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

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.


Introduction and Statement of the Main Results
We are concerned with the well-posedness of the fully nonlinear second-order uniformly elliptic problem 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: for x ∈ R n and any u ∈ R, ξ, η ∈ R n , X, Y ∈ S n .

International Journal of Differential Equations
As regards the monotonicity in the variable u, we ask something more than the usual monotonicity assumption δ 0 : 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 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 with f ∈ L 1 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 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, p 0 p 0 n, Λ/λ ∈ n/2, n is the exponent such that for p > p 0 the generalized maximum principle GMP holds true; see Escauriaza 3  Here and in the sequel by L p -strong solutions of the equation F x, u, Du, D 2 u 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 C 2 pointwise solutions.
We also call L p -strong classical subsolutions, respectively supersolutions, of 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 We establish a first result in the case of F independent of x.
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.
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 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 International Journal of Differential Equations 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 We put SC : 1.13 and 1.3 and 1.4 .

1.14
Then, we say that F satisfies C 1,1 -estimates at x 0 if for all w 0 ∈ C 0 ∂B r 0 x 0 there exists a solution u ∈ C 2 B r 0 x 0 ∩ C 0 B r 0 x 0 of the Dirichlet problem F x 0 , 0, 0, D 2 w 0 in B r 0 x 0 , w w 0 on ∂B r 0 x 0 1.15 such that for some r 0 > 0. Finally, let 1.17 In the case p > n, by Caffarelli 7 , if β F x, x 0 is sufficiently "small" in a sense that will be made precise below, then u ∈ W 2,p loc B r 0 x 0 .Such result was generalized by Escauriaza to the range p > p 0 with p 0 ∈ n/2, n introduced above.
As a consequence, the structure conditions SC can be used "pointwise" to compare F x, v x , Dv x , D 2 v x with F x, u x , Du x , D 2 u 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 r 0 > 0; A3 F is measurable in x for all u, ξ, X ∈ R × R n × S n and convex in X.
By our assumptions, the L p -strong solution of Theorem 1.3 will be also the unique L pviscosity solution.
Theorem 1.3 can be used for instance in the case of Bellman-type equations where L α is a semilinear second-order operator with bounded measurable coefficients such that for almost everywhere x ∈ R n and every α, provided the a ij α are uniformly continuous in R n with continuity modulus independent of α and inf α f 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 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 International Journal of Differential Equations 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 > p 0 the Dirichlet problem 23 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 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 > p 0 the Dirichlet problem 24 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 Labutin 13 showed that the origin is a removable singularity for the equation that is every L n -viscosity solution in the punctured ball B R \ {0} can be continued to an L nviscosity 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.

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 We say that F is uniformly elliptic with ellipticity constants λ > 0 and Λ ≥ λ if We are interested in the weaker notion of solution in the viscosity sense.Firstly suppose that F is continuous in International Journal of Differential Equations for each x ∈ Ω and ϕ ∈ C 2 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 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 in Ω, where h min f, g , respectively h max f, g . in Ω ∩ {w > 0}.
Proof.Let us suppose, for instance, that v ∈ C 2 Ω .Let ϕ be a C 2 -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 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 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.

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If one takes Proof.By the assumptions, it is sufficient to show that and the result follows by choosing C R > 0 as claimed.
In what follows p 0 ∈ n/2, n is the exponent such that GMP holds for p > p 0 see Sections 1 and 2 and u ± max ±u, 0 .

3.9
Proof.By SC we have 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

3.11
On the other hand u is an L p -viscosity subsolution of the equation Hence, by Lemma 2.2 see Remark 2.3 the function w u − Φ is an L p -viscosity solution of the equation with C 0 , 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

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 where which turns out to satisfy SC .Therefore, u − max v, 0 is an L p -viscosity solution of the equation On the other hand, by virtue of Lemma 3.1, Φ is an L p -strong supersolution of the equation Hence, we finish the proof arguing as in Lemma 3.2 to obtain the estimate

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 By the relationship between C-viscosity and L p -viscosity solutions and between L pviscosity 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 where C 0 C 0 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 almost everywhere x ∈ R n for |v| ≤ R.

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Therefore for h > k we have By C α -estimates see 14, Proposition 4.10 and 19, Theorem 2 we deduce that for a positive constant C 1 independent of h > k.
By a diagonal process, using Ascoli-Arzelà theorem we extract a subsequence h k ∈ N such that u h k → 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 Here, according to the notations of Proposition 3.
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-H ölder continuous in every subset for every x ∈ ∂Ω and ρ ≤ ρ k , and therefore u h are also equicontinuous in Ω R .
International Journal of Differential Equations 15 Thus, using a diagonal procedure as in the proof of Proposition 4.1 we find an L pviscosity solution u ∈ C Ω of the Dirichlet problem under consideration.

