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

Authors: G. Galise; A. Vitolo
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

We deal with fully nonlinear second-order equations assuming a superlinear growth inu 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.


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## Body

## 1. Introduction and Statement of the Main Results

We are concerned with the well-posedness of the fully nonlinear second-order uniformly elliptic problem(1.1)F(x,u,Du,D2u)=f(x)inRn,
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:(1.2)Pλ,Λ-(Y-X)-γ|η-ξ|≤F(x,u,η,Y)-F(x,u,ξ,X)≤Pλ,Λ+(Y-X)+γ|η-ξ|
for x∈ℝn and any u∈ℝ,ξ,η∈ℝn,X,Y∈𝒮n.As regards the monotonicity in the variableu, we ask something more than the usual monotonicity assumption (δ=0):
(1.3)F(x,u,ξ,X)-F(x,v,ξ,X)≤-δ(u-v)sifv<u
for s>1 and δ>0.We also suppose, as generally with the formulation (1.1), that
(1.4)F(x,0,0,0)=0.We collect the above assumptions in the structure condition(1.5)(SC):=(1.2)and(1.3)and(1.4),
noting that
(1.6)F(x,u,Du,D2u)=Pλ,Λ+(D2u)+γ|Du|-|u|s-1u
fits into our framework.Hence, this paper is in the wake of Brezis [1], who proved the existence and uniqueness of distributional solutions u∈Llocs(ℝn) for the semilinear equation
(1.7)Δu-|u|s-1u=f(x)inRn
with f∈Lloc1(ℝn) and of Esteban et al. [2] for the case of Ln-viscosity solutions of the fully nonlinear second order uniformly elliptic equation
(1.8)F(D2u)-|u|s-1u=f(x)
with f∈Llocn(ℝn).Our aim is to extend the result in different directions, including lower-order terms, allowing the dependence onx and going below the exponent n.Throughout the paperλ 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∈Lp(Ω) with p>p0 and u∈Wloc2,p(Ω)∩C(Ω¯) is an Lp-strong solution of the maximal equation
(1.9)Pλ,Λ+(D2u)+γ|Du|≥f,
then
(1.10)max⁡⁡Ω¯u≤max⁡⁡∂Ωu+Cd2-(n/p)‖f-‖Lp(Ω)
with d=diam(Ω) and C being a positive constant depending on n,λ,Λ,p,γd.Here and in the sequel byLp-strong solutions of the equation F(x,u,Du,D2u)=f(x) we intend the Wloc2,p 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 callLp-strong (classical) subsolutions, respectively supersolutions, of F=f the Wloc2,p- (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 ofC-viscosity and Lp-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 ofF independent of x.Theorem 1.1.
LetF:ℝn×ℝ×ℝn×𝒮n→ℝ be a continuous function satisfying the structure condition (SC) such that
A1F(x,u,ξ,X)=F¯(u,ξ,X)
for all (x,u,ξ,X)∈ℝn×ℝ×ℝn×𝒮n. If f∈C(ℝn), there exists a unique C-viscosity solution of equation
(1.11)F¯(u,Du,D2u)=f(x)inRn.To consider a dependence onx, we need to control the oscillations in the variable x, and this also requires a uniform bound of the local Lp-norms of f.Theorem 1.2.
