The Phragm´en-Lindel¨of theorem forLp-viscosity solutions of fully nonlinear second order elliptic partial differential equations with unbounded coefficients and inhomogeneous terms is established

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

REMARKS ON THE PHRAGM ´EN-LINDEL ¨OF THEOREM FOR L^p-VISCOSITY SOLUTIONS OF FULLY NONLINEAR PDES

WITH UNBOUNDED INGREDIENTS

SHIGEAKI KOIKE, KAZUSHIGE NAKAGAWA

Abstract. The Phragm´en-Lindel¨of theorem forL^p-viscosity solutions of fully nonlinear second order elliptic partial differential equations with unbounded coefficients and inhomogeneous terms is established.

1. Introduction

The notion ofL^p-viscosity solutions was introduced in [5] to study fully nonlinear second order elliptic partial differential equations (PDEs for short) with unbounded inhomogeneous terms. We refer to [3] (see also [4]) as a pioneering work for the regularity theory of viscosity solutions of fully nonlinear PDEs.

It turned out that the Aleksandrov-Bakelman-Pucci (ABP for short) maximum principle can be extended toL^p-viscosity solutions for fully nonlinear second order elliptic PDEs with unbounded coefficients and inhomogeneous terms in [14]. See also [17] for a generalization.

As an application of the ABP maximum principle in [14], it is known that the (boundary) weak Harnack inequality for L^p-viscosity solutions of the associated extremal PDEs in [15] (see also [16]) holds, which implies H¨older continuity for L^p-viscosity solutions of fully nonlinear elliptic PDEs with unbounded ingredients.

We also refer to [19] for H¨older continuity estimates onL^p-viscosity solutions by a different approach.

On the other hand, qualitative properties of viscosity solutions of fully nonlinear elliptic PDEs have been investigated as generalizations for classical elliptic PDE theory. For instance, the ABP maximum principle in unbounded domains in [7]

and [15], the Liouville property in [11, 6], the Hadamard principle in [6], and the Phragm´en-Lindel¨of theorem in [8]. We refer to references in [8, 11, 6] for these qualitative properties of strong/classical solutions.

Our aim here is to extend the Phragm´en-Lindel¨of theorem in [8] when PDEs have unbounded coefficients (i.e. µ in this paper). In view of the boundary weak Harnack inequality in [15], it is natural to relax the hypotheses on coefficients and inhomogeneous terms. However, for the weak Harnack inequality, we need

2000Mathematics Subject Classification. 35B53, 35D40, 35B50.

Key words and phrases. Phragm´en-Lindel¨of theorem;L^p-viscosity solution;

weak Harnack inequality.

c

2009 Texas State University - San Marcos.

Submitted August 28, 2009. Published November 20, 2009.

1

to suppose that the coefficient to the first derivatives is small enough inLⁿ-norm.

When we work in bounded domains, this is not a restriction. Since we are concerned with unbounded domains, we will need a bit more delicate analysis than those in [8].

Since our argument is essential to treat domains of conical type (i.e. the case for η >0 in our notation), we will mainly discuss this case. We will add corresponding results for domains of cylindrical type (i.e. the case forη= 0).

Our paper is organized as follows: section 2 is devoted to showing the definitions and known results. In section 3, we present the ABP type estimates onL^p-viscosity subsolutions of fully nonlinear PDEs with unbounded ingredients under appropriate geometric conditions. We show the Phragm´en-Lindel¨of theorem in our setting in section 4. In section 5, we give a proof of an elementary geometric property, which is needed in the proof of Lemma 3.2.

2. Preliminaries

We consider fully nonlinear second order PDEs in unbounded domains Ω⊂Rⁿ: G(x, u, Du, D²u) =f(x) in Ω, (2.1) whereG: Ω×R×Rⁿ×Sⁿ →Randf : Ω→Rare given measurable functions. We also suppose that (r, p, M)∈R×Rⁿ×Sⁿ→G(x, r, p, M) is continuous for almost allx∈Ω. Here,Sⁿdenotes the set of symmetric matrices of ordernequipped with the standard order.

We will use the standard notation from [13]. We denote byL^p₊(Ω) the set of all nonnegative functions inL^p(Ω).

Throughout this paper, we assume that p > n

2.

We recall two facts: if u ∈ W_loc^2,p(Ω) for p > ⁿ₂, then we may identify u with a continuous function on Ω, anduis twice differentiable for almost allx∈Ω.

First of all, we recall the definition ofL^p-viscosity solutions of (2.1).

Definition 2.1. We call u∈C(Ω) an L^p-viscosity subsolution (resp., supersolu- tion) of (2.1) if

ess lim inf

x→x₀ {G(x, u(x), Dφ(x), D²φ(x))−f(x)} ≤0

resp., ess lim sup

x→x₀

{G(x, u(x), Dφ(x), D²φ(x))−f(x)} ≥0

wheneverφ∈W_loc^2,p(Ω) andx0∈Ω is a local maximum (resp., minimum) point of u−φ. A functionu∈C(Ω) is called anL^p-viscosity solution of (2.1) if it is both anL^p-viscosity subsolution and anL^p-viscosity supersolution of (2.1).

