VISCOSITY SOLUTIONS TO THE INFINITY LAPLACIAN EQUATION WITH LOWER TERMS

. We establish the existence and uniqueness of viscosity solutions to the Dirichlet problem


Introduction
In this manuscript, we investigate the following inhomogeneous problem for q ∈ C(∂Ω), where ∆ h ∞ is strongly degenerate and is defined as Throughout this article, Ω is assumed to be a bounded domain of R n , n ≥ 2. Note that the operator ∆ h ∞ is not in divergence form. Hence, the solution is usually understood in the viscosity frame introduced by Crandall and Lions [23], and Crandall, Evans and Lions [20].
For the special case h = 3, the operator ∆ h ∞ is the infinity Laplacian which is often denoted by ∆ ∞ u = n i,j=1 u xi u xj u xixj . The operator ∆ ∞ was motivated in studying the absolutely minimizing Lipschitz extension (AMLE) by Aronsson [2,3,4,5] in 1960's. Jensen [25] showed the uniqueness of AMLE and the equivalence of the AMLE and infinity harmonic functions(viscosity solutions to ∆ ∞ u = 0). Crandall, Gunnarsson and Wang [21] studied the uniqueness of infinity harmonic functions in an unbounded domain. Crandall, Evans and Gariepy [19] showed that infinity harmonic functions enjoy comparison property with linear cones. Aronsson, Crandall and Juutinen [6] gave a systematic treatment of the theory of AMLEs. For more results of AMLEs, see for example Armstrong and Smart [1], Barron, Jensen and Wang [1,9], Barles and Busca [7], Barron, Evans and Jensen [8], Evans [24], Yu [44] and the references therein.
For h = 1, ∆ h ∞ u is the 1-homogeneous normalized ∞-Laplacian operator, ∆ N ∞ u := |Du| −2 D 2 uDu, Du . There is a "tug-of-war" game when approaching the normalized infinity Laplacian Dirichlet problem which was first introduced by Peres et al. [40] based on a probability view, ∆ N ∞ u(x) = H(x), in Ω, u(x) = q(x), on ∂Ω. (1. 2) The continuum value function of the game is proven to satisfy (1.2) and ∆ N ∞ is also called game infinity Laplacian (denoted also by ∆ G ∞ ). Lu and Wang [34] studied the well-posedness of (1.2) from the partial differential equation perspective. Note that the uniqueness is valid if the nonlinear source term H(x) > 0(< 0). A counterexample was shown in [37,40] that the uniqueness does not hold if H(x) changes its sign. One can see [36] for more uniqueness results of infinity Laplacian equations. We direct the reader to [27,28,29,30,31,33,36,37,39,42,44] and the references therein for the ∞-Laplacian operator.
Lu and Wang [35] proved that the inhomogeneous Dirichlet problem ∆ ∞ u = H(x), in Ω, u = q, on ∂Ω has a unique viscosity solution u ∈ C(Ω) under the assumptions that the continuous function H has one sign. They also proved the comparison property with special functions for the viscosity solutions which extended the result of Crandall, Evans and Gariepy [19]. Bhattacharya and Mohammed [10] studied the existence and nonexistence of viscosity solutions to the Dirichlet problem for f with the sign and the monotonicity restrictions and g ∈ C(∂Ω). In [11], they further removed the sign and the monotonicity restrictions and gave the existence result from the general structure condition on f . Bhattacharya and Mohammed [10] also investigated the bounds and boundary behavior of viscosity solutions to the problem (1.3). For the general boundary behavior of the viscosity solution to (1.3), one can see [38]. In [32], the existence of the viscosity solutions of the following inhomogeneous problem was obtained, And for 1 ≤ h ≤ 3, under suitable conditions on α and f , Biswas and Vo [14] studied the existence, nonexistence and uniqueness of positive viscosity solutions to the Dirichlet problem of the equation [10] otained bounds and boundary behavior of viscosity solutions to the problem when f ∈ C 1 ((0, ∞), (0, ∞)), lim t→0 + f (t) = ∞ and f is decreasing on (0, ∞). By Karamata regular variation theory, Mi [38] gave the boundary asymptotic estimate of solutions to the problem for a wide range of the functions b(x). Biset and Mohammed [13] established the existence of ground state solutions to the problem in a bounded domain and in the whole Euclidean space. The study is based on the subsolution/supersolution method and the existence of the principal Dirichlet eigenfunctions.
