Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity

Abstract

We deal with fully nonlinear second-order equations assuming a superlinear growth in u with the aim to generalize previous existence and uniqueness results of viscosity solutions in the whole space without conditions at infinity. We also consider the solvability of the Dirichlet problem in bounded and unbounded domains and show a blow-up result.

1. Introduction and Statement of the Main Results

We are concerned with the well-posedness of the fully nonlinear second-order uniformly elliptic problem 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inℝ^𝑛, (1.1) with no limitation on the growth of 𝑓 and no condition on the behaviour of 𝑢 at infinity.

We will assume the following standard structure condition, which implies the uniform ellipticity: 𝒫⁻_𝜆,Λ(𝑌−𝑋)−𝛾𝜂−𝜉≤𝐹𝑥,𝑢,𝜂,𝑌−𝐹(𝑥,𝑢,𝜉,𝑋)≤𝒫⁺_𝜆,Λ(𝑌−𝑋)+𝛾𝜂−𝜉 (1.2) for 𝑥∈ℝ^𝑛 and any 𝑢∈ℝ,𝜉,𝜂∈ℝ^𝑛,𝑋,𝑌∈𝒮^𝑛.

As regards the monotonicity in the variable 𝑢, we ask something more than the usual monotonicity assumption (𝛿=0): 𝐹(𝑥,𝑢,𝜉,𝑋)−𝐹(𝑥,𝑣,𝜉,𝑋)≤−𝛿(𝑢−𝑣)^𝑠if𝑣<𝑢 (1.3) for 𝑠>1 and 𝛿>0.

We also suppose, as generally with the formulation (1.1), that 𝐹(𝑥,0,0,0)=0. (1.4)

We collect the above assumptions in the structure condition (𝐒𝐂)∶=(1.2)and(1.3)and(1.4), (1.5) noting that 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝒫⁺_𝜆,Λ𝐷²𝑢+𝛾|𝐷𝑢|−|𝑢|^𝑠−1𝑢 (1.6) fits into our framework.

Hence, this paper is in the wake of Brezis [1], who proved the existence and uniqueness of distributional solutions 𝑢∈𝐿^𝑠_loc(ℝ^𝑛) for the semilinear equation Δ𝑢−|𝑢|^𝑠−1𝑢=𝑓(𝑥)inℝ^𝑛 (1.7) with 𝑓∈𝐿¹_loc(ℝ^𝑛) and of Esteban et al. [2] for the case of 𝐿^𝑛-viscosity solutions of the fully nonlinear second order uniformly elliptic equation 𝐹𝐷²𝑢−|𝑢|^𝑠−1𝑢=𝑓(𝑥) (1.8) with 𝑓∈𝐿^𝑛_loc(ℝ^𝑛).

Our aim is to extend the result in different directions, including lower-order terms, allowing the dependence on 𝑥 and going below the exponent 𝑛.

Throughout the paper 𝜆 and Λ are positive constants such that Λ≥𝜆, called ellipticity constants, and 𝛾≥0 will play the role of a Lipschitz constant.

Also, 𝑝₀=𝑝₀(𝑛,Λ/𝜆)∈(𝑛/2,𝑛) is the exponent such that for 𝑝>𝑝₀ the generalized maximum principle (GMP) holds true; see Escauriaza [3], Crandall and Swiech [4], and Koike and Swiech [5]: if 𝑓∈𝐿^𝑝(Ω) with 𝑝>𝑝₀ and 𝑢∈𝑊^2,𝑝_loc(Ω)∩𝐶(Ω) is an 𝐿^𝑝-strong solution of the maximal equation 𝒫⁺_𝜆,Λ𝐷²𝑢+𝛾|𝐷𝑢|≥𝑓, (1.9) then max_Ω𝑢≤max_𝜕Ω𝑢+𝐶𝑑^{2−(𝑛/𝑝)}𝑓⁻_𝐿𝑝(Ω) (1.10) with 𝑑=diam(Ω) and 𝐶 being a positive constant depending on 𝑛,𝜆,Λ,𝑝,𝛾𝑑.

Here and in the sequel by 𝐿^𝑝-strong solutions of the equation 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓(𝑥) we intend the 𝑊^2,𝑝_loc functions which are solutions almost everywhere with respect to the Lebesgue measure (almost everywhere) as well as classical solutions to be 𝐶² pointwise solutions.

We also call 𝐿^𝑝-strong (classical) subsolutions, respectively supersolutions, of 𝐹=𝑓 the 𝑊^2,𝑝_loc- (𝐶²-) solutions of the inequalities 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≥𝑓(𝑥), respectively 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)≤𝑓(𝑥), almost everywhere (everywhere).

For the definition of 𝐶-viscosity and 𝐿^𝑝-viscosity solutions, which are our main concern, we refer to Section 2. Correspondingly subsolutions, respectively supersolutions, in the viscosity sense will be referred to as solutions of the equation 𝐹≥𝑓, respectively 𝐹≤𝑓.

We establish a first result in the case of 𝐹 independent of 𝑥.

Theorem 1.1. Let 𝐹∶ℝ^𝑛×ℝ×ℝ^𝑛×𝒮^𝑛→ℝ be a continuous function satisfying the structure condition (𝐒𝐂) such that 𝐹(𝑥,𝑢,𝜉,𝑋)=𝐹(𝑢,𝜉,𝑋) (A1) for all (𝑥,𝑢,𝜉,𝑋)∈ℝ^𝑛×ℝ×ℝ^𝑛×𝒮^𝑛. If 𝑓∈𝐶(ℝ^𝑛), there exists a unique 𝐶-viscosity solution of equation 𝐹𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inℝ^𝑛. (1.11)

To consider a dependence on 𝑥, we need to control the oscillations in the variable 𝑥, and this also requires a uniform bound of the local 𝐿^𝑝-norms of 𝑓.

Theorem 1.2. Let 𝐹∈𝐶(ℝ^𝑛×ℝ×ℝ^𝑛×𝒮^𝑛) satisfy the structure condition (𝐒𝐂). Suppose also that for all 𝑅>0 there exist a constant 𝐾_𝑅>0 and a function 𝜔_𝑅∶ℝ₊→ℝ₊ such that lim_𝑡→0⁺𝜔_𝑅(𝑡)=0 and 𝐹𝑦,𝑢,𝜉,𝑋−𝐹(𝑥,𝑢,𝜉,𝑋)≤𝐾_𝑅‖𝑋‖𝑦−𝑥+𝜔_𝑅1+𝜉𝑦−𝑥 (A2.1) as 𝑥,𝑦∈ℝ^𝑛, 𝑢∈(−𝑅,𝑅) and (𝜉,𝑋)∈ℝ^𝑛×𝒮^𝑛. If 𝑝>𝑝₀, 𝑓∈𝐶(ℝ^𝑛), and 𝑓_𝑀𝑝∶=sup_{𝑥∈ℝ𝑛}𝑓_{𝐿𝑝(𝐵1(𝑥))}<+∞, (A2.2) then (1.1) has a unique 𝐶-viscosity solution.