To make easier recalling the right inequality, we will often say thatuis an L^p- viscosity solution of

G(x, u, Du, D²u)≤f(x) (2.2) resp., G(x, u, Du, D²u)≥f(x)

, (2.3)

if it is anL^p-viscosity subsolution (resp., supersolution) of (2.1).

Remark 2.2. If u is an L^p-viscosity subsolution (resp., supersolution) of (2.1), then it is also anL^q-viscosity subsolution (resp., supersolution) of (2.1) provided q≥p.

In what follows, instead of (2.1), we mainly consider PDEs which do not depend onu-variable:

F(x, Du, D²u) =f(x) in Ω. (2.4) We will assume thatF is (degenerate) elliptic:

F(x, p, M)≤F(x, p, N)

for (x, p, M, N)∈Ω×Rⁿ×Sⁿ×Sⁿ providedM ≥N. (2.5) For fixed ellipticity constants 0< λ≤Λ, we assume that

there isµ∈L^q₊(Ω) such that

P⁻(M)−µ(x)|p| ≤F(x, p, M) for (x, p, M)∈Ω×Rⁿ×Sⁿ, (2.6) where the Pucci operatorsP^±:Sⁿ→Rare defined by

P⁻(M) = min{−trace(AM) :A∈Sⁿ_λ,Λ}, P⁺(M) =−P⁻(−M).

Here,S_λ,Λⁿ :={M ∈Sⁿ :λI ≤M ≤ΛI}. We refer the reader to [8] for examples of PDEs which satisfy (2.5) and (2.6). We first recall a lemma concerning change of unknown functions.

Lemma 2.3 ([8, Lemma 1]). Assume (2.5)and (2.6) with µ∈L^q₊(Ω) forq > n.

Then, there exist constants h_j > 0 (j = 1,2) satisfying the following property: if ξ∈C²(Ω) satisfies

ξ(x)>0, |Dξ|

ξ (x)≤k1(x), |D²ξ|

ξ (x)≤k2(x) forx∈Ω

with some functions k_j ∈ C(Ω) (j = 1,2), then for L^p-viscosity subsolution w ∈ C(Ω) of (2.4)with f ∈L^p₊(Ω),u:= ^w_ξ is anL^p-viscosity solution of

P⁻(D²u)−γ1(x)|Du| −γ2(x)u≤f(x)

ξ(x) inΩ[u], (2.7)

where Ω[u] ={x∈Ω| u(x)>0},γ1(x) =h1k1(x) +µ(x) andγ2(x) =h2k2(x) + k1(x)µ(x).

We will use the constant p0=p0(n, λ,Λ)∈[ⁿ₂, n), for which we refer to [12]. It is known that forp > p0, andf ∈L^p(Br(z)), whereBr(x) ={y∈Rⁿ:|x−y|< r}, there exists a (unique) strong solutionu∈C(Br(z))∩W_loc^2,p(Br(z)) of

P⁻(D²v(x)) =f(x) a.e. inBr(z) underv(x) = 0 forx∈∂B_r(z) with estimates:

−Ckf⁻k_Lp(B_r(z))≤v(x)≤Ckf⁺k_Lp(B_r(z)) inBr(z), whereC=C(n, λ,Λ, p)>0 is a constant, and for 0< s < r,

kvk_W2,p(Bs(z))≤C⁰kfk_Lp(Br(z)), whereC⁰=C⁰(n, λ,Λ, p, r−s)>0.

We remark that to prove the ABP maximum principle [14, Theorem 2.9], which implies the boundary weak Harnack inequality [15, Theorem 6.1], it suffices to obtain the existence of strong solutions of the above extremal equation only in balls

although this fact is not clearly mentioned in [14, 15]. In fact, this existence result holds with local W^2,p-estimates for more general domains satisfying the uniform exterior cone property but thep0∈[ⁿ₂, n) associated with general domains might be bigger than the above. We also notice that we may replace cubes by balls in the (boundary) weak Harnack inequality in [15] by Cabr´e’s covering argument, which we will see in the proof of Lemma 3.2 below.

FixR >0 andz∈Rⁿ. LetT, T⁰ ⊂B_R(z) be domains such that T ⊂T⁰, and θ₀≤ |T|

|T⁰| ≤1 for someθ₀>0.

When we apply our weak Harnack inequality below, our choice ofT andT⁰ always satisfies the above condition.

For a given domainA⊂Rⁿand a functionv∈C(A), we definev⁻_T0,AonT⁰∪A by

v⁻_T0,A(x) =

(min{v(x), m} ifx∈A, m ifx∈T⁰\A, where

m= lim inf

x→T⁰∩∂Av(x).

We note that if T⁰ ∩∂A 6= ∅, then v_T⁻0,A is a real-valued function and that if T⁰∩∂A6= ∅, v is a nonnegative L^p-viscosity supersolution of (2.4) andf ≤0 in T⁰∩A, thenv_T⁻0,Ais a nonnegativeL^p-viscosity supersolution of (2.4) inT⁰.

