Regularity estimates for fully nonlinear integro-differential equations with nonhomogeneous degeneracy

We investigate the regularity of the solutions for a class of degenerate/singular fully nonlinear nonlocal equations. In the degenerate scenario, we establish that there exists at least one viscosity solution of class Cloc1,α , for some constant α∈(0,1) . In addition, under suitable conditions on degree of the operator σ, we prove regularity estimates in Hölder spaces for any viscosity solution. We also examine the singular setting and prove Hölder regularity estimates for the gradient of the solutions.


Introduction
In this paper, we examine the regularity of solutions to degenerate/singular fully nonlinear integro-differential equations of the form where 0 ≤ a(•) ∈ C(B 1 ), I σ is a fully nonlinear elliptic integro-differential operator with order σ ∈ (1, 2) and f ∈ L ∞ (B 1 ).We establish Hölder regularity estimates for the gradient of bounded viscosity solutions to (1).
More precisely, in the degenerate setting 0 < p ≤ q, we show that for any 1 < σ < 2, there exists at least one viscosity solution u to (1), so that u ∈ C 1,α loc (B 1 ) for some α ∈ (0, 1).This result is new even in the case a ≡ 0.Moreover, we prove the existence of a constant σ 0 , sufficiently close to 2, for which solutions are of class C 1,α  loc for some α ∈ (0, 1), provided σ 0 < σ < 2. In the singular framework −1 < p ≤ q < 0, we show that solutions belong to C 1,α loc (B 1 ) for some α ∈ (0, 1).In the literature, integro-differential operators are commonly referred to as nonlocal operators.This class of problems appears in many mathematical modeling processes, such as image processing and payoff models, see [12,14,27], to mention just a few.For instance, linear nonlocal operators arise naturally in discontinuous stochastic processes taking the form Lu(x) := i,j for a suitable measure µ.Meanwhile, nonlinear integro-differential equations can be found in stochastic control problems, where a player can change strategies at every step.In this case, we end up with convex nonlinear equations driven by the operator In addition, we can also obtain models of the type whenever two or more players are involved, see [32].
Integro-differential operators have been extensively studied over the years by many authors.Regarding the qualitative properties of the solutions, the authors in [7,8,33] established Harnack inequality by using probabilistic methods.It is important to mention that Hölder estimates do not follow from the Harnack inequality, since it requires the solution to be positive in the whole R d .Keeping this in mind, Hölder estimates were proved in [7], still using probabilistic techniques.In [31] L. Silvestre provides an analytical proof of Hölder continuity, and also gives more flexible assumptions on the operator than the previous works.
All these previous results enjoy the same feature: the estimates blow up as the order of the operator approaches 2. The first results that are uniform in the degree were given by L. Caffarelli and L. Silvestre in [15].In that paper, the authors produced a series of results that extended the theory for second-order operators, such as a comparison principle, a nonlocal version of the Alexandroff-Bakelman-Pucci estimate, a Harnack inequality, and C 1,α -estimates.Since the estimates are uniform as the degree of the operator goes to 2. These results can be seen as a natural extension of the regularity theory for elliptic PDEs.
In [16] the authors extended their previous results to nontranslation-invariant equations using perturbative methods.In particular, they prove C 1,α -regularity for solutions of equations that are close, in an appropriate sense, to another one with C 1, ᾱ-estimates.More or less simultaneously G. Barles, E. Chasseigne, and C. Imbert also dealt with nontranslation-invariant equations establishing Hölder regularity estimates.
It is worth noticing that their assumptions are different from the ones in [16], and the equations that they work with involve second and first-order terms, allowing some degenerate operators.For more details, see [5].
A very interesting question that remains widely open is whether or not the solutions of are of class C 1,α .In the local case, this type of problem drew the attention of many authors, see for instance [1,2,10,11,13,19,29] just to cite a few.More generally, equations with double degeneracy of the form with 0 < p ≤ q were also investigated.We refer the reader to [18,21,25], and the references therein.The main novelty in those works is that equation ( 4) is no longer homogeneous, which makes the scaling process more delicate.We stress that the operator in ( 4) is a nonvariational counterpart of the extensively studied (p, q)-Laplacian, see for instance [17,20,22,23,24,26,30].