As it can be seen in Section 4, the structure condition (𝐒𝐂) is sufficient by itself for the existence. The uniqueness, as shown in Section 5, relies on a result of Da Lio and Sirakov [6] which is fundamental in our proof for the comparison between two solutions 𝑢 and 𝑣. By virtue of this result, conditions (A2.1) and (A2.2) imply that the difference 𝑤=𝑢−𝑣 satisfies a maximal equation with a constant first-order coefficient. In particular, (A2.1) and (A2.2) are, respectively, stronger than the continuity of 𝐹 in the 𝑥-variable and than the local summability of |𝑓|^𝑝. Later on we also refer to these conditions as to the assumption: (𝐴2)∶=(𝐴2.1)and(𝐴2.2). (1.12)

In order to deal with 𝐹 merely measurable in 𝑥, we will suppose for every 𝑅>0 there exists 𝑐_𝑅>0 such that 𝒫⁻_𝜆,Λ(𝑌−𝑋)−𝛾𝜂−𝜉−𝑐_𝑅||()𝑣−𝑢≤𝐹𝑥,𝑣,𝜂,𝑌−𝐹𝑥,𝑢,𝜉,𝑋≤𝒫⁺_𝜆,Λ(𝑌−𝑋)+𝛾𝜂−𝜉+𝑐_𝑅|𝑣−𝑢| (1.13) for 𝑥∈ℝ^𝑛 and 𝑢,𝑣∈(−𝑅,𝑅),𝜉,𝜂∈ℝ^𝑛,𝑋,𝑌∈𝒮^𝑛. We put 𝐒𝐂∶=(1.13)and(1.3)and(1.4). (1.14)

Then, we say that 𝐹 satisfies 𝐶^1,1-estimates at 𝑥₀ if for all 𝑤₀∈𝐶⁰(𝜕𝐵_𝑟0(𝑥₀)) there exists a solution 𝑢∈𝐶²(𝐵_𝑟0(𝑥₀))∩𝐶⁰(𝐵_𝑟0(𝑥₀)) of the Dirichlet problem 𝐹𝑥₀,0,0,𝐷²𝑤=0in𝐵_𝑟0𝑥₀,𝑤=𝑤₀on𝜕𝐵_𝑟0𝑥₀ (1.15) such that ‖𝑢‖_𝐶^1,1_{(𝐵𝑟0/2(𝑥0))}≤𝐶𝑟₀⁻²‖𝑤‖_𝐿^∞_{(𝜕𝐵𝑟0(𝑥0))} (1.16) for some 𝑟₀>0.

Finally, let 𝛽_𝐹𝑥,𝑥₀∶=sup_{𝑋∈𝒮𝑛𝑋≠0}𝐹(𝑥,0,0,𝑋)−𝐹𝑥₀,0,0,𝑋‖𝑋‖. (1.17)

In the case 𝑝>𝑛, by Caffarelli [7], if 𝛽_𝐹(𝑥,𝑥₀) is sufficiently “small” in a sense that will be made precise below, then 𝑢∈𝑊^2,𝑝_loc(𝐵_𝑟0(𝑥₀)). Such result was generalized by Escauriaza to the range 𝑝>𝑝₀ with 𝑝₀∈(𝑛/2,𝑛) introduced above.

As a consequence, the structure conditions (𝐒𝐂) can be used “pointwise” to compare 𝐹(𝑥,𝑣(𝑥),𝐷𝑣(𝑥),𝐷²𝑣(𝑥)) with 𝐹(𝑥,𝑢(𝑥),𝐷𝑢(𝑥),𝐷²𝑢(𝑥)) almost everywhere obtaining a maximal equation for the difference 𝑢(𝑥)−𝑣(𝑥) to which GMP is applicable.

By virtue of the results of Winter [8] (see also Swiech [9]), the argument can be generalized to the case of 𝐹 merely measurable in the variable 𝑥 provided 𝐹 is convex in the matrix-variable 𝑋.

Theorem 1.3. Let 𝐹∶ℝ^𝑛×ℝ×ℝ^𝑛×𝒮^𝑛→ℝ be a function satisfying the structure condition (𝐒𝐂) almost everywhere 𝑥∈ℝ^𝑛 such that one of the following assumptions blocks holds true:(A3)𝐹 is continuous and has 𝐶^1,1-estimates for each 𝑥∈ℝ^𝑛 with some 𝑟₀>0;(A3)′𝐹 is measurable in 𝑥 for all (𝑢,𝜉,𝑋)∈ℝ×ℝ^𝑛×𝒮^𝑛 and convex in 𝑋. Let 𝑓∈𝐶(ℝ^𝑛) in the case (A3), 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) in the case (A3)′. If sup_{𝑟∈0,𝑟0}-_{𝐵𝑟(𝑥)}𝛽_𝐹𝑥,𝑦^𝑛𝑑𝑦^1/𝑛≤𝜃, (1.18) for every 𝑥∈ℝ^𝑛, with 𝜃=𝜃(𝑛,𝑝,𝜆,Λ,𝑟₀), then (1.1) has a unique 𝐿^𝑝-strong solution 𝑢∈𝑊^2,𝑝_loc(ℝ^𝑛), provided that 𝑝>𝑝₀.

By our assumptions, the 𝐿^𝑝-strong solution of Theorem 1.3 will be also the unique 𝐿^𝑝-viscosity solution.

Theorem 1.3 can be used for instance in the case of Bellman-type equations sup_𝛼𝐿_𝛼𝑢−𝑓_𝛼(𝑥)=0, (1.19) where 𝐿_𝛼 is a semilinear second-order operator 𝐿_𝛼𝑢∶=𝑎_𝛼^𝑖𝑗(𝑥)𝐷_𝑖𝑗𝑢+𝑏^𝑖_𝛼(𝑥)𝐷_𝑖𝑢+𝑐(𝑥)𝑢+𝑑(𝑥)|𝑢|^𝑠−1𝑢 (1.20) with bounded measurable coefficients such that 𝜆≤sup_𝜉≤1𝑎_𝛼^𝑖𝑗(𝑥)𝜉,𝜉≤Λ,𝑏^𝑖_𝛼(𝑥)𝑏^𝑖_𝛼(𝑥)^1/2≤𝛾,𝑐(𝑥)≤0,𝑑(𝑥)≤−𝛿<0 (1.21) for almost everywhere 𝑥∈ℝ^𝑛 and every 𝛼, provided the 𝑎_𝛼^𝑖𝑗 are uniformly continuous in ℝ^𝑛 with continuity modulus independent of 𝛼 and inf_𝛼𝑓_𝛼∈𝐿^𝑝_loc(ℝ^𝑛).