LetF∈C(ℝn×ℝ×ℝn×𝒮n) satisfy the structure condition (SC). Suppose also that for all R>0 there exist a constant KR>0 and a function ωR:ℝ+→ℝ+ such that limt→0+ωR(t)=0 and
A2.1|F(y,u,ξ,X)-F(x,u,ξ,X)|≤KR‖X‖|y-x|+ωR((1+|ξ|)|y-x|)
as x,y∈ℝn, u∈(-R,R) and (ξ,X)∈ℝn×𝒮n. If p>p0, f∈C(ℝn), and
A2.2‖f‖Mp:=sup⁡x∈Rn‖f‖Lp(B1(x))<+∞,
then (1.1) has a unique C-viscosity solution.As it can be seen in Section4, 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:
(1.12)(A2):=(A2.1)and(A2.2).In order to deal withF merely measurable in x, we will suppose for every R>0 there exists cR>0 such that
(1.13)Pλ,Λ-(Y-X)-γ|η-ξ|-cR|v-u|≤F(x,v,η,Y)-F(x,u,ξ,X)≤Pλ,Λ+(Y-X)+γ|η-ξ|+cR|v-u|
for x∈ℝn and u,v∈(-R,R),ξ,η∈ℝn,X,Y∈𝒮n. We put
(1.14)(SC¯):=(1.13)and(1.3)and(1.4).Then, we say thatF satisfies C1,1-estimates at x0 if for all w0∈C0(∂Br0(x0)) there exists a solution u∈C2(Br0(x0))∩C0(B¯r0(x0)) of the Dirichlet problem
(1.15)F(x0,0,0,D2w)=0inBr0(x0),w=w0on⁡∂Br0(x0)
such that
(1.16)‖u‖C1,1(Br0/2(x0))≤Cr0-2‖w‖L∞(∂Br0(x0))
for some r0>0.Finally, let(1.17)βF(x,x0):=sup⁡X∈SnX≠0⁡|F(x,0,0,X)-F(x0,0,0,X)|‖X‖.In the casep>n, by Caffarelli [7], if βF(x,x0) is sufficiently “small” in a sense that will be made precise below, then u∈Wloc2,p(Br0(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.
LetF:ℝn×ℝ×ℝn×𝒮n→ℝ be a function satisfying the structure condition (SC¯) almost everywhere x∈ℝn such that one of the following assumptions blocks holds true:(A3)
F is continuous and has C1,1-estimates for each x∈ℝn with some r0>0;(A3)′
F is measurable in x for all (u,ξ,X)∈ℝ×ℝn×𝒮n and convex in X.Let f∈C(ℝn) in the case (A3), f∈Llocp(ℝn) in the case (A3)′. If
(1.18)sup⁡r∈(0,r0)(–∫Br(x)|βF(x,y)|ndy)1/n≤θ,
for every x∈ℝn, with θ=θ(n,p,λ,Λ,r0), then (1.1) has a unique Lp-strong solution u∈Wloc2,p(ℝn), provided that p>p0.By our assumptions, theLp-strong solution of Theorem 1.3 will be also the unique Lp-viscosity solution.Theorem1.3 can be used for instance in the case of Bellman-type equations
(1.19)sup⁡α⁡{Lαu-fα(x)}=0,
where Lα is a semilinear second-order operator
(1.20)Lαu:=aαij(x)Diju+bαi(x)Diu+c(x)u+d(x)|u|s-1u
with bounded measurable coefficients such that
(1.21)λ≤sup⁡|ξ|≤1〈aαij(x)ξ,ξ〉≤Λ,|bαi(x)bαi(x)|1/2≤γ,c(x)≤0,d(x)≤-δ<0
for almost everywhere x∈ℝn and every α, provided the aαij are uniformly continuous in ℝn with continuity modulus independent of α and infαfα∈Llocp(ℝn).Some cases of Isaacs-type equations can be treated with Theorem1.3, as for instance (see [10]), the minimum
(1.22)min⁡⁡{inf⁡α(Lαu-fα(x)),sup⁡β(Lβu-fβ(x))}=0
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 domainΩ⊊ℝn. 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 Ln-viscosity solution in [11, Theorem  4.1].The technique of the existence part of Theorem1.3 allows to generalize such results to any regular domain, even unbounded, of ℝn. 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.
LetΩ⊊ℝn be a domain satisfying a uniform exterior cone condition, and let F(x,t,ξ,X) be measurable in x∈ℝn for all (t,ξ,X)∈ℝ×ℝn×𝒮n such that the structure condition (SC)′ holds almost everywhere x∈Ω. Then, for p>p0 the Dirichlet problem
(1.23)F(x,u,Du,D2u)=f(x)inΩ,u(x)=ψ(x)on⁡∂Ω
has an Lp-viscosity solution u∈C(Ω¯) for every f∈Llocp(ℝn) and every ψ∈C(∂Ω).Remark 1.5.
The solutionu∈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∈ℝn.Finally, a monotonicity argument can instead be used whenf is bounded from below to obtain boundary blow-up Lp-viscosity solutions.Theorem 1.6.