Inspired by the previous work, we study the Dirichlet problem (1.1). The hdegree operators ∆ h ∞ , besides their wide applications, are not only degenerate, singular for 1 < h < 3, but also not in divergence form and have no variational structure. They constitute a class of operators with particular properties. Our main results are summarized as follows.
is non-negative and non-decreasing in t and sup x∈Ω f (x, t) < ∞ for each t ∈ R. Then (1.1) admits a viscosity solution. Furthermore, if f is positive, the solution is unique. The existence of viscosity solutions is proved via Perron's method. The key is to construct a suitable viscosity subsolution. Thanks to the "good" structure of the operator ∆ h ∞ , we can use the cone functions to construct the subsolution. The uniqueness can be derived from the comparison principle. We remark that the uniqueness is still open when the function f (x, t) is nonpositive or nonnegative.
In fact, when f (x, t) does not depend on the second variable t and h = 1, Peres, Schramm, Sheffield and Wilson [40] construct a counterexample to show that the uniqueness does not hold if f changes its sign. For h = 3, Lu and Wang [35] gave a counterexample to show the uniqueness is invalid if the function f (x) changes its sign.
With Theorem 1.1 in hand, when f (x, t) is non-increasing in the variable t, we can also prove the existence of the viscosity solution to the Dirichlet problem (1.1) if the domain is of small diameter. Note that the small diameter condition (1.4) guarantees the existence of a viscosity subsolution and then we can use an iteration technique to establish the following existence result.
For h = 3, Bhattacharya and Mohammed [10] constructed a counter-example in the appendix to show that the uniqueness of the viscosity solution does not generally hold when f (x, t) is non-increasing in t.
To establish the existence of viscosity solutions to (1.1), a difficulty with respect to the degenerate operators is the lack of existence of barriers. Thanks to the particular structure of ∆ h ∞ , we can construct 'good' barriers and use the standard Perron's method to get the existence of the approximate solutions. Due to the strong degeneracy of the operator ∆ h ∞ , we combine the iteration method, Theorem 1.1 and stability method to establish Theorem 1.3. The key idea is that the existence of an appropriate viscosity subsolution leads to the existence of a viscosity solution.
Furthermore, we investigate the singular Dirichlet problem (1.5) We first construct a viscosity supersolution of (1.5) and then prove the existence of viscosity solutions to (1.5) using the comparison principle and the stability of viscosity solutions.
When the bounded domain Ω has smooth boundary, we also investigate the boundary behavior of viscosity solutions to (1.5). The functions b(·) and g(·) satisfy the following conditions: (H1) b ∈ C(Ω) is positive in Ω.
To obtain the existence of the viscosity solutions of the singular problem (1.5), we adopt the truncation method to deal with the singularity of the equation and then use the stability and compactness methods. Based on the comparison principle, the uniqueness result of the viscosity solution follows immediately.
One should notice that the distance function is a solution of ∆ h ∞ v = 0 near the boundary. Therefore, we can perturb the distance function to analyze the asymptotic behavior near the boundary of viscosity solutions to the singular boundary value problem (1.5). The idea is based on Karamata regular variation theory which was first introduced by Cîrstea and Rǎdulescu in stochastic process to study the boundary behavior and uniqueness of solutions to boundary blow-up elliptic problems. And a series of rich and significant information about the boundary behavior of solutions was obtained based on such theory [15,16,17]. Note that, unlike the case h = 1, the operator ∆ h ∞ is quasi-linear even in one-dimension for h > 1. Therefore, we must make subtle analysis.