Some cases of Isaacs-type equations can be treated with Theorem 1.3, as for instance (see [10]), the minimum mininf_𝛼𝐿_𝛼𝑢−𝑓_𝛼(𝑥),sup_𝛽𝐿_𝛽𝑢−𝑓_𝛽(𝑥)=0 (1.22) between concave and convex operators, which are realized as infimum and supremum, respectively, of two families of semilinear operators, indexed by 𝛼 and 𝛽, with the above conditions.

Next, consider a regular domain Ω⊊ℝ^𝑛. If Ω is bounded, in the case of a continuous 𝐹, condition (𝐒𝐂) is sufficient in order that the Dirichlet problem with continuous boundary conditions has a 𝐶-viscosity solution by [11, Theorem  1.1]. If 𝐹 is merely measurable, we will use the stronger condition (𝐒𝐂)^′ of Section 4 needed for the existence of 𝐿^𝑛-viscosity solution in [11, Theorem  4.1].

The technique of the existence part of Theorem 1.3 allows to generalize such results to any regular domain, even unbounded, of ℝ^𝑛. For other results in unbounded domains we refer to [12], where the case 𝑠=1 is considered limiting the growth of 𝑓.

Theorem 1.4. Let Ω⊊ℝ^𝑛 be a domain satisfying a uniform exterior cone condition, and let 𝐹(𝑥,𝑡,𝜉,𝑋) be measurable in 𝑥∈ℝ^𝑛 for all (𝑡,𝜉,𝑋)∈ℝ×ℝ^𝑛×𝒮^𝑛 such that the structure condition (𝐒𝐂)^′ holds almost everywhere 𝑥∈Ω. Then, for 𝑝>𝑝₀ the Dirichlet problem 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inΩ,𝑢(𝑥)=𝜓(𝑥)on𝜕Ω (1.23) has an 𝐿^𝑝-viscosity solution 𝑢∈𝐶(Ω) for every 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) and every 𝜓∈𝐶(𝜕Ω).

Remark 1.5. The solution 𝑢∈𝐶(Ω) of Theorem 1.4 is unique in the cases of Theorems 1.1, 1.2 and 1.3, where the structural and the additional conditions are to be intended correspondingly to hold for 𝑥∈Ω instead of 𝑥∈ℝ^𝑛.

Finally, a monotonicity argument can instead be used when 𝑓 is bounded from below to obtain boundary blow-up 𝐿^𝑝-viscosity solutions.

Theorem 1.6. Let Ω⊊ℝ^𝑛 be a domain satisfying a uniform exterior cone condition, and suppose that at least one of the assumption blocks of Theorems 1.1, 1.2 and 1.3 holds true for 𝑥∈Ω. Then, for 𝑝>𝑝₀ the Dirichlet problem 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inΩ,lim_{𝑥→𝜕Ω}𝑢(𝑥)=+∞ (1.24) has an 𝐿^𝑝-viscosity solution 𝑢∈𝐶(Ω) for every 𝑓∈𝐿^𝑝_loc(Ω) such that essinf𝑓>−∞.

Remark 1.7. Generally the existence results for the BVP of Theorems 1.4 and 1.6 fail to hold if the domain is not sufficiently regular. In fact, assuming 𝐹=𝐹(𝑋) uniformly elliptic such that Λ1≤𝜆()<𝑛−1,𝑠≥𝑛−1𝜆+Λ(𝑛−1)𝜆−Λ, (1.25) Labutin [13] showed that the origin is a removable singularity for the equation 𝐹𝐷²𝑢−|𝑢|^𝑠−1𝑢=0, (1.26) that is every 𝐿^𝑛-viscosity solution in the punctured ball 𝐵_𝑅⧵{0} can be continued to an 𝐿^𝑛-viscosity solution in 𝐵_𝑅.

The paper is organized as follows. In Section 2 we introduce the notations and recall the main features of viscosity solutions which will be used. Then in Section 3 we prove a locally uniform bound which is the basic tool to construct the solutions in unbounded domains. The proof of the existence results will be given in Section 4, while the issue of uniqueness and blow-up is dealt with in Section 5.

2. Preliminaries

We will consider functions 𝐹∶Ω×ℝ×ℝ^𝑛×𝒮^𝑛→ℝ where Ω is a domain (open connected set) of ℝ^𝑛 and 𝒮^𝑛 the space of 𝑛×𝑛 real symmetric matrices. The identity matrix will be denoted by 𝐼 and the trace of 𝑋∈𝒮^𝑛 with Tr(𝑋), while ‖𝑋‖ is one of the equivalent norms of 𝑋 in 𝒮^𝑛. For 𝑋,𝑌∈𝒮^𝑛 we put 𝑋≤𝑌 if 𝑋𝜉,𝜉≤𝑌𝜉,𝜉∀𝜉∈ℝ^𝑛. (2.1) We say that 𝐹 is uniformly elliptic with ellipticity constants 𝜆>0 and Λ≥𝜆 if 𝒫⁻_𝜆,Λ(𝑌−𝑋)≤𝐹(𝑥,𝑡,𝜉,𝑌)−𝐹(𝑥,𝑡,𝜉,𝑋)≤𝒫⁺_𝜆,Λ(𝑌−𝑋) (2.2) for all (𝑥,𝑡,𝜉)∈Ω×ℝ×ℝ^𝑛 and 𝑋,𝑌∈𝒮^𝑛, where 𝒫⁺_𝜆,Λ(𝑍)=sup_{𝜆𝐼≤𝐴≤Λ𝐼}Tr(𝐴𝑍),𝒫⁻_𝜆,Λ(𝑍)=inf_{𝜆𝐼≤𝐴≤Λ𝐼}Tr(𝐴𝑍),𝑍∈𝒮^𝑛, (2.3) are, respectively, the maximal and the minimal (Pucci) operators in the class of uniformly elliptic operators with ellipticity constants 𝜆 and Λ.