LetΩ⊊ℝn 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
(1.24)F(x,u,Du,D2u)=f(x)inΩ,lim⁡x→∂Ωu(x)=+∞
has an Lp-viscosity solution u∈C(Ω) for every f∈Llocp(Ω) such that essinff>-∞.Remark 1.7.
Generally the existence results for the BVP of Theorems1.4 and 1.6 fail to hold if the domain is not sufficiently regular. In fact, assuming  F=F¯(X) uniformly elliptic such that
(1.25)1≤Λλ<n-1,s≥(n-1)λ+Λ(n-1)λ-Λ,
Labutin [13] showed that the origin is a removable singularity for the equation
(1.26)F¯(D2u)-|u|s-1u=0,
that is every Ln-viscosity solution in the punctured ball BR∖{0} can be continued to an Ln-viscosity solution in BR.The paper is organized as follows. In Section2 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 functionsF:Ω×ℝ×ℝn×𝒮n→ℝ where Ω is a domain (open connected set) of ℝn and 𝒮n the space of n×n real symmetric matrices. The identity matrix will be denoted by I and the trace of X∈𝒮n with Tr(X), while ∥X∥ is one of the equivalent norms of X in 𝒮n. For X,Y∈𝒮n we put X≤Y if
(2.1)〈Xξ,ξ〉≤〈Yξ,ξ〉∀ξ∈Rn.
We say that F is uniformly elliptic with ellipticity constants λ>0 and Λ≥λ if
(2.2)Pλ,Λ-(Y-X)≤F(x,t,ξ,Y)-F(x,t,ξ,X)≤Pλ,Λ+(Y-X)
for all (x,t,ξ)∈Ω×ℝ×ℝn and X,Y∈𝒮n, where
(2.3)Pλ,Λ+(Z)=sup⁡λI≤A≤ΛITr⁡(AZ),Pλ,Λ-(Z)=inf⁡λI≤A≤ΛITr⁡(AZ),Z∈Sn,
are, respectively, the maximal and the minimal (Pucci) operators in the class of uniformly elliptic operators with ellipticity constants λ and Λ.Foru∈C2(Ω) denote by Du and D2u the gradient and the Hessian matrix of u. We wish to discuss the solvability of equation
(2.4)F(x,u(x),Du(x),D2u(x))=f(x)
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∈Wloc2,p(Ω) and the equation is satisfied almost everywhere in Ω, we call it an Lp-strong solution.We are interested in the weaker notion of solution in the viscosity sense.Firstly suppose thatF is continuous in Ω×ℝ×ℝn×𝒮n 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
(2.5)F(x,u(x),Dφ(x),D2φ(x))≥f(x),
respectively
(2.6)F(x,u(x),Dφ(x),D2φ(x))≤f(x),
for each x∈Ω and φ∈C2(Br(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 byBr(x) the ball of radius r centered at x, for short Br if x=0.Remark 2.1.
Ifu∈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, ifu∈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
(2.7)F(x,w,Dw,D2w)=h
in Ω, where h=min(f,g), respectively h=max(f,g).Lemma 2.2.
Letu,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
(2.8)Pλ,Λ+(D2w)+γ|Dw|-δws=f-g
in Ω∩{w>0}.Proof.
Let us suppose, for instance, thatv∈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
(2.9)f(x)-g(x)≤F(x,u(x),D(v+φ)(x),D2(v+φ)(x))-F(x,v(x),Dv(x),D2v(x))≤Pλ,Λ+(D2φ(x))+γ|Dφ(x)|-δ(w(x))s
as claimed.WhenF 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.Letf∈Llocp(Ω); then u∈C(Ω) is called an Lp-viscosity subsolution, respectively supersolution, of the equation F(x,u,Du,D2u)=f in Ω if
(2.10)esslimsup⁡x→x¯{F(x,u(x),Dφ(x),D2φ(x))-f(x)}≥0,
respectively
(2.11)essliminf⁡x→x¯{F(x,u(x),Dφ(x),D2φ(x))-f(x)}≤0,
for each x¯∈Ω and φ∈W2,p(Br(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 Lp-viscosity solution.It is important that the generalized maximum principle (GMP) forLp-strong solutions of the maximal equation (see (1.10) at the beginning of the Introduction) continues to hold for Lp-viscosity subsolutions as p>p0; see for instance [5, Theorem  3.2].Note thatLp-viscosity solutions are C-viscosity solutions of F(x,u,Du,D2u)=f because the space Wloc2,p(Ω) of test functions for Lp-viscosity solutions is larger than C2(Ω).Conversely, ifp>p0, assuming that F and f are continuous, then C-viscosity solutions are Lp-viscosity solutions; see [15, Proposition  2.9].Remark 2.3.
