐵𝑅𝑢⁺_𝑀^𝑝1/𝑝+𝑅𝑓⁻_{𝐿𝑛(𝐵𝑅∩𝐴)}, (3.25)with a positive constant 𝐶, depending on 𝑛,𝜆,Λ,𝑞,𝜏,𝑏₀𝑁^𝑞−1𝑅^2−𝑞, and 𝑝.

4. ABP-Type Estimates and Maximum Principles

Here we use boundary estimates of previous section to obtain MP in unbounded domains Ω of 𝑅^𝑛 for viscosity subsolutions 𝑢∈𝑈𝑆𝐶(Ω), bounded above, of equation 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=0 under structure condition (1.5).

We will make use of the measure-geometric condition 𝐆_𝜎, 0<𝜎<1, given in the introduction. By a continuity argument, see [12], eventually passing to a smaller 𝑅, which we will call 𝑟_𝑦, we can assume that condition 𝐆_𝜎 is satisfied with |𝐵⧵Ω_𝑦|=𝜎|𝐵| exactly.

We also recall that Ω is a 𝐰𝐆 domain (with parameter 𝜎) if each point 𝑦∈Ω satisfies condition 𝐆_𝜎 in Ω. In particular, if 𝑅(𝑦) is the radius of the ball 𝐵=𝐵(𝑦) provided by condition 𝐆_𝜎, we define domains of cylindrical and conical type as 𝐰𝐆 domains such that 𝑅(𝑦)=𝑂(1) and 𝑅(𝑦)=𝑂(|𝑦|), respectively as |𝑦|→∞.

4.1. Domains of Cylindrical Type

We start with the condition 𝐆 of Cabré [10]. Let 𝜎<1, 𝜏<1, and 𝑅₀ be positive real numbers. We say that an open connected set Ω of 𝑅^𝑛 is a 𝐆 domain if to each 𝑦∈Ω we can associate a ball 𝐵=𝐵_𝑅(𝑥_𝑦) of radius 𝑅≤𝑅₀ such that𝑦∈𝐵_𝜏𝑅𝑥_𝑦,𝐵⧵Ω_𝑦≥𝜎|𝐵|,𝐆_𝜎,𝜏 (4.1)where Ω_𝑦 is the connected component of Ω∩𝐵 containing 𝑦.

Since 𝐆_𝜎≡𝐆_𝜎,𝟏, then a 𝐆 domain of 𝑅^𝑛 is of cylindrical type, like domains of finite Lebesgue 𝑛-dimensional measure, subdomains of 𝜔×𝑅^𝑛−𝑘, where 𝜔 has finite Lebesgue 𝑘-dimensional measure, the complement of the spiral of equation 𝑟=𝜃 in polar coordinates of 𝑅².

Given a differential operator with structure conditions, like (1.4) and (1.5), Ω will be called a narrow domain when, for given 𝜏 and 𝑅₀, condition 𝐆_𝜎,𝜏 is satisfied for 𝜎 suitably close to 1, depending on the structure constants and the remaining geometric constants.

A straightforward application of Theorem 3.8 yields MP in narrow domains. Indeed, assume that 𝑢≤𝑁 and 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≥0 in Ω. Then, by (1.5), we have𝒫⁺𝐷²𝑢⁺+𝑏₀𝐷𝑢^+𝑞≥0. (4.2)Suppose that 𝑢≤0 on 𝜕Ω and set 𝑀=sup_Ω𝑢⁺. Applying Theorem 3.8 in 𝐴=Ω_𝑦 with 𝑝=1, we obtain𝑢(𝑦)≤sup_{Ω𝑦∩𝐵𝜏𝑅(𝑥𝑦)}𝐶𝑢≤|𝐵|_{Ω𝑦∩𝐵}𝑢⁺≤𝐶𝑀|Ω_𝑦∩𝐵||𝐵|≤𝐶𝑀(1−𝜎). (4.3)From this, taking the supremum over 𝑦∈Ω, we get 𝑀≤0, that is 𝑢≤0 in Ω, provided 𝜎>1−1/𝐶, and hence MP holds in this case.

