COMPLETE CLASSIFICATION OF THE POSITIVE

SOLUTIONS OF −1u + uq

By

MOSHE MARCUS

Abstract. We study the equation −1u + uq = 0, q > 1, in a bounded

C2 domain � ⊂ RN . A positive solution of the equation is moderate if it is

dominated by a harmonic function and σ-moderate if it is the limit of an in-creasing sequence of moderate solutions. It is known that in the subcritical case,

1 < q < qc = (N + 1)/(N − 1), every positive solution is σ-moderate [32]. More

recently, Dynkin proved, by probabilistic methods, that this remains valid in the

supercritical case for q ≤ 2, [15]. The question remained open for q > 2. In this

paper, we prove that for all q ≥ qc, every positive solution is σ-moderate. We

use purely analytic techniques, which apply to the full supercritical range. The

main tools come from linear and non-linear potential theory. Combined with pre-

vious results, our result establishes a one-to-one correspondence between positivesolutions and their boundary traces in the sense of [36].

1 Introduction

In this paper, we study boundary value problems for the equation

(1.1) −1u + |u|qsign u = 0, q > 1

in a bounded C2 domain � in RN . We say that u is a solution of this equation if

u ∈ Lqloc(�) and the equation holds in the sense of distributions. Every solution of

the equation is in W2,∞loc (�). In particular, every solution is in C1(�).

Let M(∂�) denote the space of finite Borel measures on the boundary. Set

ρ(x) := dist (x, ∂�)

and denote by Lqρ(�) the Lebesgue space with weight ρ.

For ν ∈ M(∂�), a weak solution of the boundary value problem

(1.2) −1u + |u|qsign u = 0 in �, u = ν on ∂�

is a function u ∈ L1(�) ∩ Lqρ(�) such that

(1.3) −

∫

�

u1φdx +

∫

�

|u|qsign (u)φdx = −

∫

∂�

∂nφdν,

JOURNAL D’ANALYSE MATHEMATIQUE, Vol. 117 (2012)

DOI 10.1007/s11854-012-0019-1

187

188 MOSHE MARCUS

for every φ ∈ C20 (�), where

(1.4) C20 (�) := {φ ∈ C2(�) : φ = 0 on ∂�}.

The boundary value problem (1.2) with data given by a finite Borel measure is

well understood. It is known that if a solution exists, it is unique. Gmira and Veron

[20] proved that if 1 < q < (N + 1)/(N − 1), the problem possesses a solution

for every ν ∈ M(∂�); if q ≥ (N + 1)/(N − 1), then the problem has no solution

for any measure ν concentrated at a point. The number qc := (N + 1)/(N − 1) is

the critical value for (1.2). The interval (1, (N + 1)/(N − 1)) is the subcritical

range; the interval [(N + 1)/(N − 1),∞) is the supercritical range.

In the early nineties, the boundary value problem (1.2) became of great interest

due to its relation to branching processes and superdiffusions (see Dynkin [11, 12],

Le Gall [24]). At first, the study of the problem concentrated on the characteri-

zation of the family of finite measures for which (1.2) possesses a solution. This

question is closely related to the characterization of removable boundary sets. A

compact set K ⊂ ∂� is removable if every positive solution u of (1.1) which has

a continuous extension to � \ K can be extended to a function in C(�).

In a succession of works by Le Gall [25, 26] (for q = 2), Dynkin and Kuznetsov

[16, 17] (for 1 < q ≤ 2), Marcus and Veron [33] (for q ≥ 2), and [34] (providing

a new proof for all q ≥ qc), the following results were established.

Theorem A. Let K be a compact subset of ∂�. Then

(1.5) K is removable if and only if C2/q,q′

(K ) = 0.

Here, q′ = q/(q − 1) and C2/q,q′

denotes the (2/q, q′) - Bessel capacity on ∂�.

Theorem B. Let ν ∈ M(∂�). Problem (1.2) possesses a solution if and only

if ν ≺ C2/q,q′

, i.e., ν vanishes on every Borel set E ⊂ ∂� such that C2/q,q′

(E) = 0.

Remark A.1. For solutions in Lqρ(�), the removability criterion applies to

signed solutions as well.

Remark A.2. For a non-negative solution u of (1.1), the removability crite-

rion can be extended to an arbitrary set E ⊂ ∂�. If u vanishes on ∂� \ E (where

E denotes the C2/q,q′-fine closure of E), then C2/q,q′

(E) = 0 implies u = 0. This is

a consequence of the capacitary estimates of [35]; for the definition of the C2/q,q′

-fine topology and related concepts, see [1] and Section 4 below.

In view of the estimates of Keller [22] and Osserman [40], equation (1.1) pos-

sesses solutions which are not in Lqρ(�). In particular, for domains of class C2, the

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 189

equation possesses solutions which blow up everywhere on the boundary. Such

solutions, called large solutions, have been studied for a long time. Among the

numerous works on the subject, we mention the paper of Loewner and Nirenberg

[29], who studied the case q = (N + 2)/(N − 2), the papers of Bandle and Marcus

[7, 8] who studied a large class of equations including (1.1) for any q > 1, and

the references therein. For domains of class C2 with compact boundary, it was

shown that the large solution is unique, and its precise asymptotic behavior at the

boundary was described.

The uniqueness of large solutions was also established for more general do-

mains, e.g., domains of class C0 and even for C2/q,q′-finely open sets (see Marcus

and Veron [30, 38]). The question of existence of large solutions and their asymp-

totic behavior at the boundary in arbitrary domains with compact boundary was

studied by Labutin [23] and Marcus and Veron [35].

The next question in the study of equation (1.1) was whether it is possible to

assign to an arbitrary positive solution a measure on the boundary, not necessarily

finite, which uniquely determines the solution. (Eventually such a measure was

called a boundary trace.) In investigating this question, attention was restricted to

positive solutions. The Herglotz Theorem for positive harmonic functions served

as a model. But, in contrast to the linear case, here one must allow unbounded

measures.

In [25], Le Gall studied (1.1) with q = 2 and � a disk in R2. He showed that,

in this case, every positive solution possesses a boundary trace which uniquely

determines the solution. The boundary trace was described in probabilistic terms,

and the proof relied mainly on probabilistic techniques.

In [31], Marcus and Veron introduced a notion of boundary trace (later Dynkin

called it the rough trace), which can be described as a (possibly unbounded) Borel

measure ν with the following properties. There exists a closed set F ⊂ ∂� such

that

(i) ν(E) = ∞ for every non-empty Borel subset of F ,

(ii) ν is a Radon measure on ∂� \ F .

190 MOSHE MARCUS

Let us denote the family of positive measures possessing these properties by

Breg(∂�). Given a positive solution u of (1.1), we say that it has (rough) bound-

ary trace ν ∈ Breg(∂�) if (with F as above)

(i’) for every open neighborhood A of F , u ∈ L1(� \ A) ∩ Lqρ(� \ A) and (1.3)

holds for every ϕ ∈ C20 (�) vanishing in a neighborhood of F ;

(ii’) if ξ ∈ F , then for every open neighborhood A of ξ ,∫

A∩� uqρdx = ∞.

The following result (announced in [31]) was proved in [32].

Theorem C. Every positive solution of (1.1) possesses a boundary trace in

Breg(∂�). If 1 < q < qc, then for every ν ∈ Breg(∂�), (1.1) possesses a unique

solution with boundary trace ν.

It was shown in [33] that in the supercritical case, under some additional con-

ditions on ν, (1.1) possesses a solution with rough trace ν. These conditions were

shown to be necessary and sufficient for existence. However, it soon became ap-

parent that in the supercritical case, the solution is no longer unique. A counterex-

ample to this effect was constructed by Le Gall in 1997. Therefore, in order to

deal with the supercritical case, a more refined definition of boundary trace was

necessary.

Kuznetsov [21] and Dynkin and Kuznetsov [18] provided such a definition,

which they called the fine trace. Their definition was similar to that of the rough

trace, but the singular set F was not required to be closed in the euclidean topol-

ogy. Instead, it was required to be closed with respect to a finer topology defined

in probabilistic terms. With this definition, they showed that if q ≤ 2, then for

any positive ‘fine trace’ ν, (1.1) possesses a solution the trace of which is equiv-

alent but not necessarily identical to ν. The equivalence was defined in terms of

polarity. Furthermore, they showed that for each positive fine trace ν, there ex-

ists a σ-moderate solution which is unique in the class of σ-moderate solutions.

The restriction to q ≤ 2 is due to the fact that the proof is based on probabilistic

techniques which do not apply to q > 2.

A σ-moderate solution was defined as the limit of an increasing sequence of

positive moderate solutions. We recall that a moderate solution is a solution in

L1(�) ∩ Lqρ(�), i.e., a solution whose boundary trace is a finite measure.

In 2002, Mselati proved in his Ph.D. thesis (under the supervision of Le Gall)

that for q = 2, every positive solution of (1.1) is σ-moderate. This work appeared

in [39]. Mselati used a combination of analytic and probabilistic techniques such

as the “Brownian snake”, developed by Le Gall [28]. Following this, Dynkin [15]

extended Mselati’s result proving the following statement.

If qc ≤ q ≤ 2, then every positive solution of (1.1) is σ-moderate.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 191

Instead of the “Brownian snake” technique, which can be applied only to the

case q = 2, Dynkin’s proof used new results on Markov processes that are appli-

cable to q ≤ 2.

At about the same time, Marcus and Veron introduced a notion of boundary

trace, denoted by F2/q,q′

(∂�) (they called it the precise trace), based on the classi-

cal notion of C2/q,q′

-fine topology; see [1]. A Borel measure ν on ∂� belongs to

this family of traces if there exists a C2/q,q′

-finely closed set F ⊂ ∂� such that

(T1) ν(E) = ∞ for every non-empty Borel subset of F ;

(T2) every point x ∈ ∂� \ F has a C2/q,q′-finely open neighborhood Qx such that

ν(Qx) < ∞;

(T3) if E is a Borel set such that ν(E) < ∞, then ν vanishes on subsets of E of

C2/q,q′

-capacity zero.

