WOLFF POTENTIAL ESTIMATES FOR ELLIPTIC

EQUATIONS WITH NONSTANDARD GROWTH AND

APPLICATIONS

TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Abstract. We study superharmonic functions related to ellipticequations with structural conditions involving a variable growthexponent. We establish pointwise estimates for such functions interms of a Wolff type potential. We apply these estimates to provea variable exponent version of the Hedberg–Wolff theorem on thedual of Sobolev spaces with zero boundary values.

1. Introduction

We study pointwise behaviour of superharmonic functions definedrelative to quasilinear elliptic equations

− divA(x,∇u) = 0, (1.1)

where A satisfies structural conditions involving a variable exponentp(x). We shall call such functions Ap(·)-superharmonic. By definition,Ap(·)-superharmonic functions obey the comparison principle with re-spect to solutions of (1.1); see Section 2 for the details. The modelexample is given by the variable exponent p-Laplace equation

− div(|∇u|p(x)−2∇u) = 0.

Roughly speaking, superharmonic functions related to a partial dif-ferential equation can be characterised as solutions of a nonhomoge-neous equation involving a positive measure µ on the right hand side.It is possible to study the pointwise behaviour of superharmonic func-tions in terms of an appropriate nonlinear potential of this right handside measure, see for example [25, 26, 28, 32, 33, 35, 38]. Nonlinearpotentials also have an interesting theory in their own right and otherapplications, see for instance [4, 5, 6, 19].

2000 Mathematics Subject Classification. 35J60, 46E35, 31C45.Key words and phrases. p(x)-Laplace equation, Nonstandard growth, variable

exponent, Wolff potential, superharmonic functions.First author supported in part by the Academy of Finland.Third author supported by the Academy of Finland and the Emil Aaltonen

foundation.1

2 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

We establish potential estimates similar to those given by Kilpeläinenand Malý [25, 26]. The appropriate potential for controlling Ap(·)-superharmonic functions turns out to be

Wµp(·)(x,R) =

∫ R

0

(µ(B(x, r))

rn−p(x)

)1/(p(x)−1)dr

r, (1.2)

which is the constant exponent Wolff potential taken pointwise. Asan application of these estimates, we prove a variable exponent versionof a theorem due to Hedberg and Wolff [20] on the dual of Sobolevspaces with zero boundary values. A characterization of the points ofcontinuity of an Ap(·)-superharmonic funtion can also be given in termsof this potential. In addition, we prove a Harnack inequality and localHölder continuity under an extra assumption on the measure µ. Thelast two results readily follow as in [26] once the pointwise estimatesare available.

The structural conditions we consider here are a particular class of so-called nonstandard growth conditions. There is an extensive literatureon the calculus of variations and partial differential equations withvarious types of such conditions, see, e.g., [1, 2, 3, 8, 12, 16, 34, 40].In particular, potential estimates sufficient for boundary regularity ofsolutions of the variable exponent p-Laplace equation in terms of aWiener criterion have been given by Alkhutov and Krashenninikova[7]. However, their estimates rely on boundedness of the functionsunder consideration. We are able to relax such restrictions.

In the case of solutions to (1.1), our potential estimates reduce toHarnack’s inequality. It is well-known that Harnack estimates andother regularity results for (1.1) do not hold without additional as-sumptions on the function p(x); see the counterexamples in [23, 40].Even the variable exponent Lebesgue and Sobolev spaces have few goodproperties for general, for instance just measurable, exponents. How-ever, there is a condition, called logarithmic Hölder continuity, whichseems to be the right one for our purposes. This condition was origi-nally introduced by Zhikov [39] in the context of the Lavrentiev phe-nomenon, and it has turned out to be very useful in regularity andother applications, see, e.g., [2, 3, 9, 10, 12, 16, 37, 40].

For our potential estimates, a lack of homogeneity is a major sourceof difficulties. We need to use the logarithmic Hölder continuity andother techniques extensively to deal with these difficulties, and the finalestimates become intrinsic in the sense that the constants depend onthe function under consideration. This feature is already present in theHarnack estimates of [7, 17].

The paper is organised as follows. Section 2 discusses the relevantbackground material, including variable exponent Sobolev spaces, loga-rithmic Hölder continuity and its implications, and the basic propertiesof Ap(·)-superharmonic functions. In Sections 3 and 4 we establish lower

WOLFF POTENTIAL ESTIMATES 3

and upper pointwise estimates, respectively, for Ap(·)-superharmonicfunctions in terms of the Wolff potential (1.2). In Section 5, we provethe variable exponent version of the Hedberg–Wolff theorem. Finally,Section 6 closes the paper with some simple consequences of the point-wise estimates.

2. Preliminaries

We call a measurable function p : Rn → (1,∞), n ≥ 2, a variable

exponent. We denote

p−E = infx∈E

p(x) and p+E = sup

x∈Ep(x),

where E is a measurable subset of Rn. We assume that 1 < p−Ω ≤ p+

Ω <∞, where Ω is an open, bounded subset of R

n.The variable exponent Lebesgue space Lp(·)(Ω) consists of all mea-

surable functions f defined on Ω for which∫

Ω

|f |p(x) dx <∞.

The Luxemburg norm on this space is defined as

‖f‖p(·) = inf

λ > 0 :

∫

Ω

∣∣∣∣f(x)

λ

∣∣∣∣p(x)

dx ≤ 1

.

Equipped with this norm Lp(·)(Ω) is a Banach space, see Kováčik andRákosník [27]. The variable exponent Lebesgue space is a special caseof a more general Orlicz–Musielak space studied in [36]. For a constantfunction p(·) the variable exponent Lebesgue space coincides with thestandard Lebesgue space.

The variable exponent Sobolev space W 1,p(·)(Ω) consists of func-tions f ∈ Lp(·)(Ω) whose distributional gradient ∇f exists and satisfies|∇f | ∈ Lp(·)(Ω). This space is a Banach space with the norm

‖f‖1,p(·) = ‖f‖p(·) + ‖∇f‖p(·).

For basic properties of the spaces Lp(·) and W 1,p(·), we refer to [27].In particular, we will use the fact that if E is an open (or, mea-surable) subset of Ω and p and q are variable exponents satisfyingq(x) ≤ p(x) for almost every x ∈ E, then Lp(·)(Ω) embeds continuouslyinto Lq(·)(E). This implies that every function f ∈ W 1,p(·)(Ω) also be-

longs to W 1,p−Ω (Ω) and to W 1,p−B(B), where B is a ball contained inΩ.

Smooth functions are not dense in W 1,p(·)(Ω) without additional as-sumptions on the exponent p(·). This was observed by Zhikov [39, 40]in the context of the Lavrentiev phenomenon, which means that min-imal values of variational integrals may differ depending on whetherone minimises over smooth functions or Sobolev functions. Zhikov has

4 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

also introduced the logarithmic Hölder continuity condition to rectifythis. The condition is

|p(x) − p(y)| ≤C

− log (|x− y|)(2.1)

for all x, y ∈ Ω such that |x − y| ≤ 1/2. If the exponent is boundedand satisfies (2.1), smooth functions are dense in variable exponentSobolev spaces and we can define the Sobolev space with zero boundary

values, W1,p(·)0 (Ω), as the completion of C∞

0 (Ω) with respect to thenorm ‖·‖1,p(·). We refer to [10, 13, 24, 37] for density results in variableexponent Sobolev spaces.