For 𝑢∈𝐶²(Ω) denote by 𝐷𝑢 and 𝐷²𝑢 the gradient and the Hessian matrix of 𝑢. We wish to discuss the solvability of equation 𝐹𝑥,𝑢(𝑥),𝐷𝑢(𝑥),𝐷²𝑢(𝑥)=𝑓(𝑥) (2.4) under the assumptions (1.2), (1.3) and (1.4), and we refer to 𝑢∈𝐶²(Ω) satisfying (2.4) for all 𝑥∈Ω as classical solutions of the equation 𝐹=𝑓 in Ω. If 𝑢∈𝑊^2,𝑝_loc(Ω) and the equation is satisfied almost everywhere in Ω, we call it an 𝐿^𝑝-strong solution.

We are interested in the weaker notion of solution in the viscosity sense.

Firstly suppose that 𝐹 is continuous in Ω×ℝ×ℝ^𝑛×𝒮^𝑛 and 𝑓 continuous in Ω. The function 𝑢∈𝐶(Ω) is a 𝐶-viscosity subsolution, respectively supersolution, of the equation 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 in Ω if 𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)≥𝑓(𝑥), (2.5) respectively 𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)≤𝑓(𝑥), (2.6) for each 𝑥∈Ω and 𝜑∈𝐶²(𝐵_𝑟(𝑥)) such that 𝑢−𝜑 has a local maximum, respectively minimum, in 𝑥. A function 𝑢∈𝐶(Ω) that is both a subsolution and a supersolution in the above sense is a 𝐶-viscosity solution.

Here and below we denote by 𝐵_𝑟(𝑥) the ball of radius 𝑟 centered at 𝑥, for short 𝐵_𝑟 if 𝑥=0.

Remark 2.1. If 𝑢∈𝐶²(Ω) is a classical subsolution (supersolution) of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓, then 𝑢 is a 𝐶-viscosity subsolution (supersolution) of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓. Conversely, if 𝑢∈𝐶²(Ω) is a 𝐶-viscosity subsolution (supersolution), then 𝑢 is a classical subsolution (supersolution); see [14, Corollary  2.6].
In the sequel we will also use the fact that, if 𝑢∈𝐶(Ω) is a 𝐶-viscosity subsolution, respectively supersolution, of 𝐹=𝑓 and 𝑣∈𝐶(Ω) is a 𝐶-viscosity subsolution, respectively supersolution, of 𝐹=𝑔 in Ω, then the function 𝑤=max(𝑢,𝑣), respectively 𝑤=min(𝑢,𝑣), is a 𝐶-viscosity subsolution, respectively supersolution, of the equation 𝐹𝑥,𝑤,𝐷𝑤,𝐷²𝑤=ℎ (2.7) in Ω, where ℎ=min(𝑓,𝑔), respectively ℎ=max(𝑓,𝑔).

Lemma 2.2. Let 𝑢,𝑣∈𝐶(Ω) be, respectively 𝐶-viscosity subsolution and supersolution of the equations 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 and 𝐹(𝑥,𝑣,𝐷𝑣,𝐷²𝑣)=𝑔 in Ω, and assume (1.2)-(1.3). If at least one between 𝑢 and 𝑣 is in 𝐶²(Ω), then the difference 𝑤=𝑢−𝑣 is a 𝐶-viscosity subsolution of the maximal equation 𝒫⁺_𝜆,Λ𝐷²𝑤+𝛾|𝐷𝑤|−𝛿𝑤^𝑠=𝑓−𝑔 (2.8) in Ω∩{𝑤>0}.

Proof. Let us suppose, for instance, that 𝑣∈𝐶²(Ω). Let 𝜑 be a 𝐶²-function such that 𝑤−𝜑 has a local maximum in 𝑥∈Ω∩{𝑤>0}, then 𝑣+𝜑 is a test function for 𝑢, and by structure conditions (1.2)-(1.3) we have 𝑓(𝑥)−𝑔(𝑥)≤𝐹𝑥,𝑢(𝑥),𝐷𝑣+𝜑(𝑥),𝐷²𝑣+𝜑(𝑥)−𝐹𝑥,𝑣(𝑥),𝐷𝑣(𝑥),𝐷²𝑣(𝑥)≤𝒫⁺_𝜆,Λ𝐷²𝜑(𝑥)+𝛾𝐷𝜑(𝑥)−𝛿(𝑤(𝑥))^𝑠 (2.9) as claimed.

When 𝐹 is merely measurable in 𝑥, we assume that the structure condition (𝐒𝐂) holds for almost every 𝑥∈Ω. Note that, if 𝐹 is continuous in 𝑥, then (1.2) implies the uniform ellipticity.

Let 𝑓∈𝐿^𝑝_loc(Ω); then 𝑢∈𝐶(Ω) is called an 𝐿^𝑝-viscosity subsolution, respectively supersolution, of the equation 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 in Ω if esslimsup_𝑥→𝑥𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)−𝑓(𝑥)≥0, (2.10) respectively essliminf_𝑥→𝑥𝐹𝑥,𝑢(𝑥),𝐷𝜑(𝑥),𝐷²𝜑(𝑥)−𝑓(𝑥)≤0, (2.11) for each 𝑥∈Ω and 𝜑∈𝑊^2,𝑝(𝐵_𝑟(𝑥)) such that 𝑢−𝜑 has a local maximum, respectively minimum, in 𝑥. A function 𝑢∈𝐶(Ω) that is both a subsolution and a supersolution is an 𝐿^𝑝-viscosity solution.

It is important that the generalized maximum principle (GMP) for 𝐿^𝑝-strong solutions of the maximal equation (see (1.10) at the beginning of the Introduction) continues to hold for 𝐿^𝑝-viscosity subsolutions as 𝑝>𝑝₀; see for instance [5, Theorem  3.2].

Note that 𝐿^𝑝-viscosity solutions are 𝐶-viscosity solutions of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓 because the space 𝑊^2,𝑝_loc(Ω) of test functions for 𝐿^𝑝-viscosity solutions is larger than 𝐶²(Ω).

Conversely, if 𝑝>𝑝₀, assuming that 𝐹 and 𝑓 are continuous, then 𝐶-viscosity solutions are 𝐿^𝑝-viscosity solutions; see [15, Proposition  2.9].

