ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 103–112 (September 27, 2004) S 1079-6762(04)00135-0

AN UPPER BOUND FOR POSITIVE SOLUTIONS OF THE EQUATION Δu = u^{α}

S. E. KUZNETSOV

(Communicated by Mark Freidlin)

Abstract. In 2002 Mselati proved that every positive solution of the equation Δu = u^{2} in a bounded domain of class C^{4} is the limit of an increasing sequence of moderate solutions. (A solution is called moderate if it is dominated by a harmonic function.) As a part of his proof, he established an upper bound (in terms of the capacity of K) for solutions vanishing off a compact subset K of ∂E. We use a different kind of capacity (we call it the Poisson capacity) and we establish in terms of this capacity an upper bound for solutions of Δu = u^{α} with 1 < α ≤ 2. This is a part of the program: to classify all positive solutions of this equation.

1. Introduction

1.1. Main result. Let E ⊂ ℝ^{d} be a bounded smooth domain of class C^{4} in ℝ^{d}. For x ∈ E, we denote by ρ(x) the distance to the boundary ∂E and by k(x, y) the Poisson kernel in E for the Laplacian Δ.

Let M(S) stand for the set of all finite measures on a measurable space S. For every ν ∈ M(∂E), we denote by h_{ν} the harmonic function h_{ν}(x) = ∫_{∂E} k(x, y)ν(dy).

For every α > 1 and every Radon measure m on E, there exists a Choquet capacity given on compact subsets of ∂E by the formula

(1.1)  Cap(K) = sup_{ν∈P(K)} E(ν)^{−1}

where P(K) is the set of all probability measures on K and

(1.2)  E(ν) = ∫_{E} h_{ν}(x)^{α}m(dx).

We call Cap the Poisson capacity.

Our goal is to establish the following theorem.

Theorem 1.1. Suppose Cap is the Poisson capacity corresponding to 1 < α ≤ 2 and the measure m(dx) = ρ(x)dx. Let K be a compact subset of ∂E. There exists a constant C(E) depending only on E, such that, for every compact K ⊂ ∂E and

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Reverts to public domain 28 years from publication

Received by the editors April 5, 2004.

2000 Mathematics Subject Classification. Primary 35J15; Secondary 35J25.

Partially supported by the National Science Foundation Grant DMS-9971009.

every solution u of the boundary value problem

(1.3)  \begin{cases}Δu = u^{α} in E, \\ u = 0 on ∂E \ K, \end{cases}

we have

(1.4)  u(x) ≤ C(E)ρ(x) dist(x, K)^{−d} Cap(K)^{1/(α−1)}.

1.2. Equivalent definitions of the Poisson capacity. Put

(1.5)  K̂f(y) = ∫_{E} f(x)m(dx)k(x, y).

The following definitions of the Poisson capacity are equivalent to (1.1):

(1.6)  Cap(K)^{1/α} = sup{ν(K) : ν ∈ M(K), E(ν) ≤ 1}

and

(1.7)  Cap(K)^{1/α} = inf{∥f∥_{α′} : K̂f ≥ 1 on K}

where α′ = α/(α − 1) and ∥f∥_{α′} stands for the norm in L_{α′}(m).

The equivalence of (1.6) and (1.7) is proved, for instance, in [1] (see Theorem 5.1 in Chapter 13). To prove the equivalence of (1.1) and (1.5), we note that ν ∈ M(K) is equal to tμ where t = ν(K) and μ = ν/t ∈ P(K) and

sup_{ν∈M(K)}{ν(K) : E(ν) ≤ 1} = sup_{μ∈P(K)} sup_{t≥0}{t : t^{α}E(μ) ≤ 1}

= sup_{μ∈P(K)} E(μ)^{−1/α} = (Cap(K))^{1/α}.