Next, we recall the boundary weak Harnack inequality when PDEs have un- bounded coefficients and inhomogeneous terms.

Lemma 2.4([15, Theorem 6.1]). LetT,T⁰,Abe as above. Assume thatT∩A6=∅ andT⁰\A6=∅ and that

q > n, q≥p > p0. (2.8) Then, there exist constants ε0 = ε0(n, λ,Λ) > 0, r = r(n, λ,Λ, p, q) > 0 and C0 = C0(n, λ,Λ, p, q) > 0 satisfying the following property: if µ ∈ L^q₊(T⁰ ∩A), f ∈L^p₊(T⁰∩A), a nonnegativeL^p-viscosity solutionw∈C(T⁰∩A)of

P⁺(D²w) +µ(x)|Dw| ≥ −f(x) inT⁰∩A, and

kµk_Ln(T⁰∩A)≤ε₀, (2.9)

then it follows that 1

|T| Z

T

(w⁻_T0,A)^rdx^1/r

≤C0

infT w⁻_T0,A+RkfkLⁿ(T⁰∩A)

provided that q > nandq≥p≥n, and 1

|T| Z

T

(w⁻_T0,A)^rdx^1/r

≤C0

inf

T w_T⁻0,A+R^2−npkfk_Lp(T⁰∩A) M

X

k=0

R^(1−nq)kkµk^k_Lq(T⁰∩A)

provided that q > n > p > p0, whereM =M(n, p, q)≥1 is an integer.

Remark 2.5. We refer to [16] for the (boundary) weak Harnack inequality for L^p-viscosity supersolutions of fully nonlinear PDEs with superlinear growth in the gradient and unbounded ingredients.

In the next section, we will establish some local and global ABP type estimates on L^p-viscosity subsolutions for (2.4). To this end, we recall the notations concerning the shape of domains from [8].

Definition 2.6 (Local geometric condition). Let σ, τ ∈ (0,1). We call y ∈ Ω a Gσ,τ point in Ω if there existR=Ry>0 andz=zy∈Rⁿ such that

y∈BR(z), and |BR(z)\Ωy,B_R(z),τ| ≥σ|BR(z)|, (2.10) where Ω_y,BR(z),τ is the connected component of BR

τ(z)∩Ω containing y. For σ, τ ∈(0,1), andR0 >0, η ≥0, we call y ∈ Ω aG^R_σ,τ^0,η point in Ω if y is a Gσ,τ

point in Ω withR=Ry>0 andz=zy satisfying

R≤R0+η|y|. (2.11)

Remark 2.7. For the sake of simplicity of notations, for a Gσ,τ pointy ∈Ω, we will writeBy forBRy

τ

(zy), whereRy>0 andzy∈Rⁿ are from Definition 2.6.

Definition 2.8 (Global geometric condition). We call Ω a ˆG^R_σ,τ^0,η domain if any y∈Ω is aG^R_σ,τ^0,η point in Ω.

We refer the reader to [20] and [8] for examples of domains Ω satisfyingG^R_σ,τ^0,η. We also refer to [1] for a generalization.

3. ABP type estimates

We present pointwise estimates on L^p-viscosity subsolutions of (2.4), which is often referred as the Krylov-Safonov growth lemma.

In what follows, we fixσ, τ ∈(0,1) andR₀>0. Lety∈Ω be aG^R_σ,τ^0,ηpoint with η ≥0. It is possible to apply our weak Harnack inequality in B_y, which is from Definition 2.6, if kµk_Ln(B_y∩Ω) ≤ε₀. Here and later, ε₀ >0 is the constant from Lemma 2.4.

Even ifkµk_Ln(B_y∩Ω)> ε₀, we may use Cabr´e’s covering argument; we divideB_y into small pieces so that we may apply the weak Harnack inequality in each piece.

We then obtain the weak Harnack inequality inBy by combining all the inequalities for small pieces.

However, since we need the estimates uniform iny∈Ω, this argument does not work immediately because of unboundedness of{Ry}_y∈Ωwhenη >0.

To avoid this difficulty, we will suppose a decay rate of µ: kµk_Lq(Ω\Bt(0)) = o(t^−(1−nq)). More precisely, for fixed q > n, we suppose that for allδ >0 there is Tδ>0 such that

kµk_Lq(Ω\Bt(0))≤δt^−(1−nq) fort≥Tδ. (3.1) Remark 3.1. It is assumed in [8] that µ(x) =O(|x|⁻¹) as|x| → ∞, which only implieskµk_Lq(Ω\Bt(0))=O(t^−(1−nq)).

Of course, ifη = 0 (henceR_y ≤R₀), then we can apply directly Cabr´e’s argu- ment.