Remark 2.3. Suppose that 𝑝>𝑝₀, as before. If 𝑢∈𝑊^2,𝑝_loc(Ω) is an 𝐿^𝑝-strong subsolution (supersolution) of 𝐹(𝑥,𝑢,𝐷𝑢,𝐷²𝑢)=𝑓, then 𝑢 is also an 𝐿^𝑝-viscosity subsolution (supersolution); see [15, Lemma  2.6].
Conversely, if 𝑢∈𝑊^2,𝑝_loc(Ω) is an 𝐿^𝑝-viscosity subsolution (supersolution), then 𝑢 is an 𝐿^𝑝-strong subsolution (supersolution); see [15, Corollary  3.7].
Also, Remark 2.1 for 𝑤=max(𝑢,𝑣), respectively 𝑤=min(𝑢,𝑣), continues to hold for 𝐿^𝑝-viscosity subsolutions, respectively supersolutions, 𝑢,𝑣∈𝐶(Ω).
Finally, assuming (1.2)-(1.3) almost everywhere in Ω, we infer that Lemma 2.2 continues to hold for 𝐿^𝑝-viscosity solutions.

For an extensive treatment of viscosity solutions see [11, 15–17].

In the existence results we need some regularity of the domain Ω of ℝ^𝑛. We say that Ω satisfies an exterior cone condition if for every point 𝑥∈𝜕Ω there exists a finite right circular cone Σ_𝑥 with vertex 𝑥 such that Σ_𝑥∩Ω={𝑥}. A uniform exterior cone condition means that all the cones Σ_𝑥 are congruent to a fixed cone Σ. For the use of these conditions see, for instance, [18] about the 𝐿^𝑝 theory and [11] in the viscosity setting.

3. Uniform Estimates

In this section, following [2], we introduce Osserman’s barrier function Φ(𝑥)=𝐶_𝑅𝑅^𝜇𝜉^−𝜇(𝑥),|𝑥|<𝑅, (3.1) where 𝜉(𝑥)=𝑅²−|𝑥|² and 𝜇=2/(𝑠−1) (recall that 𝑠>1).

Lemma 3.1. Suppose for almost everywhere 𝑥∈𝐵_𝑅 that 𝐹(𝑥,𝑣,𝜉,𝑋)≤𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉−𝛿𝑣^𝑠 (3.2) for all (𝑣,𝜉,𝑋)∈ℝ₊×ℝ^𝑛×𝒮^𝑛, where 𝛿>0 and 𝑠>1.
If one takes 𝐶_𝑅^𝑠−1=2𝜇𝛿⁻¹Λ𝑛+21+𝜇+𝛾𝑅, (3.3) then the function Φ∈𝐶²(𝐵_𝑅), defined in (3.1), is an 𝐿^𝑝-strong solution of the equation 𝐹𝑥,Φ,𝐷Φ,𝐷²Φ≤0in𝐵_𝑅. (3.4)

Proof. By the assumptions, it is sufficient to show that 𝒫⁺_𝜆,Λ𝐷²Φ+𝛾|𝐷Φ|−𝛿Φ^𝑠≤0. (3.5) Since Φ(𝑥)=𝜑(𝑟)∶=𝐶_𝑅𝑅^𝜇(𝑅²−𝑟²)^−𝜇, where 𝑟=|𝑥|, then 𝒫⁺_𝜆,Λ𝐷²Φ+𝛾|𝐷Φ|−𝛿Φ^𝑠=Λ𝜑^′′+𝑛−1𝑟𝜑^′+𝛾𝜑^′−𝛿𝜑^𝑠, (3.6) and the result follows by choosing 𝐶_𝑅>0 as claimed.

In what follows 𝑝₀∈(𝑛/2,𝑛) is the exponent such that GMP holds for 𝑝>𝑝₀ (see Sections 1 and 2) and 𝑢^±=max(±𝑢,0).

Lemma 3.2. Let Ω be a domain of ℝ^𝑛 such that Ω_𝑅∶=Ω∩𝐵_𝑅≠∅. Suppose that 𝐹 satisfies structure conditions (𝐒𝐂) almost everywhere 𝑥∈Ω_𝑅. If 𝑢∈𝐶(Ω_𝑅) is an 𝐿^𝑝-viscosity solution (𝑝>𝑝₀) of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢≥𝑓(𝑥) (3.7) with 𝑓∈𝐿^𝑝(Ω_𝑅), then for each 𝑟∈(0,𝑅) one has sup_Ω𝑟𝑢≤𝑢⁺_𝜕Ω+𝐶₀(1+𝑅)^𝜇/2𝑅^𝜇𝑅²−𝑟^2𝜇+𝐶𝑓⁻_{𝐿𝑝(Ω𝑅)} (3.8) where 𝜇=2/(𝑠−1), and 𝐶₀=𝐶₀(𝑛,Λ,𝛾,𝑠,𝛿) and 𝐶=𝐶(𝑛,𝑝,𝜆,Λ,𝛾𝑅) are positive constants.
Here, 𝑢⁺_𝜕Ω=sup_{𝐵𝑅∩𝜕Ω}𝑢⁺if𝐵_𝑅∩𝜕Ω≠∅,0if𝐵_𝑅∩Ω. (3.9)

Proof. By (SC) we have 𝐹(𝑥,𝑣,𝜉,𝑋)=𝐹(𝑥,𝑣,𝜉,𝑋)−𝐹(𝑥,𝑣,0,0)+𝐹(𝑥,𝑣,0,0)≤𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉−𝛿𝑣^𝑠 (3.10) for all (𝑣,𝜉,𝑋)∈ℝ₊×ℝ^𝑛×𝒮^𝑛. Thus, from Lemma 3.1 we deduce that Φ is an 𝐿^𝑝-strong supersolution of the equation 𝐹𝑥,Φ,𝐷Φ,𝐷²Φ=0inΩ_𝑅. (3.11) On the other hand 𝑢 is an 𝐿^𝑝-viscosity subsolution of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inΩ_𝑅. (3.12) Hence, by Lemma 2.2 (see Remark 2.3) the function 𝑤=𝑢−Φ is an 𝐿^𝑝-viscosity solution of the equation 𝒫⁺_𝜆,Λ𝐷²𝑤+𝛾|𝐷𝑤|≥𝒫⁺_𝜆,Λ𝐷²𝑤+𝛾|𝐷𝑤|−𝛿𝑤^𝑠≥𝑓(𝑥) (3.13) in 𝐴=Ω_𝑅∩{𝑢>Φ}.
Let 𝑟∈(0,𝑅) be such that 𝑢⁺(𝑥)≤Φ(𝑥) as 𝑥∈Ω and 𝑟≤|𝑥|<𝑅; then 𝐴⊂Ω_𝑟 and 𝑤≤0 on Ω∩𝜕𝐵_𝑟. Therefore, applying GMP (1.10), we get sup_𝐴𝑤≤sup_{𝐵𝑟∩𝜕Ω}𝑤+𝐶𝑓⁻_{𝐿𝑝(Ω𝑅)} (3.14) from which 𝑢(𝑥)≤𝐶_𝑅𝑅^𝜇𝑅²−|𝑥|^2𝜇+𝑢⁺_𝜕Ω+𝐶𝑓⁻_{𝐿𝑝(Ω𝑅)} (3.15) for all 𝑥∈Ω_𝑅.