In the nonlocal case, a partial result was given by D. Prazeres and E. Topp in [28].The authors suppose that σ is sufficiently close to 2, so that in the interactive process I σ approaches F , where F is a local operator.Under these assumptions, they prove that viscosity solutions to (3), with p > 0, are of class C 1,α .
Their condition on σ allows them to import some regulatity of the F -harmonic functions to the solutions of (3).In the general case, σ ∈ (1, 2), there are not regularity results available for the solution of (3).One of our contribution in this manuscript is to show that the existence of a C 1,α loc viscosity solution for a class of operators that includes the one in (3), for any σ ∈ (1, 2).This is the content of our first main result.
Theorem 1.1.Let 0 < p ≤ q.Suppose A1-A3, to be detailed further, hold true.Then, there exists at least one u ∈ C(B 1 ) bounded viscosity solution to where and C = C(d, p, λ, Λ) is a positive constant.Here, ᾱ is the exponent associated with the regularity of I σ harmonic functions.
We recall that a general regularity theory for viscosity solutions to (1) when 0 < p ≤ q remains open, even in the case where a ≡ 0. This happens because of the lack of an uniqueness result, which is the core of the arguments for the general regularity theory in local case.See also Remark 3.1.
In order to prove a general regularity result we need to restrict ourselves to the case where σ is close to 2, as in [28].In particular in extend the results in [28] to a nonhomogeneous degeneracy setting.This is the content of our second result.
Theorem 1.2.Let u ∈ C(B 1 ) be a bounded viscosity solution to with 0 < p ≤ q.Suppose A1-A4, to be detailed further, hold true.Then, there exists σ 0 ∈ (1, 2) sufficiently close to 2 such that if σ 0 < σ < 2, then u ∈ C 1,α loc (B 1 ) with the estimate where and C = C(d, p, λ, Λ) is a positive constant.Here, ᾱ is the exponent associated with the regularity of F harmonic functions.
The constant C that appears in the estimate above is uniform in σ which means that it does not blow up as σ approaches 2. Hence, we can see Theorem 1.2 as an extension of [21, Theorem 1].
In the singular case the difficulty starts with the notion of viscosity solution.Indeed, the definition considered in the local case [9, Definition 2.2], seems not to be suitable in our scenario, because whenever ) and hence one could say that 0 ≤ f (x 0 ) (in the case of supersolutions).On the other hand, in the nonlocal case, we still need to consider the quantity which depends on r(x 0 ) and may not vanish.To overcome this, we use the notion of approximated viscosity solution, which coincides with the usual notion of viscosity solution over the set {Du = 0}, and are defined as the limit of the solutions of uniformly elliptic nonlocal equations .See Definition 2.5 for the precise notion of approximated viscosity solution.Under this setting, we establish C 1,α -regularity for solutions of (1).More precisely, we prove the following: ) be a bounded approximated viscosity solution to with −1 < p ≤ q < 0. Assume A1-A3, to be detailed later, are in force.Then u ∈ C 1,α loc (B 1 ) and we have the estimates for some α ∈ (0, 1) and The remainder of this article is structured as follows: in the second section, we collect some auxiliary results and present our assumptions.In the third section, we investigate the Hölder regularity for the gradient of the solutions in the degenerate case.The last section is devoted to the proof of the regularity estimates in Hölder spaces for the singular scenario.

Preliminaries
In this section, we gather basic notions and detail our main assumptions used throughout this paper.In what follows, we present the definition of the nonlocal operator that we work with in this article.
Definition 2.1.Consider σ ∈ (1, 2), and constants 0 < λ ≤ Λ.We define the family K 0 , as the set of measurable kernels K : Definition 2.2.Given u : R d → R, Ω ⊆ R d a measurable set and K ∈ K 0 , we define the operator I K as where P.V. denotes the Cauchy principal value of the integral.
