The keller-osserman condition for quasilinear elliptic equations in sobolev spaces with variable exponent

The Keller-Osserman Condition for Quasilinear Elliptic Equations in Sobolev Spaces with

Variable Exponent

Azeddine Baalal and Abdelbaset Qabil Department of Mathematics - Laboratory MACS Faculty of Sciences A¨in Chock, University of HASSAN II

B.P. 5366, Casablanca - Morocco

Copyright c2014 Azeddine Baalal and Abdelbaset Qabil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show by proving a comparison principle that Keller-Osserman property is valid and we discuss the existence of Evans functions for solutions to the quasilinear elliptic equations in Sobolev spaces with variable exponent.

Mathematics Subject Classification: 35J62, 49N60, 35B65, 35D30, 35K55 Keywords: Quasilinear elliptic equation, variable exponent, comparison prin- ciple, p(.)-superharmonic, p(.)-supersolution, p(.)-obstacle problem, the Keller- Osserman property

1 Introduction

In this paper, we study the quasilinear elliptic equation

−divA(x,∇u) +B(x, u) = 0, (1) where A : R^d×R^d → R^d and B : R^d ×R → R, d ≥ 2, are Carath´eodory functions satisfying the structure conditions given in assumptions (H0),(H1), (H2),(H3) and (H4) below.

In [8],[12] Keller and Osserman prove that a necessary and sufficient condition for the considered problem to have an entire solution is that B.

Here we provide a more elementary proof, which in particular we are inter- ested in the nonlinear potential theory associated with variable exponent for quasilinear elliptic equations (1), and we prove a comparison principal p(.)- supersolutions and p(.)-subsolutions, existence and uniqueness of the Dirichlet problem related to the sheaf H of continuous solutions of (1).

In this paper, we consider for every openU and every measurable functionα(.), where is the measurable function mentioned in the hypothesis (A3) below, the set H(U) of all functions u ∈ W_loc^1,p(.)(U) ∩ C(U) which are solutions of the equation (1). For α(.) < p(.) −1, the Harnack inequality is valid see [3].

We define, regular Evans functions u tending to the infinity (or exploding) at the regular boundary points of U. We assume that the exponent p, A and B satisfies the following supplementary differentiability and homogeneity conditions:

(A0) For every x∈R^d, the function p is differentiable and |∇p(x)| bounded.

(A1) For every x₀ ∈ R^d, the function F from R^d to R^d defined by F(x) = A(x, x−x₀) is differentiable and divF is locally (essentially) bounded.

(A2) A(x, λξ) = λ|λ|^p(x)−2A(x, ξ) for every λ∈R and everyx, ξ∈R^d. (A3) |B(x, ζ)| b(x)|ζ|^α(x), α(x)> p(x)−1 whereb ∈L

loc^d−d (^Êd), 0< <1, with ess inf

B b(x)>0 for every Bin R^d.

These conditions are satisfied in the particular case of the p(.)-Laplace opera- tor.

A typical example for the operatorAand Bare A(x,∇u) =|∇u|^p(x)−2∇uand B(x, u) =|u|^p(x)−2urespectively, for allx∈R^dthus Δ_p(x)u= div(|∇u|^p(x)−2∇u).

For the existence and uniqueness of solutionsu∈ W^1,p(x)(Ω) where 1< p(x)<

d for all x∈Ω, of the variational Dirichlet problem associated with the quasi- linear elliptic equation (1) see [2], these solutions are obtained by the p(.)- obstacle problem .

The contribution of this paper is to extend some of the results in [1] to the equation (1) under general assumptions below.

In the first section, we introduce some generalization and position of the prob- lem. In second section we give some basic facts about variable exponent spaces and a rough overview of properties of solutions of the prototype equality.

In section 3, we generalize, with detailed proofs, comparison principle to all

quasilinear elliptic equations (1) with growth conditions of a non-standard form. In fourth section, we then prove that for every α(.) > p(.)−1, the Keller-Osserman property in (R^d,H) is valid; i.e., every open ball admits a regular Evans function, and we present the concluding remarks.