Lemma 3.2. Assume that (2.5),(2.8)and (2.6)hold with µ∈L^q₊(Ω). Letη >0 andy ∈Ωbe a G^R_σ,τ^0,η point in Ωwith R=Ry >0 and z=zy ∈Rⁿ. Then, there existκ=κ(n, λ,Λ, σ, τ, R0, η)∈(0,1)andε=ε(n, σ, η)>0satisfying the following property: ifw∈C(Ω)is anL^p-viscosity subsolution of (2.4)withf ∈L^p₊(Ω), then we have the following properties: (i) Assume that|y| ≤R0. (a) If p≥n, then

w(y)≤κ sup

By∩Ω

w⁺+ (1−κ) lim sup

x→By∩∂Ω

w⁺+R0kfk_Ln(By∩Ω). (b) Ifp0< p < n, then

w(y)≤κ sup

B_y∩Ω

w⁺+ (1−κ) lim sup

x→By∩∂Ω

w⁺

+R²⁻

n p

0 kfk_Lp(By∩Ω) M

X

k=0

R⁽¹⁻

n q)k

0 kµk^k_Lq(B_y∩Ω). (ii) Assume that (3.1)is satisfied and that|y|> R₀. (a) If p≥n, then

w(y)≤κ sup

By∩Ω

w⁺+ (1−κ) lim sup

x→By∩∂Ω

w⁺+Rkfk_Ln(By∩Ω\BεR(0)). (b) Ifp0< p < n, then

w(y)≤κ sup

B_y∩Ω

w⁺+ (1−κ) lim sup

x→By∩∂Ω

w⁺

+R^2−npkfk_Lp(B_y∩Ω\B_εR(0)) M

X

k=0

R^(1−nq)kkµk^k_Lq(B_y∩Ω\BεR(0)).

Here M =M(n, p, q)≥1 is the integer in Lemma 2.4.

Remark 3.3. To get the weak maximum principle (Lemma 4.1 below), it is impor- tant to have the termkfkL^p(B_y∩Ω\BεR(0)) instead ofkfkL^p(B_y∩Ω) in the estimates of the assertion (ii) above.

Proof. First of all, we shall omit giving the proof in the case ofkµk_Lq(Ω)= 0 because it is easy to do it, and we suppose thatkµk_Lq(Ω)>0.

It is enough to show the assertion when ˆC := lim sup_{x→By∩∂Ω}w⁺(x) = 0. In fact, after having established the assertion when ˆC = 0, we may apply the result tow−Cˆ to prove the assertion in the general case.

Due to (2.6),wis anL^p-viscosity solution of

P⁻(D²w)−µ(x)|Dw| ≤f(x) in Ω.

Setting C_w = sup_B

y∩Ωw⁺, we immediately see that v(x) := C_w −w(x) is an L^p-viscosity solution of

P⁺(D²v) +µ(x)|Dv| ≥ −f(x) in Ω.

We shall first prove (ii).

Case (ii) |y| > R₀: Fixε ∈(0,¹₂min{_1+η¹ ,(^σ₄)¹ⁿ}). Note that 2ε < 1/(1 +η) and (2ε)ⁿ < σ/4. We setT =B_R(z)\B_2εR(0) andT⁰=B_y\B_εR(0). Observe that

2εR < R

1 +η ≤R0+η|y|

1 +η <|y|

and consequentlyy∈T =BR(z)\B2εR(0). LetAbe the connected component of T⁰∩Ω which containsy. We have

|T\A| ≥ |T\Ω_y,BR(z),τ|

≥ |B_R(z)\Ω_y,BR(z),τ| − |B_2εR(0)|

≥σ|BR(0)| −(2ε)ⁿ|BR(0)|

≥ σ 2|B_R(0)|

≥ σ 2|T|.

Since

T⁰∩∂A⊂T⁰∩∂(T⁰∩Ω)⊂T⁰∩(∂T⁰∪∂Ω) =T⁰∩∂Ω, (3.2) in view of ˆC≤0, we have

lim inf

x→T⁰∩∂Av(x) =C_w− lim sup

x→T⁰∩∂A

w(x)≥C_w. (3.3)

Now, we verify (2.9). By (3.1), we can chooseTε>0 such that kµk_Lq(Ω\Bt(0))≤ ε0

|B1(0)|^1n(1−nq) τ ε

t 1−ⁿ_q

fort≥Tε. AssumeR≥A1:=Tεε⁻¹. Using the above, we see

kµkLⁿ(T⁰∩A)≤ |B1(0)|^n1(1−nq) R

τ 1−ⁿ_q

kµkL^q(Ω\BεR(0))≤ε0. Settingm= lim inf_x→T⁰_∩∂Av(x), we use (3.3) to show for anyr >0,

σ 2

1/r

Cw≤|T\A|

|T| 1/r

Cw≤ 1

|T| Z

T\A

m^rdx1/r

≤ 1

|T| Z

T

(v⁻_T0,A)^rdx1/r

.