Higher integrability [40], Hölder regularity results [2, 12], Harnackestimates [7, 17] and gradient estimates [3] for variational integrals andpartial differential equations with p(·)-growth use condition (2.1). Fur-thermore, examples show that if the logarithmic Hölder continuity con-dition is violated even slightly, higher integrability [40] and continuityof minimisers [23] may fail and the Lavrentiev phenomenon describedabove may occur [39]. Thus it is no surprise that logarithmic Höldercontinuity plays a crucial role also in this paper.

We will use logarithmic Hölder continuity in the form

R−(p+B−p−B) ≤ C, (2.2)

where B = B(x0, 2R) ⋐ Ω. It is well-known that requiring (2.2) tohold for all such balls is equivalent with condition (2.1); a proof of thisis given in [10, Lemma 3.2]. An elementary consequence of (2.2) is theinequality

C−1R−p(y) ≤ R−p(x) ≤ CR−p(y), (2.3)

which holds for any points x, y ∈ B(x0, 2R) with a constant depend-ing only on the constant of (2.2). We use phrases like “by log-Höldercontinuity” when applying either (2.2) or (2.3).

We need the following assumptions to hold for the operator A :Ω × R

n → Rn.

(1) x 7→ A(x, ξ) is measurable for all ξ ∈ Rn,

(2) ξ 7→ A(x, ξ) is continuous for all x ∈ Ω,(3) A(x, ξ) · ξ ≥ α|ξ|p(x), where α > 0 is a constant, for all x ∈ Ω

and ξ ∈ Rn,

(4) |A(x, ξ)| ≤ β|ξ|p(x)−1, where β ≥ α > 0 is a constant, for allx ∈ Ω and ξ ∈ R

n,(5) (A(x, η) −A(x, ξ)) · (η − ξ) > 0 for all x ∈ Ω and η 6= ξ ∈ R

n.

These are called the structure conditions of A.Assume from now on that p(·) is log-Hölder continuous. We say that

a function u ∈W1,p(·)loc (Ω) is a supersolution of the equation (1.1) if

∫

Ω

A(x,∇u) · ∇ϕ dx ≥ 0

WOLFF POTENTIAL ESTIMATES 5

for all nonnegative test functions ϕ ∈ C∞0 (Ω). Further, u is a solution

if equality holds. The dual of Lp(·)(Ω) is the space Lp′(·)(Ω) obtainedby conjugating the exponent pointwise, see [27]. Combining this with

the definition of W1,p(·)0 (Ω) as the completion of C∞

0 (Ω) allows us to

employ test functions ϕ ∈ W1,p(·)0 (Ω) with compact support in Ω by

the usual approximation argument.

Definition 2.1. We say that a function u : Ω → (−∞,∞] is Ap(·)-

superharmonic in Ω if

(1) u is lower semicontinuous,(2) u belongs to Lt

loc(Ω) for some t > 0, and(3) The comparison principle holds: Let U ⋐ Ω be an open set. If

h is a solution in U , continuous in U and u ≥ h on ∂U , thenu ≥ h in U .

Note that our definition is stronger than the one given in [14, 18,31]. More specifically, we require that u belongs to Lt

loc(Ω) for somet > 0, instead of just assuming that u is finite almost everywhere.This way, we avoid constant repetition of the Lt

loc assumption. Weemphasise the fact that in the definition, any exponent t > 0 willdo; having a small integrability exponent t to begin with allows usto conclude the integrability of u to certain natural exponents, seeTheorem 2.2 below. It is not known whether assumption (2) can beweakened without losing the conclusion of Theorem 2.2. However, forAp(·)-superharmonic solutions of Dirichlet problems involving measures,property (2) can be verified by using a priori estimates, see [31, proofof Theorem 4.7].

For the basic properties of Ap(·)-superharmonic functions, we refer to[14]. Most of these properties are similar to the case of p-superharmonicor A-superharmonic functions considered in, e.g., [21, 22, 29]. Oneof these properties is the fact that if u is Ap(·)-superharmonic, thenmin(u, k) is a supersolution for any constant k. Indeed, it is easy to seethat min(u, k) is Ap(·)-superharmonic, and bounded Ap(·)-superharmonicfunctions are supersolutions, see [14, Corollary 6.6]. In particular, thetruncations are weakly differentiable, hence, we can follow [25] anddefine a generalised gradient Du of u in a standard fashion as thepointwise limit

Du = limk→∞

∇min(u, k).

Note that Du is not necessarily the gradient of u in the distributionalsense, since Du might not belong to L1

loc(Ω). We adopt the followingconvention: ∇ denotes the usual weak gradient, and D refers to theabove definition.

The integrability lemma of Kilpeläinen and Malý [25, Lemma 1.11]is used to prove the following result. See [14, Theorem 7.5] and [30,Theorem 4.4].

6 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Theorem 2.2. Let u be a Ap(·)-superharmonic function in Ω. Then

there is a number q > 1 such that |u|q(p(x)−1) and |Du|q(p(x)−1) are locally

integrable.

The basis of this paper is the following theorem, which is a simpleconsequence of Theorem 2.2 and the Riesz representation theorem, seethe proof of Theorem 4.2 in [31].

Theorem 2.3. Let u be a Ap(·)-superharmonic function in Ω. Then

there is a positive Radon measure µ such that

− divA(x,Du) = µ

in the sense of distributions. Further, if uk = min(u, k) and the mea-

sures µk are given by

− divA(x,∇uk) = µk,

then µk → µ weakly as k → ∞.

We use the following lemma in order to pass to the limit in ourestimates. See [11, Theorem 1, pp. 54 – 55] for the proof.

Lemma 2.4. Let µ and µk, k = 1, 2, . . ., be positive Radon measures

such that µk → µ weakly in Ω. Then

lim supk→∞

µk(K) ≤µ(K) for all compact sets K ⊂ Ω, and

lim infk→∞

µk(U) ≥µ(U) for all open sets U ⊂ Ω.

We deal with integrals involving measures that are not necessarilyabsolutely continuous with respect to the Lebesgue measure. The fol-lowing lemma takes care of this. It can be proved in the same way as[35, Lemma 2.5]. The notion of p(·)-quasicontinuity is very similar tothat of p-quasicontinuity, see [15, Section 5] for the details.

Lemma 2.5. Let µ be a Radon measure in Ω that belongs to (W1,p(·)0 (Ω))∗.

Then

(µ, u) =

∫

Ω

u dµ

for all u ∈W1,p(·)0 (Ω), where u is a p(·)-quasicontinuous representative

of u.

If u is Ap(·)-superharmonic, then the measures µk associated to the

truncations min(u, k) are in (W1,p(·)0 (U))∗ for open sets U such that

U ⋐ Ω. Hence, the above lemma allows us to write integrals of Sobolevfunctions with respect to µk fairly freely, with the implicit understand-ing that we always work with p(·)-quasicontinuous representatives.

In the estimates below, we will use C to denote various constants,whose exact value is not important. We will not explicitly indicatethe dependencies of such constants on the dimension n, the structural

WOLFF POTENTIAL ESTIMATES 7

constants α and β of the operator A, p+Ω and p−Ω, and the log-Hölder con-

stant of p(·). Any other dependencies will be indicated; in particular,we will indicate how the constants depend on the Ap(·)-superharmonicfunction u under consideration.

The following Caccioppoli type estimates follow by standard choicesof test functions; the proofs are written out in [17, Lemma 4.3] and [31,Lemma 3.1].

Lemma 2.6. Let u be a nonnegative supersolution of (1.1) in a ball

B = B(x0, 2R) ⋐ Ω, η ∈ C∞0 (B) be such that 0 ≤ η ≤ 1, and let ε > 0.

