Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

Accepted Manuscript Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

Dongsheng Li, Shanshan Ma

PII: DOI: Reference:

S0022-1236(19)30077-1 https://doi.org/10.1016/j.jfa.2019.02.018 YJFAN 8197

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Journal of Functional Analysis

Received date: Accepted date:

28 November 2017 28 February 2019

Please cite this article in press as: D. Li, S. Ma, Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides, J. Funct. Anal. (2019), https://doi.org/10.1016/j.jfa.2019.02.018

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Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides$ Dongsheng Lia , Shanshan Maa,∗ a School

of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Abstract In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Amp`ere equations, and are more precise than obtained from Hopf lemma. Keywords: Hessian equations, Singularity, Existence, Boundary behavior

1. Introduction Let f ∈ C 1 (0, ∞) be positive and nonincreasing, and b ∈ C 1,1 (Ω) be positive in Ω, where Ω ⊆ Rn (n ≥ 2) is a bounded domain with ∂Ω ∈ C 3,1 . In this paper we investigate the following k-Hessian equation (1 ≤ k ≤ n): ⎧ ⎨Sk (D2 u) = σk (λ) = b(x)f (−u) in Ω, ⎩ u=0

(1.1)

on ∂Ω,

where λ = (λ1 , λ2 , · · · , λn ) are the eigenvalues of the Hessian matrix D2 u and 

σk (λ) =

λ i1 · · · λ i k

1≤i1 <···
is the k th elementary symmetric function of λ. For completeness, we also set σ0 (λ) = 1 and σk (λ) = 0 for k > n. $ This

work was supported by NSFC grant 11671316. author. Email addresses: [email protected] (Dongsheng Li ), [email protected] (Shanshan Ma ) ∗ Corresponding

Preprint submitted to Elsevier

March 7, 2019

To work in the realm of elliptic operators, we have to restrict the class of functions and domains. For 1 ≤ k ≤ n, following [2], a function u ∈ C 2 (Ω) ∩ C(Ω) is called a k-admissible function if for any x ∈ Ω, λ(D2 u(x)) belongs to the cone given by Γk = {λ ∈ Rn : σj (λ) > 0, j = 1, · · · , k}. From [2] and [9], Γk is an open, convex, symmetric (under the interchange of any two λj ) cone with vertex at the origin, and Γ1 ⊃ Γ2 ⊃ · · · ⊃ Γn = {λ ∈ Rn : λ1 , · · · , λn > 0}. In [2], it is also shown that ∂σk (λ) > 0 in Γk ∀ i ∂λi and 1/k

σk (λ) is a concave function in Γk . Let S(Γk ) = {A : A ∈ Sn×n , λ(A) ∈ Γk }, where Sn×n denotes the space of real n × n symmetric matrices. Recall that S(Γk ) is an open convex cone with vertex at the origin in matrix spaces. The properties of σk described above guarantee that (see [2,12])   ∂Sk > 0 ∀ A ∈ S(Γk ) ∂Aij n×n and 1/k

Sk

is concave in S(Γk ).

For an open bounded subset Ω ⊂ Rn with boundary of class C 2 and for every x ∈ ∂Ω, we denote by ρ(x) = (ρ1 (x), · · · , ρn−1 (x)) the principal curvatures of ∂Ω (relative to the interior normal). Recall that Ω is said to be l-convex (1 ≤ l ≤ n − 1) if ∂Ω, regarded as a hypersurface in Rn , is l-convex, that is, for every x ∈ ∂Ω, σj (ρ(x)) ≥ 0 with j = 1, 2, · · · , l. Respectively, Ω is called strictly l-convex if σj (ρ(x)) > 0 with j = 1, 2, · · · , l. See [3, 20]. 2

Since we will consider viscosity solutions of (1.1), we first give the following definitions (See [23]). A function u ∈ C(Ω) is said to be a viscosity subsolution (supersolution) of (1.1) if x0 ∈ Ω, A is an open neighborhood of x0 , ψ ∈ C 2 (A) is k-admissible and u − ψ has a local maximum (minimum) at x0 , then Sk (D2 ψ(x0 )) ≥ b(x0 )f (−ψ(x0 )) (≤ b(x0 )f (−ψ(x0 ))). A function u ∈ C(Ω) is said to be a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution. From now on we shall always assume that Ω is bounded and strictly (k − 1)convex and consider viscosity solutions of (1.1). Note that a viscosity solution of (1.1) is subharmonic and then, by the maximum principle, is negative in Ω. First, we state our existence and uniqueness results of viscosity solutions of (1.1) as follows. Theorem 1.1. Let Ω ⊆ Rn be a bounded and strictly (k − 1)-convex domain with ∂Ω ∈ C 3,1 . Suppose that f ∈ C 1 (0, ∞) is positive and nonincreasing, and that b ∈ C 1,1 (Ω) is positive in Ω. Then problem (1.1) admits a unique viscosity solution u ∈ C(Ω). The existence of solutions of Hessian equations with regular right-hand sides has been considered in many papers. See, for instance, [3,11,15,24,25] and the references therein. The main interest here is that of Hessian equations with singular right-hand sides. Note that we assume f (s) → ∞ as s → 0.