In order to pass from narrow domains to arbitrary cylindrical domains we will use Theorem 3.6, from which the following ABP-type estimate follows.

Theorem 4.1 (ABP estimate). Let 𝜎,𝜏<1, let 𝑅₀ and 𝑁 be positive real numbers, and 1≤𝑞≤2. Let Ω be a cylindrical domain such that condition 𝐆_𝜎 in Ω is satisfied at each 𝑦∈Ω with 𝑅(𝑦)≤𝑅₀.
Suppose that 𝑤∈𝑈𝑆𝐶(Ω) is a viscosity solution of 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)≥𝑓, under the structure condition (1.5), with 𝑏≤𝑏₀ and 𝑓∈𝐶(Ω).
If 𝑤≤𝑁 in Ω and 𝑤≤0 on 𝜕Ω, thensup_Ω𝑤≤𝐶𝑅₀sup_𝑦∈Ω‖𝑓‖_𝐿^𝑛_{(𝐵𝑅(𝑦)/𝜏(𝑥𝑦)∩Ω)}, (4.4)where 𝐶 depends on 𝑛,𝜆,Λ,𝜎,𝜏, and 𝑏₀𝑅₀^2−𝑞𝑁^𝑞−1.

Proof. It is enough to show the result for 𝜏→1⁻.
Set 𝑀=sup_Ω𝑤⁺ and 𝑢=𝑀−𝑤. Let 𝑦∈Ω and 𝐵=𝐵_𝑅 of radius 𝑅, provided by condition 𝐆_𝜎 in 𝑦. We choose 𝑅=𝑟_𝑦 such that |𝐵⧵Ω_𝑦|=𝜎|𝐵|; see the beginning of this section. We also denote by 𝐵_𝜏𝑅 the concentric ball of radius 𝜏𝑅.
Now we apply Theorem 3.6 to 𝑢 in 𝐴=Ω_𝑦 with 𝐵_𝜏𝑅(𝑥_𝑦) instead of 𝐵_𝑅 and 𝜏 close enough to 1 in such a way that |𝐵_𝜏𝑅(𝑥_𝑦)⧵Ω_𝑦|≥(𝜎/2)|𝐵| and |𝐵_𝜏𝑅(𝑥_𝑦)∩Ω_𝑦|≥((1−𝜎)/2)|𝐵|. Since 𝑤≤0 on 𝜕Ω, then 𝑚≥𝑀, hence we get𝜎2^1/𝑝01𝑀≤|𝐵_𝜏𝑅(𝑥_𝑦)|_{𝐵𝜏𝑅(𝑥𝑦)}𝑢⁻_𝑚^𝑝01/𝑝0≤𝐶𝑀−sup_{𝐵𝜏𝑅(𝑥𝑦)∩Ω𝑦}𝑤+𝑅‖𝑓‖_𝐿^𝑛_(𝐵∩Ω), (4.5)from which, for 𝑥∈𝐵_𝜏𝑅(𝑥_𝑦)∩Ω_𝑦, we obtain the pointwise inequality𝑤(𝑥)≤sup_{𝐵𝜏𝑅(𝑥𝑦)∩Ω𝑦}𝑤≤𝑡𝑀+𝑅‖𝑓‖_𝐿^𝑛_(𝐵∩Ω), (4.6)with 0<𝑡<1. On the other hand, setting 𝐾=𝑡𝑀+𝑅‖𝑓‖_𝐿^𝑛_(𝐵∩Ω) and Ω_𝐾={𝑥∈Ω/𝑤(𝑥)>𝐾}, by virtue of (4.6) we have 𝐵⧵Ω_𝐾⊃𝐵_𝜏𝑅(𝑥_𝑦)∩Ω_𝑦 and therefore, by our choice of 𝑅 and 𝜏,|𝐵⧵Ω_𝐾|≥|𝐵||𝐵_𝜏𝑅(𝑥_𝑦)∩Ω_𝑦|≥|𝐵|1−𝜎2. (4.7)A further application of Theorem 3.6 to 𝑢=𝑀−𝑤 in 𝐴=Ω_𝐾 yields1−𝜎2^1/𝑝01(𝑀−𝐾)≤|𝐵|_𝐵𝑢⁻_𝑚^𝑝01/𝑝0≤𝐶𝑀−sup_{𝐵∩Ω𝐾}𝑤+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏(𝑥𝑦)∩Ω)}, (4.8)since in this case 𝑚≤𝑀−𝐾. From this we deduce that, for 𝑥∈𝐵∩Ω_𝐾𝑤(𝑥)≤sup_{𝐵∩Ω𝐾}𝑤≤1−𝑡^′𝑀+𝑡^′𝐾+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏(𝑥𝑦)∩Ω)}≤(1−𝑡^′+𝑡𝑡^′)𝑀+2𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏(𝑥𝑦)∩Ω)}, (4.9)with 0<𝑡^′<1. From the definition of Ω_𝐾, see also (4.6), it follows that𝑤(𝑥)≤𝑡^′′𝑀+2𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/𝜏(𝑥𝑦)∩Ω)}, (4.10)with 0<𝑡^′′<1, for all 𝑥∈Ω∩𝐵 and hence also for 𝑥=𝑦.
Finally, passing to the supremum over 𝑦∈Ω, we get the result.