In the subcritical case, the C2/q,q′

-fine topology is identical to the euclidean

topology and consequently the precise trace coincides with the rough trace.

With this definition, they proved the following result.

Theorem D ([36]). For q ≥ qc,

(a) every positive solution of (1.1) possesses a boundary trace ν ∈ F2/q,q′

(∂�);

(b) for every measure ν ∈ F2/q,q′

(∂�), problem (1.2) possesses a σ-moderate

solution;

(c) the solution is unique in the class of σ-moderate solutions.

The question whether every positive solution of (1.1) with q > 2 is σ-moderate

remained open. In the present paper, we settle this question, proving the following

result.

Theorem 1. For q ≥ qc, every positive solution of (1.1) is σ-moderate.

The proof employs only analytic techniques and applies to all q ≥ qc. Of

course, the statement is also valid in the subcritical case, in which case it is an

immediate consequence of Theorem 1.

Combining Theorem C and Theorem D with Theorem 1, we obtain the follow-

ing result.

Corollary 1. For q > 1 and every non-negative ν ∈ F2/q,q′

(∂�), problem

(1.2) possesses a unique solution. If 1 < q < qc, F2/q,q′

(∂�) = Breg(∂�).

The method developed in the present paper can be adapted and applied to a

general class of problems which includes boundary value problems for equations

such as −1u + ρα|u|qsign u = 0, α > −2 and −1u + g(u) = 0, where g ∈ C(R) is

odd, monotone increasing, and satisfies the 12 condition and the Keller–Osserman

192 MOSHE MARCUS

condition. For equations of this type, the method can be adapted to boundary

value problems in Lipschitz domains as well. These results will be presented in a

subsequent paper.

The main ingredients used in the present paper are

(a) nonlinear potential theory and fine topologies associated with Bessel capac-

ities (see [1] and [36]) and

(b) the theory of boundary value problems for equations of the form

LV u := −1u + Vu = 0 in �,

where V > 0 and ρ2V is bounded.

In (b), we use mainly the results of Ancona [3], together with classical potential

theory results; see e.g., [2].

2 Preliminaries: on the equation −1u + Vu = 0

For the convenience of the reader, we collect here some definitions and results of

classical potential theory concerning operators of the form

(2.1) LV = −1 + V

that are used in the sequel. The results apply also to operators of the form −L0 +V ,

where L0 is a second order uniformly elliptic operator on a differentiable manifold

with negative curvature. However, we confine ourselves to the operator LV in a

domain � ⊂ RN . The results presented in this section apply to bounded Lipschitz

domains.

The following conditions on V are assumed throughout the paper without fur-

ther mention:

(2.2) 0 < V ≤ cρ(x)−2, V ∈ C(�).

By [3], if � is a Lipschitz domain, the Martin boundary can be identified with

the euclidean boundary ∂� and, for every ζ ∈ ∂�, there exists a positive LV -

harmonic function which vanishes on ∂� \ {ζ}. If normalized, this function is

unique. We choose a fixed reference point in �, say x0, and denote by K Vζ this

LV -harmonic function, normalized by K Vζ (x0) = 1.

The function K V (·, ζ ) = K Vζ (·) is the LV -Martin kernel in �, normalized at

x0. The L0-Martin kernel (normalized at x0) is denoted by K (·, ζ ) = Kζ (·). In

C2-domains, K (·, ζ ) = P(·, ζ )/P(x0, ζ ), where P is the Poisson kernel for −1 in

�.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 193

We observe that the positivity of V is essential for the result quoted above.

Indeed, this result depends on the weak coercivity of LV (see the definition in [3,

Section 2]), which is guaranteed in our case by Hardy’s inequality.

As a consequence, one obtains the following basic result; see Ancona [3, Theo-

rem 3 and Corollary 13].

Representation Theorem. Let � be a bounded Lipschitz domain. For each

positive LV -harmonic function u in �, there exists a unique positive measure µ on

∂� such that

(2.3) u(x) =

∫

∂�

K Vζ dµ(ζ ) for all x ∈ �.

The measure µ corresponding to an LV -harmonic function u is called the LV -

boundary measure of u. We also use the following notation:

(2.4) KVµ :=

∫

∂�

K Vζ dµ(ζ ), Kµ :=

∫

∂�

Kζdµ(ζ ).

Let D be a Lipschitz domain such that D ⊂ � and h ∈ L1(∂D). We denote by

SV (D, h) the solution of the problem

(2.5) LV w = 0 in D, w = h on ∂D .

If µ is a finite measure on ∂D , SV (D, µ) is defined in the same way.

If D is a C2 domain, a function w ∈ L1(D) is a solution of (2.5) if

(2.6)

∫

D

(−w1ϕ + Vwϕ)dx = −

∫

∂D

h∂nϕdS,

for every ϕ ∈ C20 (D). As usual, dS denotes the area element on ∂D . If the boundary

data is given by a finite measure µ on ∂D , (2.6) becomes∫

D(−w1ϕ + Vwϕ)dx =

−∫

∂D ∂nϕdµ.

If f ∈ W1,ploc (�) for some p > 1 and D ⋐ � is a Lipschitz domain, then f

possesses a Sobolev trace f ⌊∂D

belonging to Lp(∂D). In this case, the notation

S(D, f ) indicates the solution of (2.6) with h = f ⌊∂D

.

A family of domains {�n} such that �n ⊂ �n+1 and ∪�n = � is called an ex-

haustion of �. We say that {�n} is a Lipschitz (respectively C2) exhaustion

if each domain �n is Lipschitz (respectively C2).

Given positive numbers R, R′, we let

D(R, R′) = {ξ = (ξ ′, ξN ) ∈ RN−1 × R : |ξ ′| < R′, |ξN | < R}.

If � is a bounded C2 domain, there exist positive numbers R, R′, γ such that for

every y ∈ ∂�, there exist a Cartesian set of coordinates ξ centered at y such that

194 MOSHE MARCUS

the ξN axis is in the direction of n(y) ( = the outward normal vector to ∂� at y) and

a function F y ∈ C2(BN−1R (0)) (where BN−1

R (0) denotes the ball of radius R centered

at 0 in RN−1) such that ‖F y‖C2(BN−1

R(0))

≤ γ, F y(0) = 0, |F y(ξ ′)| ≤ R/2 for |ξ ′| < R′,

and � ∩ D(R, R′) = {−R ≤ ξN < F y(ξ ′), |ξ ′| < R′}. A triple (R, R′, γ) as above

is called a C2 characteristic of �.

If � is a bounded Lipschitz domain, its Lipschitz characteristic is defined in a

similar way, with obvious modifications.

An exhaustion {�n} of � is uniformly Lipschitz (resp. C2) if there exist

positive numbers R, R′, γ such that (R, R′, γ) is a Lipschitz (resp. C2) characteristic

of �n for every n.

Definition 2.1. A function u ∈ L1loc(�) is LV -superharmonic in � (in the

sense of distributions) if

(2.7) LV u ≥ 0 in �.

Lemma 2.2. Assume that u ∈ L1loc(�). Condition (2.7) holds if and only if

u ∈ W1,ploc (�) for some p ∈ (1, N/(N − 1)) and, for every C2 domain D ⋐ �,

(2.8)

∫

D

(−u1ϕ + uVϕ)dx ≥ −

∫

∂D

u∂nϕdS

for every non-negative ϕ ∈ C20 (D).

Remark. In the integral on the right hand side, u stands for u⌊∂D

, the Sobolev

trace of f on ∂D .

Proof. Obviously (2.8) implies (2.7). On the other hand, if (2.7) holds, it

follows that there exists a positive Radon measure µ in � such that

(2.9) LV u = µ in �.

Since uV ∈ L1loc(�), (2.9) implies that u ∈ W

1,ploc (�) for 1 ≤ p < N/(N − 1) and

∫

D

(−u1ϕ + uVϕ)dx =

∫

D

ϕdµ −

∫

∂D

u∂nϕdS

for every domain D as above and every ϕ ∈ C20 (D). Hence (2.8) holds for every

non-negative ϕ ∈ C20 (D). �

Corollary 2.3. Let u be LV -superharmonic in � and let D be a C2 domain

such that D ⋐ �. Then

(2.10) u ≥ SV (D, u) a.e. in D.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 195

Proof. Let µ be the measure in (2.9) and let w be the solution of the problem

LV w = µ in D, w = 0 on ∂D.

If v := SV (D, u), then u⌊D= v + w and w ≥ 0. �

Remark. A different (standard) definition of superharmonic functions is the

following. u is LV -superharmonic in � if it is l.s.c. and satisfies the inequality

in (2.10) everywhere in D for every C2 domain D ⋐ � [2, p. 11]. Therefore, if

u ∈ C(�) and is LV -superharmonic in the sense of Definition 2.1, then it is also

LV -superharmonic in the sense of [2].

More generally, assume that u is LV -superharmonic in the sense of Definition

2.1 and that u is locally bounded in �. Then

u(x) := lim infr→0

1

|Br |

∫

Br (x)

u(y)dy ≥ SV (D, u)(x) for all x ∈ D.

(As u ∈ W1,ploc (�) ∩ L∞

loc(�), it follows that u⌊∂D

equals the Sobolev trace of u on

∂D , HN−1-a.e. Since by assumption V ∈ C(�), it follows that SV (D, u) ∈ C(D).)

It is well known that if u ∈ C2(�) and u is LV -superharmonic in the sense of

[2], then LV u ≥ 0.