1.3. Notation. We denote by B_{r}(x) a ball of radius r centered at x. Let H be a compact subset of ∂E∩B_{r}(x) and let ϕ be a C^{∞} function on E such that 0 ≤ ϕ ≤ 1. We call ϕ an (R, x)-truncating function for H if ϕ = 0 in a neighborhood of H and ϕ(y) = 1 if dist(x, y) ≥ R. We call ϕ an R-localizing function if ϕ = 0 in a neighborhood of H and ϕ(y) = 1 if dist(y, H) ≥ R.

2. Bounds in a halfspace

2.1. First, we establish some bounds in the case when E = (0, ∞) × ℝ^{d−1}. A generic element of E is denoted by z or (s, x), s ∈ (0, ∞), x ∈ ℝ^{d−1}. We use the notation f(s, x) and f^{s}(x) for functions on E.

Denote by 𝔼 an infinite strip {(s, x) : 0 ≤ s < 1}. The measure m(dz) = 1_{[0,1]}(s)s ds dx is concentrated on 𝔼. We denote by ∥f∥_{α} the norm in L_{α}(m). The Poisson kernel k can be represented by the formula

k((s, x), y) = Cq^{s}(x − y)

where C is a constant depending only on the dimension, and

(2.1)  q^{s}(x) = \frac{s}{(|x|^{2} + s^{2})^{d/2}} .

Lemma 2.1. Suppose that z_{0} ∈ ∂E and H is a compact subset of ∂E ∩ B_{1}(z_{0}). If Cap(H) > 0, then there exists a (3/2, z_{0})-truncating function β for H such that β^{s}(x) = 1 for s ≥ 1 and

(2.2)  ∥∇^{2}β∥_{α′}^{α′} + ∥|∇β|^{2}∥_{α′}^{α′} + ∥∇β∥_{α′}^{α′} + ∥\frac{1}{s} \frac{∂β}{∂s} ∥_{α′}^{α′} ≤ C(d) Cap(H)^{1/(α−1)}

where the constant C(d) depends only on d. If Cap(H) = 0, then, for every ϵ > 0, there exists a (3/2, z_{0})-truncating function β for H such that β^{r}(x) = 1 for r ≥ 1 and

(2.3)  ∥∇^{2}β∥_{α′}^{α′} + ∥|∇β|^{2}∥_{α′}^{α′} + ∥∇β∥_{α′}^{α′} + ∥\frac{1}{s} \frac{∂β}{∂s} ∥_{α′}^{α′} < ϵ.

Proof. Let Cap(H) > 0. By (1.7), there exists a function f on E such that ∥f∥_{α′}^{α′} ≤ 2 Cap(H)^{α′/α} and K̂f ≥ 1 on H. We may assume that f ≥ 0 (otherwise we just replace f with f^{+}).

Let A(t), 0 ≤ t < ∞, be an increasing C^{2} function such that A(t) = 0 for t ≤ 1 and A(t) = 1 for t ≥ \sqrt{2}. We set

(2.4)  (T f)^{t}(y) = ∫_{0}^{1} s dsA(\sqrt{s/t}) ∫_{ℝ^{d−1}} f^{s}(x)q^{s}(y − x) dx for t > 0

and

(2.5)  (T f)^{0}(y) = lim_{t↓0}(T f)^{t}(y) = K̂f(y)

(cf. [1, formula (6.3), p. 175]).

By [1, Theorem 13.6.1], we have

∥T f∥_{α′} + ∥∇T f∥_{α′} + ∥∇^{2}T f∥_{α′} + ∥\frac{1}{s} \frac{∂T f}{∂s} ∥_{α′} ≤ C∥f∥_{α′}.

Let g(s, y) = a(s)b(y) be such that 0 ≤ a, b ≤ 1, a, b are C^{2} functions, a = 1 in a neighborhood of 0, b = 1 in a neighborhood of ∂E ∩ B_{1}(z_{0}) and g = 0 outside B_{3/2}(z_{0}). Let h(t) be an increasing C^{2} function on [0, ∞) such that h(t) = 0 if t ≤ 1/4 and h(t) = 1 if t ≥ 3/4. As in the proof of Lemma 13.6.5 from [1], we put

u = T f, v = gu, ϕ = h(v)

and, finally,

β = 1 − ϕ = 1 − h(gT f).