4.2. General Domains

Firstly, we consider 𝐰𝐆 domains Ω, such that condition 𝐆_𝜎 in Ω holds at each 𝑦∈Ω without bounds for the radii 𝑅(𝑦) of the balls provided by 𝐆_𝜎.

Note that in general the ABP-type estimate of Theorem 4.1 is useless unless 𝑏₀=0, see [13], since the the constant 𝐶 of ABP estimate blows up when 𝑅→+∞. This is why we assume 𝑏(𝑥)=𝑂(1/|𝑥|^2−𝑞) as |𝑥|→+∞ in the structure condition (1.5). Moreover, to take advantage from the decay of 𝑏(𝑥), it is convenient to use the boundary wH inequality for annular regions of Corollary 3.7 rather than Theorem 3.6.

Reasoning as in the proof of Theorem 4.1, but quite more carefully with the aid of (3.24) instead of (3.16), see [16], we get the following ABP-type estimate.

Theorem 4.2 (ABP). Let 𝜎 and 𝑁 be positive real numbers and 1≤𝑞≤2. Let Ω be a 𝐰𝐆 domain (with parameter 𝜎<1).
Suppose that 𝑤∈𝑈𝑆𝐶(Ω) is a viscosity solution of 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)≥𝑓, under the structure condition (1.5), with 𝑏,𝑓∈𝐶(Ω) such that𝑏_𝑞∶=sup_{𝑦∈Ω;|𝑦|≥𝜀𝑟𝑦}𝑟_𝑦^2−𝑞‖𝑏‖_𝐿^∞_{(Ω∩𝐵𝑦′𝑟𝑦𝜏𝜀𝑟,𝜏)}<+∞, (4.11)for all 𝜀>0 small enough, all 𝜏<1 sufficiently close to 1, and some 𝜏^′>1.
If 𝑤≤𝑁 in Ω and 𝑤≤0 on 𝜕Ω, thensup_Ω𝑤≤𝐶sup_{𝑦∈Ω;|𝑦|≥𝜀𝑟𝑦}𝑟_𝑦𝑓⁻_{𝐿𝑛(Ω∩𝐵𝑦′𝑟𝑦𝜏𝜀𝑟,𝜏)}+sup_{𝑦∈Ω;|𝑦|≤𝜀𝑟𝑦}𝐶_𝑦𝑟_𝑦𝑓⁻_{𝐿𝑛(Ω∩𝐵𝑦𝜀𝑟)} (4.12)for possibly smaller 𝜀>0 and larger 𝜏<1, depending on 𝑛 and 𝜎.
Here 𝐶 and 𝐶_𝑦 are positive constants depending on 𝑛, 𝑞, 𝜆, Λ, 𝑏_𝑞𝑁^𝑞−1, 𝜎, 𝜀, 𝜏, 𝜏^′, while 𝐶_𝑦 also depends on 𝑁^𝑞−1𝑟_𝑦^2−𝑞‖𝑏‖_𝐿^∞_{(Ω∩𝐵𝑦𝜀𝑟)}.