Sincey∈A, we have

infT v⁻_T0,A≤v(y) =Cw−w(y). (3.4) Thus, lettingr >0 be the constant from Lemma 2.4, we have

σ 2

1/r

C_w≤C₀ inf

T v⁻_T0,A+Rkfk_Ln(T⁰∩A)

≤C₀ C_w−w(y) +Rkfk_Ln(T⁰∩Ω) ifp≥n, and

σ 2

^1/r

Cw≤C0

Cw−w(y) +kfk_Lp(T⁰∩Ω) M

X

k=0

R^{(1−nq)k+2−np}kµk^k_Lq(T⁰∩Ω)

if p∈ (p₀, n). Therefore, we conclude that the assertion (ii) holds with κ= 1− (^σ₂)^1/rmin{C₀⁻¹,1}>0 in the case where R≥A1.

Next assume thatR < A1. We can choose constants ρ0=ρ0(n, q, τ, ε0, ε, A1,kµk_Lq(Ω)),

µ0=µ0(n, q, τ, ε0, ε, A1,kµkL^q(Ω))∈(0,1),N0=N0(n, q, τ, ε0, ε, A1,kµkL^q(Ω))∈N and a finite sequence{xi}^N_i=1⁰ ⊂T⁰ such that

T ⊂ ∪^N_i=1⁰Bρ₀R(xi)⊂ ∪^N_i=1⁰ B2ρ₀R(xi)⊂T⁰, (3.5)

|B_ρ0R(x_i)∩B_ρ0R(x_i+1)| ≥µ₀|B_ρ0R(0)|, (3.6)

whereBρ₀R(xN₀+1) =Bρ₀R(x1), and

ρ₀≤ 1

A1|B1(0)|^1/n ε₀

kµk_Lq(Ω)

q−n^q

. (3.7)

We see that

kµk_Ln(Bρ0R(xi))≤ |Bρ₀R(xi)|^n1−1qkµk_Lq(By∩Ω)≤ε0.

For the reader’s convenience, we recall Cabr´e’s covering argument whenp≥n.

Sincev⁻_T0,A is a nonnegative L^p-viscosity supersolution of P⁺(D²u) +µ(x)|Du| ≥

−f(x) inT⁰, in view of Lemma 2.4, we have kv_T⁻0,Ak_Lr(Bρ0R(xi))≤ |Bρ₀R(xi)|^1/rC0

inf

B_ρ0R(x_i)v⁻_T0,A+ρ0Rkfk_Ln(A)

for i = 1,2, . . . , N0, where r, C0 > 0 are from Lemma 2.4. Furthermore, for i ∈ {1,2, . . . , N0}, setting Bi=Bρ₀R(xi), we have

inf

Bi

v⁻_T0,A≤ inf

Bi∩Bi+1

v⁻_T0,A

≤ 1

|Bi∩B_i+1| Z

Bi∩Bi+1

(v⁻_T0,A)^rdx1/r

≤C1

Binf_i+1v⁻_T0,A+RkfkLⁿ(A)

for someC1≥1. Thus, repeating this argument, for 1≤i < N0, we have inf

Bi

v_T⁻0,A≤C₁^N0−1 inf

BN0

v⁻_T0,A+N0Rkfk_Ln(A)

.

Since we may assume that infTv⁻_T0,A= infB_N0v_T⁻0,A, there isC2>0 such that kv⁻_T0,Ak_Lr(T)≤

N₀

X

i=1

kv⁻_T0,Ak_Lr(Bi)≤R^nrC2

inf

T v_T⁻0,A+Rkfk_Ln(A)

.

Whenp₀< p < n, we can easily apply the above argument to show that kv⁻_T0,AkL^r(T)≤R^nrC2

infT v_T⁻0,A+R^2−npkfkL^p(A) M

X

k=0

R^(1−nq)kkµk^k_Lq(A)

.

What remains of the proof follows the same argument as in the case ofR≥A₁. Case (i) |y| ≤R₀: Since we haveR≤(1 +η)R₀ in this case, we may regard Ω as a bounded domain. Thus, we can use the standard covering argument by Cabr´e without using (3.1). SettingT =B_R(z),T⁰=BR

τ(z) andA= Ω_y,BR(z),τ, we have

|T\A|=|BR(z)\Ω_y,BR(z),τ| ≥σ|BR(z)| ≥ σ 2|T|.

We shall only give a proof whenkµk_Ln(T⁰∩A)≤ε0.

Following the same argument as in case (ii) with the above inequality, and new A, T, T⁰, we have

σ 2

1/r

Cw≤C0

inf

T v⁻_T0,A+R0kfk_Ln(B_y∩Ω)

≤C0 Cw−w(y) +R0kfk_Ln(B_y∩Ω) provided thatp≥n, and

σ 2

1/r

Cw≤C0

Cw−w(y) +kfk_Lp(By∩Ω) M

X

k=0

R⁽¹⁻

n q)k+2−ⁿ_p

0 kµk^k_Lq(B_y∩Ω)

provided thatp∈(p0, n). Therefore, we conclude that the assertion holds with the

sameκ∈(0,1) as in case (ii).

Remark 3.4. The above proof clearly shows thatεcan be any constant satisfying 0 < ε < ¹₂min{_1+η¹ ,(^σ₄)^1/n}. In the above proof, we have stated that N0 can be chosen independently ofz andR, which may not be trivial. We will give a proof of this fact in Appendix.

The corresponding result forη= 0 is as follows.