Proposition 3.3. Let Ω_𝑅, 𝐹, and 𝑓 be as in Lemma 3.2. If 𝑢∈𝐶(Ω_𝑅) is an 𝐿^𝑝-viscosity solution (𝑝>𝑝₀) of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥), (3.16) then for each 𝑟∈(0,𝑅) one has sup_Ω𝑟|𝑢|≤|𝑢|_𝜕Ω+𝐶₀(1+𝑅)^𝜇/2𝑅^𝜇𝑅²−𝑟^2𝜇+𝐶𝑓_{𝐿𝑝(Ω𝑅)} (3.17) with 𝐶₀, 𝐶 and |𝑢|_𝜕Ω=max(𝑢⁺_𝜕Ω,𝑢⁻_𝜕Ω) as defined in Lemma 3.2.

Proof. From Lemma 3.2 we already know that sup_Ω𝑟𝑢⁺≤𝑢⁺_𝜕Ω+𝐶₀(1+𝑅)^𝜇/2𝑅^𝜇𝑅²−𝑟^2𝜇+𝐶𝑓⁻_{𝐿𝑝(Ω𝑅)}. (3.18) The assertion will be proved showing the same inequality for 𝑢⁻.
To this end firstly observe that the function 𝑣=−𝑢 satisfies the equation 𝐺𝑥,𝑣,𝐷𝑣,𝐷²𝑣=−𝑓(𝑥)inΩ_𝑅, (3.19) where 𝐺(𝑥,𝑣,𝜉,𝑋)=−𝐹(𝑥,−𝑣,−𝜉,−𝑋), (3.20) which turns out to satisfy (SC).
Therefore, 𝑢⁻=max(𝑣,0) is an 𝐿^𝑝-viscosity solution of the equation 𝐺𝑥,𝑢⁻,𝐷𝑢⁻,𝐷²𝑢⁻≥−𝑓(𝑥). (3.21)
On the other hand, by virtue of Lemma 3.1, Φ is an 𝐿^𝑝-strong supersolution of the equation 𝒫⁺_𝜆,Λ𝐷²Φ+𝛾|𝐷Φ|−𝛿Φ^𝑠=0. (3.22)
Hence, we finish the proof arguing as in Lemma 3.2 to obtain the estimate 𝑢⁻(𝑥)≤𝑢⁻_𝜕Ω+𝐶_𝑅𝑅^𝜇𝑅²−|𝑥|^2𝜇+𝐶𝑓⁺_{𝐿𝑝(Ω𝑅)} (3.23) for all 𝑥∈Ω_𝑅.

4. Proof of the Existence Results

In this section, using the structure condition (𝐒𝐂), or the slightly stronger variant (𝐒𝐂)^′ defined below, we construct an 𝐿^𝑝-viscosity solution of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥) (4.1) in ℝ^𝑛 under the assumption that 𝑓∈𝐿^𝑝_𝑙𝑜𝑐(ℝ^𝑛) with 𝑝>𝑝₀, where 𝑝₀∈(𝑛/2,𝑛) is the exponent above which GMP holds true; see Sections 1 and 2.

By the relationship between 𝐶-viscosity and 𝐿^𝑝-viscosity solutions and between 𝐿^𝑝-viscosity and 𝐿^𝑝-strong solutions (see Section 2) the existence part of each one of Theorems 1.1, 1.2, and 1.3 will follow at once from Proposition 4.1 in rather general assumptions.

We will suppose that for all 𝑅>0 there exists a function 𝜔_𝑅∶ℝ₊→ℝ₊ such that 𝜔_𝑅(𝑡)→0 as 𝑡→0⁺ and 𝐹(𝑥,𝑣,𝜉,𝑋)−𝐹(𝑥,𝑢,𝜉,𝑋)≤𝜔_𝑅(|𝑣−𝑢|) (4.2) almost everywhere in 𝑥 for |𝑢|+|𝑣|+|𝜉|+‖𝑋‖≤𝑅, observing that it is satisfied by default if we assume that 𝐹 is a continuous function of (𝑥,𝑢,𝜉,𝑋). Then, we put (𝐒𝐂)^′∶=(𝐒𝐂)and(4.2). (4.3) It is worth to recall that condition (𝐒𝐂)^′ is equivalent to (𝐒𝐂) in the case that 𝐹 is continuous.

Proposition 4.1. Let 𝐹∶ℝ^𝑛×ℝ×ℝ^𝑛×𝒮^𝑛→ℝ be measurable in 𝑥 and satisfy the structure condition (𝐒𝐂)^′ almost everywhere 𝑥∈ℝ^𝑛 for all (𝑢,𝜉,𝑋)∈ℝ×ℝ^𝑛×𝒮^𝑛. If 𝑓∈𝐿^𝑝_loc(ℝ^𝑛), then equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥) (4.4) has an 𝐿^𝑝-viscosity solution in ℝ^𝑛 for any 𝑝>𝑝₀.