Proof of Theorem 1.2. In the case of 𝐰𝐆 domains, Theorem 1.2 follows at once letting 𝑓=0 in Theorem 4.2. Suppose now that Ω can be split by a closed set 𝐻⊂𝑅^𝑛 in components where MP holds and each 𝑦∈𝐻 satisfies condition 𝐆_𝜎 in Ω. By MP in the components, since we assume that 𝑤≤0 on 𝜕Ω, then for 𝑥∈Ω we have𝑤(𝑥)≤sup_{(Ω∩𝐻)∪𝜕Ω}𝑤⁺=sup_Ω∩𝐻𝑤⁺. (4.13)Reasoning as above for (4.10), but using Corollary 3.7 instead of Theorem 3.6 as before to obtain Theorem 4.2, from condition 𝐆_𝜎 we deduce for 𝑦∈Ω∩𝐻 that𝑤(𝑦)≤𝑡sup_Ω𝑤⁺, (4.14)where 𝑡∈]0,1[ is independent of 𝑦. Inserting this inequality in the former one, and taking the supremum over Ω, we get the result.

Examples
Provided that 𝑏(𝑥)=𝑂(1/|𝑥|^2−𝑞) as |𝑥|→∞, this last result yields MP in very general domains such as, for instance: (i)𝐰𝐆 domains, like a proper cone Ω such that Ω≠𝑅^𝑛 and in general a domain of conical type, like the complement Ω in 𝑅^𝑛 of Γ×𝑅^𝑛−2, where Γ is a logarithmic spiral of equation 𝑟=𝑒^𝜃 in polar coordinates, or also complement of a larger spiral of equation 𝑟=𝑠(𝜃), with 𝑠 a positive increasing function.(ii)Domains which can be split in 𝐰𝐆 subdomains by a suitable closed set 𝐻 of 𝑅^𝑛, like the cut plane in 𝑅² or in general the complement in 𝑅^𝑛 of a graph {(𝑥,𝑦)∈𝑅^𝑛−1×𝑅|𝑥_𝑖≥0,𝑖=1,…,𝑛−1,𝑦=𝑓(𝑥)} such that |𝑓(𝑥)|≤ℎ+𝑘|𝑥| for positive constants ℎ and 𝑘.
As a further example, we show a repeated application of Theorem 1.2. Look at the complement Ω in 𝑅² of a sequence of balls 𝐵_𝑟(𝑘), 𝑘=(𝑘_𝑥,0),𝑘_𝑥∈𝑁, with 0<𝑟<1/3. Consider the nonnegative 𝑥 -axis as 𝐻, then Ω_𝐻=Ω⧵𝐻 is connected. If 𝐾 is the half-line of equations 𝑦=(1/2)𝑥, 𝑥≥0, then we have the following: (i)Ω_𝐻⧵𝐾 has two components, which are domains of conical type, where MP holds;(ii)each point of Ω_𝐻∩𝐾 satisfies condition 𝐆_1/2 in Ω_𝐻. Thus MP holds in Ω⧵𝐻 by Theorem 1.2. Also, each point of 𝐻 satisfies condition 𝐆_𝜎 in Ω for some 𝜎∈]0,1[ depending on 𝑟. Therefore, again by Theorem 1.2, we conclude that MP holds in Ω.