Definition 2.3.We say that u : Throughout this manuscript, we consider different ranges of σ to obtain the main results.For this reason, we fix the notation I σ (u, x) to emphasize the dependence on σ for the class of nonlocal operators as defined in (5).
In the proofs of Theorem 1.1 and Theorem 1.2, we make use of some scaled functions that satisfy a variant of equation (1), namely: where ξ ∈ R d .Hence, we define the solutions and we prove some results for the equation above instead of only for equation (1).We first present the definition of viscosity solutions to (1).Let us consider a collection of kernels {K ij } i,j ⊆ K 0 .
Definition 2.4 (Viscosity solution).We say that u ∈ C(B 1 ) ∩ L 1 σ is a viscosity subsolution to (1), if for any ϕ ∈ C 2 (R d ) and for all x 0 ∈ B 1 such that u − ϕ has a local maximum at x 0 , we have where the operator I δ is given by Similarly, we say that u ∈ C(R d ) is a viscosity supersolution to (1), if for any ϕ ∈ C 2 (R d ) and for all x 0 ∈ B 1 such that u − ϕ has a local minimum at x 0 , we have Finally, we say that u ∈ C(R d ) is a viscosity solution to (1) if it is both a viscosity subsolution and supersolution.
In the next, we define the so-called approximated solutions.Its main purpose is to allow us to deal with the singular case, but it is also a fundamental notion in the proof of Theorem 1.1.
Definition 2.5 (Approximated viscosity solution).Let ξ be a vector in R d .We say that u ∈ C(B 1 )∩L 1 σ (R d ) is an approximated viscosity solution to if there are sequences such that u j is a viscosity solution of Moreover, if 0 < p ≤ q, c j has also to satisfy jc p j → ∞, as j → ∞.If ξ = 0, then we take ξ j = 0 for all j ∈ N.
) is a test function touching u from above at a point x 0 then, because u j → u locally uniformly in B 1 , we have that there exists a sequence (ϕ j ) j∈N such that ϕ j → ϕ locally uniformly in B 1 and ϕ j touches u j from above at a point x j , where x j → x 0 .Since u j is a viscosity solution, this implies, Remark 2.2 (Scaling properties).Throughout the paper, we require for some ε to be determined.The condition in (8) is not restrictive.In fact, consider the function Hence, by choosing we can assume (8) without loss of generality.

Main assumptions
In what follows, we detail the assumptions of the paper.The first one concerns the degree of the operator A 1 (Degree of the operator).We suppose that 1 < σ < 2.
To guarantee that the nonlocal operator in ( 5) is well-defined, we must impose further conditions on the growth at infinity of the function u.This is the content of our next assumption.
We remark that bounded functions satisfy the growth condition (6).The next assumption concerns the source term.
Although we require f ∈ C(B 1 ), all the estimates will only depend on the L ∞ (B 1 ) norm of f .The next assumption concerns the regulairy of a.Our last assumption ensures the convergence of the operator I σ , as σ goes to 2.
A 4. Let {K ij } i,j be a collection of kernels in K 0 .There exists a modulus of continuity ω and {k ij } i,j ∈ (λ, Λ) satisfying the estimates 3 Analysis of the degenerate case In this section, we assume 0 < p ≤ q, i. e., we are considering the degenerate scenario.We present the results in two subsections: The first one is dedicated to the proof of Theorem 1.1, while in the second one we give a proof of Theorem 1.2.

C 1,α -regularity via approximated viscosity solutions
This subsection is devoted to the proof of Theorem 1.1.We start with some properties of approximated viscosity solutions to (1).The first one states that the set of approximated viscosity solutions is contained in the set of viscosity solutions.
Proposition 3.1.Let u ∈ C(B 1 ) be an approximated viscosity solution to (7).Suppose that A1-A3 are in force.Then u is a viscosity solution to the same equation.