2 Preliminaries

We start with a brief overview of the state of the art concerning Lebesgue spaces with variable exponent and Sobolev spaces modeled upon them. We define the Lebesgue space with variable exponent L^p(.)(Ω) as the set of all measurable functions p : Ω →]1,+∞[ called a variable exponent and we denotep⁻:= ess inf_x∈Ωp(x) andp⁺:= ess sup_x∈Ωp(x). For each open bounded subset Ω of R^d (d≥2), we denote

C+( ¯Ω) ={p∈ C( ¯Ω) :p(x)>1 for any x∈Ω¯}. We introduce also the convex modular _p(x)(u) =

Ω|u|^p(x)dx.

If the exponent is bounded, i.e., if p⁺ <∞, then the expression u _p(.)= inf{λ >0 :_p(.)(u

λ)≤1} defines a norm in L^p(.)(Ω), calledthe Luxemburg norm.

One central property of L^p(.)(Ω) is that the norm and the modular topologies coincide,i.e., _p(.)(u_n)→0 if and only if u_{n p(.)}→0.

We denote byL^p(.)(Ω)the conjugate spaceofL^p(.)(Ω) where _p(¹_x)+_p¹_(x) = 1

Proposition 1 (Generalized H¨older inequality [13]) For anyu∈L^p(.)(Ω) and v ∈L^p(.)(Ω), we have

|

Ω

uvdx| ≤ 1 p⁻ + 1

p⁻

u _p(.) v _p(.).

We define the variable exponent Sobolev space (see [9], [5],[7], [13]) by W^1,p(.)(Ω) ={u∈L^p(.)(Ω) : ∇u ∈L^p(.)(Ω)}.

with the norm

u _1,p(.) = u _p(.)+ ∇u _p(.) ∀u∈ W^1,p(.)(Ω). (2) The local Sobolev space W_loc^1,p(.)(Ω) consists of functions u that belong to W_loc^1,p(.)(U) for all open sets U compactly contained in Ω. The Sobolev space with zero boundary values, W0^1,p(.)(Ω), is defined as the completion of C₀^∞(Ω)

in the norm of W^1,p(.)(Ω) .

Let p^∗(x) bethe critical Sobolev exponent of p(x) defined by

p^∗(x) =

_dp(x)

d−p(x) for p(x)< d +∞ for p(x)≥d Proposition 2 ([6])

If q ∈ C+( ¯Ω) and q(x) < p^∗(x) for any x ∈ Ω, then W^1,p(x)(Ω) → L^q(x)(Ω) is compact and continuous.

We assume further on that, there exist positive constant C such that the function p satisfies logarithmic H¨older continuity condition if :

( )

∃C >0 :|p(x)−p(y)| ≤ _−log^C_|x−y| f or |x−y|< ¹₂, 1< p⁻ ≤p⁺ < d.

Proposition 3 (The p(.)-Poincar´e inequality)

Let Ω be a bounded open set and let p: Ω → [1,∞[ satisfy ( ) There exists a constant C, depending only on p(.) and Ω, such that the inequality

u _p(.)≤C ∇u _p(.) ∀u ∈ W0^1,p(.)(Ω).

Proposition 4 (Sobolev-Poincar´e inequality) There exists a constant C >0, such that

u _p∗(x) ≤C ∇u _p(x) ∀u∈ W0^1,p(x)(Ω).

Remark 2.1 By Proposition (3), we know that ∇u _p(.) and u _1,p(.) are equivalent norms on W0^1,p(.).

Proposition 5

Assuming p⁻ > 1, the spaces W^1,p(.)(Ω) and W0^1,p(.)(Ω) are separable and re- flexive Banach spaces.

Throughout the paper we suppose that the functions A : R^d×R^d → R^d is a Carath´eodory function satisfying the following assumptions:

(H0) For all ξ,ξ ∈R^d with ξ =ξ, [A(x, ξ)− A(x, ξ)]·(ξ−ξ)>0. (H1) |A(x, ξ)| ≤β[k(x) +|ξ|^p(x)−1];

(H2) A(x, ξ)ξ≥ν|ξ|^p(x);

for a.e. x ∈ Ω, all ξ ∈ R^d, where k is a positive bounded function lying in L^p(x)(Ω) andβ, ν >0.

In this paper we suppose that the functionB :R^d×R→Ris given Carath´eodory function and the following condition is satisfied:

(H3) ζ → B(x, ζ) is increasing and B(x,0) = 0 for every x∈R^d. (H4) |B(x, ζ)| ≤g(x) +|ζ|^δ(x);

for a.e. x ∈Ω, all ζ ∈ R^d, where g is a positive bounded function lying in L^p(x)(Ω) andp(x)≤δ(x) + 1< p^∗(x).