4.3. Parabolic Shaped Domains

For a parabolic shaped 𝐰𝐆 domain, condition 𝐆_𝜎 at 𝑦∈Ω holds with 𝑅(𝑦)=𝑂(|𝑦|^𝛼) as |𝑦|→∞, for some 0≤𝛼≤1, the limit cases 𝛼=0 and 𝛼=1 representing, respectively, the cylindrical and the conical cases. Hence 𝑟_𝑦≤𝑅(𝑦)≤𝑅₀+𝛽|𝑦|^𝛼 for all 𝑦∈Ω with positive constants 𝑅₀ and 𝛽. Then, choosing 𝜀 sufficiently small in Theorem 4.2, if |𝑦|≤𝜀𝑟_𝑦, then|𝑦|≤𝜀𝑅₀+𝜀𝛽|𝑦|^𝛼≤𝜀𝑅₀+𝛽+𝜀𝛽|𝑦|, (4.15)so that the supremum in the second term of the right-hand side of (4.12) is taken over a bounded subset of 𝑦∈Ω, in which 𝑟_𝑦≤𝑅₁ for some positive constant 𝑅₁. Thussup_{𝑦∈Ω;|𝑦|≤𝜀𝑟𝑦}𝑟_𝑦𝑓⁻_{𝐿𝑛(Ω∩𝐵𝑦𝜀𝑟)}≤𝑅₁sup_{𝑦∈Ω;𝑟𝑦≤𝑅1}𝑓⁻_{𝐿𝑛(Ω∩𝐵𝑦𝜀𝑟)}. (4.16)

Proof of Theorem 1.3. Since condition 𝐰𝐆 holds with 𝑟_𝑦=𝑂(|𝑦|^𝛼), 0≤𝛼≤1, the assumption 𝑏(𝑥)=𝑂(1/|𝑥|^{𝛼(2−𝑞)}) as |𝑥|→+∞ implies the finiteness of 𝑏_𝑞 in (4.11). Taking account of (4.16), by continuity of 𝑓 the estimate (1.10) follows letting 𝜀→0.

5. Strong Maximum Principle and Liouville Theorem

The weak Harnack inequality of Theorem 3.3 can be used to show the following strong MP.

Theorem 5.1 (strong MP). Let Ω be a domain of 𝑅^𝑛. Let 𝑤∈𝐶(Ω) be such that 𝐹(𝑥,𝑤,𝐷𝑤,𝐷²𝑤)≥0 in the viscosity sense, and assume structure condition (1.5), with 𝑏∈𝐶(Ω). If 𝑥₀∈Ω and 𝑀∶=𝑤(𝑥₀)≥𝑤(𝑥), for all 𝑥∈Ω, then 𝑤≡𝑀 in Ω.

Proof. Following [33], set Ω₁=𝑤⁻¹({𝑀}) and Ω₂=Ω⧵Ω₁. By assumption Ω₁≠∅. By continuity of 𝑤, it turns out that Ω₂=𝑤⁻¹(]−∞,𝑀[) is an open subset of 𝑅^𝑛. Moreover, plainly, Ω=Ω₁∪Ω₂ and Ω₂∩Ω₁=∅.
Recall that Ω is an open connected set. Thus it is sufficient to show that Ω₁ is in turn an open subset to have Ω=Ω₁, as claimed in the statement of the theorem. Indeed, let 𝑥₁∈Ω₁, that is, 𝑤(𝑥₁)=𝑀, and set 𝑢=𝑀−𝑤, then 𝑢 is a nonnegative viscosity solution of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤0. Applying (3.7) in a ball 𝐵_𝑅∶=𝐵_𝑅(𝑥₁)⊂𝐵_𝑅/𝜏(𝑥₁)⊂⊂Ω, we get1|𝐵_𝑅|_𝐵𝑅(𝑀−𝑤)^𝑝01/𝑝0≤𝐶inf_𝐵𝑅𝑢=0, (5.1)from which, by continuity, 𝑢≡𝑀 in 𝐵_𝑅(𝑥₁). This shows that Ω₁ is an open subset of 𝑅^𝑛 and concludes the proof.