Then one has∫

B

u−1−ε|∇u|p(x)ηp+B dx ≤ C

∫

B

u−1−ε+p(x)|∇η|p(x) dx,

where the constant C depends on ε.

Lemma 2.7. Let u be a nonpositive supersolution of (1.1) in a ball

B = B(x0, 2R) ⋐ Ω, η ∈ C∞0 (B) be such that 0 ≤ η ≤ 1. Then one has

∫

B

|∇u|p(x)ηp+B dx ≤ C

∫

B

(−u)p(x)|∇η|p(x) dx.

We also need the sharp form of the weak Harnack inequality (2.4)for Ap(·)-superharmonic functions.

Lemma 2.8. Let u be a nonnegative Ap(·)-superharmonic function in

a ball B = B(x0, 2R) ⋐ Ω, let t > 0 be such that u ∈ Lt(B). If

p+B − p−B < t/n, then

(−

∫

B(x0, 32R)

us dx

)1/s

≤ C( infx∈B(x0,R)

u(x) +R) (2.4)

for any 0 < s < nn−1

(p−B − 1), with a constant of the form

C = C(1 + ‖u‖p+

B−p−BLt(B) ).

Here C is independent of u and depends on s and q, where q > n is a

constant such that p+B − p−B < t/q.

We can establish (2.4) for nonnegative supersolutions as follows. Wemodify the argument of the proof of [7, Lemma 6.3] in a fashion similarto [17, Lemma 3.5]. This gives the estimate of [7, Lemma 6.3] withthe supremum replaced by the Lt-norm in the constant, provided thatp+

B − p−B < t/n. Then (2.4) follows by an application of [17, Theorem3.7]. The case of general Ap(·)-superharmonic functions then followsby considering the truncations min(u, k) and letting k tend to infinity.Both [7] and [17] are concerned only with the p(·)-Laplacean case, butit is straightforward to modify the arguments to cover our case, too.

Note that the condition p+B−p−B < t/q restricts the size of the balls B

in which the lemma can be applied. However, once the exponents q and

8 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

t are fixed, this restriction depends only on the continuity properties ofp(·), not on u. Further, in view of Theorem 2.2, the admissible choicesof t are the same for all Ap(·)-superharmonic functions. The interestedreader can consult [17, Section 3] for the technical details.

3. Lower pointwise estimate

In this section, we prove a lower bound for Ap(·)-superharmonic func-tions in terms of the Wolff potential (1.2). We do this along the linesof Kilpeläinen and Malý [25].

Hereafter, let x0 be an arbitrary but fixed point of Ω, and denotep0 = p(x0) and BR = B(x0, R). By the continuity of p(·), we can finda radius 0 < R0 ≤ 1 such that BR0 ⋐ Ω, and

p+0 − 1 <

n

n− 1(p−0 − 1),

where we denoted p+0 = p+

BR0and p−0 = p−BR0

. Fix a number γ such

that n(p+0 − p−0 ) < γ < n(p−0 − 1)/(n − 1). We may also assume that

q(p+0 −p−0 ) < γ, where q is the technical exponent in the weak Harnack

inequality, Lemma 2.8. Indeed, q is not a priori given, so we may choosea suitable value of q once γ is fixed. We denote

M(u,B) = (1 + ‖u‖Lγ(B))p+

B−p−B

for balls B ⊂ BR0 and functions u ∈ Lγ(B). Note that M(u,B) < ∞by Lemma 2.8 for Ap(·)-superharmonic functions u due to the choice ofγ.

Let us begin with the following estimate which combines the Cac-cioppoli type estimate (Lemma 2.6) and the weak Harnack inequality(Lemma 2.8).

Lemma 3.1. Let u be a nonnegative supersolution of (1.1) in a ball

B = B2R, where 0 < R ≤ R0

2, and let η ∈ C∞

0 (B3R/2) be such that

0 ≤ η ≤ 1 and |∇η| ≤ 4/R. Assume that M(u,B) ≤ L <∞. Then∫

B3R/2

|∇u|p(x)−1ηp+B−1|∇η| dx ≤ CRn−p0

(ess infx∈BR

u(x) +R

)p0−1

,

where the constant C depends on γ and L.

Proof. Let p+ = p+B and p− = p−B for simplicity, and b = ess infx∈BR

u(x).We note that for x ∈ B

bp+−p(x) ≤ C

(−

∫

B

uγ dx

)(p+−p(x))/γ

≤ CM(u,B) (3.1)

by the log-Hölder continuity of p(·). Set

ε =1

2min

(n

n− 1

p−0 − 1

p+0 − 1

− 1, p0 − 1

).

WOLFF POTENTIAL ESTIMATES 9

Then 0 < ε < 1, and

0 < p+ − 1 − ε < (p+ − 1)(1 + ε) <n

n− 1(p− − 1).

In particular, this means that the exponents p+−1−ε and (p+−1)(1+ε)are admissible in the weak Harnack inequality.

We use Young’s inequality and obtain∫

B3R/2

|∇u|p(x)−1ηp+−1|∇η| dx

≤(b+R)ε

∫

B3R/2

|∇u|p(x)(u+R)−1−εηp+

dx

+

∫

B3R/2

(b+R)−ε(p(x)−1)(u+R)(p(x)−1)(1+ε)|∇η|p(x) dx

=:I1 + I2.

The proof will be completed by showing that both I1 and I2 can bebounded by CRn−p0(b+R)p0−1.

By the log-Hölder continuity of p(·), we have |∇η|p(x) ≤ CR−p0. This,together with the Caccioppoli estimate for supersolutions (Lemma 2.6),gives

I1 ≤ C(b+R)εR−p0

∫

B3R/2

(u+R)−1−ε+p(x) dx.

Thanks to log-Hölder continuity, one has (u+ R)p(x)−p+≤ Rp(x)−p+

≤C, and by the weak Harnack inequality we have

I1 ≤ C(b+R)εRn−p0(b+R)p+−1−ε = CRn−p0(b+R)p+−1.

Furthermore, since (3.1) implies that (b+R)p+−p0 ≤ CM(u,B) ≤ CL,we obtain

I1 ≤ CRn−p0(b+R)p0−1,

where the constant C depends on L.To estimate I2, we note that (3.1) yields

(b+R)−ε(p(x)−1) ≤ C(b+R)−ε(p+−1)

and log-Hölder continuity implies

(u+R)(p(x)−1)(1+ε) ≤ C(u+R)(p+−1)(1+ε).

Using these inequalities, the weak Harnack inequality and (3.1), onehas

I2 ≤C(b+R)−ε(p+−1)Rn−p0(b+R)(p+−1)(1+ε)

=CRn−p0(b+R)p+−1 ≤ CRn−p0(b+R)p0−1

with a constant C depending on L.

To prove the lower estimate we need the following lemma.

10 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Lemma 3.2. Let u be a nonnegative Ap(·)-superharmonic function in

B = B2R, where 0 < R ≤ R0

2, and µ = −divA(x,Du). Then we have

Rp0−nµ(BR) ≤ C

(inf

x∈BR

u(x) − infx∈B2R

u(x) +R

)p0−1

,

where the constant depends on M(u,B).