By (2.5), T f = K̂f ≥ 1 on H, and β = 0 in a neighborhood of H by the choice of g and h. By direct computation,^{1} we get

|v| + |∇v| + |∇^{2}v| ≤ C(|u| + |∇u| + |∇^{2}u|), (2.6)

|\frac{1}{s} \frac{∂v}{∂s} | = (|\frac{a}{s} \frac{∂u}{∂s} + \frac{a′}{s} u|) b ≤ C (|u| + \frac{1}{s} |\frac{∂u}{∂s}|) . (2.7)

More computation yields

|∇ϕ| ≤ C|∇v|, (2.8)

|∇^{2}ϕ| ≤ C (|∇^{2}v| + \frac{|∇v|^{2}}{v} ) , (2.9)

|∇ϕ|^{2} ≤ C (\frac{|∇v|^{2}}{v} ) . (2.10)

Therefore

∥∇^{2}β∥_{α′} + ∥|∇β|^{2}∥_{α′} + ∥∇β∥_{α′} + ∥\frac{1}{s} \frac{∂β}{∂s} ∥_{α′} ≤ C∥f∥_{α′} ≤ C Cap(H)^{1/(α−1)}.

1See [1, pp. 181–182], or [2, Section 3].

If Cap(H) = 0, then ∥f∥_{α′} can be made arbitrary small, and the same construction yields (2.3). □

2.2. For a set H ⊂ R^{d−1}, we put λH = {λx : x ∈ H}.

Lemma 2.2. For every compact set H ⊂ R^{d−1} and every 0 < λ < 1,

(2.11)  Cap(λH)^{1/(α−1)} ≤ λ^{d−2α′+1} Cap(H)^{1/(α−1)}.

Proof. Let λ > 0 and ν ∈ P(H). Then ν_{λ}(A) = ν(A/λ) is concentrated on λH. Note that q^{λs}(λx) = λ^{d−1}q^{s}(x) and therefore

(2.12)  h_{ν}(s, x) = λ^{d−1}h_{ν_{λ}}(λs, λx).

Formula (2.12) and change of variables t = λs, y = λx yield

E(ν) = ∫_{0}^{1} ∫_{ℝ^{d−1}} h_{ν}^{α}(s, x)sdsdx
= ∫_{0}^{1} ∫_{ℝ^{d−1}} λ^{(d−1)α}h_{ν_{λ}}^{α}(λs, λx)sdsdx
= ∫_{0}^{λ} ∫_{ℝ^{d−1}} λ^{(d−1)α}λ^{−(d+1)}h_{ν_{λ}}^{α}(t, y)tdtdy
≤ λ^{(d−1)α}λ^{−(d+1)}E(ν_{λ})

and (1.1) implies

(2.13)  Cap(H) ≥ λ^{d+1−(d−1)α} Cap(λH).

Formula (2.11) follows from (1.1) because d−2α′+1 = −[d+1−(d−1)α]/(α−1). □

Lemma 2.3. Let z_{0} ∈ ∂E, 0 < δ < 1 and let Γ be a compact subset of ∂E ∩ B_{δ}(z_{0}). Suppose Cap(Γ) > 0. There exists a (3δ/2, z_{0})-truncating function γ = γ_{Γ,δ} for Γ such that

(2.14)  ∥∇^{2}γ∥_{α′}^{α′} + ∥|∇γ|^{2}∥_{α′}^{α′} + ∥\frac{1}{δ}∇γ∥_{α′}^{α′} + ∥\frac{1}{s} \frac{∂γ}{∂s} ∥_{α′}^{α′} ≤ C(d) Cap(Γ)^{1/α−1}

where the constant C(d) depends only on d. If Cap(Γ) = 0, then the left side of (2.14) can be made smaller than any ε > 0.