The Liouville type result of Theorem 1.5 is instead based on Harnack inequality (3.11) of Theorem 3.5. It is convenient to consider its version in annular regions 𝐵_𝑅,2𝑅=𝐵_2𝑅(0)⧵𝐵_𝑅(0) to take advantage of the decay of 𝑏(𝑥), obtained in standard way, using inequality (3.11) in a chain of linked balls. This yields, for continuous solutions 𝑢∈𝐶(𝐵_𝑅/2,4𝑅) of equation 𝐹=𝑓, 0≤𝑢≤𝑁, under the structure conditions (1.4) and (1.5), with 𝑏,𝑓∈𝐶(𝐵_𝑅/2,4𝑅), the following inequality:sup_{𝐵𝑅,2𝑅}𝑢≤𝐶inf_{𝐵𝑅,2𝑅}𝑢+𝑅‖𝑓‖_𝐿^𝑛_{(𝐵𝑅/2,4𝑅)}, (5.2)with a positive constant 𝐶=𝐶(𝑛,𝜆,Λ,𝑞,𝜏,‖𝑏‖_𝐿^∞_{(𝐵𝑅/2,4𝑅)}𝑁^𝑞−1𝑅^2−𝑞).

Proof of Theorem 1.5. By the strong MP of Theorem 5.1, we know that 𝑤 can achieve neither a maximum nor a minimum at a point of 𝑅^𝑛 unless it is constant, in which case we should be done.
Suppose for instance that 𝑤≤𝑀∶=sup𝑤<+∞. Let 𝑅_𝑘 be an increasing sequence of positive numbers such that lim_𝑘→∞𝑅_𝑘=∞. Set 𝑀_𝑘=sup_{𝜕𝐵𝑅𝑘}𝑤 and 𝑚_𝑘=inf_{𝜕𝐵𝑅𝑘}𝑤. By weak maximum principle, 𝑀_𝑘 is increasing and 𝑚_𝑘 is decreasing; thuslim_𝑘→∞𝑀_𝑘=𝑀,lim_𝑘→∞𝑚_𝑘=𝑚∈[−∞,+∞[. (5.3)Then, using Harnack inequality (5.2), with 𝑢=𝑀−𝑤, we get𝑀−𝑚_𝑘=sup_{𝜕𝐵𝑅𝑘}(𝑀−𝑤)≤𝐶inf_{𝜕𝐵𝑅𝑘}(𝑀−𝑤)=𝐶𝑀−𝑀_𝑘, (5.4)from which𝑀≤𝐶𝑀−𝑀_𝑘+𝑚_𝑘≤𝐶𝑀−𝑀_𝑘+𝑚_𝑘 (5.5)and, letting 𝑘→∞, we get 𝑀=𝑚, as we wanted to show.

Appendix

Proof of Lemmas 2.2 and 2.3

Although the proof of Lemma 2.2 is already contained in previous papers also in the case of an almost-quadratic gradient term, see for instance [17], here, for the sake of completeness, we give a sketch of the simple version in the case of linear gradient term, following [3], where the fundamental case of a second-order uniformly elliptic operator is treated, with no lower-order terms.

However, it seems useless to repeat the nice proof of [3], to which we refer for the ideas and details. We only outline the steps which are influenced by the first-order term. For this reason, we keep the same notations of [3].

Also, for the sake of brevity, we will refer to constants depending only on 𝑛,𝜆,Λ,𝑏₀ as to structural constants.

Firstly, we introduce a test function; see [3, Lemma 4.1].

Lemma 5. There exist positive structural constants 𝑀, 𝐶 and a function 𝜑∈𝐶²(𝑅^𝑛) such that𝜑≤−2𝑖𝑛𝐵_√(3/2)𝑛,𝜑≥0𝑖𝑛ℝ^𝑛⧵𝐵_2√𝑛,𝜑≥−𝑀𝑖𝑛ℝ^𝑛,𝒫⁺_𝜆,Λ(𝐷²𝜑)+𝑏₀𝐷𝜑≤𝐶𝜉𝑖𝑛ℝ^𝑛, (A.1)(A.2)(A.3)where 𝜉∈𝐶(𝑅^𝑛), 0≤𝜉≤1, 𝑠𝑢𝑝𝑝𝜉⊂𝐵_1/2.

