Continuity of solutions to the G-Laplace equation involving measures

Continuity of solutions to the G-Laplace equation involving measures

Yan Zhang

and Jun Zheng

1School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China Received 7 January 2019, appeared 9 June 2019

Communicated by Dimitri Mugnai

Abstract. We establish local continuity of solutions to theG-Laplace equation involving measures, i.e.,

−div

g(|∇u|)

|∇u| ∇u

=µ,

where µ is a nonnegative Radon measure satisfying µ(Br(x₀)) ≤ Cr^m for any ball Br(x0) ⊂⊂ _Ω with r ≤ 1 and m > n−1−δ ≥ 0. The function g is supposed to be nonnegative andC¹-continuous on[0,+_∞), satisfyingg(0) =0 and

δ≤ ^tg

0(t)

g(t) ≤g0,∀t>0

with positive constants δ and g₀, which generalizes the structural conditions of Ladyzhenskaya–Ural’tseva for an elliptic operator.

Keywords: G-Laplace, Radon measure, Orlicz space, Hölder continuity.

2010 Mathematics Subject Classification: 35J60, 35B65, 35J70, 35J75.

1 Introduction

Let Ω be an open bounded domain of Rⁿ(n ≥ 2), and µ a nonnegative Radon measure in Ωwith µ(B_r(x₀)) ≤ Cr^m for some constant C > 0 whenever B_r(x₀) ⊂⊂ Ω. We consider the equation

−_∆Gu =−div

g(|∇u|)

|∇u| ∇u

=µ inD⁰(_Ω), (1.1)

where gis a nonnegativeC¹-function on [0,+_∞), satisfyingg(0) =0 and the following struc- tural condition

0<δ ≤ ^tg

0(t)

g(t) ≤ g₀, ∀ t>0, δ,g₀ are positive constants. (1.2)

BCorresponding author. Email: zhengjun@swjtu.edu.cn

The structural condition of gwas introduced by Tolksdorf in 1983 [14], which is a natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations (see [10]). The conditions of g imply that the operator ∆G includes not only the p-Laplace operator∆p where g(t) = t^p−1 andδ = g₀ = p−1, but also the case of a variable exponent p= p(t)>0:

−_∆Gu= −div(|∇u|^{p(|∇u|)−2}∇u),

corresponding to setg(t) =t^p(t)−1, for which (1.2) holds if δ≤t(lnt)p⁰(t) +p(t)−1 ≤g₀for allt > 0. Another typical example of gis g(t) = t^plog(at+b)with p,a,b > 0 where in this caseδ= p andg0= p+1. More examples can be found in [2,3,6,17] etc.

Let G(t) = Rt

0g(s)ds. Under assumption (1.2), G is an increasing, C²-continuous and convex function, which is an N-function satisfying ∆2-condition (see [1]). Thus our class of operators will be considered in the setting of Orlicz spaces. We recall the definitions of Orlicz and Orlicz–Sobolev spaces together with their respective norms (see [1])

L^G(_Ω) ={u∈L¹(_Ω); Z

ΩG(|u(x)|)dx <+_∞}, kuk_LG(_Ω)=_inf

k>_0;

Z

ΩG

|u(x)|

k

dx≤₁

, W^1,G(_Ω) ={u∈L^G(_Ω);|∇u| ∈L^G(_Ω)},

kuk_W1,G(_Ω)= kuk_LG(_Ω)+k∇uk_LG(_Ω).

Under the assumption (1.2),W^1,G(_Ω)is a reflexive and separable Banach space (see [1]).

We shall call a solution of (1.1) any functionu∈W_loc^1,G(_Ω)that satisfies Z

Ω

g(|∇u|)

|∇u| ∇u· ∇ϕdx=

Z

Ωϕdµ ∀ϕ∈ D(_Ω). Ifµ≡0 in a domainD⊂_Ω, we say thatuisG-harmonic inD.