In the sequel the term LV -superharmonic is always understood in the sense of

Definition 2.1 unless otherwise indicated.

Lemma 2.4. Let u be a non-negative LV -superharmonic and {�n} a Lipschitz

exhaustion of �. Then

(2.11) u := lim SV (�n, u)

exists and is the largest LV -harmonic function dominated by u.

Proof. The sequence {SV (�n, u)} is non-increasing. Consequently, the limit

exists and is LV -harmonic. Every LV -harmonic function v dominated by u must

satisfy v ≤ SV (�n, u) in �n. Therefore u is the largest LV -harmonic function

dominated by u. �

Definition 2.5. A non-negative LV -superharmonic function is called an LV -

potential if its largest LV -harmonic minorant is zero.

The following is an immediate consequence of Lemma 2.4.

Lemma 2.6. Let p be a non-negative LV -superharmonic function. If, for some

Lipschitz exhaustion {�n} of �,

(2.12) limn→∞

SV (�n, p) = 0,

196 MOSHE MARCUS

then p is an LV -potential in �. Conversely, if p is an LV -potential, then (2.12)

holds for every Lipschitz exhaustion {�n} of �.

Remark. Lemmas 2.4 and 2.6 are based only on the fact that a superharmonic

function has a well-defined trace in L1(∂D) and satisfies (2.10). Therefore, these

lemmas remain valid if u and p are replaced with real valued LV -superharmonic

functions in the sense of [2].

Riesz Representation Theorem. Every non-negative LV -superharmonic

function u in � can be written in a unique way in the form u = p + h, where p

is an LV -potential and h a non-negative LV -harmonic function in �.

Remark. In fact, h = u as defined in (2.11).

For further results concerning the notion of LV -potential, see [2, Ch.I sec.4].

Definition 2.7. Let A ⊂ � and let s be a positive LV -superharmonic function

in � in the sense of [2]. Then RAs , the reduction of s relative to A, is the lower

envelope of the set

{ f : 0 ≤ f is LV -superharmonic in the sense of [2] and s ≤ f on A.}

If A is open, then RAs itself is LV -superharmonic in the above sense, so the lower

envelope is in fact the minimum [2, p.13].

Definition 2.8. Let ζ ∈ ∂�. A set A ⊂ � is LV -thin at ζ (in French, “A est

ζ-effile”) if RAK V

ζ6≡ K V

ζ .

By a theorem of Brelot, if A is open, then

(2.13) RAK V

ζ6≡ K V

ζ if and only if RAK V

ζis an LV -potential.

Furthermore, if A is LV -thin at ζ , there exists an open set O such that A ⊂ O and

O is LV -thin at ζ ; see [2, p. 26].

As a consequence of Lemma 2.6 and (2.13), we have the following result.

Lemma 2.9. Assume that A ⊂ � is open. Then A is LV -thin at a point ζ ∈ ∂�

if and only if there exists a Lipschitz exhaustion {�n} of � such that

SV (�n, K Vζ χ

An) → 0,

where An = ∂�n ∩ A.

Remark. For a discussion of the notions defined in the last two definitions,

see [2, p. 13, 26–27]. In the sequel, we do not use the notion of “reduction”. The

notion of an LV -thin set is used only with respect to open sets, in which case the

statement of Lemma 2.9 can serve as a definition.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 197

Definition 2.10. Let ζ ∈ ∂� and f be a real function on �. We say that f

admits an LV -fine limit ℓ at ζ if there exists a set E ⊂ � such that E is LV -thin

at ζ and limx→ζ,x∈�\E f (x) = ℓ. To indicate this type of convergence, we write

limx→ζ f (x) = ℓ, LV -finely.

In view of Brelot’s theorem, one may assume that E is open.

For the next two theorems, see [2, Prop.1.6 and Thm. 1.8].

Theorem 2.11. If p is an LV -potential, then for every positive LV -harmonic

function v , limx→ζ p/v = 0 LV -finely, µv -a.e. on ∂�, where µv is the LV -boundary

measure of v .

Theorem 2.12 (Fatou-Doob-Naim). If u, v are positive LV -harmonic func-

tions in �, then limx→ζ u/v = dµu/dµv LV -finely µv -a.e. on ∂�, where µu and

µv are the LV -boundary measures of u and v , respectively, and the term on the

right hand side denotes the Radon-Nikodym derivative.

The following statement is proved in [3].

Lemma 2.13. Let f be a positive function in � satisfying the Harnack in-

equality; i.e., for every a ∈ (0, 1), there exists a constant c(a) such that c(a) ↓ 1

as a ↓ 0, and, for every x ∈ �,

sup{ f (y) : y ∈ Baρ(x)(x)} ≤ c(a) inf{ f (y) : y ∈ Baρ(x)(x)}.

If f has an LV -fine limit at a point ζ ∈ ∂�, then it has a non-tangential limit at ζ

and

(2.14) limx→ζ

f = ℓ LV -finely implies limx→ζ

f = ℓ non-tangentially.

The case ℓ = ∞ is included.

Remark. This assertion is established in the proof of [3, Theorem 4]. In

fact, that theorem is an immediate consequence of Theorem 2.12 and the above

assertion.

3 Moderate solutions of LV u = 0

We recall some definitions from [15], following the notation of [6].

Definitions 3.1. We say that a boundary point ζ is LV -regular if K V (·, ζ )

(the largest LV -harmonic dominated by the LV -superharmonic function K (·, ζ ))

198 MOSHE MARCUS

is positive. The point ζ is LV -singular if K V (·, ζ ) = 0, i.e., K (·, ζ ) is an LV -

potential.

We denote by Sing(V ), respectively Reg(V ), the set of singular points, respec-

tively regular points, of LV .

Remark. If ζ is LV -regular, then there exists a positive constant cζ such that

K V (·, ζ ) = cζK V (·, ζ ). In fact, cζ = K V (x0, ζ ), where x0 is a fixed reference point

in � such that K V (x0, ζ ) = 1 for all ζ ∈ ∂�.

Notation. The family of finite Borel measures on a set A is denoted by M(A).

For A = ∂�, we simply write M. If µ ∈ M(A), we denote by |µ| the total variation

measure and by∥

∥µ∥

∥

M(A)the total variation norm.

In the sequel, we assume that � is a bounded domain of class C2.

The results of the present section – with some modifications – extend to the

case of bounded Lipschitz domains. However, in the following sections, we use

results that, at present, are known only for domains of class C2. Therefore – and

for the sake of simplicity – we confine ourselves to such domains in this section

as well.

Definition 3.2. An LV -harmonic function u is defined to be LV -moderate if

u ∈ L1(�) ∩ L1(�; Vρ) and there exists a measure ν ∈ M such that

(3.1)

∫

�

(−u1ϕ + uVϕ)dx = −

∫

∂�

∂nϕdν,

for every ϕ ∈ C20 (�).

A measure ν ∈ M is LV -moderate if there exists a moderate LV -harmonic

satisfying (3.1).

The space of LV -moderate measures is denoted by MV .

Definition 3.3. Let u ∈ W1,ploc (�) for some p > 1. We say that u possesses

an m-boundary trace ν ∈ M(∂�) if for every uniform C2 exhaustion of �, say

{�n}, u⌊∂�n

dHN−1 ⇀ ν, weakly with respect to C(�), i.e.,

(3.2)

∫

∂�n

uhdS →

∫

∂�

hdν for all h ∈ C(�).

If ν is the m-boundary trace of u, we write tr u := ν.

Remark. If u possesses an m-boundary trace ν, then u ∈ L1(�) and

(3.3) sup

∫

∂�n

|u|dS < ∞.

This follows immediately from the definition. It is easily verified that if u is LV -

moderate and satisfies (3.1), then ν is the m-boundary trace of u.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 199

Notation. Let ρ(x) := dist (x, ∂�). It is known that if � is of class C2, there

exists a number β0 > 0 such that ρ ∈ C2(�0), where

�0 := {x ∈ � : 0 < ρ(x) < β0}

and |∇ρ(x)| = 1. Furthermore, for every x ∈ �0, there exists a unique point

on ∂�, to be denoted by σ(x), such that |x − σ(x)| = ρ(x). Thus σ ∈ C1(�0) and

x = σ(x)−ρ(x)nσ(x)

, where nζ denotes the unit normal at ζ ∈ ∂� pointing outwards.

Let Dβ = [x ∈ �, ρ(x) > β], �β = � − Dβ, 6β = [x ∈ �, ρ(x) = β]. If

{βn} is a strictly monotone sequence decreasing to zero, then {Dβn} is a uniformly

C2 exhaustion of �.

Lemma 3.4. The following hold.

(i) Let v be a positive LV -superharmonic function in � and assume that

(3.4)

∫

�

vVρdx < ∞.

Then v is in L1(�) and possesses an m-boundary trace ν ∈ M. The supre-

mum of LV -harmonic functions dominated by v , say v ′, is LV -harmonic and

has the same m-boundary trace.

(ii) If v is a positive LV -superharmonic function and possesses an m-boundary

trace ν ∈ M, then v ′, defined as in (i), is LV -moderate harmonic and tr v ′ ≤

ν. If v is not a potential, then v ′ is positive.

(iii) If v is an LV -harmonic (not necessarily positive) and v possesses an m-

boundary trace ν, then v is LV -moderate.

Proof. (i). Let w ∈ L1(�) be the (unique) solution of the problem

(3.5) −1w = Vv in �, w = 0 on ∂�.

The solution exists because v satisfies (3.4). Then −1(w + v) ≥ 0, and conse-

quently w + v is an element of L1(�) and possesses an m-boundary trace

ν ∈ M(∂�). As w is in L1(�) and has m-boundary trace zero, v is also in L1(�)

and has m-boundary trace ν.