The following lemma plays a fundamental role in the proof of main result it corresponds to Lemma 3.2 and 3.3 of [2] adapted with our conditions.

Lemma 2.1 Assume that (H1) and (H2) are satisfied and let (u_n)_n be a sequence in W0^1,p(.)(Ω) and let u∈ W0^1,p(.)(Ω). If u_n u in W0^1,p(.)(Ω), then for some subsequence denoted (u_ϕn),

we have A(x,∇u_ϕn)→ A(x,∇u)in

L^p(.)(Ω)_d .

Lemma 2.2 The mappingx→ B(x, u)is continuous from the spaceW0^1,p(x)(Ω) to the space L^p(x)(Ω).

Definition 2.1 We say that a u ∈ W_loc^1,p(.)(Ω) is a p(.)-solution of (1) in Ω provided that for all ϕ∈ W0^1,p(.)(Ω) if ,

ΩA(x,∇u)· ∇ϕdx+

ΩB(x, u)ϕdx= 0. (3) Definition 2.2 A function u ∈ W_loc^1,p(.)(Ω) is termed p(.)-supersolutions of (1), if and only if, for all non-negative functions ϕ ∈ W₀^1,p(.)(Ω) we have,

ΩA(x,∇u)· ∇ϕdx+

ΩB(x, u)ϕdx≥0.

A function u is a p(.)-subsolution in Ω if −u is a p(.)-supersolution in Ω, and a solution in Ω if it is both a super- and a p(.)-subsolution in Ω.

The following Theorem plays a fundamental role in the proof of main result it corresponds to Theorem 3.2 of [2]

Theorem 2.1 Let Ωbe a bounded open set of R^d, ψ ∈ W^1,p(.)(Ω)∩L^∞(Ω).

Then there is a unique function u ∈ W^1,p(.)(Ω) with u−ψ ∈ W0^1,p(.)(Ω) such

that

ΩA(x,∇u)· ∇ϕdx+

ΩB(x, u)ϕdx= 0, whenever ϕ∈ W0^1,p(.)(Ω).

3 Comparison Principle and Dirichlet Prob- lem

In this section, we are concerned with the continuity of solutions of (1) at the boundary. By regularity theory, any bounded solution of (1) can be redefined in a set of measure zero so that it becomes continuous.

Definition 3.1 A relatively compact open set U is called p(.)−regular if, for each function f ∈ W^1,p(.)(U)∩ C(U), the continuous solution uof (1) in U with u−f ∈ W^1,p(.)(U) satisfies lim_x→yu(x) =f(y) for all y∈∂U.

A relatively compact open setU is called regular, if for every continuous func- tion f on ∂U, there exists a unique continuous solution u of (1) on U such that lim_x→yu(x) =f(y) for all y∈∂U.

If U is p(.)-regular and f ∈ W^1,p(.)(U)∩ C(U), then the solution u given by Theorem 2.1 satisfies

x∈U,x→zlim u(x) = f(z) for all z ∈∂U .

The followingcomparison principle is useful for the potential theory associated with equation (1):

Lemma 3.1 Suppose thatuis a p(.)-supersolution andvis a p(.)-subsolution on Ω such that

lim sup

x→y v(x)lim inf

x→y u(x) (4)

for all y ∈ ∂Ω and if both sides of the inequality are not simultaneously +∞ or −∞, then v u in Ω.

Proof 1 By virtue of (4), we may assume thatu is lower semicontinuous and v is upper semicontinuous on Ω.

For fixed ε >0, the set K_ε={x∈Ω :v(x)u(x) +ε} is a compact subset of Ω and therefore ϕ = (v−u−ε)⁺ ∈ W0^1,p(.)(R^d). Testing by ϕ, we obtain

{v>u+ε}[A(x,∇(u+ε))− A(x,∇v)]· ∇ϕdx+

{v>u+ε}[B(x, u+ε)− B(x, v)]ϕdx0

Using assumptions (H0) and (H3) we have

{v>u+ε}[A(x,∇u+ε)− A(x,∇v)]· ∇(v−u−ε)dx= 0

and again by (H0) we infer that v u+ε on Ω. Letting ε→0 we have v u on Ω.

Theorem 3.1 Every p(.)-regular set is regular in the sense of definition 3.1.