Suppose thatp>p0, as before. If u∈Wloc2,p(Ω) is an Lp-strong subsolution (supersolution) of F(x,u,Du,D2u)=f, then u is also an Lp-viscosity subsolution (supersolution); see [15, Lemma  2.6].
Conversely, ifu∈Wloc2,p(Ω) is an Lp-viscosity subsolution (supersolution), then u is an Lp-strong subsolution (supersolution); see [15, Corollary  3.7].
Also, Remark2.1 for w=max(u,v), respectively w=min(u,v), continues to hold for Lp-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 Lp-viscosity solutions.For an extensive treatment of viscosity solutions see [11, 15–17].In the existence results we need some regularity of the domainΩ of ℝn. We say that Ω satisfies an exterior cone condition if for every point x¯∈∂Ω there exists a finite right circular cone Σx¯ with vertex x¯ such that Σ¯x¯∩Ω¯={x¯}. A uniform exterior cone condition means that all the cones Σx¯ are congruent to a fixed cone Σ. For the use of these conditions see, for instance, [18] about the Lp theory and [11] in the viscosity setting.


## 3. Uniform Estimates

In this section, following [2], we introduce Osserman’s barrier function
(3.1)Φ(x)=CRRμξ-μ(x),|x|<R,
where ξ(x)=R2-|x|2 and μ=2/(s-1) (recall that s>1).Lemma 3.1.
Suppose for almost everywherex∈BR that
(3.2)F(x,v,ξ,X)≤Pλ,Λ+(X)+γ|ξ|-δvs
for all (v,ξ,X)∈ℝ+×ℝn×𝒮n, where δ>0 and s>1.
If one takes(3.3)CRs-1=2μδ-1(Λ(n+2(1+μ))+γR),
then the function Φ∈C2(BR), defined in (3.1), is an Lp-strong solution of the equation
(3.4)F(x,Φ,DΦ,D2Φ)≤0inBR.Proof.
By the assumptions, it is sufficient to show that(3.5)Pλ,Λ+(D2Φ)+γ|DΦ|-δΦs≤0.
Since Φ(x)=φ(r):=CRRμ(R2-r2)-μ, where r=|x|, then
(3.6)Pλ,Λ+(D2Φ)+γ|DΦ|-δΦs=Λ[φ′′+n-1rφ′]+γφ′-δφs,
and the result follows by choosing CR>0 as claimed.In what followsp0∈(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.
LetΩ be a domain of ℝn such that ΩR:=Ω∩BR≠∅. Suppose that F satisfies structure conditions (SC) almost everywhere x∈ΩR. If u∈C(Ω¯R) is an Lp-viscosity solution (p>p0) of the equation
(3.7)F(x,u,Du,D2u)≥f(x)
with f∈Lp(ΩR), then for each r∈(0,R) one has
(3.8)sup⁡Ωr⁡u≤u∂Ω++C0(1+R)μ/2Rμ(R2-r2)μ+C‖f-‖Lp(ΩR)
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,(3.9)u∂Ω+={sup⁡BR∩∂Ωu+ifBR∩∂Ω≠∅,0ifBR∩Ω.Proof.
By (SC) we have
(3.10)F(x,v,ξ,X)=F(x,v,ξ,X)-F(x,v,0,0)+F(x,v,0,0)≤Pλ,Λ+(X)+γ|ξ|-δvs
for all (v,ξ,X)∈ℝ+×ℝn×𝒮n. Thus, from Lemma 3.1 we deduce that Φ is an Lp-strong supersolution of the equation
(3.11)F(x,Φ,DΦ,D2Φ)=0inΩR.
On the other hand u is an Lp-viscosity subsolution of the equation
(3.12)F(x,u,Du,D2u)=f(x)inΩR.