Proof.Let ϕ ∈ C 2 (R d ) and x 0 ∈ B 1 such that u − ϕ has a local maximum at x 0 .From Definition 2.5 and Remark 2.1 we obtain where ϕ j → ϕ and u j → u locally uniformly in B 1 , x j → x 0 , ξ j → ξ and c j → 0. By passing the limit as j → ∞, we get that which implies that u is a viscosity subsolution to (7).We recall that the convergence I δ (u j , ϕ j , x j ) → I δ (u, ϕ, x 0 ) follows from [16, Lemma 5].Similarly, we can prove that u is also a viscosity supersolution.This finishes the proof.
Remark 3.1.Due to the absence of a uniqueness result, the reverse statement may not be true.
Next, we present a stability type result for approximated viscosity solutions of (7).
in the approximated viscosity solution sense.Assume that A1-A3 hold true.Suppose further that there exists Proof.By Definition 2.5, there exist sequences (u m j as m → ∞, such that u m j is a viscosity solution of for each fixed j.Now, we consider the sequence (u j j ) j∈N .We have that u j j → u locally uniformly in B 1 , and u j j is a viscosity solution to Therefore, u j j is also a viscosity solution to Since by definition jc p j → ∞, as j → ∞, we infer that u is a viscosity subsolution to Similarly, we show that u is also a viscosity supersolution.This finishes the proof.Now, for each j ∈ N, we consider the nonlocal uniformly elliptic equation We are going to show that a sequence (u j , where u is an approximated viscosity solution to (1) with a suitable boundary data.In order to do that, we stablish a compactness result for the solutions of (11).
Lemma 3.1.Let u ∈ C(B 1 ) be a viscosity solution to (11).Suppose that A1-A3 are in force.Then u j is locally Lipschitz continuous, i. e., where the constant C > 0 does not depend on j.
Proof.We consider ψ : R d → R a nonnegative and smooth function such that and we define ψ = (osc B1 u j + 1) ψ.
It follows that Observe that if ȳ ∈ B c 3/4 , then ψ ≡ 1 which implies which is a contradiction, hence ȳ ∈ B 3/4 .Since Lϕ(|x − ȳ|) ≤ osc B1 u, by taking L sufficiently large, we may assume that x ∈ B 7/8 .Finally, it is straightforward to notice that x = ȳ, otherwise we would have Φ(x, ȳ) < 0. Now, we compute D x φ and D y φ at (x, ȳ) and Let us denote We have that u j −Φ ȳ attains its maximum at x, where Φ ȳ (x) := u j (ȳ)+φ(x, ȳ).Hence, since u j is a viscosity solution to (11), we obtain the following viscosity inequality Now, we take L sufficiently large, only depending on osc B1 u, so that Similarly we can obtain −I δ (u, −Φ x, ȳ) ≥ −2 f L ∞ (B1) , and consequently At this point, we concentrate in estimating the left hand side of the inequality above, that consists of uniformly elliptic nonlocal operators.The next step follows the arguments in [3,6] and we present here for the sack of completeness.
For |z| ≤ 1/10, we have x + z, ȳ + z ∈ B 1 .Then, there exists a kernel K in the family K 0 such that where and the constant C > 0 depends on the ellipticity constants.
We denote e = x − ȳ, ê = e/|e| and for constants η ∈ (0, 1) and ρ ∈ (0, 1/10) to be fixed.Since (x, ȳ) is a maximum point of Φ, for all z ∈ B 1/10 we have the following inequalities and hence, it is possible to get that where the constant C > 0 is uniformly bounded from above and from below as δ → 0.Moreover, for every set O ⊂ R d such that x + z, ȳ + z ∈ B 1 , we have where the latter inequality follows from the smoothness of ψ.

Hence by taking
where C 3 is a large enough constant, depending only on λ, Λ, d we get a contradiction.This finishes the proof.

Proof. Consider the equation
for any 1 + α ∈ (0, σ).Since the operator is nonlocal uniformly elliptic, the existence of a viscosity solution u j is assured by [4].By Lemma 3.1, we have that the solution u j ∈ C 0,1 loc (B 1 ), with estimates that are independent of j.Hence, there exists a function u ∞ ∈ C α loc (B 1 ), for some α ∈ (0, 1), such that u j → u ∞ locally uniformly in B 1 , and u ∞ ≡ g outside B 1 .By taking c j := j 1/2p , we have directly by definition that u ∞ is an approximated viscosity solution to (1).