Proof. We set p+ = p+B, a = infx∈B2R

u(x), b = infx∈BRu(x), uj =

min(u, j), and µj = − divA(x,∇uj) for j ≥ b. Choose a cut-offfunction η ∈ C∞

0 (B3R/2) such that 0 ≤ η ≤ 1, |∇η| ≤ 4/R, and

set v = min(u, b) − a+R. We use w = vηp+as a test function, noting

that 0 ≤ w ≤ b− a +R and w = b− a + R in BR. Applying Lemmas2.7 and 3.1 to uj − a, we obtain

(b− a+R)µj(BR) ≤

∫

B3R/2

w dµj =

∫

B3R/2

A(x,∇(uj − a)) · ∇w dx

≤

∫

B3R/2

A(x,∇(uj − a)) · ∇vηp+

dx

+ p+

∫

B3R/2

|A(x,∇(uj − a))|ηp+−1|∇η|v dx

≤C

∫

B3R/2

|∇(v − (b− a+R))|p(x)ηp+

dx

+ C(b− a+R)

∫

B3R/2

|∇(uj − a)|p(x)−1ηp+−1|∇η| dx

≤C

∫

B3R/2

|b− a− v +R|p(x)|∇η|p(x) dx

+ C(b− a+R)

∫

B3R/2

|∇(uj − a)|p(x)−1ηp+−1|∇η| dx

≤C(b− a+R)p+

∫

B3R/2

|b− a +R|p(x)−p+

|∇η|p(x) dx

+ CRn−p0(b− a+R)p0

≤C((b− a+R)p+

+ (b− a+R)p0)Rn−p0.

Note that M(uj − a,B) ≤ M(u,B) for all j and a, so we can takeL = M(u,B) in Lemma 3.1 to ensure that the constant is independentof j and a. In the last inequality, we used the fact that b − a ≥ 0 toobtain

(b− a +R)p(x)−p+

≤ Rp(x)−p+

≤ C

by log-Hölder continuity, and the fact that |∇η|p(x) ≤ CR−p0 .Since u is nonnegative, we have a ≥ 0, and thus

(b− a+R)p+−p0 ≤ (b+R)p+−p0 ≤ C(bp+−p0 + 1) ≤ C(1 +M(u,B))

WOLFF POTENTIAL ESTIMATES 11

by (3.1). Hence, we have obtained

µj(BR) ≤ CRn−p0(b− a+R)p0−1.

Due to Lemma 2.4, letting j → ∞ completes the proof, since µj tendsto µ weakly.

We are now ready to prove the lower pointwise estimate for Ap(·)-superharmonic functions in terms of a Wolff-type potential.

Theorem 3.3. Let u be a nonnegative Ap(·)-superharmonic function

in B2R, where 0 < R ≤ R0/2, and let

µ = − divA(x,Du).

Then

u(x0) ≥ infx∈B2R

u(x) + CWµp(·)(x0, R) − 2R,

where

Wµp(·)(x0, R) =

∫ R

0

(µ(B(x0, r))

rn−p(x0)

)1/(p(x0)−1)dr

r,

and the constant C > 0 depends on M(u,B2R).

Proof. Let Rj = 21−jR and aj = infx∈BRju(x). Lemma 3.2 and the

lower semicontinuity of u imply that

C

∞∑

j=1

(Rp0−n2j(n−p0)µ(BRj

)

)1/(p0−1)

≤∞∑

j=1

(aj − aj−1 +Rj)

= limk→∞

(ak − a0) +

∞∑

j=1

Rj

=u(x0) − infx∈B2R

u(x) + 2R.

The theorem follows from the inequality∫ R

0

(µ(Br)

rn−p0

)1/(p0−1)dr

r≤ C

∞∑

j=1

(Rp0−n2j(n−p0)µ(BRj

))1/(p0−1)

.

4. Upper pointwise estimate

In this section, we prove a pointwise upper bound for Ap(·)-super-harmonic functions in terms of the Wolff potential (1.2). Our approachis an adaptation of the method in Kilpeläinen and Malý [26]. In oursetting, a similar estimate has been proved by Alkhutov and Krashenin-nikova (see [7, Theorem 8.1 and (8.34)]). However, they consider onlycapacitary potentials u, in which case 0 ≤ u ≤ 1, and their argumentsrely on this boundedness of u. Our goal is to give an estimate forgeneral, i.e., not necessarily bounded, Ap(·)-superharmonic functions.

12 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Throughout this section, let x0 be an arbitrary but fixed point of Ωand denote p0 = p(x0) and BR = B(x0, R). By the continuity of p(·),there is a radius 0 < R0 ≤ 1 such that BR0 ⋐ Ω, and for p+

0 = p+BR0

and p−0 = p−BR0, one has

p+0 − 1 <

κp−0 (p0 − 1)

p0 − 1 + κ, (4.1)

where κ = n/(n − 1). Note that this choice of R0 is slightly differentfrom that made in the previous section. Observe that (4.1) impliesp0 − 1 + κ < κp−0 , and furthermore,

n(p+0 − p−0 ) < p+

0 − 1 <κp−0 (p0 − 1)

p0 − 1 + κ< κ(p−0 − 1), (4.2)

and we may also find a q > n such that q(p+0 − p−0 ) < p+

0 − 1.We begin with the following Caccioppoli type estimate.

Lemma 4.1. Let 0 < R ≤ R0 and set p+ = p+BR

, p− = p−BR. Let u be

a supersolution of (1.1) in BR, and let

µ = − div A(x,∇u). (4.3)

Let σ0 > 1, λ > 0 and let η ∈ C∞0 (BR) be such that 0 ≤ η ≤ 1. Then

there exists a constant C, depending on p+0 and σ0, such that

∫

BR∩u>0

|∇u|p(x)(1 + λu)−σηp+

dx

≤ C

(max(λ−p+

, λ−p−)

∫

BR∩u>0

(1 + λu)σ(p+−1)|∇η|p(x) dx

+1

λµ(supp η)

)

for σ ≥ σ0.

Proof. For σ ≥ σ0, let

Ψ(τ) =

∫ τ

0

(1 + λs)−σ ds =1

λ(σ − 1)

(1 − (1 + λτ)1−σ

)

for τ ≥ 0. Set v = (Ψ u+)ηp+. Then v ∈W

1,p(·)0 (BR) and

∇v = (1 + λu)−σχu>0ηp+

∇u+ p+(Ψ u+)ηp+−1∇η.

Note that

0 ≤ Ψ(τ) ≤1

λ(σ − 1)≤

1

λ(σ0 − 1).

WOLFF POTENTIAL ESTIMATES 13

Hence, using v as a test function in (4.3), we have

α

∫

BR∩u>0

|∇u|p(x)(1 + λu)−σηp+

dx (4.4)

≤

∫

BR∩u>0

(A(x,∇u) · ∇u

)(1 + λu)−σηp+

dx

= −p+

∫

BR

(A(x,∇u) · ∇η

)(Ψ u+)ηp+−1 dx+

∫

BR

v dµ

≤1

λ(σ0 − 1)

(p+

0 β

∫

BR∩u>0

|∇u|p(x)−1|∇η|ηp+−1 dx+ µ(supp η)

).

By Young’s inequality, for δ > 0 we have

|∇u|p(x)−1|∇η|ηp+−1 ≤δ|∇u|p(x)(1 + λu+)−σηp+

+ δ1−p(x)(1 + λu+)σ(p(x)−1)|∇η|p(x)

≤δ|∇u|p(x)(1 + λu+)−σηp+

+ δmax(δ−p−, δ−p+

)(1 + λu+)σ(p+−1)|∇η|p(x).