Hence, by Lemma 2.2 (see Remark 2.3) the function w=u-Φ is an Lp-viscosity solution of the equation
(3.13)Pλ,Λ+(D2w)+γ|Dw|≥Pλ,Λ+(D2w)+γ|Dw|-δws≥f(x)
in A=ΩR∩{u>Φ}.
Letr∈(0,R) be such that u+(x)≤Φ(x) as x∈Ω and r≤|x|<R; then A⊂Ωr and w≤0 on Ω∩∂Br. Therefore, applying GMP (1.10), we get
(3.14)sup⁡Aw≤sup⁡Br∩∂Ωw+C‖f-‖Lp(ΩR)
from which
(3.15)u(x)≤CRRμ(R2-|x|2)μ+u∂Ω++C‖f-‖Lp(ΩR)
for all x∈ΩR.Proposition 3.3.
LetΩR, F, and f be as in Lemma 3.2. If u∈C(Ω¯R) is an Lp-viscosity solution (p>p0) of the equation
(3.16)F(x,u,Du,D2u)=f(x),
then for each r∈(0,R) one has
(3.17)sup⁡Ωr|u|≤|u|∂Ω+C0(1+R)μ/2Rμ(R2-r2)μ+C‖f‖Lp(ΩR)
with C0, C and |u|∂Ω=max(u∂Ω+,u∂Ω-) as defined in Lemma 3.2.Proof.
From Lemma3.2 we already know that
(3.18)sup⁡Ωru+≤u∂Ω++C0(1+R)μ/2Rμ(R2-r2)μ+C‖f-‖Lp(ΩR).
The assertion will be proved showing the same inequality for u-.
To this end firstly observe that the functionv=-u satisfies the equation
(3.19)G(x,v,Dv,D2v)=-f(x)inΩR,
where
(3.20)G(x,v,ξ,X)=-F(x,-v,-ξ,-X),
which turns out to satisfy (SC).
Therefore,u-=max(v,0) is an Lp-viscosity solution of the equation
(3.21)G(x,u-,Du-,D2u-)≥-f(x).
On the other hand, by virtue of Lemma3.1, Φ is an Lp-strong supersolution of the equation
(3.22)Pλ,Λ+(D2Φ)+γ|DΦ|-δΦs=0.
Hence, we finish the proof arguing as in Lemma3.2 to obtain the estimate
(3.23)u-(x)≤u∂Ω-+CRRμ(R2-|x|2)μ+C‖f+‖Lp(ΩR)
for all x∈ΩR.


## 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 Lp-viscosity solution of the equation
(4.1)F(x,u,Du,D2u)=f(x)
in ℝn under the assumption that f∈Llocp(ℝn) with p>p0, where p0∈(n/2,n) is the exponent above which GMP holds true; see Sections 1 and 2.By the relationship betweenC-viscosity and Lp-viscosity solutions and between Lp-viscosity and Lp-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 allR>0 there exists a function ωR:ℝ+→ℝ+ such that ωR(t)→0 as t→0+ and
(4.2)|F(x,v,ξ,X)-F(x,u,ξ,X)|≤ωR(|v-u|)
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
(4.3)(SC)′:=(SC)and(4.2).
It is worth to recall that condition (SC)′ is equivalent to (SC) in the case that F is continuous.Proposition 4.1.
LetF:ℝn×ℝ×ℝn×𝒮n→ℝ be measurable in x and satisfy the structure condition (SC)′ almost everywhere x∈ℝn for all (u,ξ,X)∈ℝ×ℝn×𝒮n. If f∈Llocp(ℝn), then equation
(4.4)F(x,u,Du,D2u)=f(x)
has an Lp-viscosity solution in ℝn for any p>p0.Proof.
Consider a smooth approximationfk∈C∞(ℝn) of f such that
(4.5)lim⁡k→∞‖fk-f‖Lp(Ω)=0
for every bounded domain Ω in ℝn.
By [11, Theorem  4.1, Remark  4.8] we can solve in the Lp-viscosity sense any Dirichlet problem for the equation F=fk in the ball B2k with continuous boundary condition.