This article is organized in the following way. In Section 2, we prove the comparison principle to ∆ h ∞ u = f (x, u), based on the double variables method. In Section 3, by Perron's method and the comparison principle, we prove the existence and uniqueness of viscosity solutions to (1.1) when f (x, t) is non-decreasing in t. In Section 4, when f (x, t) is non-increasing in t, we use an iteration technique to obtain the existence of the viscosity solution to (1.1) in domains with small diameter. In Section 5, we establish the existence and boundary behavior of viscosity solutions to (1.5).

Comparison principles
In this section, we give the comparison principles via the perturbation method for the equation ∞ has no divergence structure, we define the viscosity solution by the semicontinuous extension. See for example [22,30,32,34].
where S denotes the set of n × n real symmetric matrices, (2.1) can be rewritten as Since h > 1, we have lim p→0 F h (M, p) = 0 for all M ∈ S. Therefore, we define the following continuous extension of F h , Now we state the definition of viscosity solutions to the equation (2.1).
then we say that u is a viscosity subsolution to (2.1) in Ω.
Similarly, suppose that u : Ω → R is a lower semi-continuous function. If, for any x 0 ∈ Ω and ϕ ∈ C 2 (Ω) such that u( , then we say that u is a viscosity supersolution to (2.1) in Ω.
A function u ∈ C(Ω) is called a viscosity solution to (2.1) in Ω if it is both a viscosity subsolution and viscosity supersolution of (2.1).
Next, we state the strong maximum principle for infinity subharmonic functions (see for example [6,18]).
. Then u attains its maximum only on the boundary ∂Ω unless u is a constant.
We also need the comparison principle which was established by Li and Liu [26].
Now we present a comparison result applicable to singular problems and prove it by a truncating function.
Proof. We set v ε := v + ε, ε > 0. We claim that u ≤ v ε in Ω for each ε > 0. We assume to the contrary. Set We define 3. Existence when f (x, t) non-decreasing in t We first construct the viscosity subsolution to the problem (1.1), and then establish the existence result using Perron's method and the Lipschitz continuity of infinity harmonic functions.
Proof. Let and then a constant d 1 such that That is, u is a desired viscosity subsolution to (1.1).
3. If f is non-negative andũ is a viscosity subsolution of (2.1), thenũ is locally Lipschitz continuous in Ω (see for example [18,Lemma 4.1]). Hence, we have the function u defined in (3.3) is locally Lipschitz continuous.
Proof of Theorem 1.1. The existence is an application of standard Perron's method.
Since f is non-negative and sup x∈Ω f (x, t) < ∞, Lemma 3.1 implies that problem (1.1) has a viscosity subsolution u ∈ C(Ω). Indeed, the function u defined in (3.3) is a viscosity solution of (1.1).
Now we show that u ∈ C(Ω). Let z ∈ ∂Ω and B r (z) be as above, and {x k } a sequence in Ω such that lim k→∞ x k = z. Since the lower semi-continuity of u, we have lim inf k→∞ u(x k ) ≥ u(z) = q(z).
Step 3. Next, we will prove that u is a viscosity supersolution. We assume to the contrary. Then there exist x 0 ∈ Ω and ϕ ∈ C 2 (Ω) such that For 0 < ε ≤ ε 0 , we define For x = x 0 , a direct calculation yields Hence, we obtain Then, by (3.9), we have ). We claim that there exists an ε 1 , with 0 < ε 1 ≤ ε 0 , such that We argue by contradiction. Then, for each ε > 0 with ε small enough, there exists an (3.10) We define It is obvious that u * ∈ C(Ω) is in ℵ + . However, by (3.10), we see that which is impossible due to the definition of u. Thus, u is a viscosity supersolution, and this completes the proof that u is a viscosity solution to (1.1) in Ω. The uniqueness follows by the comparison principle, Theorem 2.4.