Choose δ = λ(σ0 − 1)α/(2p+0 β). Then

p+0 β

λ(σ0 − 1)|∇u|p(x)−1|∇η|ηp+

≤α

2|∇u|p(x)(1 + λu)−σηp+

+ C max(λ−p−, λ−p+

)(1 + λu)σ(p+−1)|∇η|p(x) (4.5)

on BR ∩ u > 0 with C > 0 depending on p+0 and σ0. Thus we obtain

the required estimate by absorbing the first term in (4.5) to the lefthand side of (4.4).

Recall our notation from the previous section

M(u,B) = (1 + ‖u‖Lγ(B))p+

B−p−B ,

where γ is introduced in the following lemma.

Lemma 4.2. Let u be an Ap(·)-superharmonic function in B2R, where

0 < R ≤ R0

2, and let µ = − div A(x,Du). Assume that M(u+, B2R) ≤

L <∞, and let γ satisfy

p+0 − 1 < γ <

κp−0 (p0 − 1)

p0 − 1 + κ. (4.6)

Then there exist a constant C, depending on p0, p+0 , p

−0 , γ, L, and

ρj > 0, j = 1, 2, 3, depending on p0, p+0 , p

−0 , and γ, such that

(−

∫

BR

u+γ dx

)1/γ

≤ C

(θR

ρ1R+ θRρ2

(−

∫

B2R

u+γ dx

)1/γ

+ θRρ3

(Rp0

µ(B2R)

|B2R|

)1/(p0−1)), (4.7)

14 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

where

θR = |B2R ∩ u > 0|/|B2R|.

Proof. Let 0 < R ≤ R0/2 and write p+ = p+B2R

, p− = p−B2Rand θ = θR

for simplicity. First, assume that u is bounded. Then u is a supersolu-tion of (1.1) in B2R. Let σ = γ/(p+−1). Then σ ≥ σ0 := γ/(p+

0 −1) >1, and furthermore,

σ ≤γ

p0 − 1<

κp−0p0 − 1 + κ

< p−.

Consider w = (1 + λu+)1−σ/p− − 1 for λ > 0. Then, 0 ≤ w ≤

(λu+)1−σ/p− and w ∈ W1,p(·)loc (B2R). If η ∈ C∞

0 (B2R) is such that 0 ≤η ≤ 1, η = 1 on BR, and |∇η| ≤ 2/R, then the function ηp+/p−w

belongs to W1,p(·)0 (B2R) and

|∇(ηp+/p−w)|p−

≤ C(|∇η|p

−

wp− + |∇w|p−

ηp+)

≤C(R−p−(λu+)p−−σ + λp−|∇u|p

−

(1 + λu)−σχu>0ηp+)

≤C(R−p−

(χu>0 + (λu+)γ

)

+ λp−(1 + |∇u|p(x)(1 + λu)−σηp+)

χu>0

).

By Sobolev’s inequality, we have

(−

∫

B2R

(ηp+/p−w)κp− dx

)1/κ

≤ CRp−−

∫

B2R

|∇(ηp+/p−w)|p−

dx

≤C

(θ + λγ−

∫

B2R

uγ+ dx

+ λp−Rp−(θ +

1

|B2R|

∫

B2R∩u>0

|∇u|p(x)(1 + λu)−σηp+

dx

)).

Hence, by Lemma 4.1,(−

∫

B2R

(ηp+/p−w)κp− dx

)1/κ

≤ C

((1 + λp−Rp−)θ + λγ−

∫

B2R

uγ+ dx

+ max(λp−−p+

, 1) Rp−

|B2R|

∫

B2R∩u>0

(1 + λu)γ|∇η|p(x) dx

+ λp−−1Rp− µ(supp η)

|B2R|

)(4.8)

≤ C

((λp−Rp− + max

(λp−−p+

, 1))θ

+ max(λp−−p+

, 1)λγ−

∫

B2R

uγ+ dx+ λp−−1Rp− µ(supp η)

|B2R|

),

WOLFF POTENTIAL ESTIMATES 15

where we used log-Hölder continuity to estimate |∇η|p(x) ≤ CR−p−.

We have w ≥ C(λu+)1−σ/p− for some C > 0 in case λu+ ≥ 1. By (4.6),

κp−(

1 −σ

p−

)≥ κp−0 −

κγ

p0 − 1≥ γ.

Thus, since η = 1 on BR,

(λγ−

∫

BR

uγ+ dx

)1/κ

≤ C

(θ1/κ +

(−

∫

B2R

(ηp+/p−w)κp− dx

)1/κ).

Hence, by (4.8)

(λγ−

∫

BR

uγ+ dx

)1/κ

≤ C0

(θ1/κ +

(λp−Rp− + max

(λp−−p+

, 1))θ

+ max(λp−−p+

, 1)λγ−

∫

B2R

uγ+ dx+ λp−−1Rp− µ(supp η)

|B2R|

). (4.9)

By log-Hölder continuity, Rp− can be replaced by Rp0 . Note that C0

depends on γ, but does not depend on R, u, or λ.Hereafter, let

AR :=

(−

∫

BR

uγ+ dx

)1/γ

.

Then by log-Hölder continuity one has

Ap+−p−

R ≤ C1M(u+, B2R) ≤ C1L and

Ap0−p−

R ≤ C1M(u+, B2R) ≤ C1L, (4.10)

where the constant C1 ≥ 1 depends on γ. Next, we set

M = 1 + C0 + C0C1L ≥ 1

and

λ = Mκ/γθ1/γA−1R .

Since (λγAγR)1/κ = Mθ1/κ and 0 ≤ θ ≤ 1, (4.9) and (4.10) imply

(M − C0)θ1/κ ≤ C0

(C ′

2θp−0 /γA−p0

R Rp0 + C1Lθ1−(p+

0 −p−0 )/γ

+C ′3θ

1−(p+0 −p−0 )/γA−γ

R Aγ2R +C ′

4θ(p−0 −1)/γA1−p0

R Rp0µ(supp η)

|B2R|

)

with the primed constants C ′j depending on γ and L. Since γ > n(p+

0 −p−0 ) by (4.2) and 0 ≤ θ ≤ 1,

θ1−(p+0 −p−0 )/γ ≤ θ1/κ.

16 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Hence, the above estimate yields

3θ1/κ ≤ C ′

(θp−0 /γA−p0

R Rp0

+ θ1−(p+0 −p−0 )/γA−γ

R Aγ2R + θ(p−0 −1)/γA1−p0

R Rp0µ(supp η)

|B2R|

)

with a constant C ′ > 0 depending on γ and L. Therefore we haveeither

θ1/κ ≤ C ′θp−0 /γRp0A−p0

R

orθ1/κ ≤ C ′θ1−(p+

0 −p−0 )/γA−γR Aγ

2R

or

θ1/κ ≤ C ′θ(p−0 −1)/γA1−p0

R Rp0µ(supp η)

|B2R|.

This means that we have three possibilities: either

AR ≤ Cθ

(p−0γ

− 1κ

)1

p0R

or

AR ≤ Cθ

(1−

p+0

−p−0

γ

)1γ− 1

κγA2R

or

AR ≤ Cθ

(p−0

−1

γ− 1

κ

)1

p0−1Rp0/(p0−1)

(µ(supp η)

|B2R|

)1/(p0−1)

,

where the constant C > 0 depends on γ and L. Let us now define

ρ1 :=

(p−0γ

−1

κ

)1

p0

, ρ2 :=1

γ−p+

0 − p−0γ2

−1

κγ,

and

ρ3 :=

(p−0 − 1

γ−

1

κ

)1

p0 − 1.