Proof 2 Let Ω be a p(.)-regular set in R^d and f be a continuous function on

∂Ω.

We shall prove that there exists a unique continuous solution u of 1 onΩsuch that lim_x→yu(x) =f(y) for all y∈∂Ω.

The uniqueness is given by Lemma 3.1 and we have the continuity of u see [3].

For the existence, we may suppose that f ∈ Cc(R^d) (Tietze’s extension theo- rem).

Letf_i be a sequence of functions from C_c¹(R^d)such that|f_i−f|2⁻ⁱ and|f_i|+

|f|M onΩfor the same constantM and for alli. Letu_i ∈ W^1,p(.)(Ω)∩C(Ω) be the unique solution for the Dirichlet problem with boundary dataf_i (Theorem 2.1). Then from Lemma 3.1 we deduce that |u_i−u_j|2⁻ⁱ+ 2^−j and|u_i|M on Ω for all i and j. We denote by u the limit of the sequence (u_i)_i.

We will show that u is a local solution of the equation. For this, we prove that the sequence (∇u_i)_i is locally uniformly bounded in

L^p(.)(Ω)_d .

Let ϕ = −η^p+u_i, η ∈ C_c^∞(B), 0 η 1 and η = 1 on B ⊂ B ⊂ Ω. Since ϕ ∈ W0^1,p(.)(Ω) then

∇ϕ=−(p⁺η^p+−1∇ηu_i+η^p+∇u_i).

Therefore and by (H2), (H1), and (H4) we have 0 =

ΩA(x,∇u_i)· ∇ϕdx+

ΩB(x, u_i)ϕdx

=

ΩA(x,∇u_i).(−η^p+∇u_i−p⁺u_iη^p+−1∇η)dx−

Ω

η^p+B(x, u_i)u_idx

= −

Ω

η^p+A(x,∇u_i).∇u_idx−p⁺

Ω

η^p+−1∇ηA(x,∇u_i).u_idx−

Ω

η^p+B(x, u_i)u_idx

−ν

Ω

η^p+|∇u_i|^p(x)dx+βp⁺

Ω

η^p+−1∇η(k(x) +|∇u_i|^p(x)−1)|u_i|dx

+

Ω

η^p+g(x)|u_i|dx+

Ω

η^p+|u_i|^δ(x)+1dx, Therefore

ν

Ωη^p+|∇ui|^p(x)dx≤βp⁺

=I1

Ωη^p+−1|∇η|ui| |k(x)dx+βp⁺

=I2

Ωη^p+−1|∇η| |∇ui|^p(x)−1)|ui|dx +

=I3

Ωη^p+g(x)|u_i|dx+

=I4

Ωη^p+|u_i|^δ(x)+1dx

Next we estimate the three integrals I₁, I₃ and I₄.

By H¨older’s inequality and p is Log-H¨older, for x₀ ∈B , we have (see[10]) ∇η _p(.) ≤CR^{d−p(xp(x0)0)}.

Then for the first estimation

I1 ≤C( ∇η _p(.), M, k _p(.))

For the two integrals I₃ and I₄, by H¨older’s inequality we have I₃+I₄ ≤C(M, g _p(.),|B|)

Using Young’s inequality, 0< ε≤1, we obtain the first estimate

I₂≤

B(1 ε)^p(x)−1

η^{p+− p}

+

p(x)−1|u_i||∇η| _p(x)

+ε

η ^p

+

p(x)|∇u_i|^p(x)−1 _p(x)

dx

≤(1 ε)^p+−1

B|ui|^p(x)|∇η|^p(x)dx+ε

Bη^p+|∇ui|^p(x)dx

If 0< ε < _βp^ν₊, then

B

|∇u_i|^p(x)dx νC(M, d,|B|, ∇η _∞, ε) ν−βp⁺ε

It follows that the sequence (u_i)_i is locally uniformly bounded in W^1,p(.)(Ω).

Fix B Ω, by lemma 2.1 and 2.2, we obtain 0 = lim

i→∞

ΩA(x,∇u_i)· ∇ψdx+

ΩB(x, u_i)ψdx

=

ΩA(x,∇u)· ∇ψdx+

ΩB(x, u)ψdx.