Choose a solutionuk for each k∈ℕ. Using Proposition 3.3, for h>k we have
(4.6)sup⁡B2k|uh|≤C0+C‖f‖Lp(B2k+1),
where C0=C0(n,Λ,γ,s,δ) and C=C(n,p,λ,Λ,γ2k+1) are positive constants as defined in Lemma 3.2.
By the structure condition(SC)′ we have
(4.7)F(x,v,ξ,X)≤Pλ,Λ+(X)+γ|ξ|+F(x,-R,0,0)≤Pλ,Λ+(X)+γ|ξ|+ωR(R),F(x,v,ξ,X)≥Pλ,Λ-(X)-γ|ξ|+F(x,R,0,0)≥Pλ,Λ-(X)-γ|ξ|-ωR(R),
almost everywhere x∈ℝn for |v|≤R.
Therefore forh>k we have
(4.8)Pλ,Λ-(D2uh)-γ|Duh|-ωR(R)≤f≤Pλ,Λ+(D2uh)+γ|Duh|+ωR(R)
in B2k with R=C0+C∥f∥Lp(B2k+1).
ByCα-estimates (see [14, Proposition  4.10] and [19, Theorem  2]) we deduce that
(4.9)‖uh‖Cα(B2k)≤C1(1+‖f‖Lp(B2k+1)),
for a positive constant C1 independent of h>k.
By a diagonal process, using Ascoli-Arzelà theorem we extract a subsequencehk∈ℕ such that uhk→u∈C(ℝn) uniformly on every bounded domain.
From the stability results forLp-viscosity solutions, see [15, Theorem  3.8], u is a solution of the equation.Proof of Theorem1.4.
In the caseΩ⊊ℝn we proceed along the same lines of the proof of Proposition 4.1 constructing Lp-viscosity solutions uk∈C(Ω¯2k), k∈ℕ, of the approximating Dirichlet problems
(4.10)F(x,uk,Duk,D2uk)=fk(x)inΩ2k,uk(x)=ψk(x)on⁡∂Ω2k
for a sequence fk∈C(Ω) such that ∥fk-f∥Lp(A)→0 for all bounded subsets A⊂Ω.
Here, according to the notations of Proposition3.3, Ω2k=Ω∩B2k, while ψk is a continuous extension to ℝn of ψ|∂Ω∩B¯2k; see for instance [20, Section  1.2].
LetR=2k. Since ΩR satisfies in turn a uniform exterior cone property, the existence of such uk follows from the assumptions on F and the already mentioned [11, Theorem  4.1].
Furthermore, by Proposition3.3, for h>k we get
(4.11)sup⁡ΩR|uh|≤max⁡∂Ω∩B¯R|ψ|+C0+C‖f‖Lp(Ω2R).
The argument of the proof of Lemma 3.2 leads to inequality and therefore uh are equibounded in ΩR.
As a consequence, byCα-estimates they are equi-Hölder continuous in every subset
(4.12){x∈ΩR∣dist⁡(x,∂Ω)>ɛ}
with ɛ>0. By [19, Theorem  2]
(4.13)oscΩR∩Bρ(x)⁡uh≤C0(ραk+osc∂ΩR∩Bρ(x)⁡ψh)
for every x∈∂Ω and ρ≤ρk, and therefore uh are also equicontinuous in Ω¯R.
Thus, using a diagonal procedure as in the proof of Proposition4.1 we find an Lp-viscosity solution u∈C(Ω¯) of the Dirichlet problem under consideration.


## 5. Uniqueness and Blow-Up

In this section we begin noticing that from Section3 we get at once the following maximum principle.Proposition 5.1.
Letδ>0, s>1, and let Ω be a domain of ℝn.
Suppose for almost everywherex∈Ω that
(5.1)F(x,u,ξ,X)≤Pλ,Λ+(X)+γ|ξ|-δ|u|s-1u
for all (u,ξ,X)∈ℝ×ℝn×𝒮n and u∈C(Ω¯) is an Lp-viscosity solution (p>p0) of the equation
(5.2)F(x,u,Du,D2u)≥0inΩ.