Proof. We search for a function of type 𝜑(𝑥)=𝐴₁−𝐴₂𝑒^−𝛼𝑟, for 𝑟=|𝑥|≥1/4, where 𝐴₁ and 𝐴₂ are positive constants to be chosen in order that 𝜙((3/2)√𝑛)=−2 and 𝜙(2√𝑛)=0. Next, we extend 𝜑 to 𝑅^𝑛 in such a way that 𝜑≥−𝑀. By calculations, choosing 𝛼=4Λ(𝑛−1)+1/𝜆 we have𝒫⁺_𝜆,Λ𝐷²𝜑+𝑏₀𝐷𝜑=𝛼𝐴₂𝑒^−𝛼𝑟Λ(𝑛−1)𝑟−𝛼𝜆+1≤0 (A.4)for 𝑟≥1/4. Also, for 𝑟≤1/4,𝒫⁺_𝜆,Λ𝐷²𝜑+𝑏₀𝐷𝜑≤𝐶 (A.5)and therefore (A.3) holds taking a cut-off function 𝜉∈𝐶(𝑅^𝑛) such that 𝜉=1 in 𝐵_1/4 and 𝜉=0 outside 𝐵_1/2.

Next, we get a lower bound for the size of level sets of supersolutions. Denoting by 𝑄_𝑙 a cube of side 𝑙, consider a nonnegative viscosity solution 𝑢∈𝐿𝑆𝐶(𝑄_4√𝑛) of the differential inequality 𝒫⁻_𝜆,Λ(𝐷²𝑢)−𝑏₀|𝐷𝑢|≤𝑓.

Setting 𝑤=𝑢+𝜑 and observing that𝒫⁻_𝜆,Λ𝐷²𝑤−𝑏₀|𝐷𝑤|≤𝒫⁻_𝜆,Λ𝐷²𝑢+𝒫⁺_𝜆,Λ𝐷²𝜑−𝑏₀|𝐷𝑢|+𝑏₀𝐷𝜑≤𝑓⁺+𝐶𝜉, (A.6)a positive lower bound{𝑢≤𝑀}∩𝑄₁>𝜇, (A.7)with a structural positive constant 𝜇<1, see (A.2), follows, in the same way as in [3, Lemma 4.5], using ABP estimate (2.5), provided that inf_𝑄3𝑢≤1 and ‖𝑓⁺‖_𝐿^𝑛_{(𝑄4√𝑛)}≤𝜀₀ for a positive structural constant 𝜀₀. Moreover, under the same assumptions, [3, Lemma 4.6] says that𝑢>𝑀^𝑘∩𝑄₁≤(1−𝜇)^𝑘 (A.8)for all 𝑘∈𝑁, which for 𝑘=1 agrees with (A.7). Then we point out that (A.8) follows by an induction process, based on the Calderón-Zygmund decomposition of the cube 𝑄₁, centered at the origin, supposing (A.8) to hold for 𝑘−1.