By an application of [11, Corollay 4.18] for each u_i we obtain

x∈limΩ,x→zu_i(x) = f_i(z)

for all z ∈∂Ω. From the following estimation, of u on all Ω, u_i−2⁻ⁱ uu_i+ 2⁻ⁱ for all i we deduce that for all i

f_i(z)−2⁻ⁱ lim inf

x→z_x∈Ω u(z)lim sup

x→zx∈Ω

u(z)f_i(z) + 2⁻ⁱ. Letting i→ ∞ we obtain

x→zlimu(x) = f(z) for all z ∈∂Ω which finishes the proof.

For every open set V and for every f ∈ C(∂V) we shall denote by HVf the solution of the Dirichlet problem for the equation (1) on V with the boundary data f.

4 Keller-Osserman Property

For every open set U we shall denote byU(U) the set of all relatively compact open regular subset V in U with V ⊂U.

By previous section and in order to obtain an axiomatic nonlinear potential theory, we shall investigate the harmonic sheaf associated with (1) and defined as follows: For every open subset U of R^d (d1), we set

H(U) =

u∈ C(U)∩ W_loc^1,p(.)(U) :uis a solution of (1)

=

u∈ C(U) :HVu=u for every V ∈ U(U) . Element in the set H(U) are calledharmonic onU.

Let H be the sheaf of continuous solutions related to the equation (1) and we consider ball B=B(x₀, R) with centerx₀ and radiusRand for everyu∈ H(B), Definition 4.1 A function u ∈ H⁺(B) is called p(.)-regular Evans function for H and B if lim

Bx→zu(x) = +∞ for every regular point z in the boundary of B.

For an investigation of regular Evans functions see [4].

Definition 4.2 We shall say that H satisfies the Keller-Osserman property, denoted (KO), if every ball admits a p(.)-regular Evans function for H.

Theorem 4.1 Assume that A and B satisfies the following supplementary conditions

(A0) For every x∈R^d, the function p is differentiable and |∇p(x)| bounded.

(A1) For every x₀ ∈ R^d, the function F from R^d to R^d defined by F(x) = A(x, x−x₀) is differentiable and divF is locally (essentially) bounded.

(A2) A(x, λξ) = λ|λ|^p(x)−2A(x, ξ) for every λ ∈R and every x, ξ ∈R^d. (A3) |B(x, ζ)| b(x)|ζ|^α(x), α(x)> p(x)−1 where b ∈L

loc^d−d (^Êd), 0< <1, with ess inf

B b(x)>0 for every B in R^d. Then the (KO) property is valid by H.

Proof 3 Let B be the ball with center x₀ ∈R^d and radius R.

Put f(x) = R²− x−x₀ ² andψ(x) = cf^−β(x), we obtain the desired property if we find a constant c >0 such that ψ is a p(.)-supersolution of the equation 1.

Then ∇f(x) = −2(x−x₀) and ∇ψ(x) = 2cβf(x)^−(β+1)(x−x₀) and by the condition (A2) we have

A(x,∇ψ(x)) = (2cβf(x)^−(β+1))^p(x)−1A(x, x−x₀).

We take g(x) = (2cβf(x)^−(β+1))^p(x)−1, we obtain

∇g(x) = g(x)

∇p(x) ln

2cβf(x)^−(β+1)

+ 2(p(x)−1)(β+ 1)f(x)⁻¹(x−x₀)

then

A(x,∇ψ(x)) =g(x)A(x, x−x₀).

Therefore

divA(x,∇ψ(x)) = A(x, x−x₀)∇g(x) +g(x) divA(x, x−x₀)

=

∇p(x) ln

2cβf(x)^−(β+1)

+ 2(p(x)−1)(β+ 1)f(x)⁻¹(x−x₀)

A(x, x−x₀) + divA(x, x−x₀)

g(x)

=

∇p(x) ln

2cβf(x)^−(β+1)

+ 2(p(x)−1)(β+ 1)f(x)⁻¹(x−x₀)

A(x, x−x₀) + divA(x, x−x₀)

2cβf(x)^−(β+1) _p(x)−1

≤

∇p(x)

2cβf(x)^−(β+1)

+ 2(p(x)−1)(β+ 1)f(x)⁻¹(x−x₀)

A(x, x−x₀) + divA(x, x−x₀)

2cβf(x)^−(β+1) _p(x)−1

It follows from (A0) and (A1) that A(x, x−x₀).(x−x₀) is locally bounded, there are two positive constants M1 and M2 such that the following inequality is

divA(x,∇ψ(x)) ≤

2cβf(x)^−(β+1)