To study the boundary behavior of the viscosity solution of (1.1), we need more conditions. Precisely, we assume that f satisfies: (f1 ) f ∈ C 1 (0, ∞), f (s) > 0, f (s) → ∞ as s → 0, and is nonincreasing on (0, ∞); (f2 ) There exists Cf > 0 such that 

ˆs

lim H (s)

s→0+

0

dτ = −Cf , H(τ )

3

(1.2)

where H(τ ) = ((k + 1)F (τ ))1/(k+1) ∀ 0 < τ < a and

ˆa f (s)ds ∀ 0 < τ < a

F (τ ) = τ

for some constant a > 0. Since we will investigate the boundary behavior of function u and u = 0 on the boundary, we only need to concern the behavior of F (τ ) and f (τ ) as τ → 0. That is, the choice of a is not essential to our results. For convenience, we define ϕ by ϕ(t) ˆ

0

dτ = t ∀ 0 < t < α, H(τ )

where ϕ(α) = a. Actually, the existence of ϕ is obvious since

(1.3) 1 H

is nondecreasing

and integrable on (0, a). We also assume that b satisfies: (b1 ) b ∈ C 1,1 (Ω) is positive in Ω; (b2 ) There exist a positive and nondecreasing function m(t) ∈ C 1 (0, δ0 ) (for some δ0 > 0), and two positive constants b and b such that b = lim inf x∈Ω

d(x)→0

b(x) mk+1 (d(x))

≤ lim sup x∈Ω

b(x) mk+1 (d(x))

= b,

d(x)→0

where d(x) = dist(x, ∂Ω), and there exists Cm ∈ [0, ∞) such that   M (t) = Cm , lim m(t) t→0+ where M (t) =

´t 0

m(s)ds < ∞ for any 0 < t < δ0 .

The boundary behavior of the solution of (1.1) may involve the curvatures of ∂Ω. We set L0 = max σk−1 (ρ(x)) and l0 = min σk−1 (ρ(x)), x∈∂Ω

x∈∂Ω

(1.4)

where ρ(x) = (ρ1 (x), ρ2 (x), · · · , ρn−1 (x)) are the principal curvatures of ∂Ω at x. Observe that 0 < l0 ≤ L0 < +∞ since Ω is bounded and strictly (k − 1)convex. The boundary estimates of the solution of (1.1) are related to L0 and l0 . 4

Now, we state our boundary behavior results as follows. Theorem 1.2. Let Ω ⊆ Rn be a bounded and strictly (k − 1)-convex domain with ∂Ω ∈ C 3,1 . Suppose that f satisfies (f1 ) and (f2 ), and b satisfies (b1 ) and (b2 ). If C f > 1 − Cm ,

(1.5)

where Cf and Cm are the constants defined in (f2 ) and (b2 ) respectively, then the viscosity solution u of (1.1) satisfies 1 ≤ lim inf x∈Ω

d(x)→0

u(x) u(x) and lim sup ≤ 1, −ϕ(ξM (d(x))) x∈Ω −ϕ(ξM (d(x)))

(1.6)

d(x)→0

where ϕ is defined by (1.3),   1/(k+1) 1/(k+1) b b and ξ = . ξ= L0 (1 − Cf−1 (1 − Cm )) l0 (1 − Cf−1 (1 − Cm )) (1.7) Here L0 and l0 are the constants in (1.4). Remark 1.3. Theorem 1.2 contains two interesting cases. (i) b ≡ 1 and f (s) = s−γ , γ > 1. In this case, choose m(t) = 1, and we get M (t) = t and Cm = 1. We also obtain Cf =

γ−1 k+γ ,



(k + γ)k+1 ϕ(t) = (γ − 1)(k + 1)k  1/(k+1) 1 and ξ= L0

1/(k+γ)

ξ=

t(k+1)/(k+γ) ,

 1/(k+1) 1 . l0

Note that in this case (1.5) holds for any γ > 1. Hence, by (1.6), the solution of (1.1) satisfies 1 ≤ lim inf x∈Ω

d(x)→0

−



u(x) 1/(k+γ)

(k+γ)k+1

L0 (γ−1)(k+1)k

d(x)(k+1)/(k+γ)

and lim sup x∈Ω

d(x)→0

−



(k+γ)k+1 l0 (γ−1)(k+1)k

u(x) 1/(k+γ)

5

≤ 1. d(x)(k+1)/(k+γ)

(ii) b = d(x)α(k+1) , α > 0, near ∂Ω and f (s) = s−γ , γ > 1. In this case, choose m(t) = tα , and we obtain M (t) = We still have Cf =

γ−1 k+γ ,

 ϕ(t) =  ξ=

1 tα+1 and Cm = . α+1 α+1

(k + γ)k+1 (γ − 1)(k + 1)k

(α + 1)(γ − 1) L0 (γ − 1 − αk − α)

1/(k+γ) 

1/(k+1) and

ξ=

t(k+1)/(k+γ) ,

(α + 1)(γ − 1) l0 (γ − 1 − αk − α)

1/(k+1) .