Proof. Let H = Γ/δ and let β(s, x) be the function constructed in Lemma 2.1 applied to H and z_{0}/δ. Put γ(s, x) = β(r/δ, x/δ). Since β(s, x) = 1 if s ≥ 1, γ(s, x) = 1 if s ≥ δ. Also,

∇γ(s, x) = (1/δ)∇β(s/δ, x/δ), ∇^{2}γ(s, x) = (1/δ^{2})∇^{2}β(s/δ, x/δ)

and therefore

∥∇γ∥_{α′}^{α′} = ∫_{0}^{δ} ∫_{ℝ^{d−1}} |∇γ(s, x)|^{α′} s ds dx
= ∫_{0}^{δ} ∫_{ℝ^{d−1}} δ^{−2α′}|∇β(s/δ, x/δ)|^{α′} s ds dx
= ∫_{0}^{1} ∫_{ℝ^{d−1}} δ^{d+1−2α′}|∇β(s, x)|^{α′} s ds dx = δ^{d+1−2α′}∥β∥_{α′}^{α′}.

In a similar way,

∥|∇γ|^{2}∥_{α′}^{α′} = δ^{d+1−2α′}∥|∇β|^{2}∥_{α′}^{α′},  ∥\frac{1}{δ}∇γ∥_{α′}^{α′} = δ^{d+1−2α′}∥∇β∥_{α′}^{α′}

and

∥\frac{1}{s} \frac{∂γ}{∂s} ∥_{α′}^{α′} = ∥\frac{1}{s} \frac{∂β}{∂s} ∥_{α′}^{α′} .

Therefore (2.14) follows from (2.2) and Lemma 2.2. □

3. Bounds in a unit ball

3.1. Now let E be a ball of radius 1 in ℝ^{d} centered at a point z_{0} with coordinates s = 1, x = 0. As before, let 𝔼 = {(s, x) : 0 ≤ s < 1}. For a point z = (s, x) ∈ 𝔼, we denote by

ϕ(z) = x/(1 − s)

a projection of z to ℝ^{d−1} with center at z_{0}. For a point z ∈ E ∩ 𝔼, we put

ψ(z) = (1 − |z − z_{0}|, ϕ(z))

(cf. [3], Section 3.1.1). The mapping ψ defines a 1-1 correspondence between E ∩ 𝔼 and 𝔼.

For a set H ⊂ ∂E, denote by Cap_{E}(H) the Poisson capacity of H with respect to the domain E and the measure m(dz) = dist(z, ∂E) dz. For a set K ⊂ ℝ^{d−1}, we denote by Cap_{𝔼}(K) the Poisson capacity with respect to the halfspace and the measure m(ds, dx) = 1_{[0,1)}(s) s ds dx.

Lemma 3.1. Let H be a compact subset of ∂E that is contained in a ball of radius 1/4 centered at zero, and let K = ψ(H). There exists a constant C depending only on the dimension, such that

C^{−1} Cap_{𝔼}(K) ≤ Cap_{E}(H) ≤ C Cap_{𝔼}(K).

Proof. Let μ be a probability measure on H and let ν be a measure on K defined by the formula ν(Γ) = μ(ψ^{−1}(Γ)). It is enough to show that

(3.1)  C^{−1}E_{E}(μ) < E_{𝔼}(ν) < CE_{E}(μ)

for some constant C depending only on the dimension.

Denote by D a ball of radius 1/2 centered at zero. Let D′ = ψ(D). Put

I_{E} = ∫_{E\D} ρ(z)h_{μ}^{α}(z) dz, J_{E} = ∫_{E∩D} ρ(x)h_{μ}^{α}(x) dx

and

I_{𝔼} = ∫_{𝔼\D′} ρ(z)h_{ν}^{α}(z) dz, J_{𝔼} = ∫_{𝔼∩D′} ρ(z)h_{ν}^{α}(z) dz.

Note that

C^{−1}ρ(z)/(|z| + 1/4)^{d} < h_{μ}(z) ≤ Cρ(z)/(|z| − 1/4)^{d}

on E \ D and therefore C^{−1} < I_{E} < C. For the same reason, C^{−1} < I_{𝔼} < C and therefore

(3.2)  C^{−1}I_{E} < I_{𝔼} < CI_{E}.