Given ϕ ∈ C20 (�), for each β ∈ (0, β0/2), we denote by ϕβ ∈ C2

0 (Dβ) the

function satisfying

(3.6) −1ϕβ = −1ϕ in Dβ, ϕβ = 0 on ∂Dβ .

Then ϕβ → ϕ in C2(�) and supβ

∥

∥ϕβ

∥

∥

C2(Dβ )

< ∞. Let vβ = SV (Dβ, v). Then

0 ≤ vβ ≤ v and vβ ↓ v ′ as β ↓ 0. Furthermore,∫

Dβ

(−vβ1ϕβ + vβVϕβ)dx = −

∫

∂Dβ

∂nϕβvdS.

200 MOSHE MARCUS

Since v is in L1(�; Vρ) ∩ L1(�) and possesses m- boundary trace ν on ∂�, we

obtain (by going to the limit as β → 0)

(3.7)

∫

�

(−v ′1ϕ + v ′Vϕ)dx = −

∫

∂�

∂nϕdν

for every ϕ ∈ C20 (�).

(ii). If v is a positive LV -superharmonic function, then there exists a Radon

measure λ > 0 in � such that LV u = λ in the sense of distributions. Therefore,

v ∈ W1,ploc (�) for some p > 1. Let vβ = SV (Dβ, v). Then vβ ≤ v in Dβ and

(3.8)

∫

Dβ

(−vβ1ϕ + vβVϕ)dx = −

∫

∂Dβ

(∂nϕ)vdS

for every ϕ ∈ C20 (Dβ). Choosing ϕ = φβ , where φβ satisfies −1φβ = 1 in Dβ ,

φβ = 0 on ∂Dβ , we obtain

(3.9)∥

∥vβ

∥

∥

L1(Dβ )

+∥

∥vβ

∥

∥

L1(Dβ ;Vρβ )

≤ C

∫

∂Dβ

vdS,

with a constant C independent of β. Here, ρβ is the first eigenfunction of −1 in

Dβ , normalized by ρβ(x0) = 1. Since by assumption, v has an m-boundary trace,

the right hand side of (3.9) is bounded. In addition, ρβ tends to the first normalized

eigenfunction of −1 in �. Therefore, vβ ↓ v ′ as β ↓ 0, locally uniformly in �,

and v ′ ∈ L1(�) ∩ L1(�; Vρ).

Now consider (3.8), with ϕ = ϕβ defined by (3.6). Taking the limit as β → 0

and applying Fatou’s lemma to the second integral on the left hand side, we obtain

(3.10)

∫

�

(−v ′1ϕ + v ′Vϕ)dx ≤ −

∫

∂�

∂nϕdν

for every non-negative ϕ ∈ C20 (�). Here we have used the facts that vβ ↓ v ′,

vβ ≤ v ∈ L1(�), and ϕβ → ϕ in C2(�). Consequently (by a standard argument),

v ′ has an m-boundary trace, say ν′, such that ν′ ≤ ν.

If, in addition, v is not an LV -potential, then v ′ > 0.

(iii). The proof is essentially the same as that of part (ii) except that, in the

present case, (3.9) is replaced by

(3.11)∥

∥vβ

∥

∥

L1(Dβ )

+∥

∥vβ

∥

∥

L1(Dβ ;Vρβ )

≤ C

∫

∂Dβ

|v |dS.

This inequality is proved by a standard argument as in, e.g., [33]. Since v is LV -

harmonic, vβ = v in Dβ . Therefore, we obtain v ∈ L1(�) ∩ L1(�; Vρ) and

∫

Dβ

(−v1ϕβ + vVϕβ)dx = −

∫

∂Dβ

∂nϕβvdS.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 201

Finally, taking the limit as β → 0, we obtain

∫

�

(−v1ϕ + vVϕ)dx = −

∫

∂�

∂nϕdν.�

Remark. If u is a positive LV -superharmonic function in � satisfying con-

dition (3.4), we say that u is LV -moderate. By Lemma 3.4(ii), if u is positive

LV -harmonic, this definition agrees with Definition 3.2.

Lemma 3.5. The following hold.

(i) If ν ∈ MV , then the solution of (3.1) is unique. It is denoted by SV (�, ν) or,

briefly, by SVν .

(ii) The space MV is linear, and

(3.12) 0 ≤ ν if and only if 0 ≤ SVν .

(iii) Let τ ∈ M and ν ∈ MV . Then

|τ| ≤ ν implies τ ∈ MV ,(3.13)

ν ∈ MV implies |ν| ∈ M

V .(3.14)

(iv) If ν ∈ MV , then |SVν | ≤ SV

|ν|.

Proof. (i), (ii) and (3.13) are classical. We turn to the proof of (3.14). Let

u = SVν . Since uV ∈ L1

ρ(�), there exist v+ and v− in L1(�) ∩ L1(�; Vρ) such

that −1v± + Vu± = 0 in � and v± = ν± on ∂�. It follows that u = v+ − v−,

|u| ≤ v+ + v− =: w, and −1w + Vw ≥ −1w + V |u| = 0. Thus w is a positive

LV -superharmonic with m-boundary trace |ν| and w ∈ L1(�) ∩ L1(�; Vρ). By

Lemma 3.4(i), the largest LV -harmonic dominated by w, say w′, is LV -moderate,

and tr w′ = tr w = |ν|. Thus |ν| ∈ MV and w′ = SV

|ν|.

Since ±ν ≤ tr w′, it follows that ±u ≤ w′, which is precisely assertion (iv). �

Let

(3.15) MV0 := {ν ∈ M : K|ν|V ∈ L1

ρ(�)}.

The following is an immediate consequence of Lemma 3.4.

Lemma 3.6. MV0 ⊂ M

V .

Proof. If ν ∈ MV0 , then K|ν| is an LV -superharmonic satisfying (3.4). �

202 MOSHE MARCUS

Remark. In general, there exist positive measures ν such that KVν ∈ L1(�; Vρ)

but Kν 6∈ L1(�; Vρ).

Lemma 3.7. If V ∈ Lq′

ρ (�) for some q′ > 1, then every positive measure in

W−2/q,q belongs to MV0 .

Proof. If ν ∈ W−2/q,q, then Kν ∈ Lqρ(�); see [41, 1.14.4.] or [37]). Therefore,

VKν ∈ L1ρ(�). If, in addition, ν ≥ 0, then ν ∈ M

V0 . �

Remark. There are signed measures ν ∈ W−2/q,q such that |ν| 6∈ W−2/q,q.

Therefore, in general, W−2/q,q may not be contained in MV0 .

Proposition 3.8. Let v be a positive, LV-moderate harmonic with m-boun-

dary trace ν. Let ν′ be the LV -boundary measure of v , i.e.,

(3.16) v = SVν = KV

ν′ .

Then, for every Borel set E ⊂ ∂�,

(3.17) ν′(E) = 0 if and only if ν(E) = 0

and

(3.18) SVνχ

E= KV

ν′χE.

Furthermore,

(3.19) K V (·, ζ )V ∈ L1ρ(�) ν′ − a.e.

Proof. Since vV ∈ L1ρ(�), it follows (by Fubini) that

∫

∂�

(∫

�

K V (·, ζ )Vρ

)

dν′(ζ ) < ∞,

which implies (3.19).

If E ⊂ ∂� is a Borel set, take

(3.20) vE = SVνχ

E, v ′

E = KVν′χ

E.

Let F be a compact subset of ∂�. The function v ′F is non-negative and LV -

moderate harmonic. Therefore, by Lemma 3.4, it has an m-boundary trace, say

µF . It is easy to verify that µF (∂� \ F ) = 0. As v ′F ≤ v = SV

ν , this fact implies

that

(3.21) µF ≤ νχF.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 203

If ν′(F ) > 0, then v ′F > 0 and consequently its m-boundary trace µF is not the

zero measure. Therefore ν′(F ) > 0 implies ν(F ) > 0.

The function vF is a non-negative, LV -moderate harmonic. Let µ′F denote its

LV -boundary measure. It is easy to verify that µ′F (∂� \ F ) = 0. As vF ≤ v = KV

ν′ ,

it follows that

(3.22) µ′F ≤ ν′χ

F.

If ν(F ) > 0, then vF > 0 and consequently µ′F is not the zero measure. Therefore,

ν(F ) > 0 implies ν′(F ) > 0. This yields (3.17).

If µ∂�\F denotes the m-boundary trace of v ′∂�\F , then µ∂�\F ≤ νχ

∂�\F. This is

obtained by representing ∂�\F as the union of an increasing sequence of compact

subsets and applying (3.21) to each set in this sequence. Since v ′F + v ′

F\∂� = v , we

have µF + µ∂�\F = νF . Therefore, in view of (3.21), µF = νχF. This equality and

(3.20) yield (3.18). �

Proposition 3.9. The following hold.

(i) For every ζ ∈ ∂� ,

(3.23) K V (·, ζ )V ∈ L1ρ(�) if and only if ζ is LV -regular.

(ii) If ν is a positive measure in MV , then K V (·, ζ ) is LV -moderate ν-a.e. and

(3.24) ν(Sing(V )) = 0.

(iii) If V ∈ Lq′

(ω) for some q′ > 1, then C2/q,q′-a.e. point ζ ∈ ∂� is LV -regular.

(Here, 1/q + 1/q′ = 1.)

Proof. (i). Assume that K V (·, ζ )V ∈ L1ρ(�). Then by Lemma 3.4, K V

ζ is LV -

moderate. Its m-boundary trace τζ ∈ M is concentrated at ζ . Thus τζ = a(ζ )δζ for

some a > 0. It follows that K Vζ is a subsolution of the boundary value problem

−1z = 0 in �, z = τζ on ∂�.