As we mentioned before, we will need to deal with equations of the form ( 7), hence we first prove some level of compactness for the solutions of such equations.Lemma 3.2.Let u ∈ C(B 1 ) be a viscosity solution to (7).Suppose that A1-A3 are in force.There exists a constant c 0 > 1 such that if |ξ| ≥ c 0 , then u is locally Lipschitz continuous, i. e., where the constant C > 0 does not depend on ξ.
After this, we continue by computing D x φ and D y φ at (x, ȳ), and we observe that u − Φ ȳ attains a maximum at x. Hence, since u is a viscosity solution to (14), we obtain Similarly, we have the viscosity inequalities for u at ȳ, In the sequel, by choosing c 0 such that b ≤ 1/(2L(2 + α)), we may conclude that Therefore, the former inequalities imply and consequently From here we can proceed as in Lemma 3.1 to finish the proof.
The next lemma deals with the other alternative of the norm of ξ, i.e., the case where |ξ| ≤ c 0 .
Lemma 3.3.Let u ∈ C(B 1 ) be a viscosity solution to (7).Suppose that A1-A3 hold true.If |ξ| ≤ c 0 , where c 0 is as in the previous lemma, then u is locally Lipschitz continuous, i. e., where the constant C > 0 does not depend on ξ.
Proof.The proof follows the same general lines as in the proof of Lemma 3.2.We define ψ, φ and Φ as in the previous lemma.As before, we can assume that Φ attains its maximum at (x, ȳ) ∈ B 7/8 × B 7/8 and we argue by contradiction assuming that Φ(x, ȳ) < 0. By taking L sufficient large, depending only on osc B1 u and c 0 we can infer that Since u is a viscosity solution to (7), we have which implies From here we can proceed as in Lemma 3.1 to finish the proof.
By combining Lemma 3.2 and Lemma 3.3, we obtain a compactness result for solutions of (7).In particular, we obtain Lipschitz estimates for the viscosity solutions of (1).where the constant C > 0 does not depend on ξ.
The next lemma is the core of the arguments needed for the proof of Theorem 1.1.
Lemma 3.4.Let u ∈ C(B 1 ) be a normalized approximated viscosity solution to (7).Suppose that A1-A3 are in force.Given M, δ > 0 and α ∈ (0, 1), there exists ε > 0 such that if and Proof.We argue by contradiction.That is, there are M 0 , δ 0 > 0, α 0 ∈ (0, 1) and sequences (ξ j ) j∈N , (u j ) j∈N , (f j ) j∈N such that u j is an approximated viscosity solution to and for all h ∈ C 1, ᾱ loc (B 1 ).Thanks to Proposition 3.4, we can guarantee the existence of a function u ∞ such that u j → u ∞ locally uniformly in B 1 , recall that approximated viscosity solutions are also viscosity solutions by Proposition 3.1.
The contradiction assumptions combined with Proposition 3.2 allow us to pass the limit in ( 16) and conclude that u ∞ is a viscosity solution to Hence, we infer that u ∞ is of class C 1,α loc .Finally, by taking h = u ∞ , we arrive in a contradition with (17) for j sufficiently large.Proposition 3.5.Let u ∈ C(B 1 ) be an approximated viscosity solution to (1).Assume that A1-A3 hold true.Given M > 0, there exists ε > 0 such that, if one can find a constant 0 < ρ ≪ 1/2 and an affine function ℓ for which for every α ∈ (0, ᾱ).
Proof.Let h ∈ C 1, ᾱ(B 3/4 ) be the function from the Lemma 3.4 such that From the regularity available for h, we get where C 0 = C 0 (λ, Λ, d, p) is a positive constant.Set ℓ(x) := h(0) + Dh(0) • x, by using the triangular inequality, we obtain provided we choose ρ and δ such that Notice that the universal choice of δ determines the value of ε through the Lemma 3.4.