If γ > α(k + 1) + 1, (1.5) holds. Therefore, by (1.6), the solution of (1.1) satisfies 1 ≤ lim inf x∈Ω

d(x)→0

−



u(x) 1/(k+γ)

(k+γ)k+1

L0 (γ−αk−α−1)(k+1)k (α+1)k

d(x)(k+1)(α+1)/(k+γ)

and lim sup x∈Ω

d(x)→0

−



u(x) 1/(k+γ)

(k+γ)k+1 l0 (γ−αk−α−1)(k+1)k (α+1)k

≤ 1. d(x)(k+1)(α+1)/(k+γ)

As far as we know, up to now, there has been no paper to study boundary behavior of solutions of general Hessian equations with singular right-hand sides. Only for the special cases k = 1, Poisson equations, and k = n, Monge-Amp`ere equations, boundary behavior of solutions has been considered. In [8], Crandall, Rabinowitz and Tartar studied the existence of classical solution of Poisson equations and the boundary behavior of the solution with f only satisfying (f1 ) and b = 1. They showed that the unique solution u satisfies c1 p(d(x)) ≤ −u(x) ≤ c2 p(d(x)) in Ωα , where c1 and c2 are two positive constants, Ωα = {x ∈ Ω : d(x) ≤ α} for some α > 0, and p satisfies ⎧  ⎪ ⎪ −p (s) = g(p(s)) for 0 < s ≤ α, ⎪ ⎨ p(0) = 0, ⎪ ⎪ ⎪ ⎩ p(s) > 0

for 0 < s ≤ α. 6

This is actually the generalization of Hopf lemma for singular elliptic equations. Later, Lazer and McKenna [14] generalized their results to Monge-Amp`ere equations with f (s) = s−γ , γ > 1 and a positive b ∈ C ∞ (Ω). They showed that there exist two positive constants k1 and k2 such that n+1

n+1

k1 d(x) n+γ ≤ −u(x) ≤ k2 d(x) n+γ in Ω. Lazer and McKenna’s results were generalized by Mohammed [18] with f only satisfying (f1 ). Mohammed’s results were stated as follows: there are two positive constants C1 and C2 such that C1 ϕ(d(x)) ≤ −u(x) ≤ C2 ϕ(d(x)) in Ωα , where ϕ is defined by (1.3) with k being replaced by n. Recently, the authors of this paper proved that for Monge-Amp`ere equations, the boundary behavior results in Theorem 1.2 hold under the corresponding conditions (f1 ), (f2 ), (b1 ) and (b2 ) (see [16]). In this paper, we generalize the above boundary behavior results to k-Hessian equations with 1 ≤ k ≤ n. In particular, if k = n, they are consistent with those in [16]. Theorem 1.2 shows us that the precise rate of the solution u tending to zero near the boundary could be described by the constructed functions −ϕ(ξM (d(x))) and −ϕ(ξM (d(x))). From Lemma 3.2(i3 ) below, which is proved by Karamata regular variation theory (see[1,13,17,19]), we see that both rates of −ϕ(ξM (d(x))) and −ϕ(ξM (d(x))) tending to zero are slower than d(x). That is, our results are more precise than those obtained directly from Hopf lemma. Actually, by Hopf lemma, we only have lim inf x∈Ω

u(x) −d(x)

> 0 (see [10, Lemma 3.4]).

d(x)→0

Theorem 1.1 and Theorem 1.2 will be proved in Section 2 and Section 3 respectively.

2. Existence The following comparison principle is a basic tool for proofs of both Theorem 1.1 and Theorem 1.2. 7

Lemma 2.1 (The comparison principle). Let Ω be a bounded domain in Rn . Suppose that g(x, η) is positive and continuously differentiable, and is nondecreasing only with respect to η. If u, v ∈ C(Ω) are respectively viscosity subsolution and supersolution of Sk (D2 u) = g(x, u) and u ≤ v on ∂Ω, then we have u≤v

in Ω.

Proof. We refer to [23, Proposition 2.3] for the detailed proof.

Proof of Theorem 1.1. Since b(x) ∈ C 1,1 (Ω) and is positive in Ω, by [20, Theorem 4.1], we see that the following equation ⎧ ⎨Sk (D2 w) = b(x) in Ω, ⎩

w=0

(2.1)

on ∂Ω,

3,β admits a k-admissible solution w ∈ Cloc (Ω) ∩ C(Ω) for some 0 < β < 1. Define

v(x) = −η(−w(x)) in Ω, and 1 Ωj = {x ∈ Ω : v(x) < − } j for j = 1, 2, · · · , where η is given by η(t) ˆ

1

t= 0

f (τ )

1/k

dτ .