Existence when f (x, t) non-increasing in t
In this section, we investigate the existence of viscosity solutions to the problem (1.1), when f (x, t) is non-increasing in t. We first prove a stability result of the viscosity solutions. Then under the assumption that the problem (1.1) has a viscosity subsolution with f replaced by f + ε, ε > 0, we prove the existence of the viscosity solution to (1.1). Finally, we use the iteration method and Theorem 1.1 to establish the existence of the viscosity solution to (1.1).
k=1 be a sequence of non-negative functions in C(Ω) such that ξ k → ξ locally uniformly in Ω for some ξ ∈ C(Ω). For each positive integer k, let u k ∈ C(Ω) be a viscosity solution to the problem such that u 0 ≤ u k ≤ u ∞ in Ω, for some functions u 0 and u ∞ in C(Ω), with u 0 = u ∞ = q on ∂Ω. Then {u k } has a subsequence that converges locally uniformly in Ω to a viscosity solution u ∈ C(Ω) to the problem Proof. Set M := sup Ω u ∞ − inf Ω u 0 . Clearly, we have sup Ω u k − inf Ω u k ≤ M , for every k = 1, 2, · · · . Let K be any compact subset of Ω and d := dist(K, ∂Ω). We take R > 0 such that 4R < d. Since ∆ h ∞ u k ≥ 0 in Ω, by [6, Lemma 2.9], we obtain By compactness, we obtain {u k } are equicontinuous in K. On taking an exhaustion of Ω by subdomains compactly contained in Ω, we apply the standard method of Cantor diagonalization to extract a subsequence of {u k } that converge uniformly on compact subsets of Ω. For simplicity we will continue to denote such subsequence by We extend this definition to the closure Ω by defining u = q on ∂Ω. By the assumption, we have u 0 ≤ u ≤ u ∞ in Ω. This means that u ∈ C(Ω).
Next, we show that ∆ h ∞ u = ξ in the viscosity sense. Suppose that ϕ ∈ C 2 (Ω) and u − ϕ has a local maximum at some x 0 ∈ Ω, i.e. Particularly, Since x k ∈ B r (x 0 ), by passing to a subsequence, x k →x, for somex ∈ B r (x 0 ), letting k → ∞ in (4.1), we have Then we havex = x 0 . Thus, x k ∈ B r/2 (x 0 ) for sufficiently large k. Since u k is a viscosity subsolution and x k is a point of local maximum of Taking the limit in (4.2) and recalling that ξ k → ξ locally uniformly in Ω, we obtain Similarly, we can prove that u is a viscosity supersolution. Now we first give an existence result under the condition that problem (1.1) has a viscosity subsolution with f replaced by f + ε. Then we combine the iteration method and Theorem 1.1 to establish the existence result. The idea is that the existence of an appropriate viscosity subsolution leads to the existence of a viscosity solution. If the problem (1.1) has a viscosity subsolution with f replaced by f + ε, for some ε > 0, then there exists a viscosity solution u ∈ C(Ω) to (1.1).
Proof. By the assumption, let η 0 ∈ C(Ω) satisfy Then we define a sequence The existence of η k is ensured by Theorem 1.
Theorem 4.2 provides us with an approach to the existence problem, but it suffers from the shortcoming that we need a viscosity subsolution for the function f + ε, for some ε > 0. Next we impose the condition (4.3) on the domain to remove the assumption on the existence of the viscosity subsolution. where γ = 1 h+1 h (h+1)/h , λ 0 < 1 := inf ∂Ω q, and C ≥ sup Ω f (x, λ 0 ) 1/h , then (1.1) has a viscosity subsolution in C(Ω).
Proof. If f (x, t) ≡ g(x), the existence follows immediately by Theorem 1.1. Thus we consider the inhomogeneous term depending on the variable t. We choose d satisfying (4.5) Clearly W ∈ C ∞ (Ω) and one can verify that With the choice of d as in (4.4), we have λ 0 ≤ W ≤ 1 in Ω. Since f (x, t) is non-increasing in t, we obtain that W satisfies in Ω. Recalling that W ≤ q on ∂Ω, we conclude that W is a viscosity subsolution of (1.1) in Ω.
Note that the existence of viscosity subsolution depends on the size of the domain when f is non-increasing. Based on this point, we are ready to prove the existence result with the iteration method and Theorem 1.1.