Proposition 3.6.Let u ∈ C(B 1 ) be an approximated viscosity solution to (1).Suppose that A1-A3 are in force.Given M > 0, there exists ε > 0 such that, if then we can find a sequence of affine functions such that for all where Proof.We proceed by induction argument on j.The case j = 1 is the content of Proposition 3.5.Suppose the statement have been verified for j = 1, . . ., k.We shall prove the case j = k + 1. Define the auxiliary 1+α)  .
By the induction hypotheses v k L ∞ (B1) ≤ 1.In addition, v k solves where ã(x) = a(ρ k x)ρ kα(q−p) and f (x) = ρ kσ−k(α+αp+1) f (ρ k x).Observe that ã(x) ≥ 0 and ã ∈ C(B 1 ).Let α be as in (20), so that σ − α(1 Once we have verified (21), we can apply the Proposition 3. We again resort to an induction argument.The case k = 0, we take v 0 = u.Suppose that we have proved the statement in the case k = 0, ..., m.We shall verify the case k = m + 1.Notice that If |x|ρ > 1/2, we have where in the last inequality we used (18).On the other hand, if |x|ρ ≤ 1/2, we obtain where h comes from the Lemma 3.4 and we used (18) again in the last inequality.This finishes the proof.Moreover, we have the estimates Fix 0 < ρ ≪ 1, and let k ∈ N be such that ρ k+1 < r < ρ k .We estimate from the previous computations ≤ Cr (1+α) .
To conclude the proof, we characterize the coefficients a ∞ and b ∞ .In fact, by taking the limit in the inequality sup evaluated at 0, we obtain that a ∞ = u(0, 0).We can also conclude that b ∞ = Du(0, 0) see for instance [13].
We have proved that approximated viscosity solutions are of class C 1,α loc , and that they exist by Proposition 3.3.Since approximated viscosity solutions are also viscosity solutions (Proposition 3.1), we have the existence of a C 1,α loc (B 1 ) viscosity solution of (1).
3.2 C 1,α regularity via a smallness condition on 2 − σ In this subsection, we give a proof of Theorem 1.2.We start with an approximation lemma, which allows us to relate our model with a second order local equation.
Given M, δ > 0 and α ∈ (0, 1), there exists ε > 0 such that if and Proof.We argue by contradiction.That is, there are M 0 , δ 0 > 0, α 0 ∈ (0, 1) and sequences (ξ j ) j∈N , (a j ) j∈N , (u j ) j∈N , (f j ) j∈N and (σ j ) j∈N such that and for all h ∈ C 1, ᾱ loc (B 1 ).Since σ j → 2, thanks to A4 we have I σj → F ∞ , where F ∞ is a uniformly elliptic operator.In addition, from Proposition 3.4 we can guarantee the existence of a function u ∞ such that u j → u ∞ in B 3/4 .We shall prove that u ∞ solves in the viscosity sense.touches u j from below at xj in a neighborhood of the origin.Since the sequence {x j } is bounded, up to a subsequence, we obtain that xj → x∞ for some x∞ ∈ B 1 .Now, we study two cases: Π Γ (x ∞ ) = 0 and Π Γ (x ∞ ) = 0.In the case Π Γ (x ∞ ) = 0, we have consequently, we can rewrite the function φ k as Notice that In addition, This implies that −F ∞ (M ) ≥ 0, which is a contradiction with (27).Hence, for all j sufficiently large, we get If Γ ≡ R d , then we can find e ∈ S d−1 ∩ Γ ⊥ for which by (28) it holds Thus, for all j large enough, we have In both cases, we obtain that

Existence of approximated viscosity solutions
For each j ∈ N, we consider the nonlocal uniformly elliptic equation where (c j ) j∈N is a sequence of positive real numbers such that c j → 0 and c j ≤ 1 for every j.We prove that the sequence (u j ) j∈N of viscosity solutions for (30), with boundary data in which is an approximated viscosity solution to equation (1).We start with a compactness result, independent of j, for the solutions of (30).Lemma 4.1.Let u j ∈ C(B 1 ) be a viscosity solution to (30).Suppose that A1-A3 are in force.Then u j is locally Lipschitz continuous, i. e., where M > 0 does not depend on j.