+ 2(p(x)−1)(β+ 1)f(x)⁻¹

M₁+M₂

2cβf(x)^−(β+1) _p(x)−1

On the other hand and since of hypothesis (A3), we have B(x, ψ(x))b(x)(cf(x)^−β)^α(x) Let ϕ∈ C_c^∞(B), ϕ 0 and we set

I_ϕ =

A(x,∇ψ)∇ϕdx+

B(x, ψ)ϕdx

then

I_ϕ = − divA(x,∇ψ)dx− B(x, ψ)

ϕdx

− 2cβf(x)^−(β+1)+ 2(p(x)−1)(β+ 1)f(x)⁻¹

M₁+M₂

2cβf(x)^−(β+1)

_p(x)−1

−c^α(x)b(x)f(x)^−βα(x)

ϕdx

− 2cβf(x)^−(β+1)+ 2(p(x)−1)(β+ 1)f(x)⁻¹

M1+M2

c^p(x)−1(2β)^p(x)−1f(x)^{−(β+1)(p(x)−1)}−c^α(x)b(x)f(x)^−βα(x)

ϕdx

= − 2βf(x)^−(β+1)+ 2(p(x)−1)(β+ 1)f(x)⁻¹

M1+M2

c^{p(x)−α(x)−1}(2β)^p(x)−1f(x)^{−(β+1)(p(x)−1)}−b(x)f(x)^−βα(x)

c^α(x)ϕdx

= − 2βf(x)^−(β+1)+ 2(p(x)−1)(β+ 1)f(x)⁻¹

M1+M2

c^{p(x)−α(x)−1}(2β)^p(x)−1f(x)−b(x)f(x)^{(β+1)(p(x)−1)+1−βα(x)}

c^α(x)f(x)^{−(β+1)(p(x)−1)−1}ϕdx

Putting β = _α(x)^p_−p^(x₍⁾_x)+1, we obtain

I_ϕ − p(x)f(x)⁻α(x)−p(x)+1^α(x)+1

α(x)−p(x) + 1 +(p(x)−1)(α(x) + 1)f(x)⁻¹ α(x)−p(x) + 1

2M₁+M₂

cp(x)−α(x)−1( 2p(x)

α(x)−p(x) + 1)^p(x)−1f(x)−b(x)

c^α(x)f(x)⁻⁽(α(x)+1)(p(x)−1) α(x)−p(x)+1 +1)ϕdx

= p(x)f(x)⁻α(x)−p(x)+1^p(x)

α(x)−p(x) + 1 +(p(x)−1)(α(x) + 1) α(x)−p(x) + 1

2M₁+f(x)M₂

( 2p(x)

α(x)−p(x) + 1)^p(x)−1−cα(x)−p(x)+1b(x)

c^p(x)−1f(x)⁻⁽(α(x)+1)(p(x)−1) α(x)−p(x)+1 +1)ϕdx

cα(x)−p(x)+1

p(x)f(x)⁻α(x)−p(x)+1^p(x)

α(x)−p(x) + 1 +(p(x)−1)(α(x) + 1) α(x)−p(x) + 1

2M₁

b(x)+M₂f(x) b(x)

2p(x)

α(x)−p(x) + 1 _p(x)−1

then

c sup

x∈B

p(x)

α(x)−p(x) + 1+(p(x)−1)(α(x) + 1) α(x)−p(x) + 1

2M₁

b(x)+M₂R² b(x)

2p(x)

α(x)−p(x) + 1

_p(x)−1α(x)−p(x)+1¹

Then I_ϕ 0 holds for every ϕ ∈ C_c^∞(B) with ϕ0.

Thus the function ψ(x) = c(R²− x−x₀ ²)α(x)−p(x)+1^p(x) is a p(.)-supersolution satisfying lim

x→zψ(x) = +∞ for every z ∈ ∂B. By the comparison principle we have H Bn ψ for every n ∈ N and therefore, the increasing sequence (HBn)_n of harmonic functions is locally uniformly bounded on B. The Bauer convergence property implies that u = sup

n H_Bn ∈ H(B) see([1]), therefore we have lim inf

x→z u(x) n for every z in ∂B, thus lim

x→zu(x) = +∞ for every z in

∂B and uis a regular Evans function. Since B is an arbitrary ball, we get the desired property.

References

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Received: February 21, 2014