By (4.6) and (4.2), we see that ρj > 0, j = 1, 2, 3. By the above threeinequalities for AR, we have

AR ≤ C

(θρ1R+ θρ2A2R + θρ3

(Rp0

µ(supp η)

|B2R|

)1/(p0−1)). (4.11)

This completes the proof in the bounded case.For an unbounded Ap(·)-superharmonic function u on B2R, apply

(4.11) to um = min(u,m), m = 1, 2, . . . and let m→ ∞. If

µm = −divA(x,∇um) and µ = −divA(x,Du),

thenlim sup

m→∞µm(supp η) ≤ µ(supp η) ≤ µ(B2R)

by Lemma 2.4. Hence, we obtain the required estimate (4.7) for u.

Finally, we are ready to prove the upper estimate.

WOLFF POTENTIAL ESTIMATES 17

Theorem 4.3. Let u be an Ap(·)-superharmonic function in B2R, where

0 < R ≤ R0

2, and

µ = − divA(x,Du).

Let γ satisfy (4.6). Then there exists a positive constant C depending

on γ and M(u+, B2R) such that

u(x0) ≤ C

(R+

(−

∫

BR

uγ+ dx

)1/γ

+ Wµp(·)(x0, 2R)

). (4.12)

Proof. Let B0 = BR, Rj = 2−jR, Bj = BRjand set

Mj =

(Rp0

j

µ(Bj−1)

|Bj−1|

)1/(p0−1)

,

j = 1, 2, . . .. For 0 < δ < 1, let l0 = 0, and

lj+1 = lj +1

δ

(−

∫

Bj

(u− lj)γ+ dx

)1/γ

, j = 0, 1, . . . .

Set θj = |Bj−1 ∩ u > lj|/|Bj−1|, j = 1, 2, . . .. Since

|Bj ∩ u > lj+1| ≤ (lj+1 − lj)−γ

∫

Bj∩u>lj+1

(u− lj)γ+ dx

≤ δγ

(−

∫

Bj

(u− lj)γ+ dx

)−1 ∫

Bj

(u− lj)γ+ dx = δγ|Bj |, (4.13)

θj ≤ δγ for all j = 1, 2, . . . .Now, applying Lemma 4.2 to (u− lj)+ and R = Rj , j ≥ 1, we obtain

lj+1 − lj =1

δ

(−

∫

Bj

(u− lj)γ+ dx

)1/γ

≤C

δ

(θρ1

j Rj + θρ2

j

(−

∫

Bj−1

(u− lj)γ+ dx

)1/γ

+ θρ3

j Mj

)

≤ C(δγρ1−1Rj + δγρ2(lj − lj−1) + δγρ3−1Mj

)

with C depending on γ and M(u+, B2R). Note above that

M((u − lj)+, B2R) ≤M(u+, B2R)

for all j, so we can take L = M(u+, B2R) in Lemma 4.2 to ensure thatthe constant is independent of j. Hence, for k ≥ 2 we have

lk − l1 =

k−1∑

j=1

(lj+1 − lj) ≤ C(δγρ1−1R+ δγρ2lk + δγρ3−1

k−1∑

j=1

Mj

).

18 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Thus, taking δ so small that Cδγρ2 ≤ 1/2, we have

lk ≤ l1 + C

(R+

∞∑

j=1

Mj

)≤ C

(R+

(−

∫

B0

uγ+ dx

)1/γ

+∞∑

j=1

Mj

).

By (4.13),

|Bj ∩ u ≤ lj+1| ≥ (1 − δγ)|Bj | > 0,

so that infx∈Bju(x) ≤ lj+1. Hence by the lower semicontinuity of u, we

have

u(x0) ≤ limj→∞

lj ≤ C

(R+

(−

∫

B0

uγ+ dx

)1/γ

+∞∑

j=1

Mj

).

Since∑∞

j=1Mj ≤ CWµp(·)(x0, 2R), we obtain (4.12).

Observe that if we combine Theorem 4.3 and Lemma 2.8, we readilyobtain the following corollary.

Corollary 4.4. Let u be a nonnegative Ap(·)-superharmonic function

in B2R, where 0 < R ≤ R0/2, and µ = − divA(x,Du). Then there

exists a positive constant C depending on M(u,B2R) such that

u(x0) ≤ C

(R+ inf

x∈BR

u(x) + Wµp(·)(x0, 2R)

).

5. The Hedberg–Wolff theorem

In this section we prove a variable exponent version of the Hedberg–Wolff theorem, see [20], by applying the pointwise estimates in sections3 and 4. The constant exponent case can also be found in [41, Theorem4.7.5].

Let µ be a nonnegative Radon measure on Rn with compact support

and let Ω be a bounded open set containing supp µ. Let p+ = p+Ω and

p− = p−Ω in this section. We first prove a version of the maximumprinciple for the variable exponent Wolff potential.

Proposition 5.1. Let 0 < R < min(dist(∂Ω, supp µ), 1). Then there

exist constants A1 ≥ 1, depending on µ(Ω)p+−p−, and A2 ≥ 0, depend-

ing on µ(Ω) and R, such that

Wµp(·)(x,R) ≤ A1 sup

x′∈supp µWµ

p(·)(x′, R) + A2

for all x ∈ Ω.

Proof. We may assume

M := supx′∈supp µ

Wµp(·)(x

′, R) <∞.

WOLFF POTENTIAL ESTIMATES 19

Let x 6∈ supp µ and let δ = dist(x, supp µ). If δ ≥ R, then Wµp(·)(x,R) =

0. So we consider the case δ < R. Choose x′ ∈ supp µ such that|x− x′| = δ. Then

Wµp(·)(x,R) =

∫ R

δ

(µ(B(x, r))

rn−p(x)

)1/(p(x)−1)dr

r

≤

∫ R

δ

(µ(B(x′, 2r))

rn−p(x)

)1/(p(x)−1)dr

r

≤ C

∫ 2R

2δ

(µ(B(x′, r))

rn−p(x)

)1/(p(x)−1)dr

r.

We note that ∫ 2R

R

r(p(x)−n)/(p(x)−1)−1 dr ≤ C,

with a constant depending on R. Hence we obtain

Wµp(·)(x,R) ≤ C1

∫ R

δ

(µ(B(x′, r))

rn−p(x)

)1/(p(x)−1)dr

r+ C2, (5.1)

where C2 depends on R and µ(Ω). At this point we need to distinguishtwo cases. If p(x) ≥ p(x′), then

(µ(B(x′, r))

rn−p(x)

)1/(p(x)−1)1

r=

(µ(B(x′, r))

rn−1

)1/(p(x)−1)

≤ 1 +

(µ(B(x′, r))

rn−1

)1/(p(x′)−1)

= 1 +

(µ(B(x′, r))

rn−p(x′)

)1/(p(x′)−1)1

r,

so that∫ R

δ

(µ(B(x′, r))

rn−p(x)

)1/(p(x)−1)dr

r≤ R + Wµ

p(·)(x′, R) ≤ 1 +M.

On the other hand, if p(x) < p(x′), then

µ(B(x′, r))1/(p(x)−1) ≤ µ(Ω)1/(p(x)−1)

(µ(B(x′, r))

µ(Ω)

)1/(p(x′)−1)

≤(1 + (µ(Ω)p+−p−)1/(p−−1)2

))µ(B(x′, r))1/(p(x′)−1).