Therefore, K V (·, ζ ) ≤ a(ζ )K (·, ζ ). This implies that K Vζ > 0, i.e., ζ is regular.

Assume that ζ is LV -regular. Then, by definition, K (·, ζ ) is not a potential and

has m-boundary trace δζ . By Lemma 3.4(ii), the largest LV -harmonic dominated

by K (·, ζ ), which we denote by K V (·, ζ ), is LV -moderate and its m-boundary trace,

say λ, is a positive measure bounded by δζ . By uniqueness of the positive, nor-

malized LV -harmonic vanishing on ∂�\{ζ}, K V (·, ζ ) = K V (·, ζ )/K V (x0, ζ ). Thus

K V (·, ζ )V ∈ L1ρ(�).

204 MOSHE MARCUS

(ii). By (3.19) and (3.17),

(3.25) K V (·, ζ )V ∈ L1ρ(�) ν-a.e.

By (i), this implies the second assertion.

(iii). In this case, every positive measure ν ∈ W−2/q,q(∂�) is in MV0 which

is contained in MV . It follows that the set of singular points of LV must have

C2/q,q′-capacity zero. �

4 Preliminaries: on the equation −1u + uq = 0

In this section, we collect some definitions and known results on positive solutions

of (1.1) that are needed for the proof of our main result.

A basic concept in this theory is that of C2/q,q′ -fine topology on ∂�, which is

defined relative to the C2/q,q′ capacity on this manifold. For the general theory of

Cm,p capacity and Cm,p-fine topology, we refer the reader to [1]. For more special

results used in the sequel, see the summary in [36, Section 2].

We denote the closure of a set A ⊂ ∂� in C2/q,q′-fine topology by A. We say that

two sets A, B are C2/q,q′ equivalent (or, briefly, q-equivalent) if C2/q,q′(A1B) = 0.

There exists a constant c such that for every set A, C2/q,q′(A) ≤ cC2/q,q′(A).

We recall the definition of regular and singular boundary points of a positive

solution u of (1.1). A point ζ ∈ ∂� is a q-regular point of u if there exists a

C2/q,q′-fine neighborhood of ζ in ∂�, say Aζ , such that

(4.1)

∫

Oζ

uqρdx < ∞, where Oζ = {x ∈ � : ρ(x) < β0, σ(x) ∈ Aζ }.

We say ζ is q-singular if it is not q-regular. The set of q-regular points is denoted

by R(u), and the set of q-singular points by S(u). Evidently, R(u) is C2/q,q′-finely

open.

If F is a C2/q,q′-finely closed subset of ∂�, then there exists an increasing se-

quence of compact subsets {Fn} such that C2/q,q′(F \ Fn) → 0.

If u is a positive solution of (1.1), we say that u vanishes on a C2/q,q′-finely

open set O = ∂� \ F if it is the limit of an increasing sequence of positive

solutions {un} such that un ∈ C(� \ Fn) and un = 0 on ∂� \ Fn. The q-boundary

support of u, denoted by suppq∂� u, is the complement of the largest C2/q,q′-finely

open subset of ∂�, where u vanishes. A related concept is the q-support of a

measure τ ∈ M, denoted by q-suppτ, which is defined as follows: q-suppτ =

∂� \ O where O is the union of all C2/q,q′ -finely open sets Q ⊂ ∂� such that

|τ|(Q) = 0.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 205

Let ν ∈ M. We say that u is a solution of the problem

(4.2) −1u + |u|qsign u = 0 in D, u = ν on ∂D

if u ∈ L1(�) ∩ Lqρ(�) and

(4.3) −

∫

�

u1ϕdx +

∫

�

|u|q(sign u)ϕdx = −

∫

∂�

∂nϕdν

for every ϕ ∈ C20 (�).

If a solution exists it is unique; we denoted it by Sq(ν,�). Where there is

no danger of confusion, we abbreviate this to Sq(ν); when several domains are

involved, the full notation is used.

If ν is a measure for which a solution exists, we say that it is q-good. The

family of q-good measures is denoted by Gq. It is known that ν is q-good if and

only if it vanishes on sets of C2/q,q′ capacity zero; see [9]. Furthermore, a positive

measure ν ∈ M is q-good if and only if it is the limit of an increasing bounded

sequence of measures in W−2/q,q+ (∂�). In particular, a measure ν ∈ M such that

|ν| ∈ W−2/q,q+ (∂�) is a q-good measure.

A solution u of (1.1) is moderate if u ∈ L1(�) ∩ Lqρ(�). A moderate solution

possesses a m-boundary trace ν ∈ M such that (4.3) holds.

Denote by Uq the set of positive solutions of (1.1). A solution u ∈ Uq is σ-

moderate if it is the limit of an increasing sequence of moderate solutions.

A compact set F ⊂ ∂� is q-removable if every non-negative solution of (1.1)

vanishing on ∂� \ F vanishes in �. An arbitrary set A ⊂ ∂� is q-removable if

every compact subset is q-removable. It is known that A is q-removable if and only

if C2/q,q′(A) = 0; see [34] and the references therein.

By [35], if {un} is a sequence of positive solutions of (1.1), then

(4.4) C2/q,q′ (suppq∂� un) → 0 implies un → 0 locally uniformly in �.

For a C2/q,q′-finely closed subset F of ∂�, let

UF = sup{u ∈ Uq : suppq∂� u ⊂ F}.

It is well known that UF is a solution of (1.1) and vanishes on ∂� \ F . We call it

the maximal solution relative to F .

For an arbitrary Borel set A ⊂ ∂�, let

WA = sup{Sq(ν) : ν ∈ W−2/q,q+ (∂�), ν(∂� \ A) = 0}.

It is proved in [35] that WA = WA and if F is C2/q,q′ -finely closed, then UF = WF .

In particular UF is σ-moderate. To simplify notation, if A ⊂ ∂� is a Borel set, we

define UA := UA = WA.

206 MOSHE MARCUS

If v is a positive supersolution of (1.1), then the set of solutions dominated

by it contains a maximal solution v# := sup{u ∈ Uq : u ≤ v} ∈ Uq. If v is a

positive subsolution of (1.1), then the set of solutions dominating it is non-empty

and contains a minimal solution v# := inf{u ∈ Uq : u ≥ v} ∈ Uq.

If u, v ∈ Uq, then u + v is a supersolution, (u − v)+ is a subsolution, and we

define u ⊕ v = [u + v]# and u ⊖ v = [(u − v)+]#. For u ∈ Uq and a C2/q,q′-finely

closed subset F of ∂�, we define [u]F = inf(u, UF )#.

Let u ∈ Uq and ζ ∈ ∂�. Then ζ ∈ R(u) if and only if there exists a C2/q,q′-finely

open neighborhood Q of ζ in ∂� such that

(4.5)

∫

�

[u]q

Qρdx < ∞.

(In [36, Definition 5.1], a regular point is defined as above. The fact that this

definition is equivalent to the one based on (4.1) follows from [36, Lemma 5.2 and

Theorem 5.7].) Thus

(4.6) R(u) =⋃

{Q : Q ⊂ ∂�, Q is C2/q,q′-finely open , [u]Q

is moderate}.

Recall that if D is a bounded C2 domain and h ∈ L1(∂D), Sq(D, h) denotes the

solution of the problem −1u + |u|qsign u = 0 in D , u = h on ∂D .

If v is either a supersolution or a subsolution of (1.1), then v ∈ W1,ploc (�) for

every p ∈ [1, N/(N −1)). Therefore, if D is a C2 subdomain of �, v has a Sobolev

trace in Lp(∂D).

Let {�n} be a C2 exhaustion of �. Then

(4.7)

Sq(�n, v) ↓ v# if v is a positive supersolution,

Sq(�n, v) ↑ v# if v is a positive subsolution.

The following definitions were introduced in [36]. A positive Borel measure

τ on ∂� (not necessarily bounded) is called a perfect measure if it satisfies the

following conditions.

(a) τ is outer regular relative to C2/q,q′-fine topology; i.e., for every Borel set E ,

τ(E) = inf{τ(Q) : Q is C2/q,q′-finely open , E ⊂ Q}.

(b) If Q is a C2/q,q′-finely open set and A is a Borel set such that C2/q,q′(A) = 0,

then τ(Q) = τ(Q \ A).

The space of perfect measures is denoted by Mq.

Let Qτ denote the union of all C2/q,q′-finely open sets Q ⊂ ∂� such that

τ(Q) < ∞ and let Fτ := ∂� \ Oτ. Then (a) implies

(a ′) if E ⊂ ∂� is Borel and E ∩ Fτ 6= ∅, then τ(E) = ∞;

Further, (b) implies

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 207

(b ′) if Q is a C2/q,q′-finely open set, A a Borel subset such that C2/q,q′(A) = 0, and

τ(Q \ A) < ∞, then τ(A) = 0 and τχQ is a q-good measure.

Therefore, a positive Borel measure on ∂� satisfies conditions (a) and (b) if and

only if it satisfies conditions (T1)–(T3) – as formulated in the introduction – rela-

tive to F = Fτ.

For u ∈ Uq, we say that u has m-boundary trace τ ∈ Mq if

(i) R(u) = Qτ; and

(ii) for every ξ ∈ Qτ, there exists a C2/q,q′-finely open neighborhood Q such that

[u]Q is a moderate solution with m-boundary trace τχQ.

The m-boundary trace of u in this sense is called the precise trace and is denoted

by tr u.

By [36, Theorem 5.11], for every u ∈ Uq, there exists a sequence {Qn} of

C2/q,q′-finely open subsets of R(u) such that

(4.8) Qn ⊂ Qn+1, [u]Qnis moderate for all n, C2/q,q′ (R(u) \ ∪nQn) = 0.