We now introduce regularities of related elliptic equations involving measures. In 1994, Kilpeläinen considered the situation of thep-Laplacian and proved that ifµsatisfiesµ(B_r)≤ Cr^{n−p+α(p−1)} for some positive constantsC andα∈ (0, 1], then any solution to the p-Laplace equation

−_∆pu=−div(|∇u|^p−2∇u) =µ (1.3) isC^0,β_loc-continuous for each β ∈ (0,α)(see [7]). This result was improved by Kilpeläinen and Zhong in 2002, showing that every solution of (1.3) is in fact Hölder continuous with the same exponent α as the one in the assumption µ(B_r) ≤ Cr^{n−p+α(p−1)} (see [8]). In 2010, the p-Laplace problem (1.3) was extended by Lyaghfouri to the case with variable exponents, i.e., considering

−div(|∇u|^p(x)−2∇u) =µ. (1.4) Under certain assumptions on the functionp(x)and the assumptionµ(B_r)≤Cr^{n−p(x)+α(p(x)−1)} for some positive constantsCandα∈ (_{0, 1}], the author proved that any bounded solution of (1.4) isC_loc^0,α-continuous with the same exponentα(see [11]).

When focusing on the problem governed by G-Laplacian, if µ(B_r(x₀)) ≤ Cr^m with m ∈ [n−_1,n), Challal and Lyaghfouri proved that any solution of (1.1) is C^0,α_loc-continuous with

α= ^m−₁ⁿ₊⁺_g^1+δ

0 (see [3]). Particularly, ifm=n−1, any bounded solution isC_loc^0,α-continuous with anyα ∈ 0,_g^δ

0

(see Theorem 3.3 in [3]). In 2011, these regularities were improved by Challal and Lyaghfouri in [5], showing that any local bounded solution of (1.1) is C_loc^0,α-continuous with any α ∈ 0,^m−n_g^+1+δ

0

provided m > n−1−δ. Note that under the assumption of non-decreasing monotonicity on ^g(_t^t), Zheng, Feng and Zhang obtained local C^1,α-continuity of solutions form> n and local Hölder continuity with a small exponent for somem< nin 2015 (see [15]).

In this paper, we continue the work of Challal, Lyaghfouri and Zheng et al. by improving the regularity of solutions of the equation (1.1). Particularly, we prove theC^0,α_loc-continuity of solutions with anyα∈(0, 1)ifm=n−1. More precisely, for anym>n−1−δ and without any monotonicity assumption on ^g(_t^t), we have the following results.

Theorem 1.1. Assume thatµsatisfies(1.1)with m>n−1−δ ≥0. For any local bounded solution u∈W_loc^1,G(_Ω)of (1.1), we have the following regularities:

(i) If m > n, then u ∈ C^1,α_loc(_Ω)with anyα ∈ (0, min{₁₊^σ_g

0,₂₍^m₁₊⁻_gⁿ

0)}), where σ is the same as in Lemma2.5.

(ii) If m∈[n−1,n), then u∈ C_loc^0,α(_Ω)with anyα∈(0, 1).

(ii) If n−1−δ<_m<_n−1, then u∈ C_loc^0,α(_Ω)_{with any}α∈ (_0,^m−n+1+δ

δ )_.

Remark 1.2. In [7], the author proved for thep-Laplacian problem thatu ∈C^0,α_loc(_Ω)with any α∈(0, 1)providedm=n−1. In this paper we not only improve the results of [3,5] and [15], but also extend the problem in [7] to general equations governed by a large class of degenerate and singular elliptic operators.

Throughout this paper, without special states, byB_R andB_rwe denote the balls contained in Ωwith the same center. Moreover, Br ⊂⊂B_R⊂⊂_Ωandkuk_L∞(BR)≤ M for some constant M >0. (u)_r = _|B¹

r|

R

Brudxbe the average value ofuon the ball Br.

2 Preliminary

In this section, we state some auxiliary results which will be used throughout this paper. We begin with some properties of the function G.

Lemma 2.1([13, Lemma 2.1, Remark 2.1]). The function G has the following properties:

(G₁) G is convex and C²-continuous.

(G₂) ₁^tg₊⁽_g^t)

0 ≤ G(t)≤ tg(t), ∀t≥0.

(G₃) min{s^δ+1,s^g0+1}₁^G₊^(t_g⁾

0 ≤G(st)≤(1+g₀)max{s^δ+1,s^g0+1}G(t), ∀s,t≥0.

(G₄) G(a+b)≤2^g0(1+g₀)(G(a) +G(b)), ∀a,b≥0.