On the other hand,

C^{−1}h_{ν}(ψ(z)) < h_{μ}(z) < Ch_{ν}(ψ(z))

on D (this follows from a similar relation for the Poisson kernels). Since the derivatives of ψ and ψ^{−1} are bounded on D and D′, we conclude that

(3.3)  C^{−1}J_{E} < J_{𝔼} < CJ_{E}.

Since E_{E}(μ) = I_{E} + J_{E} and E_{𝔼}(ν) = I_{𝔼} + J_{𝔼}, (3.1) follows from (3.2) and (3.3). □

Lemma 3.2. Let H be a compact subset of ∂E such that H ⊂ B_{δ/8}(0), where δ ≤ 2. Suppose Cap_{E}(H) > 0. There exists a (3δ/8, 0)-truncating function γ such that

(3.4) ∥∇^{2}γ∥_{α′}^{α′} + ∥|∇γ|^{2}∥_{α′}^{α′} + ∥\frac{1}{δ}∇γ∥_{α′}^{α′} + ∥\frac{1}{ρ} \frac{∂γ}{∂ρ} ∥_{α′}^{α′} ≤ C(d) Cap_{E}(H)^{1/(α−1)},

where the constant C(d) depends only on d. If Cap_{E}(H) = 0, then the left side of (3.4) can be made smaller than any ε > 0.

Proof. We apply Lemma 2.3 to the set K = ψ(H). Let γ_{K} be the function constructed in Lemma 2.3. We put γ(s, x) = γ_{K}(ψ(s, x)) if s < 1, and γ(s, x) = 0 otherwise. Similarly to the proof of [3, Sublemma 3.1.2], we show that

\begin{aligned}∥∇^{2}γ∥_{E,α′}^{α′} & + ∥|∇γ|^{2}∥_{E,α′}^{α′} + ∥\frac{1}{δ}∇γ∥_{E,α′}^{α′} + ∥\frac{1}{ρ} \frac{∂γ}{∂ρ} ∥_{E,α′}^{α′} \\ & ≤ ∥∇^{2}γ_{K}∥_{𝔼,α′}^{α′} + ∥|∇γ_{K}|^{2}∥_{𝔼,α′}^{α′} + ∥\frac{1}{δ}∇γ_{K} ∥_{𝔼,α′}^{α′} + ∥\frac{1}{s} \frac{∂γ_{K}}{∂s} ∥_{𝔼,α′}^{α′} \end{aligned} ; (3.5)

where ∥ · ∥_{E} and ∥ · ∥_{𝔼} stand for L_{α′}-norms in E and 𝔼. Finally, we apply Lemma 3.1. □

3.2. Localizing functions. Let H be a subset of ∂E and let γ be a C^{2}-function on E. We call γ an ε-localizing function for H if 0 ≤ γ ≤ 1, γ = 1 in a neighborhood of H and γ(z) = 0 if dist(z, H) > ε.

Lemma 3.3. There exists a constant C(d) such that, for every compact subset K of ∂E with Cap(K) > 0 and diam(K) ≤ 4δ, there exists a δ/2-localizing function γ = γ_{δ,K} for K such that

(3.6) ∥∇^{2}γ∥_{α′}^{α′} + ∥|∇γ|^{2}∥_{α′}^{α′} + ∥\frac{1}{δ}∇γ∥_{α′}^{α′} + ∥\frac{1}{ρ} \frac{∂γ}{∂ρ} ∥_{α′}^{α′} ≤ C(d) Cap(K)^{1/(α−1)}.

Proof. As in [3, Lemma 3.1.2], we cover the set K by finitely many balls B_{δ/8}(y_{k}) (the number n of the balls depends only on the dimension d). We apply Lemma 3.2 to each of the sets H_{k} = K ∩ B_{δ/8}(y_{k}). Denote by γ_{k} the corresponding truncating function constructed in Lemma 3.2 (we choose ϵ = Cap(Γ)^{1/(α−1)} if Cap(H_{k}) = 0 for some k). We set

γ = γ_{1} ⋯ γ_{n}.