Such a sequence is called a regular decomposition of R(u). We let

R0(u) =⋃

n

Qn, νn = tr [u]Qn,

uR

= lim[u]Qn, u

S= [u]

S(u).

(4.9)

Note that uR

does not depend on the specific sequence {Qn}. In fact (as a conse-

quence of the theorem cited above),

(4.10) uR

= sup{[u]Q

: Q ⊂ ∂�, Q is C2/q,q′-finely open , [u]Q

is moderate}

and

(4.11) S(uS) = {ζ ∈ S(u) : ζ is a C2/q,q′-thick point of S(u)}.

The set of C2/q,q′-thin points of any Borel set A ⊂ ∂� has C2/q,q′ capacity zero.

Therefore,

(4.12) C2/q,q′ (S(u) \ S(uS)) = 0.

Let F ⊂ ∂� be a C2/q,q′-finely closed set such that C2/q,q′ (F ) > 0 and let

F ′ denote the set of C2/q,q′-thick points of F . Let UF be the maximal solution

vanishing on ∂� \ F . Then F ′ is C2/q,q′-finely closed, UF = UF ′ , and S(UF ) = F ′.

A solution u ∈ Uq is purely singular if it vanishes on R(u). The maximal

solution UF defined above is purely singular.

The following result is proved in [36]; see Theorem 5.16 and the remark fol-

lowing it.

208 MOSHE MARCUS

Theorem 4.1. Every positive solution of (1.1) possesses a m-boundary trace

ν ∈ Mq. Conversely, for every ν ∈ Mq, there exists a solution of (1.1) with m-

boundary trace ν. Furthermore, there exists a unique σ-moderate solution of (1.1)

whose m-boundary trace equals ν. This solution is given by u = uR⊕ US(u), where

uR

is the σ-moderate solution defined in (4.9). Finally, u is the largest solution of

(1.1) with m-boundary trace ν.

By [36, Theorem 4.4], we also have the following result.

Theorem 4.2. Let u ∈ Uq and E, F be C2/q,q′-finely closed subsets of ∂�.

Then

(4.13) [u]E ≤ [u]E∩F + [u]E\F

.

Remark. Note that in [36], [u]A is defined for an arbitrary Borel set A ⊂ ∂�

by setting [u]A = [u]A.

Corollary 4.3. Let u ∈ Uq and E, F be disjoint C2/q,q′ -finely closed subsets

of ∂�. If suppq∂� u ⊂ E ∪ F, then

(4.14) max([u]E, [u]F ) ≤ u ≤ [u]E + [u]F .

We finish this section with two additional lemmas needed in the proof of our

main result.

Lemma 4.4. Let u ∈ Uq and A, B be two disjoint C2/q,q′-finely closed subsets

of ∂�. If suppq∂� u ⊂ A ∪ B and [u]A, [u]B are σ-moderate, then u is σ-moderate.

Furthermore,

(4.15) u = [u]A ⊕ [u]B = [max(uA, uB)]#.

Proof. Let τ and τ′ be q-good positive measures such that q-suppτ∩q-suppτ′ =

∅ and denote uτ = Sq(τ), uτ′ = Sq(τ′). Then

(4.16) [max(uτ, uτ′)]# = uτ ⊕ uτ′ = uτ+τ′ .

Let {τn} and {τ′n} be increasing sequences of q-good measures such that

uτn↑ [u]A, uτ′

n↑ [u]B.

Since q-suppτn ∩ q-suppτ′n = ∅, applying (4.16), we see that max(uτn

, uτ′n) ≤ u

implies uτn+τ′n≤ u. Thus

(4.17) v = lim uτn+τ′n≤ u.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 209

Let D be a C2 domain such that D ⋐ �. Given ǫ > 0, choose nǫ so large that

supD([u]A − uτn) < ǫ, supD([u]B − uτ′

n) < ǫ for all n ≥ nǫ. By (4.14), it follows

that for n ≥ nǫ, (u − 2ǫ)+ ≤ uτn+ uτ′

nin D . Using (4.16) again, we obtain

(4.18) (u − 2ǫ)+ ≤ uτn+τ′n

implies u ≤ v in D.

Since D is arbitrary, u ≤ v in �. This together with (4.17) implies u = v . By its

definition, v is σ-moderate.

Assertion (4.15) is equivalent to the statements

(a) u is the largest solution dominated by [u]A + [u]B , and

(b) u is the smallest solution dominating max(uA, uB).

Suppose that w ∈ Uq and u ≤ w ≤ [u]A + [u]B. Since [w]A vanishes on ∂� \ A

(which is a C2/q,q′-finely open neighborhood of B), it follows that [w]A ≤ [u]A.

Similarly, [w]B ≤ [u]B . Therefore, since u ≤ w, we obtain [w]A = [u]A and

[w]B = [u]B . Since [u]A, [u]B are σ-moderate, these equalities and the previous

argument show that v = w. This proves (a); statement (b) is proved in a similar

way. �

Lemma 4.5. If u ∈ Uq, then

(4.19) max(uR, uS) ≤ u ≤ u

R⊕ u

S

and

(4.20) u ⊖ uR

≤ uS.

Furthermore, uS

is a purely singular solution.

Proof. We start with (4.20). Let {Qn} be a regular decomposition of R(u) and

let vn = [u]Qn

. Then u ⊖ vn vanishes on Qn. Therefore, u ⊖ uR

vanishes on Qn for

every n. In view of (4.8), we conclude that u ⊖ uR

vanishes on R(u). This implies

(4.20).

The first inequality in (4.19) is obvious. The second inequality follows from

(4.20) and the fact that (by definition) u − uR

< u ⊖ uR.

By (4.19),

(4.21) S(u) = S(uS) ∪ S(u

R).

For brevity, set F = S(u), FS = S(uS), and FR = S(u

R). The solution u

Ris σ-

moderate. Consequently, by Theorem 4.1,

(4.22) [uR]

FR= UFR

.

210 MOSHE MARCUS

Let ES := FS \ FR. By (4.21), (4.22) and Theorem 4.2,

(4.23) uS≤ [u]FR

+ [u]ES= UFR

+ [u]ES

and max(UFR, [u]ES

) ≤ uS.

Let F ′R

denote the set of C2/q,q′-thick points of FR. Then UFR= UF ′

R, and

S(UFR) = F ′

R. Therefore, F ′

R⊂ S(u

S).

Every point ζ ∈ F \ FR is a q-singular point of u but a q-regular point of uR.

Therefore, ζ must be a q-singular point of uS. Since the set of q-singular points is

C2/q,q′-finely closed, it follows that ES ⊂ S(uS). Thus F ′

R∪ ES ⊂ S(u

S).

On the other hand, (4.23) and the fact that UFR= UF ′

Rimply that u

Svanishes

outside F ′R

∪ ES.

In conclusion, uS

is a purely singular solution, and

(4.24) S(uS) = F ′

R ∪ FS \ FR.�

5 Characterization of positive solutions of −1u + uq = 0

In this section, we prove the main result of the paper.

Theorem 5.1. Every positive solution of (1.1) is σ-moderate.

The proof is based on several lemmas.

Throughout the section, u is a positive solution of (1.1), V := uq−1, and LV v =

−1v + Vv . Thus V satisfies (2.2) and LV u = 0. Therefore, there exists a positive

measure µ ∈ M such that u = KVµ . For any Borel set E ⊂ ∂�, let µE = µχ

Eand

(u)E = KVµE

.

Lemma 5.2. Let D be a C2 domain such that D ⋐ � and let h ∈ L1(∂D),

0 ≤ h ≤ u. Then

(5.1) SV (D, h) ≤ Sq(D, h).

Proof. Let w := Sq(D, h) and v := SV (D, h). Then w ≤ u and consequently

(recall that V = uq−1) 0 = −1w+wq ≤ −1w+Vw. Thus w is a LV -superharmonic

function on D such that u = h on ∂D . On the other hand, v is an LV -harmonic

function on D satisfying the same boundary condition. This implies (5.1). �

Lemma 5.3. If F is a compact subset of ∂�, then

(5.2) (u)F ≤ [u]F .

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 211

Proof. Let A be a Borel subset of ∂�. Set Aβ = {x ∈ � : ρ(x) = β, σ(x) ∈ A},

vAβ = SV (Dβ, uχ

Aβ), and wA

β = Sq(Dβ, uχAβ

). By Lemma 5.2, vAβ ≤ wA

β ≤ u. For

any sequence {βn} decreasing to zero, one can extract a subsequence {βn′} such

that {wAβn′

} and {vAβn′

} converge locally uniformly; we denote the limits by wA and

vA, respectively. (The limits may depend on the sequence.) Then wA is a solution

of (1.1) while vA is an LV -harmonic, and

(5.3) vA ≤ wA ≤ [u]Q for all Q open, A ⊂ Q.

The second inequality follows from the fact that wA ≤ u and wA vanishes on

∂� \ Q.

We apply the same procedure to the set B = ∂� \ A, extracting further a subse-

quence of {βn′}, in order to obtain the limits vB and wB . Thus

vB ≤ wB ≤ [u]Q′ for all Q′ open, B ⊂ Q′.

Note that vA + vB = u, vA ≤ KVµQ

, and vB ≤ KVµQ′

. Therefore,

(5.4) vA = u − vB ≥ KVµ

∂�\Q′.

Now, given compact F , let A be a closed set and O, O′ open sets such that

F ⊂ O ⊂ A ⊂ O′, and let B = ∂�\A. Note that B ∩F = ∅. By (5.4) with Q′ = B ,

vA ≥ KVµ

O. By (5.3) and the previous inequality,

(5.5) (u)F ≤ KVµ

O≤ vA ≤ [u]O′.

If O′ shrinks to F , then [u]O ↓ [u]F ; see [36, Theorem 4.4]. Therefore, (5.5)

implies (5.2). �

Lemma 5.4. The following hold.