Proof. Consider a smooth approximation 𝑓_𝑘∈𝐶^∞(ℝ^𝑛) of 𝑓 such that lim_𝑘→∞𝑓_𝑘−𝑓_𝐿𝑝(Ω)=0 (4.5) for every bounded domain Ω in ℝ^𝑛.
By [11, Theorem  4.1, Remark  4.8] we can solve in the 𝐿^𝑝-viscosity sense any Dirichlet problem for the equation 𝐹=𝑓_𝑘 in the ball 𝐵₂^𝑘 with continuous boundary condition.
Choose a solution 𝑢_𝑘 for each 𝑘∈ℕ. Using Proposition 3.3, for ℎ>𝑘 we have sup_𝐵2𝑘𝑢_ℎ≤𝐶₀+𝐶𝑓_{𝐿𝑝(𝐵2𝑘+1)}, (4.6) where 𝐶₀=𝐶₀(𝑛,Λ,𝛾,𝑠,𝛿) and 𝐶=𝐶(𝑛,𝑝,𝜆,Λ,𝛾2^𝑘+1) are positive constants as defined in Lemma 3.2.
By the structure condition (𝐒𝐂)^′ we have 𝐹(𝑥,𝑣,𝜉,𝑋)≤𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉+𝐹(𝑥,−𝑅,0,0)≤𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉+𝜔_𝑅(𝑅),𝐹(𝑥,𝑣,𝜉,𝑋)≥𝒫⁻_𝜆,Λ(𝑋)−𝛾𝜉+𝐹(𝑥,𝑅,0,0)≥𝒫⁻_𝜆,Λ(𝑋)−𝛾𝜉−𝜔_𝑅(𝑅), (4.7) almost everywhere 𝑥∈ℝ^𝑛 for |𝑣|≤𝑅.
Therefore for ℎ>𝑘 we have 𝒫⁻_𝜆,Λ𝐷²𝑢_ℎ−𝛾𝐷𝑢_ℎ−𝜔_𝑅(𝑅)≤𝑓≤𝒫⁺_𝜆,Λ𝐷²𝑢_ℎ+𝛾𝐷𝑢_ℎ+𝜔_𝑅(𝑅) (4.8) in 𝐵₂^𝑘 with 𝑅=𝐶₀+𝐶‖𝑓‖_𝐿^𝑝_{(𝐵2𝑘+1)}.
By 𝐶^𝛼-estimates (see [14, Proposition  4.10] and [19, Theorem  2]) we deduce that 𝑢_{ℎ𝐶𝛼(𝐵2𝑘)}≤𝐶₁1+𝑓_{𝐿𝑝(𝐵2𝑘+1)}, (4.9) for a positive constant 𝐶₁ independent of ℎ>𝑘.
By a diagonal process, using Ascoli-Arzelà theorem we extract a subsequence ℎ_𝑘∈ℕ such that 𝑢_ℎ𝑘→𝑢∈𝐶(ℝ^𝑛) uniformly on every bounded domain.
From the stability results for 𝐿^𝑝-viscosity solutions, see [15, Theorem  3.8], 𝑢 is a solution of the equation.

Proof of Theorem 1.4. In the case Ω⊊ℝ^𝑛 we proceed along the same lines of the proof of Proposition 4.1 constructing 𝐿^𝑝-viscosity solutions 𝑢_𝑘∈𝐶(Ω₂^𝑘), 𝑘∈ℕ, of the approximating Dirichlet problems 𝐹𝑥,𝑢_𝑘,𝐷𝑢_𝑘,𝐷²𝑢_𝑘=𝑓_𝑘(𝑥)inΩ₂^𝑘_,𝑢_𝑘(𝑥)=𝜓_𝑘(𝑥)on𝜕Ω₂^𝑘 (4.10) for a sequence 𝑓_𝑘∈𝐶(Ω) such that ‖𝑓_𝑘−𝑓‖_𝐿^𝑝_(𝐴)→0 for all bounded subsets 𝐴⊂Ω.
Here, according to the notations of Proposition 3.3, Ω₂^𝑘=Ω∩𝐵₂^𝑘, while 𝜓_𝑘 is a continuous extension to ℝ^𝑛 of 𝜓|_{𝜕Ω∩𝐵2𝑘}; see for instance [20, Section  1.2].
Let 𝑅=2^𝑘. Since Ω_𝑅 satisfies in turn a uniform exterior cone property, the existence of such 𝑢_𝑘 follows from the assumptions on 𝐹 and the already mentioned [11, Theorem  4.1].
Furthermore, by Proposition 3.3, for ℎ>𝑘 we get sup_Ω𝑅𝑢_ℎ≤max_{𝜕Ω∩𝐵𝑅}𝜓+𝐶₀+𝐶𝑓_{𝐿𝑝(Ω2𝑅)}. (4.11) The argument of the proof of Lemma 3.2 leads to inequality and therefore 𝑢_ℎ are equibounded in Ω_𝑅.
As a consequence, by 𝐶^𝛼-estimates they are equi-Hölder continuous in every subset 𝑥∈Ω_𝑅∣dist(𝑥,𝜕Ω)>𝜀 (4.12) with 𝜀>0. By [19, Theorem  2] osc_{Ω𝑅∩𝐵𝜌(𝑥)}𝑢_ℎ≤𝐶₀𝜌^𝛼𝑘+osc_{𝜕Ω𝑅∩𝐵√𝜌(𝑥)}𝜓_ℎ (4.13) for every 𝑥∈𝜕Ω and 𝜌≤𝜌_𝑘, and therefore 𝑢_ℎ are also equicontinuous in Ω_𝑅.
Thus, using a diagonal procedure as in the proof of Proposition 4.1 we find an 𝐿^𝑝-viscosity solution 𝑢∈𝐶(Ω) of the Dirichlet problem under consideration.

5. Uniqueness and Blow-Up

In this section we begin noticing that from Section 3 we get at once the following maximum principle.

Proposition 5.1. Let 𝛿>0, 𝑠>1, and let Ω be a domain of ℝ^𝑛.
Suppose for almost everywhere 𝑥∈Ω that 𝐹(𝑥,𝑢,𝜉,𝑋)≤𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉−𝛿|𝑢|^𝑠−1𝑢 (5.1) for all (𝑢,𝜉,𝑋)∈ℝ×ℝ^𝑛×𝒮^𝑛 and 𝑢∈𝐶(Ω) is an 𝐿^𝑝-viscosity solution (𝑝>𝑝₀) of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢≥0inΩ. (5.2)
(M1)If Ω=ℝ^𝑛, then 𝑢≤0 in ℝ^𝑛.(M2)If Ω⊊ℝ^𝑛 and 𝑢≤0 on 𝜕Ω, then 𝑢≤0 in Ω.Analogously, suppose for almost everywhere 𝑥∈Ω that 𝐹(𝑥,𝑣,𝜉,𝑋)≥𝒫⁻_𝜆,Λ(𝑋)−𝛾𝜉−𝛿|𝑣|^𝑠−1𝑣 (5.3) for all (𝑣,𝜉,𝑋)∈ℝ×ℝ^𝑛×𝒮^𝑛 and 𝑣∈𝐶(Ω) is an 𝐿^𝑝-viscosity solution (𝑝>𝑝₀) of the equation 𝐹𝑥,𝑣,𝐷𝑣,𝐷²𝑣≤0inΩ. (5.4)(m1)If Ω=ℝ^𝑛, then 𝑣≥0 in ℝ^𝑛.(m2)If Ω⊊ℝ^𝑛 and 𝑣≥0 on 𝜕Ω, then 𝑣≥0 in Ω.

Proof. Let 𝑥∈ℝ^𝑛 and 𝑟=|𝑥|. Firstly consider the cases (M1) and (M2). Since 𝐹(𝑥,𝑢,𝜉,𝑋)=𝒫⁺_𝜆,Λ(𝑋)+𝛾𝜉−𝛿|𝑢|^𝑠−1𝑢 (5.5) satisfies (SC), we can apply Lemma 3.2. Letting 𝑅→+∞ in (3.8) with 𝑓=0, we get 𝑢(𝑥)≤0 as asserted.
The other cases (m1) and (m2) can be treated by means of (M1) and (M2) considering the function 𝑢(𝑥)=−𝑣(𝑥) and the operator 𝐺(𝑥,𝑢,𝜉,𝑋)=−𝐹(𝑥,−𝑢,−𝜉,−𝑋) (5.6) are as in the proof of Proposition 3.3.