For much more properties of G and problems governed by the operator ∆G, please see [2–6,13,15,16,18,19] etc.

Lemma 2.2([9, Lemma 2.7]). Letφ(s)be a non-negative and non-decreasing function. Suppose that φ(r)≤C₁r

R α

φ(R) +C₁R^β,

for all r≤ R≤ R0, with positive constantsα,βand C₁. Then, for anyτ <min{α,β}, there exists a constant C₂=C₂(C₁,α,β,τ)such that for any r ≤R≤R₀we have

φ(r)≤C₂r^τ.

The following lemmas are some properties ofG-harmonic functions.

Lemma 2.3([13, Theorem 2.3]). Assume u∈W_loc^1,G(_Ω). Let h be a weak solution of

∆Gh=0 in B_R, h−u ∈W₀^1,G(B_R), then

Z

B_R

(G(|∇u|)−G(|∇h|))dx≥C

Z

A₂G(|∇u− ∇h|)dx+

Z

A₁

g(|∇u|)

|∇u| |∇u− ∇h|²dx

, where A₁ = {x ∈ B_R;|∇u− ∇h| ≤ 2|∇u|}, A₂ = {x ∈ B_R;|∇u− ∇h| > 2|∇u|}, and C = C(δ,g₀)>0.

Lemma 2.4 ([13, Lemma 2.7]). Let h ∈ W^1,G(B_R) be a weak solution of ∆Gh = 0 in B_R. Then h∈C_loc^1,α(BR). Moreover, for everyλ∈(0,n), there exists C=C(λ,n,δ,g0)>0such that

Z

Br

G(|∇h|)dx≤Cr^λ, ∀r∈(0,R]. Proof. Indeed, we have (see [10, p. 345])

Z

Br

G(|∇h|)dx≤ C

r

R nZ

BR

G(|∇h|)dx

≤ C

r

R nZ

BR

G(|∇h|)dx+CRⁿ, ∀r∈(0,R]. Then for anyλ∈(0,n), we obtain by Lemma2.3

Z

Br

G(|∇h|)dx ≤Cr^λ, ∀r ∈(0,R], which completes the proof.

Lemma 2.5(Comparison withG-harmonic functions [15, Lemma 3.1]). Assume u∈W^1,G(BR). Let h ∈ W^1,G(B_R) be a weak solution of ∆Gh = 0 in B_R. Then there exist σ ∈ (0, 1) and C = C(n,δ,g₀)>0such that

Z

Br

G(|∇u−(∇u)_r|)dx≤C

r

R n+σZ

B_RG(|∇u−(∇u)_R|)dx+C Z

B_RG(|∇u−∇h|)dx, ∀r ∈(0,R]. Lemma 2.6. Assume u∈W_loc^1,G(_Ω). Let B_R⊂⊂_Ωand h∈W^1,G(B_R)be a weak solution of

∆Gh= 0 in B_R, h−u∈W₀^1,G(B_R).

Then for anyλ∈(0,n), there exists C =C(λ,n,δ,g₀,kuk_L∞(BR))>0such that Z

BR

G(|∇u− ∇h|)dx≤CR^m+CR^m+2λ.

Proof. Firstly, convexity ofGgives Z

BR

(G(|∇u|)−G(|∇h|))dx≤

Z

BR

g(|∇u|)

|∇u| ∇u(∇u− ∇h)dx

=

Z

BR

(u−h)dµ (2.1)

≤Cµ(BR)

≤CR^m, (2.2)

where we used the boundedness ofuwhich forces hto be bounded too.

Let A₁ and A₂ be defined as in Lemma 2.3. By Lemma 2.3, there exists a constant C = C(δ,g₀)>0 such that

Z

B_R

(G(|∇u|)−G(|∇h|))dx≥C Z

A₂G(|∇u− ∇h|)dx (2.3) and

Z

B_R

(G(|∇u|)−G(|∇h|))dx≥C Z

A₁

g(|∇u|)

|∇u| |∇u− ∇h|²dx. (2.4) By (G₂), ^G(_t^t) is increasing int>0. It follows from (G₂), (G₃), (2.2), (2.3), (2.4), Lemma2.3and 2.4 that