(M1)
IfΩ=ℝn, then u≤0 in ℝn.(M2)
IfΩ⊊ℝn and u≤0 on ∂Ω, then u≤0 in Ω.Analogously, suppose for almost everywhere x∈Ω that
(5.3)F(x,v,ξ,X)≥Pλ,Λ-(X)-γ|ξ|-δ|v|s-1v
for all (v,ξ,X)∈ℝ×ℝn×𝒮n and v∈C(Ω¯)  is an Lp-viscosity solution (p>p0) of the equation
(5.4)F(x,v,Dv,D2v)≤0inΩ.(m1)
IfΩ=ℝn, then v≥0 in ℝn.(m2)
IfΩ⊊ℝn and v≥0 on ∂Ω, then v≥0 in Ω.Proof.
Letx∈ℝn and r=|x|. Firstly consider the cases (M1) and (M2). Since
(5.5)F(x,u,ξ,X)=Pλ,Λ+(X)+γ|ξ|-δ|u|s-1u
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 functionu(x)=-v(x) and the operator
(5.6)G(x,u,ξ,X)=-F(x,-u,-ξ,-X)
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 ℝn.Concerning the uniqueness for the inhomogeneous equationF=f in ℝn, 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 Theorem1.1 (uniqueness).
Letu and v be solutions of the equation F=f. Set Ω={x∈ℝn∣u>v}. We claim that Ω=∅, so that u≤v in ℝn.
Suppose on the contrary thatΩ≠∅. Since F is continuous, arguing as in [21] and observing that u,v are  in  Cloc1,α, we can use the structure condition (SC) to have
(5.7)Pλ,Λ+(D2w)+γ|Dw|-δws≥0
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 ofu and v, we also get v≤u in ℝn, and we are done.Proof of Theorem1.2 (uniqueness).
Here we observe that, iff∈Llocp(ℝn) and ∥f∥Mp<+∞, then Proposition 3.3 implies that u is bounded. In fact, if x0∈ℝn and we consider balls centered at x0, choosing r→0+ and R=1 in (3.17), we get
(5.8)|u(x0)|≤2μ/2C0+C(n,p,λ,Λ,γ)‖f‖Mp,
which is finite and independent of x0, by (A2.2).
Thus, ifu 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
(5.9)Pλ,Λ+(D2w)+b|Dw|-δws≥0
in Ω={x∈ℝn∣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 Theorem1.1 (uniqueness).Proof of Theorem1.3 (uniqueness).
In this case we observe that both the assumptions (SC¯) and (A3) or (A3)′ of the Theorem imply that Lp-viscosity solutions u,v of F=f are in Wloc2,p(ℝn) (see [14, Theorem  7.1], [3, Theorem  1], and [8, Theorem  4.2]). Hence u,v are Lp-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∈ℝn∣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 Proposition5.1. The same method will be used to prove by monotonicity the existence of blow-up solutions.Proof of Theorem1.6.
Following [2] we consider a nondecreasing sequence of fk∈C(Ω¯) such that
(5.10)lim⁡k→∞‖fk-f‖Lp(K)
for all compact set K of Ω¯. Then by Theorem 1.4 we solve the problem
(5.11)F(x,uk,Duk,D2uk)=fk(x)inΩ,uk(x)=kon⁡∂Ω.
As in the proof of Proposition 4.1, by a diagonal process, using (SC), respectively (SC)′, we find an Lp-viscosity solution u∈C(Ω) of the equation
(5.12)F(x,u,Du,D2u)=f(x)inΩ.
To compare uk and uk+1, we use (SC) with the additional assumptions (A1) or (A2), respectively (SC¯) with (A3) or (A3)′, to get a maximal equation for w=uk-uk+1. Since fk is nondecreasing, then w satisfies the boundary value problem
(5.13)Pλ,Λ+(D2w)+b|Dw|≥0inΩ∩{uk>uk+1},w≤0on⁡∂(Ω∩{uk>uk+1}).
Hence, using the maximum principle of Proposition 5.1 we get w≤0, that is uk≤uk+1.
Thereforeuk is nondecreasing, and for x¯∈∂Ω we have
(5.14)lim⁡x→x¯u(x)≥lim⁡x→x¯uk(x)=k
for all k∈ℕ; whence the assertion follows.



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*Source: 453727-2011-11-15.xml*