Note that

|∇γ| ≤ ∑_{k} |∇γ_{k}|

and

\begin{aligned}|∇^{2}γ| & ≤ ∑_{k} |∇^{2}γ_{k}| + ∑_{k≠l} |∇γ_{k}| |∇γ_{l}| \\ & ≤ ∑_{k} |∇^{2}γ_{k}| + \frac{(n − 1)}{2} ∑_{k} |∇γ_{k}|^{2}. \end{aligned}

By applying Minkowski inequality, we get

\begin{aligned}∥∇^{2}γ∥_{α′}^{α′} & + ∥|∇γ|^{2}∥_{α′}^{α′} + ∥\frac{1}{δ}∇γ∥_{α′}^{α′} + ∥\frac{1}{ρ} \frac{∂γ}{∂ρ} ∥_{α′}^{α′} \\ & ≤ C(d, n) ∑_{k} Cap(H_{k})^{1/(α−1)} + nC(n)ϵ \\ & ≤ n(C(d, n) + C(n)) Cap(Γ)^{1/(α−1)}. \end{aligned} ; (3.7)

□

Lemma 3.4. Let K be a compact subset of ∂E such that diam(K) ≤ 4δ and Cap(K) > 0. Let γ be the δ/2-localizing function constructed in Lemma 3.3. Then

∫_{E} uγ^{2α′−1}|Δγ|ρ ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α},

∫_{E} uγ^{2α′−2}|∇γ|^{2}ρ ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α},

∫_{E} uγ^{2α′−1}|∇γ|ρ ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} δ Cap(K)^{1/α},

∫_{E} uγ^{2α′−1} |\frac{∂γ}{∂ρ} | ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α}

(3.8)

whenever u satisfies (1.3).

Proof. The assertion follows from the Hölder inequality, Lemma 3.3, the identity (α − 1)α′ = α and the inequality γ^{(2α′−1)α} ≤ γ^{(2α′−2)α} = γ^{2α′}. For instance, for the last line in (3.8), we have

∫_{E} uγ^{2α′−1} |\frac{∂γ}{∂ρ} | = ∫_{E} uγ^{2α′−1} |\frac{1}{ρ} \frac{∂γ}{∂ρ} | ρ

≤ (∫_{E} u^{α}γ^{(2α′−1)α}ρ)^{1/α} (∫_{E} |\frac{1}{ρ} \frac{∂γ}{∂ρ} |^{α′} ρ)^{1/α′}

≤ (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} C(d) Cap(K)^{1/[α′(α−1)]}.

□

Lemma 3.5. Let K, γ, u be as in Lemma 3.4. There exists a constant C(d) such that

(3.9)  ∫_{E} u^{α}γ^{2α′}ρ ≤ C(d) Cap(K)^{1/(α−1)}

whenever u satisfies (1.3).

Proof. This is an adaptation of Lemma 3.1.3 in [3]. Let E_{s} = B(s, z_{0}) and r = |z − z_{0}|. By replacing u^{2} and γ^{4} with u^{α} and γ^{2α′} in the arguments of [3], we get a bound

\begin{aligned}∫_{E} u^{α}γ^{2α′}ρ ≤ & ∫_{E} uΔ(γ^{2α′}(1 − r^{2})) − 4 ∫_{E} u\frac{∂(γ^{2α′})}{∂r}r \\ & + lim inf_{s→1−} ∫_{∂E_{s}} \frac{∂}{∂r} (uγ^{2α′}(1 − r^{2})). \end{aligned} ; (3.10)

As in [3, Sublemma 3.1.3], one can show that the last term in (3.10) is negative and can be dropped.