Uniqueness and Blow-Up
In this section we begin noticing that from Section 3 we get at once the following maximum principle.
, Crandall and Swiech 4 , and Koike and Swiech 5 : if f ∈ L p Ω with p > p 0 and u ∈ W 2,p loc Ω ∩ C Ω is an L p -strong solution of the maximal equation P λ,Λ D 2 u γ|Du| ≥ f, d diam Ω and C being a positive constant depending on n, λ, Λ, p, γd.
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 ∈ C 2 Ω denote by Du and D 2 u the gradient and the Hessian matrix of u.We wish to discuss the solvability of equation F x, u x , Du x , D 2 u x f x 2.4 under the assumptions 1.2 , 1.3 and 1.4 , and we refer to u ∈ C 2 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.

Lemma 2 . 2 .
Let u, v ∈ C Ω be, respectively C-viscosity subsolution and supersolution of the equations F x, u, Du, D 2 u f and F x, v, Dv, D 2 v g in Ω, and assume 1.2 -1.3 .If at least one between u and v is in C 2 Ω , then the difference w u − v is a C-viscosity subsolution of the maximal equation P λ,Λ D 2 w γ|Dw| − δw s f − g 2.8

F x, u, Du, D 2 u f x 4 . 1 in
R n under the assumption that f ∈ L p loc R n with p > p 0 , where p 0 ∈ n/2, n is the exponent above which GMP holds true; see Sections 1 and 2.

Proposition 5 . 1 . 2 M1 4 m1 5 satisfies
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 γ|ξ| − δ|u| s−1 u 5.1 for all u, ξ, X ∈ R × R n × S n and u ∈ C Ω is an L p -viscosity solution p > p 0 of the equationF x, u, Du, D 2 u ≥ 0 in Ω. 5.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 thatF x, v, ξ, X ≥ P − λ,Λ X − γ|ξ| − δ|v| s−1 v 5.3 for all v, ξ, X ∈ R × R n × S n and v ∈ C Ω is an L p -viscosity solution p > p 0 of the equation F x, v, Dv, D 2 v ≤ 0 in Ω. 5.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 .SinceF x, u, ξ, X P λ,Λ X γ|ξ| − δ|u| s−1 u5.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 operatorG x, u, ξ, X −F x, −u, −ξ, −X 5.6are as in the proof of Proposition 3.3.Proof of Theorem 1.6.Following 2 we consider a nondecreasing sequence off k ∈ C Ω such that lim k → ∞ f k − f L p K 5.10for all compact set K of Ω.Then by Theorem 1.4 we solve the problemF x, u k , Du k , D 2 u k f k x in Ω, u k x k on ∂Ω.5.11As 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 equationF x, u, Du, D 2 u f x in Ω.5.12To 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 problemP λ,Λ D 2 w b|Dw| ≥ 0 in Ω ∩ {u k > u k 1 }, w ≤ 0 on ∂ Ω ∩ {u k > u k 1 } .5.13Hence, 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 k ∈ N; whence the assertion follows.
∈ 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 > p 0 ; see for instance 5, Theorem 3.2 .Note that L p -viscosity solutions are C-viscosity solutions of F x, u, Du, D 2 u f because the space W 2,p loc Ω of test functions for L p -viscosity solutions is larger than C 2 Ω .Conversely, if p > p 0 , assuming that F and f are continuous, then C-viscosity solutions are L p -viscosity solutions; see 15, Proposition 2.9 .Suppose that p > p 0 , as before.If u ∈ W x → x F x, u x , Dϕ x , D 2 ϕ x − f x ≥ loc Ω is an L p -viscosity subsolution supersolution , then u is an L p -strong subsolution supersolution ; see 15, Corollary 3.7 .
It is worth to recall that condition SC is equivalent to SC in the case that F is continuous.Let F : R n × R × R n × S n → R be measurable in x and satisfy the structure condition SC almost everywherex ∈ R n for all u, ξ, X ∈ R × R n × S n .If f ∈ L 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 B 2 k with continuous boundary condition.Choose a solution u k for each k ∈ N. Using Proposition 3.3, for h > k we have sup 3, Ω 2 k Ω ∩ B 2 k , while ψ k is a continuous extension to R n of ψ| ∂Ω∩B 2 k ; 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