To perform the induction step it is crucial that, for a supersolution 𝑢(𝑥) of (2.4), the rescaled function 𝑢(𝑦)=𝑢(𝑥)/𝑀^𝑘−1, where 𝑥=𝑥₀+2^−𝑖𝑦 runs in the dyadic cube 𝑄₂^−𝑖_4√𝑛, centered at 𝑥₀, is in turn a supersolution of (2.4) with a correspondingly scaled 𝑓, namely𝒫⁻_𝜆,Λ𝐷²𝑢(𝑦)−𝑏₀𝐷𝑢≤𝑓(𝑦)⁺(𝑥)2^2𝑖𝑀^𝑘−1 (A.9)for 𝑦∈𝑄_4√𝑛. From (A.8) it follows that{𝑢>𝑡}∩𝑄₁≤𝑑𝑡^−𝜀 (A.10)for all 𝑡>0, with 𝑑 and 𝜀 positive structural constants. Then, following the proof of Theorem 4.8(1) of [3], we use (A.8) in the identity_𝑄1𝑢^𝑝0=𝑝₀₀^+∞{𝑢≥𝑡}∩𝑄₁𝑑𝑡, (A.11)see [1], with 𝑝₀=𝜀/2 and, by rescaling, remove the normalization conditions inf_𝑄3𝑢≤1 and ‖𝑓⁺‖_𝐿^𝑛_{(𝑄4√𝑛)}≤𝜀₀ to get‖𝑢‖_𝐿^𝑝_0(𝑄1)≤𝐶inf_𝑄3𝑢+𝑓⁺_{𝐿𝑛(𝑄4√𝑛)}. (A.12)From this, with a covering argument as in [10, Theorem 3.1], we obtain (2.8).

We argue in the same manner for Lemma 2.3. Suppose again that 𝑄₁ is centered at the origin. Following the proof of Theorem 4.8(2) of [3], firstly we consider a subsolution 𝑢 of (2.6) such that ‖𝑓⁺‖_𝐿^𝑛_{(𝑄4√𝑛)}≤𝜀₀ and ‖𝑢⁺‖_𝐿^𝜀_(𝑄1)≤𝑑^1/𝜀 to get, even in this case, (A.10). Then, arguing as in [3, Lemma 4.7], there exist structural constants 𝑀₀>1 and 𝜎>0 such that, for all 𝑗∈𝑁 large enough,𝑥₀<14,𝑢𝑥₀≥𝜈^𝑗−1⟹𝑄^𝑗∶=𝑄_𝑙𝑗𝑥₀⊂𝑄₁,sup_𝑄𝑗≥𝜈^𝑗𝑀₀, (A.13)where 𝜈=𝑀₀/(𝑀₀−1/2) and 𝑙_𝑗=𝜎𝑀₀^−𝜀/𝑛𝜈^{−𝜀𝑗/𝑛}. As above, to get this result we use the invariance of equation by scale transformations, namely that the function 𝑣(𝑦)=𝜈/(𝜈−1)−𝑢(𝑥)/𝜈^𝑗−1(𝜈−1)𝑀₀, where 𝑥=𝑥₀+(4√𝑛)⁻¹𝑙_𝑗𝑦 runs in the small cube 𝑄^𝑗, is in turn a supersolution of (2.4) with a correspondingly scaled 𝑓, that is,𝒫⁻_𝜆,Λ𝐷²𝑣(𝑦)−𝑏₀𝐷𝑣≤𝑓(𝑦)⁻(𝑥)𝜈^𝑗−1(𝜈−1)𝑀₀ (A.14)for 𝑦∈𝑄_4√𝑛, provided that 𝑗>1+log(2−1/𝑀₀)/log𝜈.

On the base of (A.13), reasoning as in the proof of Lemma 4.4 of [3], we infer that sup_𝑄1/4𝑢≤𝐶, from which, by rescaling to remove normalization conditions ‖𝑓⁺‖_𝐿^𝑛_{(𝑄4√𝑛)}≤𝜀₀ and ‖𝑢⁺‖_𝐿^𝜀_(𝑄1)≤𝑑^1/𝜀, we getsup_𝑄1/4𝑢≤𝐶𝑢⁺_{𝐿𝜀(𝑄1)}+𝑓⁺_{𝐿𝑛(𝑄4√𝑛)}. (A.15)as in the proof of Theorem 4.8 (2) of [3]. By a covering argument, as above for supersolutions, we get (2.9) for 𝑝=𝜀. Note that (A.10) a fortiori holds replacing 𝜀 with 𝑝<𝜀. Thus (2.9) follows for all 0<𝑝<𝜀. Finally, by Hölder inequality, we obtain (2.9) for all 𝑝>0.