We note that the function f(t) = (t− n)/(t− 1) is L-Lipschitz on theinterval [p−, p+] with L = (n− 1)/(p− − 1)2, and obtain

r(p(x)−n)/(p(x)−1) = r(p(x′)−n)/(p(x′)−1)r(p(x)−n)/(p(x)−1)−(p(x′)−n)/(p(x′)−1)

≤ r(p(x′)−n)/(p(x′)−1)r−(n−1)C/((p−−1)2 log(1/δ))

≤ e(n−1)C/(p−−1)2r(p(x′)−n)/(p(x′)−1),

20 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

for δ ≤ r ≤ 1, where the constant C is from inequality (2.1), i.e.depends only on the log-Hölder constant of p(·).

Hence in this case one has∫ R

δ

(µ(B(x′, r))

rn−p(x)

)1/(p(x)−1)dr

r≤ C3W

µp(·)(x

′, R) ≤ C3M,

where C3 depends on µ(Ω)p+−p−. Therefore, in view of (5.1), we obtainthe required result.

Before proving the next theorem, we note the following comparisonprinciple which can be proved in the same way as [22, Lemma 3.18].We omit the proof.

Lemma 5.2. Let u1, u2 ∈W 1,p(·)(Ω). If∫

Ω

A(x,∇u1) · ∇ϕ dx ≤

∫

Ω

A(x,∇u2) · ∇ϕ dx

for all nonnegative ϕ ∈ C∞0 (Ω) and max(u1 − u2, 0) ∈W

1,p(·)0 (Ω), then

u1 ≤ u2 a.e. in Ω.

As an application of the pointwise estimates, Theorem 3.3 and Theo-rem 4.3, we prove the following version of the Hedberg–Wolff theorem.Note that the theorem is trivial for p−Ω > n since in this case functions

in W1,p(·)0 (Ω) are continuous by Sobolev embedding theorem. Thus

µ ∈ (W1,p(·)0 (Ω))∗ as well as (5.2) hold for every compactly supported

µ.

Theorem 5.3. Let Ω be bounded and µ a compactly supported Radon

measure in Ω. Then µ ∈ (W1,p(·)0 (Ω))∗ if and only if

∫

Ω

Wµp(·)(x,R) dµ(x) <∞ (5.2)

for some R > 0.

Proof. In this proof let

A(x, ξ) = |ξ|p(x)−2ξ

and Ap(·)-superharmonic functions will be called p(·)-superharmonic.Suppose first that (5.2) holds true. Choose 0 < R0 < 1 for which

(4.1) holds for all x0 ∈ supp µ and R0 < dist(∂Ω, supp µ). First,consider the case supp µ ⊂ B(x0, R0/5) for some x0 ∈ supp µ. Wemay assume R < R0/5. Let B = B(x0, R0/5). For j = 1, 2, . . ., let

Kj = x ∈ supp µ : Wµp(·)(x,R) ≤ j

and µj = µχKj. Note that Kj is a compact set since Wµ

p(·)(x,R) is

lower semicontinuous in x. By [31, Theorem 4.7], there exists a non-negative p(·)-superharmonic function uj on 5B = B(x0, R0) such that

WOLFF POTENTIAL ESTIMATES 21

min(uj, k) ∈W1,p(·)0 (5B) for all k > 0 and

− divA(x,Duj) = µj

on 5B. Further, we can see that uj is a continuous solution of

− divA(x,∇u) = 0

in 5B \ supp µj (cf. the arguments given in the proof of Theorem 5.1in [31], or Proposition 6.2 below), so that it is locally bounded there.In particular, uj(y) ≤Mj <∞ for y ∈ ∂B. For k > 0, we have

∫

5B

A(x,∇uj) · ∇min(uj, k) dx =

∫

5B

min(uj, k) dµj .

Hence ∫

5B

|∇min(uj, k)|p(x) dx ≤ kµ(Ω),

which in turn implies that∫

5B

|∇min(uj, k)|p− dx ≤ |5B| + kµ(Ω)

for all k > 0. Then, as in the proof of [25, Lemma 1.11], also found in[22, Lemma 7.43], we see that

∫

5B

usj dx ≤ C (5.3)

for some 0 < s < κ(p−0 − 1) with a constant C independent of jby the choice of R0. In view of (4.2) and (4.6) we can take s = γ.Thus M(uj , 5B) is uniformly bounded, and therefore the constant inCorollary 4.4 can be taken to be independent of j.

Let x ∈ B. By Corollary 4.4,

uj(x) ≤ C

(R0 + inf

y∈B(x,R0/4)uj(y) + W

µj

p(·)(x,R0/2)

), (5.4)

where the constant C can be chosen independent of j by (5.3). SinceB(x,R0/4) ∩ ∂B 6= ∅, we have that

infy∈B(x,R0/4)

uj(y) ≤Mj .

On the other hand, thanks to Proposition 5.1, one has

Wµj

p(·)(x,R) ≤ A1j + A2.

As in the proof of the Proposition 5.1, we obtain

Wµj

p(·)(x,R0/2) ≤ Wµj

p(·)(x,R) + C ′ (5.5)

for all x ∈ B with a constant C ′ depending on R, R0, and µ(Ω),but independent of j. Hence, uj is bounded on B, so that it is locally

bounded in 5B. Therefore, uj ∈W1,p(·)loc (5B), which in turn implies that

µj ∈ (W1,p(·)0 (5B))∗. Thus, we could choose uj so that uj ∈W

1,p(·)0 (5B).

22 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

Note that ujj is non-decreasing by the comparison principle andeach uj is a continuous solution of − divA(x,∇u) = 0 in 5B \ supp µ.Thus, in view of (5.3), the Harnack principle [14, Theorem 5.6] impliesthat the sequence uj is uniformly bounded on ∂B, and hence thesequence Mj is bounded. Let Mj ≤ M0 for all j. Using (5.4) and(5.5) we have∫

5B

|∇uj|p(x) dx =

∫

5B

uj dµj ≤

∫

B

uj dµ

≤ C

∫

B

(R0 + inf

y∈B(x,R0/4)uj(y) + W

µj

p(·)(x,R) + C ′)

dµ

≤ C

∫

B

(R0 +M0 + Wµ

p(·)(x,R) + C ′)

dµ

≤ C(1 +

∫

Ω

Wµp(·)(x,R) dµ

)<∞,

with a constant C independent of j. Let

I = supj

∫

5B

|∇uj|p(x) dx <∞.

Choose ψ ∈ C∞0 (Ω) such that ψ = 1 on B and ψ = 0 on Ω \ 4B. For

every ϕ ∈ C∞0 (Ω),∣∣∣∣∫

Ω

ϕ dµ

∣∣∣∣ = limj→∞

∣∣∣∣∫

5B

ϕψ dµj

∣∣∣∣

= limj→∞

∣∣∣∣∫

5B

A(x,∇uj) · ∇(ϕψ) dx

∣∣∣∣

≤ lim supj→∞

∫

5B

|∇uj|p(x)−1|∇(ϕψ)| dx

≤ lim supj→∞

(1 +

∫

5B

|∇uj|p(x) dx

)‖∇(ϕψ)‖p(·)

≤ (1 + I)(‖ψ‖∞‖∇ϕ‖p(·) + ‖|∇ψ|‖∞‖ϕ‖p(·)

).

This shows that µ ∈ (W1,p(·)0 (Ω))∗ in the case when supp µ ⊂ B.