Then we see that 1/k

η(0) = 0, η  (t) = f (η(t))

and η  (t) =

1 (2−k)/k  f (η(t)) f (η(t)), k

and Ωj is strictly (k − 1)-convex since w is k-admissible (see [21]). 8

Consider

⎧ ⎪ ⎨Sk (D2 u) = b(x)f (−u)

in Ωj ,

1 ⎪ ⎩u = − j

on ∂Ωj .

(2.2)

We show that (2.2) has a k-admissible solution uj . By [15, Theorem 4.1], we only need to show that (2.2) has a k-admissible subsolution and we will show that v is actually a k-admissible subsolution of (2.2). By direct computation, v ij = η  (−w)wij − η  (−w)wi wj . Since f is positive and nonincreasing, η  ≤ 0. Then, according to that the matrix {wi wj } is nonnegative definite, we have D2 v ≥ η  (−w)D2 w. This combing with w being the k-admissible solution of (2.1) imply that v is k-admissible and Sk (D2 v) ≥ (η  (−w))k Sk (D2 w) = (η  (−w))k b(x) = b(x)f (−v) in Ω. By the construction of Ωj , v=−

1 on ∂Ωj . j

That is, we have shown that v is a k-admissible subsolution of (2.2). Therefore, (2.2) has a k-admissible solution uj . Furthermore, by Lemma 2.1, v ≤ uj

in Ωj .

(2.3)

Since uj = v ≤ uj+1

on ∂Ωj ,

we also derive, by lemma 2.1, uj ≤ uj+1

in Ωj .

Let v = αw in Ω 9

(2.4)

with α > 0 be a constant and we will show that v is a supersolution of (2.2) if α is sufficiently small. Actually, set Lw = max{−w(x) : x ∈ Ω}. Since f is nonincreasing, we have f (t)t−1 → ∞ as t → 0+ . Then we can find α small enough such that αk ≤ f (αLw ) and αk ≤ f (1). Therefore v satisfies σk (λ(D2 (v))) = αk b(x) ≤ b(x)f (αLw ) ≤ b(x)f (−v) in Ω. Note that (see the definition of v) t≤

η(t)

for 0 < η(t) < 1.

1/k

f (η(t))

Thus η −1 (s) ≤

s f (s)

1/k

≤

s f (1)

for 0 < s < 1.

1/k

We see that 1 α 1 1 v = αw = −αη −1 ( ) ≥ − ≥− j j f (1)1/k j

on ∂Ωj .

That is, we have shown that v is a k-admissible supersolution of (2.2). Furthermore, by Lemma 2.1, uj ≤ v

in Ωj .

(2.5)

For each x ∈ Ω, there exists a positive integer j0 so that x ∈ Ωj0 . From (2.3)-(2.5), we have, for any j ≥ j0 , v ≤ uj ≤ uj+1 ≤ v

in Ωj0 .

Thus the limit function u(x) = lim uj (x) j→∞

exists. Moreover, Since for any j ≥ j0 + 1, b(x)f (−uj (x)) ∈ L∞ loc (Ωj0 +1 ), we have, by [22, Theorem 4.1], uj ∈ C β (Ωj0 +1 ) with 0 < β < 1 and

uj C β (Ωj 10

0

)

≤ C,

where C is a positive constant depending only on n, k, min v, max v, Ωj0 , b and f . Ωj 0

Ωj 0

Hence, the convergence is uniform in every compact subset of Ω and u ∈ C(Ω). Note that v≤u≤v

in Ω.

This implies that u ∈ C(Ω). By the definition of viscosity solutions, we deduce that u is a viscosity solution of (1.1). Using Lemma 2.1, we see that this solution is unique. Remark 2.2. Theorem 1.1 is only used to guarantee the existence of solution in Theorem 1.2, which makes Theorem 1.2 meaningful. The case with more general right-hand sides is out of the main concern of this paper. However, from the proof of Theorem 1.1, it is not difficult to get that Theorem 1.1 still holds if we replace b(x)f (−u) to g(x, u) and g(x, u) satisfies appropriate conditions, for example, 0 < g(x, u) ≤ b1 (x)f1 (−u) for some positive b1 (x) ∈ C 1,1 (Ω), and positive and nonincreasing f1 (s) ∈ C 1 (0, ∞). 3. Boundary behavior To study the boundary behavior of solutions of (1.1), we need the asymptotic estimate of functions in (f2 ) and (b2 ) as t → 0. The following two lemmas describe those asymptotic behaviors. Lemma 3.1. Let m and M be the functions given by (b2 ). Then lim

t→0+

and lim+

t→0

M (t) =0 m (t)

M (t)m (t) = 1 − Cm . m2 (t)

Lemma 3.2. Assume that f satisfies (f1 ) and (f2 ), and ϕ satisfies (1.3). Then we have (i1 ) ϕ(0) = 0, ϕ(t) > 0, ϕ (t) = ((k + 1)F (ϕ(t))) 

and ϕ (t) = − ((k + 1)F (ϕ(t)))