Proof of Theorem 1.3. Since C ≥ (sup Ω f (x, λ 0 )) 1/h > 0 and Ω satisfies the condition (4.3), Theorem 4.2 implies the existence of a viscosity subsolution w defined in (4.5) satisfying and λ 0 ≤ W ≤ 1 . Then we have f (x, W ) ≤ sup Ω f (x, λ 0 ) ≤ C h . Denote u 0 := W , and we recursively define a sequence {u k } in C(Ω) as follows for k ≥ 1. By Theorem 1.1, we let u k satisfy ∆ h ∞ u k = f (x, u k−1 ), in Ω, u k = q, on ∂Ω. (4.6) By induction, we show that W ≤ u k in Ω for all k ≥ 1. Note that Therefore, by Lemma 2.3, we have W ≤ u 1 in Ω. Suppose W ≤ u k in Ω for some k ≥ 1. Then Lemma 2.3 again implies W ≤ u k+1 in Ω. This proves the claim.
Let v 1 ∈ C(Ω) be the viscosity solution to the problem Since ∆ h ∞ u k ≥ 0 in Ω, by the comparison principle in [7,21], we see that u k ≤ v 2 in Ω for all k. In summary, we have constructed a sequence {u k } of viscosity solutions to (4.6) such that Therefore, by Lemma 4.1, we can get the existence of the viscosity solution to (1.1).

Singular boundary value problem
In this section, we show the existence and the uniqueness of the viscosity solution to the singular problem (1.5). Moreover, when the domain satisfies some regular condition, we analyze the asymptotic behavior near the boundary of the viscosity solution. We now prove that the singular problem (1.5) has a viscosity supersolution.
Let ψ(x) := η(ϕ(x)) ∈ C 2 (Ω) such that w − ψ has a local minimum at z. Since w is a viscosity solution to (5.1), we have in the viscosity sense. In B δ (z), by a simple calculation, where we have used and . Thus, problem (1.5) has a viscosity supersolution v. Now, we prove the existence of a solution to (1.5) though the truncation method and the stability theory.
Proof of Theorem 1.5. For some µ > 0, let By Theorem 1.1, the problem has a viscosity solution u. By Lemma 2.2, we have u ≥ 0. And then we actually have Then for each positive integer k, the perturbed Dirichlet problem has a viscosity solution λ k . And by Lemma 2.2, we see that λ k > 0 for all k in Ω. Let w be a viscosity solution of (5.1) and v := η −1 (w), where η is as in (5.2). By Lemma 5.1, we have for every k. Theorem 2.4 shows that λ k ≤ v in Ω. Note that , where we have used that g is non-increasing. Theorem 2.4 implies λ k ≤ λ k+1 for all k. Then one has 0 < λ 1 ≤ · · · λ k ≤ λ k+1 ≤ · · · ≤ v, in Ω.
Then {λ k } are locally uniformly Lipschitz continuous and locally uniformly bounded. And the limit lim k→∞ λ k = λ is also locally Lipschitz continuous in Ω. In Lemma 4.1, let , and u k := −λ k .
Since λ ∈ C(Ω), we also have locally uniformly, and the limit function is continuous in Ω. We set Obviously, u 0 , u ∞ ∈ C(Ω). From Lemma 4.1, we see that the problem (1.5) has a viscosity solution λ. Finally, the uniqueness can be obtained by Theorem 2.4.
Next, we give some definitions and properties of Karamata's regular variation theory which was first introduced by Karamata in 1930's (see [15,16,17] and the references therein), and then based on Karamata's regular variation theory, we proceed to discuss the boundary behavior of viscosity solutions to the singular boundary value problem (1.5). Now we recall some definitions and properties of regularly varying functions (see [12,41,43]).

Definition 5.2.
A positive measurable function f defined on (0, a), for some a > 0, is called regularly varying at zero with index ρ ∈ R, written f ∈ RV Z ρ , if for each ξ > 0, In particular, when ρ = 0, f is called slowly varying at zero.
Proposition 5.5 (Asymptotic behavior). If a function L is slowly varying at zero, then for a > 0 and s → 0 + , Now we state some important results that we can use to prove Theorem 1.6.