The above implies that, assuming (𝐒𝐂)^′, the function 𝑢=0 is the unique viscosity solution of the problem 𝐹=0 in ℝ^𝑛.

Concerning the uniqueness for the inhomogeneous equation 𝐹=𝑓 in ℝ^𝑛, since solutions are considered in the viscosity sense, we need additional assumptions in order to use “pointwise” the structure conditions and thus to use the above maximum principle.

Also, we cannot in general employ the usual comparison arguments for Dirichlet problems (see for instance [16, Section  3]) not having in principle boundary conditions or bounds at infinity.

Proof of Theorem 1.1 (uniqueness). Let 𝑢 and 𝑣 be solutions of the equation 𝐹=𝑓. Set Ω={𝑥∈ℝ^𝑛∣𝑢>𝑣}. We claim that Ω=∅, so that 𝑢≤𝑣 in ℝ^𝑛.
Suppose on the contrary that Ω≠∅. Since 𝐹 is continuous, arguing as in [21] and observing that 𝑢,𝑣 are in 𝐶^1,𝛼_loc, we can use the structure condition (SC) to have 𝒫⁺_𝜆,Λ𝐷²𝑤+𝛾|𝐷𝑤|−𝛿𝑤^𝑠≥0 (5.7) in Ω. Using the maximum principle of Proposition 5.1 (M2), we should have 𝑤=𝑢−𝑣≤0 in Ω, a contradiction which proves our claim.
Interchanging the role of 𝑢 and 𝑣, we also get 𝑣≤𝑢 in ℝ^𝑛, and we are done.

Proof of Theorem 1.2 (uniqueness). Here we observe that, if 𝑓∈𝐿^𝑝_loc(ℝ^𝑛) and ‖𝑓‖_𝑀𝑝<+∞, then Proposition 3.3 implies that 𝑢 is bounded. In fact, if 𝑥₀∈ℝ^𝑛 and we consider balls centered at 𝑥₀, choosing 𝑟→0⁺ and 𝑅=1 in (3.17), we get 𝑢𝑥₀≤2^𝜇/2𝐶₀+𝐶𝑛,𝑝,𝜆,Λ,𝛾𝑓_𝑀𝑝, (5.8) which is finite and independent of 𝑥₀, by (A2.2).
Thus, if 𝑢 and 𝑣 are solutions of the equation 𝐹=𝑓, by (SC) and (A2.1) we can use of [6, Proposition  2.1] so that the difference 𝑤=𝑢−𝑣 satisfies a maximal equation 𝒫⁺_𝜆,Λ𝐷²𝑤+𝑏|𝐷𝑤|−𝛿𝑤^𝑠≥0 (5.9) in Ω={𝑥∈ℝ^𝑛∣𝑢>𝑣} for some positive constant 𝑏 depending on 𝑛,𝑝,𝜆,Λ,𝛾, and 𝑠.
Therefore, we can conclude as in the proof of Theorem 1.1 (uniqueness).

Proof of Theorem 1.3 (uniqueness). In this case we observe that both the assumptions (𝐒𝐂) and (A3) or (A3)′ of the Theorem imply that 𝐿^𝑝-viscosity solutions 𝑢,𝑣 of 𝐹=𝑓 are in 𝑊^2,𝑝_loc(ℝ^𝑛) (see [14, Theorem  7.1], [3, Theorem  1], and [8, Theorem  4.2]). Hence 𝑢,𝑣 are 𝐿^𝑝-strong solutions (see Remark 2.3), and we can use the structure condition (𝐒𝐂) to get the maximal equation (5.7) for the difference 𝑤=𝑢−𝑣 in Ω={𝑥∈ℝ^𝑛∣𝑢>𝑣}, from which we conclude again using the maximum principle of Proposition 5.1 as in the proof Theorems 1.1 and 1.2 (uniqueness).

We insist on observing that the uniqueness of the solution of the Dirichlet problem depends on the fact that the difference between two solutions is a solution of a homogeneous maximal equation, and then we can invoke the maximum principle of Proposition 5.1. The same method will be used to prove by monotonicity the existence of blow-up solutions.

Proof of Theorem 1.6. Following [2] we consider a nondecreasing sequence of 𝑓_𝑘∈𝐶(Ω) such that lim_𝑘→∞𝑓_𝑘−𝑓_{𝐿𝑝(𝐾)} (5.10) for all compact set 𝐾 of Ω. Then by Theorem 1.4 we solve the problem 𝐹𝑥,𝑢_𝑘,𝐷𝑢_𝑘,𝐷²𝑢_𝑘=𝑓_𝑘(𝑥)inΩ,𝑢_𝑘(𝑥)=𝑘on𝜕Ω. (5.11) As in the proof of Proposition 4.1, by a diagonal process, using (𝐒𝐂), respectively (𝐒𝐂)^′, we find an 𝐿^𝑝-viscosity solution 𝑢∈𝐶(Ω) of the equation 𝐹𝑥,𝑢,𝐷𝑢,𝐷²𝑢=𝑓(𝑥)inΩ. (5.12) To compare 𝑢_𝑘 and 𝑢_𝑘+1, we use (𝐒𝐂) with the additional assumptions (A1) or (A2), respectively (𝐒𝐂) with (A3) or (A3)′, to get a maximal equation for 𝑤=𝑢_𝑘−𝑢_𝑘+1. Since 𝑓_𝑘 is nondecreasing, then 𝑤 satisfies the boundary value problem 𝒫⁺_𝜆,Λ𝐷²𝑤+𝑏|𝐷𝑤|≥0inΩ∩𝑢_𝑘>𝑢_𝑘+1,𝑤≤0on𝜕Ω∩𝑢_𝑘>𝑢_𝑘+1. (5.13) Hence, using the maximum principle of Proposition 5.1 we get 𝑤≤0, that is 𝑢_𝑘≤𝑢_𝑘+1.
Therefore 𝑢_𝑘 is nondecreasing, and for 𝑥∈𝜕Ω we have lim_𝑥→𝑥𝑢(𝑥)≥lim_𝑥→𝑥𝑢_𝑘(𝑥)=𝑘 (5.14) for all 𝑘∈ℕ; whence the assertion follows.