(1−k)/(k+1)

11

1/(k+1)

f (ϕ(t));

,

(i2 ) lim

t→0+

ϕ (t) tϕ (t)

= − C1f ;

(i3 ) If (1.5) holds, lim

t→0+

t =0 ϕ(ξM (t))

for ξ ∈ [c1 , c2 ] with 0 < c1 < c2 . For detailed proofs of Lemma 3.1 and Lemma 3.2, we refer to [16, Lemma 2.9] and [16, Lemma 2.11] respectively, where Karamata regular variation theory was used. Karamata regular variation theory is a tool to describe the precise rate of functions tending to zero or infinity. It was first introduced by Cˆırstea and R˘ adulescu [4-6] to study the boundary behavior and uniqueness of solutions of boundary blow-up elliptic problems and then was developed by several authors (see [7,16,26] and references therein) to study boundary behavior of solutions of singular elliptic problem.

We also need to recall some results of the distance function. Let d(x) = dist(x, ∂Ω) = inf |x − y|. For any δ > 0, we define y∈∂Ω

Ωδ = {x ∈ Ω : 0 < d(x) < δ}.

(3.1)

If Ω is bounded and ∂Ω ∈ C 2 , by [10, Lemma 14.16], there exists δ1 > 0 such that d ∈ C 2 (Ωδ1 ). Let x ∈ ∂Ω, satisfying dist(x, ∂Ω) = |x − x|, be the projection of the point x ∈ Ωδ1 on ∂Ω, and ρi (x)(i = 1, · · · , n − 1) be the principal curvatures of ∂Ω at x. Then, in terms of a principal coordinate system at x, we have, by [10, Lemma 14.17], ⎧ ⎨ Dd(x) = (0, 0, · · · , 1),

−ρ1 (¯ x) ⎩ D 2 d(x) = diag

 −ρn−1 (¯ x) , · · · , , 0 . 1−d(x)ρ1 (¯ x) 1−d(x)ρn−1 (¯ x)

(3.2)

We will prove Theorem 1.2 by the comparison principle. The key is to construct supersolution and subsolution of (1.1). Actually, the functions −ϕ(ξM (d(x))) 12

and −ϕ(ξM (d(x))) in (1.6), are “quasi-supersolution” and “quasi-subsolution” respectively. That is, after perturbing ξ and ξ to ξ ε and ξ ε , −ϕ(ξ ε M (d(x))) and −ϕ(ξ ε M (d(x))) are supersolution and subsolution near the boundary respectively. The following Lemma 3.3 and 3.4 give the details. Lemma 3.3. For any 0 < ε < b/2, let  ξε =

b − 2ε (1 + ε)L0 (1 − Cf−1 (1 − Cm ))

1/(k+1) ,

(3.3)

where b, Cm , L0 and Cf are given by (b2 ), (1.4) and (f2 ) respectively. Then for sufficiently small δε > 0, the following function uε (x) = −ϕ(ξ ε M (d(x)))

(3.4)

is k-admissible in Ωδε and satisfies Sk (D2 uε (x)) ≤ b(x)f (−uε (x)) in Ωδε ,

(3.5)

where ϕ, M and Ωδε are given by (1.3), (b2 ) and (3.1) respectively. Proof. First, we show that uε is a k-admissible function in Ωδε with sufficiently small δε > 0. That is, for 1 ≤ j ≤ k, Sj (D2 uε ) > 0 in Ωδε .

(3.6)

By direct computation, (uε (x))αβ = (−ϕ(ξ ε M (d(x))))αβ



   = −ξ ε ξ ε ϕ ξ ε M (d (x)) m2 (d (x)) + ϕ ξ ε M (d (x)) m (d (x)) dα dβ

 − ξ ε ϕ ξ ε M (d (x)) m (d (x)) dαβ .

13

Using (3.2) and Lemma 3.2(i1 ), we derive that for 1 ≤ j ≤ k, Sj (D2 uε )



   = − ξ 2ε ϕ ξ ε M (d (x)) m2 (d (x)) + ξ ε ϕ ξ ε M (d (x)) m (d (x))



  j−1

ρn−1 (x) ρ1 (x) σj−1 1−d(x)ρ , · · · , × ξ ε ϕ ξ ε M (d (x)) m (d (x)) (x) 1−d(x)ρ (x) 1 n−1



 j

ρn−1 (x) ρ1 (x) + ξ ε ϕ ξ ε M (d (x)) m (d (x)) σj 1−d(x)ρ , · · · , (x) 1−d(x)ρ (x) 1 n−1



(j−k)/(k+1) j+1 j+1 = ξ ε m (d(x))f (ϕ(ξ ε M (d(x)))) (k + 1) F ϕ ξ ε M (d (x))   k/(k+1) M (d(x))m (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) × 1− m2 (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) ε ε