Finally, we note that 1 − r^{2} ≤ 2ρ and therefore

\begin{matrix}|∫_{E} uΔ(γ^{2α′}(1 − r^{2}))| \\ ≤ 2α′ ∫_{E} uγ^{2α′−1}|Δγ|(1 − r^{2}) + 2α′(2α′ − 1) ∫_{E} uγ^{2α′−2}|∇γ|^{2}ρ \\ ≤ 4α′ ∫_{E} uγ^{2α′−1}|Δγ|(1 − r^{2}) + 4α′(2α′ − 1) ∫_{E} uγ^{2α′−2}|∇γ|^{2}ρ \\ ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α}. \end{matrix} ; (3.11)

In a similar way,

\begin{aligned}|∫_{E} u\frac{∂(γ^{2α′})}{∂r}r| & = 2α′ |∫_{E} uγ^{2α′−1} \frac{∂γ}{∂r} r| ≤ 2α′ |∫_{E} uγ^{2α′−1} \frac{∂γ}{∂r} | \\ & ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α}. \end{aligned} ; (3.12)

From (3.10), (3.11) and (3.12), we get

∫_{E} u^{α}γ^{2α′}ρ ≤ C (∫_{E} u^{α}γ^{2α′}ρ)^{1/α} Cap(K)^{1/α},

which implies (3.9). □

Combining (3.8) with Lemma 3.5, we get:

Lemma 3.6. Let K, γ, u be as in Lemma 3.4. Then

∫_{E} uγ^{2α′−1}|Δγ|ρ ≤ C Cap(K)^{1/(α−1)},
∫_{E} uγ^{2α′−2}|∇γ|^{2}ρ ≤ C Cap(K)^{1/(α−1)},
∫_{E} uγ^{2α′−1}|∇γ|ρ ≤ Cδ Cap(K)^{1/(α−1)},
∫_{E} uγ^{2α′−1} |\frac{∂γ}{∂ρ} | ≤ C Cap(K)^{1/(α−1)}.

(3.13)

3.3. The rest of the proof of Theorem 1.1 is very close to the corresponding part of the proof of [3, Theorem 3.1.1]. We begin with

Lemma 3.7. Let K, γ, u be as in Lemma 3.4, and let G_{E} be the Green operator of E. For every y ∈ E, β > 0,

γ^{β}(y)u(y) ≤ \frac{1}{2}G_{E} (γ^{β}Δu − Δ(γ^{β}u)) (y).

This is a version of [3, Lemma 3.1.6] (in [3], β = 4). Bounds from [3, Sublemma 3.1.5] must be replaced with those of [1, Theorem 7.1]. Other modifications are obvious.

As a first step, we establish

Lemma 3.8. There exists a constant C(d) such that the inequality (1.4) holds whenever

dist(x, K) ≥ \frac{diam(K)}{4}.

Proof. The proof is an appropriate modification of the proof of [3, Lemma 3.1.5]. Let δ = dist(x, K) and let γ be the δ/2-localizing function constructed in Lemma 3.3. Clearly, γ = 1 in a neighborhood of x and therefore

(3.14)  u(x) = γ^{2α′}(x)u(x) ≤ \frac{1}{2}G_{E}(γ^{2α′}Δu − Δ(γ^{2α′}u))(x)

by Lemma 3.7. As in [3], the right side can be evaluated by means of Green’s formula applied to the domain B(z_{0}, r) \ B(x, ε) and passage to the limit as ε → 0, r → 1. Namely, we get

\begin{aligned}G_{E}(γ^{2α′}Δu & − Δ(γ^{2α′}u))(x) \\ & = ∫_{E}(2∇_{y}g_{E}(x, y)∇(γ^{2α′}(y) − g_{E}(x, y)Δ(γ^{2α′})(y))u(y) dy). \end{aligned}

Together with (3.14), this implies

\begin{aligned}u(x) & ≤ 2α′ ∫_{E} u(y)γ^{2α′−1}(y)∇γ(y)∇_{y}g_{E}(x, y) dy \\ & + α′ ∫_{E} u(y)γ^{2α′−1}(y)Δγ(y)g_{E}(x, y) dy \\ & + α′(2α′ − 1) ∫_{E} u(y)γ^{2α′−2}(y)|∇γ(y)|^{2}g_{E}(x, y) dy \end{aligned} ; (3.15)