(i) If E ⊂ ∂� is a Borel set and C2/q,q′(E) = 0, then µ(E) = 0.

(ii) µ is the limit of an increasing sequence (µk) ⊂ W−2/q,q+ (∂�)+.

Proof. If F is a compact subset of E , C2/q,q′(F ) = 0; therefore, the Remov-

ability Theorem [34] implies that [u]F = 0. It follows from Lemma 5.3 that

(u)F = 0. Consequently, µ(F ) = 0. As this holds for every compact subset of

E , we conclude that µ(E) = 0.

By a theorem of Feyel and de la Pradelle [19] or Dal Maso [10], (ii) is a con-

sequence of (i). �

Lemma 5.5. Let ν ∈ W−2/q,q(∂�) be a positive measure. Suppose that there

exists no positive solution of (1.1) dominated by the supersolution v = inf(u,Kν).

Then µ ⊥ ν.

212 MOSHE MARCUS

Proof. First we show

Assertion 1: if V ′ := vq−1, then v is an LV ′

superharmonic and furthermore is an

LV ′

potential.

Since v is a supersolution of (1.1), 0 ≤ −1v + vq = −1v + V ′v . Thus, v is

an LV ′

superharmonic. Suppose that there exists a positive LV ′

harmonic w such

that w ≤ v . Then w is a subsolution of (1.1), i.e., −1w + wq ≤ −1w + V ′w = 0.

This implies that there exists a positive solution of (1.1) dominated by v , contrary

to assumption. Thus v is an LV ′

-potential.

Note that∫

� KνV ′ρdx ≤∫

�(Kν)qρdx < ∞. Therefore, Kν is an LV ′

super-

harmonic function satisfying (3.4). By Lemma 3.4(i), the largest LV ′

harmonic

dominated by Kν, say w, is LV ′

moderate and has m-boundary trace ν. This im-

plies that Kν − w =: p is an LV ′

-potential. The function w can be represented in

the form w = KV ′

ν′ , where ν′ is a positive finite measure on ∂�, and by Proposition

3.8, ν, ν′ are mutually absolutely continuous.

By Theorem 2.11 and the Relative Fatou Theorem, since v, p are LV ′

potentials

and w is an LV ′

harmonic,

(5.6) v/w → 0, Kν/w → 1 LV ′

-finely ν′-a.e.

Since v = inf(u,Kν), (5.6) implies that

(5.7) u/w → 0 LV ′

-finely ν′-a.e.

Furthermore, by (5.6) and (5.7),

(5.8)u

Kν

→ 0 LV ′

-finely ν′-a.e.

The measures ν and ν′ are mutually a.c., so that ’ν-a.e.’ is equivalent to ’ν′-a.e.’.

The function u/Kν satisfies the Harnack inequality. Therefore, by Lemma 2.13,

(5.8) implies

(5.9)u

Kν

→ 0 n.t. ν-a.e.

Here and in the sequel, we use the abbreviation n.t. for ‘non-tangential.

However, Kν is also LV -superharmonic. Therefore Kν can be represented in

the form Kν = w∗ + p∗, where w∗ is LV -harmonic and p∗ an LV -potential in �.

Let τ ∈ M be the LV -boundary measure of w∗, i.e., w∗ = KVτ . Then, by Theorem

2.11 and the Relative Fatou Theorem, Kν/u → dτ/dµ =: h LV -finely µ-a.e., and

therefore, by Lemma 2.13,

(5.10) lim supKν

u= h n.t. µ-a.e.

Since 0 ≤ h < ∞ µ-a.e., (5.9) and (5.10) imply that ν ⊥ µ. �

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 213

Lemma 5.6. Suppose that for every positive measure ν ∈ W−2/q,q(∂�), the

largest positive solution of (1.1) dominated by v = inf(u,Kν) is the trivial solution.

Then u = 0.

Proof. By Lemma 5.4, µ is the limit of an increasing sequence of measures

(µk) ⊂ W−2/q,q+ (∂�)+. By Lemma 5.5, the assumption of the present lemma im-

plies that µk ⊥ µ. Since 0 ≤ µk ≤ µ, it follows that µk = 0, and consequently,

µ = 0. �

Recall that if ν is a q-good measure, the solution of (1.1) with m-boundary

trace ν is denoted by Sq(ν). Let u be a positive solution of (1.1) and

(5.11) u∗ := sup{Sq(ν) : ν ∈ W−2/q,q+ (∂�), 0 < Sq(ν) ≤ u}.

By Lemma 5.6, the family over which the supremum is taken is non-empty. There-

fore, u∗ is a positive solution of (1.1), and it is known that it is σ-moderate. In fact,

u∗ is the largest σ-moderate solution dominated by u.

Lemma 5.7. Let u and u∗ be as above. (By our standard notation, u = KVµ .)

Let {νn} be an increasing sequence of measures in W−2/q,q+ (∂�) such that un :=

Sq(νn) ↑ u∗. Then

(5.12) µ ≺ ν := lim νn,

i.e., µ is absolutely continuous with respect to ν and

(5.13) suppq∂� u = supp

q∂� u∗ = q-suppµ.

Proof. Let E ⊂ ∂� be a compact set such that µ(E) > 0. Then, by Lemma

5.4, there exists a measure τ ∈ W−2/q,q+ (∂�) such that τ(E) > 0, τ is concen-

trated on E , and τ ≤ µ. By Lemma 5.5, there exists a positive solution of (1.1),

say u′, such that u′ < inf(u,Kτ). If ν′ denotes the m-boundary trace of u′, then

ν′ ∈ W−2/q,q+ (∂�), ν′ ≤ τ. Since 0 < u′ < u, we have ν′(E) > 0 and ν′ ≤ ν. Thus

ν(E) > 0, and (5.12) follows.

By (5.12),

(5.14) q-suppµ ⊂ q-suppν = suppq∂� u∗ ⊂ supp

q∂� u.

Let F = q-suppµ. If E is a C2/q,q′-finely closed subset of ∂� \ F , then [u]E ≤

(u)∂�\F = 0. There exists a decreasing sequence of C2/q,q′-finely open subsets of

∂� such that Qn+1 ⊂ Qn and C2/q,q′(∩Qn\F ) = 0; see [36, Lemma 2.6]. Therefore,

[u]∂�\Qn= 0, which implies supp

q∂� u ⊂ Qn ⊂ Qn−1 for each n. Hence

suppq∂� u ⊂ q-suppµ.

This relation and (5.14) imply (5.13). �

214 MOSHE MARCUS

Notation. Let ζ ∈ ∂�, α ∈ (0, 1), and

(5.15) Cζ (α) = {x ∈ � : α|x − ζ | < ρ(x)}

For a Borel set A ⊂ ∂�, define

(5.16) CA(α) =⋃

ζ∈A

Cζ (α) = {x ∈ � : α dist (x, A) < ρ(x)}.

Let f be a positive Borel function in � and ζ ∈ ∂�. For 0 < r and α ∈ (0, 1),

let Fr(ζ ; α) = sup{ f (y) : y ∈ Cζ (α) ∩Br(ζ )}. Then the non-tangential convergence

limx→ζ f (x) = 0 n.t. can be expressed in the form

(5.17) limr→0

Fr(ζ ; α) = 0 for all α ∈ (0, 1).

Let A ⊂ ∂� be compact. We say that f converges uniformly, non-tangent-

ially to zero at A if for every fixed α ∈ (0, 1),

(5.18) limr→0

Fr(ζ ; α) = 0 uniformly with respect to ζ ∈ A.

Lemma 5.8. Let f be a positive Borel function in � and A ⊂ ∂� compact.

Suppose that τ is a positive Borel measure on ∂� such that supp τ ⊂ A and that

(5.19) limx→ζ

f (x) = 0 n.t., τ-a.e. in A.

Then for every ǫ > 0, there exists a (relatively) open set Qǫ ⊂ ∂� such that

τ(Qǫ) < ǫ and

(5.20) limx→ζ

f (x) = 0 uniformly n.t. in A \ Qǫ.

Proof. By Lusin’s theorem, there exists an open set Ok ⊂ ∂� such that

τ(Ok) < ǫ/2k and limr→0 Fr(ζ ; 2−k) = 0 uniformly with respect to ζ ∈ A \ Ok .

Then (5.20) holds if Qǫ =⋃

Ok. �

Lemma 5.9. Let v be a non-negative subsolution of (1.1). Suppose that v is

dominated by a positive harmonic function w with boundary trace τ ∈ M+(∂�). If

(5.21) limx→ζ

v(x) = 0 n.t. τ-a.e. on ∂�,

then v ≡ 0.

Remark. Ancona [5] proved that if v is a non-negative solution of (1.1), then

(5.21) holds. In this case, there is no need for the condition “v is dominated by a

harmonic function”. The proof was based on an application of the Relative Fatou

Theorem to the operator LW = −1 + W , W = vq−1, with respect to the ratio of the

functions f ≡ 1 and v .

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 215

Proof. Since v is a subsolution, v ∈ W1,ploc (�) for p ∈ (1, N/(N −1)); therefore,

v possesses a Sobolev trace v⌊6β

in Lp(6β) for every β ∈ (0, β0).

Let vβ = Sq(Dβ , v⌊6β

), and v# = limβ→0 vβ. Then vβ increases as β decreases,

and v# is the smallest solution of (1.1) dominating v . Since w is a supersolution of

(1.1) and v ≤ w, it follows that v# ≤ w.

By Lemma 5.8, for every ǫ > 0, there exists a relatively open subset of ∂�, say

Qǫ, such that τ(Qǫ) < ǫ and

(5.22) limx→ζ

v(x) = 0 uniformly n.t. in Aǫ := ∂� \ Qǫ.