 ρn−1 (x) ρ1 (x) × σj−1 1−d(x)ρ1 (x) , · · · , 1−d(x)ρ (x) n−1

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) σj 1−d(x)ρ1 (x) , · · · , 1−d(x)ρn−1 (x) . + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) ( (ε )) ε (3.7) This suggests us that to show (3.6), we only need to prove that for 1 ≤ j ≤ k,   k/(k+1) M (d(x))m (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) 1− m2 (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) ε ε

 ρn−1 (x) ρ1 (x) ×σj−1 1−d(x)ρ1 (x) , · · · , 1−d(x)ρ (x) n−1

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) , · · · , σj 1−d(x)ρ + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) 1−d(x)ρn−1 (x) 1 (x) ( (ε )) ε > 0 in Ωδε (3.8) as δε > 0 being sufficiently small. Actually, by Lemma 3.1, lim

x∈Ω

d(x)→0

M (d(x)) = 0 and m(d(x))

lim

x∈Ω

d(x)→0

M (d(x))m (d(x)) = 1 − Cm . m2 (d(x))

(3.9)

Since M ∈ C 2 (0, δ0 ) ∩ C[0, δ0 ), M (0) = 0 and, by (1.3), ϕ(Mˆ(d(x)))

((k + 1)F (τ ))

M (d(x)) =

−1/(k+1)

dτ ,

0

we see, by Lemma 3.2(i1 ) and (i2 ), k/(k+1)

lim

x∈Ω

d(x)→0

((k + 1) F (ϕ (M (d (x))))) M (d (x)) f (ϕ (M (d (x)))) 14

=

1 . Cf

(3.10)

From (3.9), (3.10) and (1.5), it follows that k/(k+1)



M (d (x)) (k + 1) F ϕ ξ M (d(x)) ε



 =0 lim x∈Ω m(d(x)) ξ M (d (x)) f ϕ ξ M (d (x))

d(x)→0

ε

and

k/(k+1)

M (d(x))m (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) m2 (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) x∈Ω ε ε

1 − lim

d(x)→0

=1−

(3.11)

ε

1−Cm Cf

(3.12)

> 0.

Therefore,





k/(k+1) (k + 1) F ϕ ξ M (d (x)) M (d (x)) m (d (x)) ε



 1− > 0 in Ωδε m2 (d (x)) ξ M (d (x)) f ϕ ξ M (d (x)) 

ε

ε

(3.13) for sufficiently small δε > 0. For 1 ≤ j ≤ k − 1, since, by Ω being strictly (k − 1)-convex,   ρn−1 (x) ρ1 (x) ,··· , = σj (ρ1 (x), · · · , ρn−1 (x)) > 0, lim σj x∈Ω 1 − d(x)ρ1 (x) 1 − d(x)ρn−1 (x)

d(x)→0

where x is the point on ∂Ω such that d(x) = |x − x|, and ρi (x)(i = 1, · · · , n − 1) are the principal curvatures of ∂Ω at x, we deduce   ρn−1 (x) ρ1 (x) ,··· , > 0 in Ωδε σj 1 − d(x)ρ1 (x) 1 − d(x)ρn−1 (x)

(3.14)

as δε > 0 being sufficiently small. Furthermore,   ρn−1 (x) ρ1 (x) ,··· , σk is bounded in Ωδε . 1 − d(x)ρ1 (x) 1 − d(x)ρn−1 (x)

(3.15)

Then it follows from (3.11) and (3.13-3.15) that (3.8) holds. Therefore, by (3.7), we have (3.6).

Next, we prove (3.5). In fact, since, by (b2 ), (b − ε)mk+1 (d(x)) ≤ b(x) in Ωδε

15

for sufficiently small δε > 0, we see that (3.5) is an easy consequence of Sk (D2 uε (x)) ≤ (b − ε) mk+1 (d (x)) f (−uε (x)) in Ωδε .

(3.16)

By (3.7), Sk (D2 uε (x)) − (b − ε) mk+1 (d (x)) f (−uε (x)) mk+1 (d(x))f (ϕ(ξ ε M (d(x)))) = ξ k+1 ε   k/(k+1)  (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) × 1 − M (d(x))m 2 m (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) ε ε

 ρn−1 (x) ρ1 (x) × σk−1 1−d(x)ρ1 (x) , · · · , 1−d(x)ρn−1 (x)

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) σk 1−d(x)ρ1 (x) , · · · , 1−d(x)ρn−1 (x) + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) ( ( )) ε

ε  −(b − ε)mk+1 (d (x)) f ϕ ξ ε M (d (x)) . (3.17) Then to show (3.16), we only need to prove   k/(k+1) M (d(x))m (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) k+1 1− ξε m2 (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) ε ε

 ρn−1 (x) ρ1 (x) × σk−1 1−d(x)ρ1 (x) , · · · , 1−d(x)ρ n−1 (x)