Corollary 3.5. Assume that (2.5), (2.8) and (2.6) with µ ∈ L^q₊(Ω). Let y ∈ Ω be a G^R_σ,τ^0,0 point in Ω with R = Ry > 0 and z = zy ∈ Rⁿ. Then, there exist κ=κ(n, λ,Λ, σ, τ, R₀)∈(0,1)andε=ε(n, σ)>0satisfying the following property:

ifw∈C(Ω) is anL^p-viscosity subsolution of (2.4)withf ∈L^p₊(Ω), then the same estimates as in Lemma 3.2 (i) hold.

In the case ofη= 0, we always have|y| ≤R₀ unlike Lemma 3.2. For the proof of the above corollary, we just follow the steps in the proof of Lemma 3.2 (i).

When Ω ⊂ Rⁿ is a ˆG^R_σ,τ^0,η domain, we derive the ABP maximum principle for L^p-viscosity subsolutions bounded from above of (2.4).

Theorem 3.6 (ABP maximum principle in unbounded domains). Assume (2.8), (2.5)and (2.6)with µ∈L^q₊(Ω) satisfying (3.1). Letη >0 andΩ⊂Rⁿ be aGˆ^R_σ,τ^0,η domain. Assume also

sup

y∈Ω,|y|>R0

R_ykfkLⁿ(A_y∩Ω)<∞ if p≥n,

sup

y∈Ω,|y|>R0

R²⁻

n

y pkfk_Lp(A_y∩Ω)<∞ if p₀< p < n. (3.8) Let 0< ε <min{_1+η¹ ,(^σ₄)^1/n}. Then, there exists

C=C(n, λ,Λ, p, q, ε, σ, τ, R0, η)>0

satisfying the following properties: if w ∈ C(Ω) is an L^p-viscosity subsolution bounded from above of (2.4) withf ∈L^p₊(Ω), then it follows that

sup

Ω

w≤lim sup

x→∂Ω

w⁺(x) +C sup

y∈Ω,|y|>R0

RykfkLⁿ(A_y∩Ω)

+CR0 sup

y∈Ω,|y|≤R0

kfk_Ln(By∩Ω), (3.9)

provided that p≥n, and

sup

Ω

w≤lim sup

x→∂Ω

w⁺(x) +C sup

y∈Ω,|y|>R₀

R²⁻

n p

y kfk_Lp(Ay∩Ω) M

X

k=0

R⁽¹⁻

n q)k

y kµk^k_Lq(Ay∩Ω)

+CR²⁻

n p

0 sup

y∈Ω,|y|≤R0

kfkL^p(B_y∩Ω) M

X

k=0

R⁽¹⁻

n q)k

0 kµk^k_Lq(By∩Ω)

(3.10) provided that p∈(p0, n). Here, Ay=BRy

τ

(zy)\BεRy(0)andBy=BRy τ

(zy).

Proof. We take the supremum over y ∈ Ω with the estimates in Lemma 3.2 to

conclude the inequalities (3.9) and (3.10).

Remark 3.7. By following our proof of Lemma 3.2 (ii), it is easy to show that (3.1) implies

sup

y∈Ω,|y|>R0

R¹⁻

n

y qkµkL^q(A_y∩Ω)<∞. (3.11) To show the ABP maximum principle in unbounded domains corresponding to the caseη= 0, we do not need to assume (3.8) sinceRy≤R0.

Corollary 3.8. Assume (2.8), (2.5) and (2.6) with µ ∈ L^q₊(Ω). Let Ω⊂ Rⁿ be a Gˆ^R_σ,τ^0,0 domain. Then, there exists C = C(n, λ,Λ, p, q, ε, σ, τ, R₀) >0 satisfying the following properties: if w∈C(Ω) is anL^p-viscosity subsolution bounded from above of (2.4)withf ∈L^p₊(Ω), then it follows that (3.9)holds providedp≥n, and that (3.10) holds providedp∈(p0, n).

4. Phragm´en-Lindel¨of theorem

In this section, we show that the weak maximum principle holds for PDEs with zero-order terms. As before, assuming that Ω is a ˆG^R_σ,τ^0,η domain, for each y ∈Ω, we use the notationsR_y >0 andz_y ∈Rⁿ. Also, B_y and A_y, respectively, denote BRy

τ

(zy) andBRy τ

(zy)\BεRy(0) forε∈(0,¹₂min{_1+η¹ ,(^σ₄)^1/n}).

Lemma 4.1. Assume (2.5),(2.8)and (2.6)with µ∈L^q₊(Ω) satisfying (3.1). Let η >0andΩbe aGˆ^R_σ,τ^0,ηdomain. Then, there existsc₀=c₀(n, λ,Λ, p, q, σ, τ, R₀, η)>

0 satisfying the following property: if c∈Lⁿ₊(Ω),w∈C(Ω) is an L^p-viscosity so- lution bounded from above of

F(x, Dw, D²w)−c(x)w⁺≤0 inΩ (4.1) such that

lim sup

x→∂Ω

w(x)≤0, (4.2)

and

K0:= max sup

y∈Ω,|y|>R0

kˆck_Ln(Ay∩Ω), sup

y∈Ω,|y|≤R0

kck_Ln(By∩Ω) ≤c0, (4.3) wherec(x) = (1 +ˆ |x|²)^1/2c(x), thenw≤0 inΩ.