For α ∈ (0, 1), let

Ŵα,β = 6β \ CAǫ(α), Ŵ′

α,β = 6β ∩ CAǫ(α)

vα,β = Sq(Dβ, v⌊Ŵα,β

), v ′α,β = Sq(Dβ, v⌊

Ŵ′α,β

),

wα,β = Sq(Dβ, w⌊Ŵα,β

), w′α,β = Sq(Dβ, w⌊

Ŵ′α,β

).

Then

max(vα,β, v′α,β) ≤ vβ ≤ vα,β + v ′

α,β

vα,β ≤ wα,β, v ′α,β ≤ w′

α,β .(5.23)

By (5.22), limβ→0

∥

∥v⌊Ŵ′α,β

∥

∥

L1(6β )= 0, and consequently

(5.24) limβ→0

∫

Dβ

v ′α,βdx = 0.

Hence, by (5.23), for every α ∈ (0, 1),

(5.25) v# = limβ→0

vβ = limβ→0

vα,β ≤ lim infβ→0

wα,β .

We proceed to estimate wα,β . Let P denote the Poisson kernel for −1 in �.

Then

(5.26) P(x, y) ≤ cρ(x)|x − y|−N ≤ cα|x − y|1−N for all (x, y) ∈ (�\CAǫ(α)) ×Aǫ.

Let wǫ =∫

QǫP(x, y)dτ(y) and w′

ǫ =∫

AǫP(x, y)dτ(y). Then, by (5.26),

(5.27) w(x) = w′ǫ + wǫ ≤ cα

∫

Aǫ

|x − y|1−N dτ(y) + wǫ for all x ∈ � \ CAǫ(α),

where c depends only on N and �. Since Ŵα,β ⊂ � \ CAǫ(α), it follows that for

β ∈ (0, β0),∫

Dβ

wα,β(x)dx ≤ c

∫

Ŵα,β

w(x)dSx ≤

≤ c

∫

Ŵα,β

(

α

∫

Aǫ

|x − y|1−N dτ(y) + wǫ

)

dSx.

(5.28)

216 MOSHE MARCUS

Here, c depends only on N , � and β0.

Recall that vβ ↑ v# as β ↓ 0 and v ≤ Kτ; therefore,

limβ→0

∫

Dβ

vβdx =

∫

�

v#dx ≤ c ‖τ‖M .

Assuming that v# > 0 in � and using (5.23) and (5.24), we see that for sufficiently

small β1 ∈ (0, β0),( ∫

� v#dx)

/2 ≤∫

Dβvα,βdx ≤

∫

Dβwα,βdx for all β ∈ (0, β1).

Integrating over β in (0, β1) and using (5.28), we obtain

β1

2

∫

�

v#dx ≤ c

∫ β1

0

∫

Ŵα,β

(

α

∫

Aǫ

|x − y|1−N dτ(y) + wǫ(x)

)

dSxdβ =: I.

Since Ŵα,β = 6β \ CAǫ(α),

I ≤ c

∫

�β1\CAǫ (α)

(

α

∫

Aǫ(α)

|x − y|1−N dτ(y) + wǫ(x)

)

dx

≤ cα

∫

Aǫ

∫

�

|x − y|1−N dxdτ(y) + c

∫

�

wǫ(x)dx

≤ c(ατ(Aǫ) + τ(Qǫ)) ≤ c(α ‖τ‖M + ǫ).

Letting first ǫ → 0 and then letting α → 0, we conclude that v# = 0, and therefore

v = 0. �

Proof of Theorem 5.1. We assume that u is not a moderate solution; oth-

erwise, there is nothing to prove.

The function u∗ defined by (5.11) is a positive σ-moderate solution of (1.1)

dominated by u. Therefore it is LV -superharmonic. By the Riesz Decomposition

Theorem, it can be written in the form u∗ = v + p, where v is LV -harmonic and p is

an LV -potential in �. Let τ be the LV -boundary measure of v , i.e., v = KVτ . Then,

by Theorem 2.11 and the Relative Fatou Theorem,

(5.29) limx→ζ

u∗

u= lim

x→ζ

v

u=

dτ

dµ=: h LV -finely

for µ a.e. ζ ∈ ∂�. Therefore, by Lemma 2.13,

(5.30) limx→ζ

u∗

u= h n.t. for µ-a.e. ζ ∈ ∂�.

As 0 < u∗ ≤ u, we have 0 ≤ h ≤ 1.

Assertion 1. If h = 1 µ-a.e. on ∂�, then u = u∗.

Under the above assumption, (5.29) implies that v = u. Since u∗ = v + p for

some p ≥ 0, it follows that u∗ ≥ u. But, by definition, u∗ ≤ u. Therefore, u = u∗

is σ-moderate.

CLASSIFICATION OF SOLUTIONS OF −1u + uq = 0 217

Assertion 2. u∗ is not a moderate solution of (1.1).

By Assertion 1, if h = 1 µ-a.e., then u = u∗, so that u∗ is not a moderate

solution.

It remains to consider the case in which there exists a compact set E ⊂ ∂� and

some 0 < a < 1 such that µ(E) > 0 and

(5.31) limx→ζ

u∗

u= h(ζ ) ≤ a n.t. for all ζ ∈ E.

By negation, assume that u∗ is a moderate solution of (1.1). Then it has an

m-boundary trace ν ∈ M+. Since u∗ is positive, ν is not the zero measure. (Note

that the above assumption does not imply that u∗ is an LV -moderate superhamonic

function.)

By Lemma 5.7,

(5.32) µ ≺ ν.

Pick a number b such that a < b < 1. Then (5.31) implies

(5.33) limx→ζ

(

1

bu∗ − u

)

+

= 0 n.t. for all ζ ∈ E.

The function u∗/b is a supersolution of (1.1), and the largest solution which it

dominates is Sq(ν/b). Since µ(E) > 0, (5.32) implies that ν(E) > 0. Therefore,

the solution wλ := Sq(λ), where λ := νχE/b, is positive.

Since wλ vanishes on ∂� \ E , (5.33) implies limx→ζ (wλ − u)+ = 0 n.t. for all

ζ ∈ ∂�. By Lemma 5.9, we conclude that (wλ − u)+ = 0, i.e., wλ ≤ u. This in

turn implies that wλ ≤ u∗. But this is impossible, because λ(E) = ν(E)/b > ν(E).

This contradiction proves Assertion 2.

Assertion 3. Suppose that u is purely singular, i.e., uR

= 0. Then u = u∗ = UF ,

where F := suppq∂� u and UF is the maximal solution of (1.1) vanish-

ing on ∂� \ F .

First we show that u∗ is purely singular. By negation, assume that (u∗)R

> 0.

Then there exists a C2/q,q′-finely open set Q ⊂ ∂� such that Q ⊂ R(u∗) and [u∗]Q

is a positive moderate solution. By (5.11),

[u∗]Q : = sup{Sq(ν) : ν ∈ W−2/q,q+ (∂�), 0 < Sq(ν) ≤ u, ν(∂� \ Q) = 0}

= sup{Sq(ν) : ν ∈ W−2/q,q+ (∂�), 0 < Sq(ν) ≤ [u]Q} = ([u]Q)∗.

Thus ([u]Q)∗ is moderate, and by Assertion 2, [u]Q is moderate. As uR

= 0, it

follows that [u]Q = 0, and consequently [u∗]Q = 0. This contradiction proves that

u∗ is purely singular.

218 MOSHE MARCUS

Thus (u∗)R = 0, and consequently S(u∗) = suppq∂� u∗.

By Lemma 5.7, suppq∂� u∗ = F . Thus u∗ is σ-moderate and tr u∗ = tr UF . But

UF is σ-moderate; see [35]. Therefore, by uniqueness in the class of σ-moderate

solutions, u∗ = UF . Since u∗ ≤ u ≤ UF , we conclude that u = u∗ = UF .

Assertion 4. Every positive solution u is σ-moderate.

By Lemma 4.5, uS

is purely singular. Therefore, by Assertion 3, uS

is σ-

moderate and uS

= UFS, where FS := supp

q∂� u

S= S(u

S).

Let {Qn} be a regular decomposition of R(u) (see (4.8)) and let vn = [u]Qnand

un := vn ⊕ UFS. By Lemma 4.4, as Qn ∩ S(u) = ∅, it follows that un is σ-moderate

and that

(5.34) un = [max(vn, UFS)]# = vn ⊕ UFS

.

Thus, un is the smallest solution dominating the subsolution max(vn, UFS). Since

u itself is a solution which dominates this subsolution, it follows that un ≤ u. As

{un} is an increasing sequence, it follows that u = lim un is a σ-moderate solution

of (1.1) and, by the previous inequality,

(5.35) u ≤ u.

In addition, (5.34) implies that

u ≥ limn→∞

max(vn, UFS) = max(uR, UFS

),

u ≤ limn→∞

(vn + UFS) = uR + UFS

.(5.36)

Since R(u) ⊂ ∂�\FS and UFSvanishes outside FS, (5.36) implies that tr u⌊

R(u)=

tr u⌊R(u)

and S(u) = FS ∪ S(uR) = S(uS) ∪ S(uR) = S(u); see (4.21). Thus tr u = tr u.

By Theorrem 4.1, the σ-moderate solution u is the largest solution with this

trace. Therefore u ≤ u. This, together with (5.35), yields u = u. �

Acknowledgments. The author thanks Professor Alano Ancona for several

stimulating discussions and for two very helpful personal communications [4] and

[5] (later included in [6]).

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Moshe Marcus

DEPARTMENT OF MATHEMATICS

TECHNION

HAIFA 32000, ISRAEL

email: marcusm@math.technion.ac.il

(Received December 19, 2010 and in revised form November 30, 2011)