Proof.The proof follows the same lines as in Lemma 3.1, hence we will only provide a few details.Consider Since Φ is a continuous function, we have Φ attains its maximum in B 1 × B 1 at (x, ȳ).We argue through a contradiction argument, suppose that Φ(x, ȳ) > 0, and with the same notation as Lemma 3.1, we obtain

An analogous reasoning yields
, where C 1 and C 2 are universal constants and consequently where C is a universal constant.
From here, we can proceed exactly as in Proposition 3.1 to finish the proof.
In the next result, we prove existence of an approximated viscosity solutions for the equation (1).where 1 + α ∈ (0, σ).Since the operator is nonlocal uniformly elliptic, the existence of a viscosity solution u j is assured by [4].By Lemma 4.1, we have that the solution u j ∈ C 0,1 , with estimates that are independent of j.Hence, there exists u ∞ ∈ C loc (B 1 ) such that u j → u ∞ locally uniformly in B 1 , and u ∞ ≡ g outside B 1 .
By taking c j := 1/j, we have directly by definition that u ∞ is an approximated viscosity solution to (1).

C 1,α -regularity estimates
This subsection is devoted to the proof of Theorem 1.3.We start with a compactness result to approximated viscosity solutions of (1).
Notice that the operator in ( 30) is a nonlocal uniformly elliptic operator, hence for each j ∈ N, we have u j ∈ C 1,α loc (B 1 ).Keep in mind that the C 1,α -norm of u j could degenerate as j → ∞.In what follows, we will verify that this does not happen.Proposition 4.2.Let u j ∈ C(B 1 ) be a viscosity solution to (30).Suppose that A1 -A4 are in force.Then u j ∈ C 1,α loc (B 1 ) with α ∈ (0, 1).In addition, there exists a positive constant C = C(λ, Λ, d, p) such that Proof.Let ϕ j ∈ C 2 (R d ) be a test function that touches u j from below at x j .We write the viscosity inequality − [(|Dϕ j (x j )| + c j ) p + a(x j )(|Dϕ j (x j )| + c j ) q ] I δ (u j , ϕ j , x j ) ≤ f (x j ).
Since u j is of class C 1,α loc (B 1 ) and ϕ j touches u j from below at x j , we can conclude from Lemma 4.1 that |Dϕ j (x j )| = |Du j (x j )| ≤ M .Hence −I σ (u j , x) ≤ M −p f L ∞ (B1) .
Therefore by [16,Theorem 52] we have that u j ∈ C 1,α loc (B 1 ) and where C > 0 is a universal constant.
At this point, we are able to prove the Theorem 1.3.
Proof of Theorem 1.3.From Definition 2.5, there exist sequences (u j ) j∈N ∈ C(B 1 ) ∩ L 1 σ (R d ), and (c j ) j∈N ∈ R + fulfilling u j converges locally uniformly to u in B 1 and c j → 0, such that u j is a viscosity solution to − ((|Du j | + c j ) p + a(x)(|Du j | + c j ) q ) I σ (u j , x) = f in B 1 .
It follows from Proposition 4.2 that u j is of class C 1,α loc with estimates where C > 0 is a constant that does not depend on j.
By applying the limit in the estimate above as j → ∞ combined with the fact that u j converges locally uniformly to u in B 1 , we obtain the following estimate This completes the proof of the theorem.

Now we are ready to present the proof of Theorem 1. 1 .
Proof of the Theorem 1.1.From (19), we conclude that the sequences (a j ) j∈N and (b j ) j∈N are Cauchy sequences.Hence, there exist constants a ∞ and b ∞ such that lim j→∞ a j = a ∞ and lim j→∞ b j = b ∞ .