Remark 4.2. Instead of (4.3), it is assumed in [8] that c(x)≤ c0

1 +|x|² forx∈Ω. (4.4)

Setc(x) = _1+|x|¹ 2. We easily see by following an argument in the proof of Lemma 2.4 (ii) that theK0associated with this cis finite.

Proof. Note that by (2.6) together with Remark 2.2,w is anLⁿ-viscosity solution of

P⁻(D²w)−µ(x)|Dw| −c(x)w⁺≤0.

We apply Theorem 3.6 withf =cw⁺to obtain that when |y| ≤R0, R0kcw⁺k_Ln(By∩Ω)≤R0sup

Ω

w⁺kck_Ln(By∩Ω)≤R0K0sup

Ω

w⁺.

On the other hand, when|y|> R₀, we have Rykcw⁺kLⁿ(A_y∩Ω) ≤ R_y

p1 + (εRy)²sup

Ω

w⁺kˆckLⁿ(A_y∩Ω)≤ K₀ ε sup

Ω

w⁺. (4.5)

Choosingε1= ¹₄min{_1+η¹ ,(^σ₄)^1/n}for instance, we have sup

Ω

w≤C3max R0, 1

ε1

c0sup

Ω

w⁺

for some constant C3 > 0. Taking c0 < 1/(C3max{R0,_ε¹

1}), we conclude the

proof.

The next Corollary can be proved exactly same as above by using Corollary 3.8 instead of Theorem 3.6.

Corollary 4.3. Assume (2.5),(2.8)and (2.6)with µ∈L^q₊(Ω). Let Ω be aGˆ^R_σ,τ^0,0 domain. Then, there existsc0=c0(n, λ,Λ, p, q, σ, τ, R0)>0satisfying the following property: ifc∈Lⁿ₊(Ω)andw∈C(Ω)is anL^p-viscosity solution bounded from above of (4.1) such that (4.2)and (4.3)hold, thenw≤0 inΩ.

Theorem 4.4 (Phragm´en-Lindel¨of theorem). Assume (2.5),(2.8) and (2.6) with µ∈L^q₊(Ω) satisfying (3.1). Let η >0 andΩ be a Gˆ^R_σ,τ^0,η domain. If w∈C(Ω) is anL^p-viscosity solution of

F(x, Dw, D²w)≤0 inΩ (4.6)

such that (4.2)holds and

w⁺(x) =O(log|x|) as|x| → ∞, (4.7) thenw≤0 inΩ.

Remark 4.5. In [8], it is assumed thatw⁺(x) =O(|x|^α) with a constantα >0 as

|x| → ∞, which is weaker than (4.7). In fact, to deal with unbounded coefficients (i.e. µ), we will have to use a different functionξto apply Lemma 2.3. This is the reason why we suppose a restrictive growth rate (4.7) in comparison with that in [8].

Proof of Theorem 4.4. Define a positive smooth function ξ(x) = log(1 + (1 +|x|²)^β/2),

whereβ >0 will be fixed later, and setu=w/ξ, which is bounded from above. A straightforward calculation shows that

|Dξ|

ξ (x)≤ β

(1 +|x|²)^1/2log(1 + (1 +|x|²)^β/2)=:k1(x),

|D²ξ|

ξ (x)≤ βC4

(1 +|x|²) log(1 + (1 +|x|²)^β/2)=:k2(x)

for some C4 > 0. Thus, in view of Lemma 2.3, we see that uis an Lⁿ-viscosity solution of

P⁻(D²u)−γ1(x)|Du| −γ2(x)u⁺≤0 in Ω, where

γ₁(x) = h₁β

(1 +|x|²)^1/2log(1 + (1 +|x|²)^β/2)+µ(x) =:γ₁₁(x) +γ₁₂(x) γ2(x) = h2βC4

(1 +|x|²) log(1 + (1 +|x|²)^β/2)+ βµ(x) (log 2)(1 +|x|²)^1/2

=:γ₂₁(x) +γ₂₂(x)

We first show thatγ1satisfies (3.1). Note that we only need to show thatγ11satisfies (3.1). Setting g(x) = (|x|log|x|)⁻¹ for |x| > 1, we easily show kgkL^q(B_t^c(0)) = o(t^−(1−nq)) ast→ ∞, which implies thatγ₁₁satisfies (3.1).

We next show that (4.3) holds forγ₂. We shall observe that K₀⁰ := max

sup

y∈Ω,|y|>R0

kˆγ2k_Ln(Ay∩Ω), sup

y∈Ω,|y|≤R0

kγ2k_Ln(By∩Ω) (4.8) is small whenβ →0, where ˆγ2(x) =p

1 +|x|²γ2(x).