(cf. [3, (3.13)]). Now, γ(y) = 1 for all y such that dist(y, K) > δ/2, in particular for all y such that |x−y| < δ/2, and therefore the integrands are equal to 0 for such y. Following [3], from the bounds for the Green’s function and its gradient, we get bounds for the integrals on the right side of (3.15) in terms of integrals (3.13). For instance,

g_{E}(x, y) ≤ Cρ(x)ρ(y)|x − y|^{−d},

and therefore

∫_{E} u(y)γ^{2α′−2}(y)|∇γ(y)|^{2}g_{E}(x, y) dy

≤ Cρ(x) ∫_{E} u(y)γ^{2α′−2}(y)|∇γ(y)|^{2}ρ(y)|x − y|^{−d} dy

= ∫_{E\B(x,δ/2)} u(y)γ^{2α′−2}(y)|∇γ(y)|^{2}ρ(y)|x − y|^{−d} dy

≤ Cρ(x)δ^{−d} ∫_{E} uγ^{2α′−2}(y)|∇γ|^{2}(y)ρ(y) dy.

In a similar way, we get

\begin{matrix}∫_{E} u(y)γ^{2α′−1}(y)∇γ(y)∇_{y}g_{E}(x, y) dy \\ ≤ Cρ(x)δ^{−d} ∫_{E} uγ^{2α′−1} |\frac{∂γ}{∂ρ} | + Cρ(x)δ^{−d−1} ∫_{E} uγ^{2α′−1}|∇γ|ρ, \\ ∫_{E} u(y)γ^{2α′−1}(y)Δγ(y)g_{E}(x, y) dy ≤ Cρ(x)δ^{−d} ∫_{E} uγ^{2α′−1}|Δγ|ρ. \end{matrix} ; (3.16)

It remains to use the bounds of Lemma 3.6. □

Theorem 1.1 can be derived from this by using the same construction as in [3]. Let x ∈ E and K, u be as in Theorem 1.1. Let δ = dist(x, K). We set

K_{1} = K ∩ \overline{B(x, 2δ)},

and

K_{n} = K ∩ (\overline{B(x, 2^{n}δ)} \ B(x, 2^{n−1}δ)) , n ≥ 2.

Since K = ⋂ K_{n}, we have

u ≤ u_{K} ≤ ∑ u_{K_{n}}

where u_{K} stands for the maximal solution of problem (1.3). By construction, we have dist(x, K_{n}) ≥ 2^{n−1}δ and diam(K_{n}) ≤ 2^{n+1}δ. Therefore Lemma 3.8 is applicable to every K_{n} and we get

u_{K_{n}} ≤ Cρ(x)2^{−nd}δ^{−d} Cap(K_{n})^{1/(α−1)} ≤ Cρ(x)2^{−nd}δ^{−d} Cap(K)^{1/(α−1)},

which implies

u(x) ≤ ∑ u_{K_{n}} ≤ Cρ(x)δ^{−d} Cap(K)^{1/(α−1)} ∑ 2^{−nd}.

The extension of the theorem to arbitrary C^{4} domains is a simple modification of the arguments by Mselati [3].

References

1. E. B. Dynkin, Diffusions, superdiffusions and partial differential equations, American Mathematical Society, Providence, RI, 2002. MR1883198 (2003c:60001)

2. S. E. Kuznetsov, Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs, Comm. Pure Appl. Math. 51 (1998), 303–340. MR1488517 (99c:35111)

3. B. Mselati, Classification et représentation probabiliste des solutions positives de Δu = u^{2} dans un domaine, Thése de Doctorat de l’Université Paris 6, 2002.

Department of Mathematics, University of Colorado, Boulder, CO 80309-0395

   E-mail address: Sergei.Kuznetsov@Colorado.edu