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) , · · · , σk 1−d(x)ρ + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) 1−d(x)ρn−1 (x) 1 (x) ( (ε )) ε −(b − ε) ≤ 0 in Ωδε (3.18) for sufficiently small δε > 0. Since, by (3.3),   (1 + ε)L0 (1 − Cf−1 (1 − Cm )) − (b − ε) = −ε, ξ k+1 ε

(3.19)

we see from (3.11), (3.12) and (3.15) that    k/(k+1)  (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) k+1 ξε (1 + ε)L0 1 − M (d(x))m 2 m (d(x)) ξ M (d(x))f (ϕ(ξ M (d(x)))) ε ε 

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) σk 1−d(x)ρ1 (x) , · · · , 1−d(x)ρn−1 (x) ( (ε )) ε −(b − ε) ≤ 0 in Ωδε (3.20) for sufficiently small δε > 0. 16

Furthermore, it follows from (1.4) that,   ρn−1 (x) ρ1 (x) ,··· , ≤ (1 + ε)L0 in Ωδε . σk−1 1 − d(x)ρ1 (x) 1 − d(x)ρn−1 (x) Combining this with (3.20), we obtain (3.18). Therefore, by (3.17), we have (3.16). Lemma 3.4. For any ε > 0, let  ξε =

b + 2ε (1 − ε)l0 (1 − Cf−1 (1 − Cm ))

1/(k+1) ,

(3.21)

where b, Cm , l0 and Cf are given by (b2 ), (1.4) and (f2 ) respectively. Then for sufficiently small δε > 0, the following function uε (x) = −ϕ(ξ ε M (d(x)))

(3.22)

is k-admissible in Ωδε and satisfies Sk (D2 uε (x)) ≥ b(x)f (−uε (x)) in Ωδε ,

(3.23)

where ϕ, M and Ωδε are given by (1.3), (b2 ) and (3.1) respectively. Proof. The proof is very similar to that of Lemma 3.3. For completeness, we still give the details. First, we show that uε is k-admissible in Ωδε for sufficiently small δε > 0. That is, for 1 ≤ j ≤ k, Sj (D2 uε ) > 0 in Ωδε .

(3.24)

As in the proof of Lemma 3.3, by direct computation, we have (3.7) with uε and ξ ε being replaced by uε and ξ ε respectively. Using (3.9), (3.10) and (1.5), we derive (3.11-3.13), where ξ ε is replaced by ξ ε , and consequently, (3.8) holds. From (3.7), we obtain (3.24).

Next, we prove (3.23). Since, by (b2 ), (b + ε)mk+1 (d(x)) ≥ b(x) in Ωδε 17

for sufficiently small δε > 0, we see, (3.23) is an easy consequence of   Sk (D2 uε (x)) ≥ b + ε mk+1 (d (x)) f (−uε (x)) in Ωδε .

(3.25)

By (3.7), we have (3.17), where uε , b − ε and ξ ε are replaced by uε , b + ε and ξ ε respectively. Then to show (3.25), we only need to prove   k/(k+1) k+1 M (d(x))m (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ξε 1− m2 (d(x)) ξ ε M (d(x))f (ϕ(ξ ε M (d(x))))

 ρn−1 (x) ρ1 (x) × σk−1 1−d(x)ρ1 (x) , · · · , 1−d(x)ρ (x) n−1

 k/(k+1) ρn−1 (x) M (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) ρ1 (x) + m(d(x)) ξ M (d(x))f ϕ ξ M (d(x)) σk 1−d(x)ρ1 (x) , · · · , 1−d(x)ρn−1 (x) ( (ε )) ε −(b + ε) ≥ 0 in Ωδε (3.26) as δε > 0 being sufficiently small. Since, by (3.21), k+1 

ξε

 (1 − ε)l0 (1 − Cf−1 (1 − Cm )) − (b + ε) = ε,

(3.27)

we see from (3.11), (3.12) (where ξ ε is replaced by ξ ε ) and (3.15) that    k/(k+1)  k+1 (d(x)) ((k+1)F (ϕ(ξ ε M (d(x))))) (1 − ε)l0 1 − M (d(x))m ξε m2 (d(x)) ξ ε M (d(x))f (ϕ(ξ ε M (d(x)))) 

 k/(k+1) (k+1)F ϕ ξ M (d(x)) ( ( ( ))) ρ (x) (d(x)) ρ1 (x) ε n−1 , · · · , 1−d(x)ρ +M σk 1−d(x)ρ m(d(x)) 1 (x) n−1 (x) ξ ε M (d(x))f (ϕ(ξ ε M (d(x)))) −(b + ε) ≥ 0 in Ωδε (3.28) for sufficiently small δε > 0. Furthermore, it follows from (1.4) that,   ρn−1 (x) ρ1 (x) ,··· , ≥ (1 − ε)l0 in Ωδε . σk−1 1 − d(x)ρ1 (x) 1 − d(x)ρn−1 (x) This and (3.28) imply (3.26). Therefore, by (3.17), we have (3.25).