Wheny ∈ Ω satisfies|y| ≤R₀, we see that B_y ⊂B_R0(2+η+τ−1(1+η))(0). Thus, the second term in (4.8) can be small whenβ >0 is small enough.

To estimate the first term of (4.8), we note thatAy=By\BεRy(0)⊂BεRy(0)^c provided ε < _2(1+η)¹ . Setting ˆγ₂₂(x) =p

1 +|x|²γ₂₂(x), by (3.1), we can choose T0>1 such that

kˆγ22k_Lq(Ω\Bt(0))≤βt^−(1−nq) fort≥T0. Hence, forRy> A2:= ^T_ε⁰, we have

kˆγ22kLⁿ(A_y∩Ω)≤C5R¹⁻

n q

y kˆγ22kL^q(A_y∩Ω)≤C5

β ε¹⁻

n q

1

for someC5>0, whereε1= ¹₄min{_1+η¹ ,(^σ₄)^1/n}. IfRy≤A2, then we have kˆγ22k_Ln(Ay∩Ω)≤C6βR¹⁻

n q

y kµk_Lq(Ω)≤C6βA¹⁻

n q

2 kµk_Lq(Ω)

for someC6>0. Thus, in this case, we may suppose thatkˆγ22k_Ln(Ay∩Ω) is small by taking smallβ >0.

The remaining case is to prove that supy∈Ω,|y|>R0kˆγ21k_Ln(Ay∩Ω) is small, where ˆ

γ21(x) =p

1 +|x|²γ21(x). To this end, we shall show that for anyc0>0, there is smallβ >0 such thatkˆγ21kLⁿ(Rⁿ)≤c0. Since

Z ∞

t

1

r(logr)ⁿdr= 1

(n−1)(logt)ⁿ⁻¹ fort >1,

we can choose ˆT > 1 independent of β > 0 such that kˆγ21kLⁿ(B_Tˆ(0)^c) ≤ c0/2.

For this fixed ˆT > 0, we can find small β > 0 such that kˆγ₂₁kLⁿ(B_Tˆ(0)) ≤ c₀/2.

Therefore, using Lemma 4.1 withµ=γ1 andc=γ2, we getu≤0. This concludes

the proof.

Our Phragm´en-Lindel¨of theorem forη= 0 is as follows.

Corollary 4.6(Phragm´en-Lindel¨of theorem). Assume (2.5),(2.8)and (2.6)with µ∈L^q₊(Ω). Let Ωbe a Gˆ^R_σ,τ^0,0 domain. If w∈C(Ω) is an L^p-viscosity solution of (4.6)such that (4.2)and (4.7)hold, thenw≤0in Ω.

Proof. The only difference from the proof of Theorem 4.4 is how to estimate ˆγ₂₂. However, sinceR_y ≤R₀, we can show it immediately.

5. Appendix: A proof of an elementary geometric property In the proof of Lemma 3.2, the integer N0 might depend on y ∈ Ω such that

|y| > R0 and R := Ry < A1. We shall show that the integer N0 has an upper bound independent of suchy∈Ω. To this end, we recall our domainsT and T⁰ in this case: T =BR(z)\B2εR(0) andT⁰ =BR

τ(z)\BεR(0).

We note that the position of (T, T⁰) varies depending on the distance of two centers;|z|.

Fort∈[0,1], we denote by (Tt, T_t⁰) the couple (T, T⁰) when|z|= (1−t)(¹_τ+ 2ε).

For instance, T₁ and T₁⁰ are annuli with the common center at z = 0 whileT₀ = B_R(z) and T₀⁰ = BR

τ(z). All the possible positions of (T, T⁰) can be found in {(Tt, T_t⁰) : t ∈ [0,1]}. For each (Tt, T_t⁰), it is easy to find an integer N0,t ∈ N satisfying (3.5), (3.6), (3.7) withN0=N0t.

For any fixed t ∈ [0,1], we can choose {xi,t}^N_i=1^0,t ⊂ T_t⁰ such that (3.5), (3.6), (3.7) with N0 =N0,t, xi = xi,t, T = Tt and T⁰ = T_t⁰. We can find δt > 0 such that (3.5) holds for T = Ts and T⁰ = T_s⁰ for s ∈ It := (t−δt, t+δt)∩[0,1]

because (Tt, T_t⁰) changes continuously in t. Since [0,1]⊂ ∪_t∈[0,1]It, we can choose a finite set {t_k ∈ [0,1]}^L_k=1 such that [0,1] ⊂ ∪^L_k=1I_tk. Therefore, we can take Nˆ := max{N_0,tk:k= 1,2, . . . , L}as an upper bound forN₀.

Acknowledgements. The authors want to thank the anonymous referee for sev- eral suggestions and comments on the first draft of this article.

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Shigeaki Koike

Department of Mathematics, Saitama University, 255 Shimo-Okubo, Sakura, Saitama 338-8570, Japan

E-mail address:[email protected]

Kazushige Nakagawa

Department of Mathematics, Saitama University, 255 Shimo-Okubo, Sakura, Saitama 338-8570, Japan

E-mail address:[email protected]