In the general case, by using a partition of unity, we can expressµ =

∑i0i=1 µ

(i) with a finite number of nonnegative measures µ(i) suchthat supp µ(i) is contained in some ball B(xi, R0/5) for each i. Since

∫

Ω

Wµ(i)

p(·) (x,R) dµ(i) ≤

∫

Ω

Wµp(·)(x,R) dµ <∞,

the above result shows that µ(i) ∈ (W1,p(·)0 (Ω))∗ for each i, and hence

µ ∈ (W1,p(·)0 (Ω))∗. This proves the first part of the theorem.

Suppose now that µ ∈ (W1,p(·)0 (Ω))∗ is compactly supported positive

Radon measure in Ω. Then by Theorem 4.5 in [31] there exists a

WOLFF POTENTIAL ESTIMATES 23

unique nonnegative p(·)-superharmonic function u ∈ W1,p(·)0 (Ω) such

that − divA(x,∇u) = µ in Ω. Moreover, we have∫

Ω

u dµ =

∫

Ω

A(x,∇u) · ∇u dx ≤

∫

Ω

|∇u|p(x) dx <∞.

Observe now that the Wolff potential Wµp(·)(x,R) is lower semicontin-

uous, and thus µ-measurable. Let R < min(R0

2, 1

2dist(∂Ω, supp µ)),

where R0 is chosen to satisfy

p+B(x,R0) − 1 <

n

n− 1(p−B(x,R0) − 1)

for all x ∈ suppµ. Then we have by Theorem 3.3 that

Wµp(·)(x,R) ≤ C(u(x) + 2R)

for all x ∈ suppµ, where C is the constant from Theorem 3.3. Theclaim now follows by integrating.

6. Other applications

We finish by recording some easy corollaries of the pointwise esti-mates.

Theorem 6.1. Let u be an Ap(·)-superharmonic function in Ω and

µ = − divA(x,Du). Then u is finite and continuous at x0 ∈ Ω if and

only if for every ε > 0 there is a number R > 0 such that

Wµp(·)(x,R) < ε

for all x ∈ B(x0, R).

Proof. Suppose first that u(x0) < ∞ and that u is continuous at x0.Let R0 be chosen as in Section 3. Since u is locally bounded frombelow, we can assume that u ≥ 0 in B(x0, R0). If x ∈ B(x0, R0/3),then B(x, 2R0/3) ⊂ B(x0, R0). Thus by Theorem 3.3

CWµp(·)(x,R) ≤ u(x) − inf

y∈B(x,2R)u(y) + 2R (6.1)

for 0 < R < R0/3, with a constant C > 0 independent of x. Given ε >0, choose 0 < Rε < min(R0/3, Cε/6) such that |u(x) − u(x0)| < Cε/3for every x ∈ B(x0, 3Rε). Now if x ∈ B(x0, Rε), (6.1) gives

Wµp(·)(x,Rε) < 2ε/3 + 2Rε/C < ε,

since B(x, 2Rε) ⊂ B(x, 3Rε).For the converse, observe first that Theorem 4.3 implies that u(x0) <

∞. Fix ε > 0, let R0 be chosen as in Section 4, and pick 0 < Rε ≤R0/3 such that Wµ

p(·)(x, 2Rε) < ε for all x ∈ B(x0, 3Rε). Since u

is lower semicontinuous, we may also assume that u(x) ≥ u(x0) − εin B(x0, 3Rε). Let x ∈ B(x0, Rε). Since B(x, 2Rε) ⊂ B(x0, 3Rε) ⊂

24 TEEMU LUKKARI, FUMI-YUKI MAEDA, AND NIKO MAROLA

B(x0, R0), an application of Corollary 4.4 to the function u−u(x0)+ εon B(x, 2Rε) gives

0 ≤u(x) − u(x0) + ε

≤C

(Rε + inf

y∈B(x,Rε)u(y)− u(x0) + ε+ Wµ

p(·)(x, 2Rε)

)

<C(Rε + 2ε)

with a constant independent of x ∈ B(x0, Rε). This shows that u iscontinuous at x0.

Proposition 6.2. Let u be an Ap(·)-superharmonic function in Ω and

µ = − divA(x,Du). Then u is a solution of (1.1) in Ω \ supp µ.

Proof. By the previous proposition u is finite and continuous, hence,u is locally bounded from above in Ω \ supp µ. By [14, Corollary 6.6]

u ∈W1,p(·)loc (Ω \ supp µ). Moreover,

∫

Ω\supp µ

A(x,∇u) · ∇ϕ dx =

∫

Ω\supp µ

ϕ dµ = 0

for all ϕ ∈ C∞0 (Ω \ supp µ), and the claim follows.

We close this paper with the following Harnack inequality for Ap(·)-superharmonic functions.

Theorem 6.3. Let x0 ∈ Ω and let 0 < R0 < 1 be chosen as in Section

4. Let u be a nonnegative Ap(·)-superharmonic function in B(x0, 5R),where 0 < R ≤ R0/5, and µ = − divA(x,Du). If there exist ε > 0 and

H > 0 such that

µ(B(x, r)) ≤ Hrn−p(x)+ε

whenever x ∈ B(x0, R) and 0 < r < 4R, then

supx∈B(x0,R)

u(x) ≤ C1( infx∈B(x0,R)

u(x) +R) + C2Rε/(p(x0)−1),

where the constant C1 depends on M(u,B(x0, 5R)), and C2 on

M(u,B(x0, 5R)), H, and ε.

Proof. If x ∈ B(x0, R),

Wµp(·)(x, 4R) =

∫ 4R

0

(µ(B(x, r))

rn−p(x)

)1/(p(x)−1)dr

r

≤ H1/(p(x)−1)

∫ 4R

0

r(ε−p(x)+1)/(p(x)−1) dr

=H1/(p(x)−1)(p(x) − 1)

ε(4R)ε/(p(x)−1)

≤ C2Rε/(p(x0)−1),

WOLFF POTENTIAL ESTIMATES 25

where we used the log-Hölder continuity of p(·) in the last inequality.Hence by Corollary 4.4

u(x) ≤ C1( infy∈B(x,2R)

u(y) +R+ C2Rε/(p(x0)−1)),

with a constant C1 depending on M(u,B(x0, 5R)) since B(x0, 4R) ⊂

B(x0, 5R). Now the claim follows readily with C2 = C1C2, since

infy∈B(x,2R)

u(y) ≤ infx∈B(x0,R)

u(x).

Local Hölder continuity of Ap(·)-superharmonic functions follows fromTheorem 6.3 by a standard iteration argument, see, e.g., [22, Proof ofTheorem 6.6, pp. 111-112].

Corollary 6.4. Let u and µ satisfy the assumptions of Theorem 6.3.

Then u is Hölder continuous in B(x0, R/2) with an exponent depending

on ε, C1, and C2. Here C1 and C2 are the constants of Theorem 6.3.

Acknowledgement

Part of this paper was written while the third author was visitingPurdue University in 2007–2008. He wishes to thank the Departmentof Mathematics for the hospitality and several of its faculty for fruitfulconversations. The authors also wish to thank the referee for carefullyreading the manuscript and for very useful suggestions.

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(Teemu Lukkari) Department of Mathematics and Systems Analysis,

Helsinki University of Technology, P.O. Box 1100, FI-02015 TKK, Fin-

land

E-mail address : teemu.lukkari@tkk.fi

(Fumi-Yuki Maeda) 4-24 Furue-Higashi-Machi, Nishiku, Hiroshima 733-

0872, Japan

E-mail address : fymaeda@h6.dion.ne.jp

(Niko Marola) Department of Mathematics and Systems Analysis, Helsinki

University of Technology, P.O. Box 1100, FI-02015 TKK, Finland

E-mail address : niko.marola@tkk.fi