Proof of Theorem 1.2. We divide the proof of Theorem 1.2 into two parts. Part (i). We prove the first inequality of (1.6). Let uε be the supersolution defined in Lemma 3.3. We choose v ∈ C 2 (Ω) to be a k-admissible function and v = 0 on ∂Ω, for example, v is the k-admissible 18

solution of Sk (D2 v) = 1 in Ω with v = 0 on ∂Ω. Since Δv > 0

in Ω,

we see v ≤ 0 in Ω.

(3.29)

Then, according to v ∈ C 2 (Ω), there exists a negative constants c such that cd(x) ≤ v(x) on Ω.

(3.30)

We claim that for sufficiently large T > 0, u + T v ≤ uε in Ωδε .

(3.31)

Here δε is the sufficiently small constant in Lemma 3.3. In fact, by (3.29), we have, for sufficiently large T , u + T v ≤ uε on Λ = {x ∈ Ω : d(x) = δε }

(3.32)

u + T v = uε = 0 on ∂Ω.

(3.33)

and

Since, D2 u(x)(in the sense of distribution), T D2 v(x) ∈ S(Γk ) for any x ∈ Ω (see Section 1, S(Γk ) is an open convex cone with vertex at the origin in matrix spaces), we have D2 u + T D2 v ∈ S(Γk ) in Ω. 1/k

Using the concavity of Sk on S(Γk ),   1 1/k 1 1/k 1 1/k 1 2 1/k Sk D (u + T v) ≥ Sk (D2 u) + Sk (T D2 v) ≥ Sk (D2 u) in Ω. 2 2 2 2 Therefore, Sk (D2 (u + T v)) ≥ Sk (D2 u) = b(x)f (−u) ≥ b(x)f (−(u + T v)) in Ω.

(3.34)

By Lemma 2.1, we deduce (3.31) from (3.5) and (3.32-3.34). Substituting (3.4) into (3.31), we have u + T v ≤ −ϕ(ξ ε M (d(x))) in Ωδε . 19

(3.35)

Note that −ϕ(ξ ε M (d(x))) < 0 in Ωδε and v satisfies (3.30). Thus, divide both sides of (3.35) by −ϕ(ξ ε M (d(x))), and we get 1−

u cT d(x) ≤ in Ωδε . −ϕ(ξ ε M (d(x))) −ϕ(ξ ε M (d(x)))

Since, by Lemma 3.2(i3 ), lim

x∈Ω

d(x)→0

d(x) = 0, ϕ(ξ ε M (d(x)))

(3.36)

we obtain 1 ≤ lim inf x∈Ω

d(x)→0

u(x) . −ϕ(ξ ε M (d(x)))

Let ε → 0 and then we conclude 1 ≤ lim inf x∈Ω

d(x)→0

u(x) . −ϕ(ξM (d(x)))

That is, the first inequality of (1.6) holds.

Part (ii). We turn to prove the second inequality of (1.6). Let uε be the subsolution defined in Lemma 3.4 and v be the function selected in Part (i). We show that for sufficiently large T > 0, uε + T v ≤ u in Ωδε .

(3.37)

Here δε is the sufficiently small constant in Lemma 3.4. Actually, by (3.29), for large enough T , we have uε + T v ≤ u on Λ = {x ∈ Ω : d(x) = δε }

(3.38)

uε + T v = u = 0 on ∂Ω.

(3.39)

and

By the same arguments of Part (i), we deduce D2 uε + T D2 v ∈ S(Γk ) in Ωδε 20

and then   1 2 1 1/k 1 1/k 1 1/k 1/k D (uε + T v) ≥ Sk (D2 uε )+ Sk (T D2 v) ≥ Sk (D2 uε ) in Ωδε . Sk 2 2 2 2 Therefore, Sk (D2 (uε + T v)) ≥ Sk (D2 uε ) ≥ b(x)f (−uε ) ≥ b(x)f (−(uε + T v)) in Ωδε . (3.40) Note that Sk (D2 u) = b(x)f (−u) in Ω. Therefore, by Lemma 2.1, we deduce (3.37) from (3.38-3.40). Substituting (3.22) into (3.37), we have −ϕ(ξ ε M (d(x))) + T v ≤ u in Ωδε .

(3.41)

Then as in Part (i), divide both sides of (3.41) by −ϕ(ξ ε M (d(x))), and we get u cT d(x) ≤1+ in Ωδε . −ϕ(ξ ε M (d(x))) −ϕ(ξ ε M (d(x))) Note that (3.36) holds with ξ ε being replaced by ξ ε . We derive lim sup x∈Ω

d(x)→0

u(x) ≤ 1. −ϕ(ξ ε M (d(x)))

Let ε → 0, and we obtain lim sup x∈Ω

d(x)→0

u(x) ≤ 1. −ϕ(ξM (d(x)))

That is, the second inequality of (1.6) holds. The proof of Theorem 1.2 is complete.

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