Z

A1

G(|∇u−∇h|)dx=

Z

A1

G(|∇u−∇h|)

|∇u−∇h| (|∇u−∇h|)dx

≤

Z

A₁

G(2|∇u|)

2|∇u| |∇u−∇h|dx

≤C Z

A₁

G(|∇u|)

|∇u| |∇u−∇h|dx

≤C

Z

A1

G(|∇u|)

|∇u|² |∇u−∇h|²dx ¹₂Z

A1

G(|∇u|)dx ¹₂

≤C

_Z

A1

g(|∇u|)|∇u|

|∇u|² |∇u−∇h|²dx ¹_2Z

A1

G(|∇u|)dx ¹₂

=C

_Z

A₁

g(|∇u|)

|∇u| |∇u−∇h|²dx ¹_2Z

B_RG(|∇u|)dx ¹₂

≤C

Z

BR

(G(|∇u|)−G(|∇h|))dx ¹₂Z

BR

G(|∇u|)dx ¹₂

=C

_Z

BR

(G(|∇u|)−G(|∇h|))dx ¹₂

·

_Z

BR

(G(|∇u|)−G(|∇h|) +G(|∇h|))dx ¹₂

≤C Z

BR

(G(|∇u|)−G(|∇h|))dx +C

_Z

BR

(G(|∇u|)−G(|∇h|))dx ¹_2Z

BR

G(|∇h|)dx ¹₂

,

≤CR^m+CR^m+2λ, (2.5)

where in the last inequality but one we used (a+b)^γ ≤ a^γ+b^γ for any a ≥ 0,b ≥ 0 and γ∈(0, 1). By (2.2), (2.3) and (2.5), we have

Z

BR

G(|∇u−∇h|)dx=

Z

A2

G(|∇u−∇h|)dx+

Z

A1

G(|∇u−∇h|)dx

≤ C Z

BR

(G(|∇u|)−G(|∇h|))dx+CR^m+CR^m+2λ

≤ CR^m+CR^m+2λ.

3 Proof of Theorem 1.1

Proof of Theorem1.1. Lethbe aG-harmonic function inB_Rthat agrees withuon the boundary, i.e.,

div g(|∇h|)

|∇h| ∇h=0 in B_R and h−u∈W₀^1,G(B_R). By Lemma2.5and Lemma 2.6, for anyr ≤Rthere holds

Z

Br

G(|∇u−(∇u)_r|)dx≤ C

r

R n+σZ

BR

G(|∇u−(∇u)_R|)dx+C Z

BR

G(|∇u− ∇h|)dx

≤ C

r

R n+σZ

BR

G(|∇u−(∇u)_R|)dx+CR^m+CR^m+2λ, whereλis an arbitrary constant in(0,n).

(i) Ifm>n, then we have Z

Br

G(|∇u−(∇u)_r|)dx ≤C

r

R n+σZ

BR

G(|∇u−(∇u)_R|)dx+CR^m+2λ.

Sincem> nandλis an arbitrary constant in(0,n), one may chooseλsatisfying ^m+₂^λ > n. In view of Lemma2.2, we conclude that for anyτ<min{σ,^m+₂^λ−n}there holds

Z

Br

G(|∇u−(∇u)_r|)dx≤Cr^n+τ, ∀r≤ R. (3.1) Now we claim that

Z

Br

|∇u−(∇u)_r|dx≤ Cr^n+1+τg0, ∀r ≤R. (3.2) Indeed, forr satisfyingr⁻ⁿR

Br|∇u−(∇u)_r|dx ≤ r^1+τg0, (3.2) holds with C = 1. Now for rsatisfyingr⁻ⁿR

Br |∇u−(∇u)_r|dx >r^1+τg0, we infer from the increasing monotonicity of ^G(_t^t) int>_0,

G r⁻ⁿR

Br|∇u−(∇u)_r|dx r⁻ⁿR

Br|∇u−(∇u)_r|dx ≥ ^{G r}

τ 1+g0

r^1+τg0 .

It follows from (G₂) and (G₃) Z

Br

|∇u−(∇u)_r|dx≤ ^r

n+₁₊^τ_g

0

G r^1+τg0G

r⁻ⁿ Z

Br

|∇u−(∇u)_r|dx

≤ ^Cr

n+₁₊^τ_g

0

r^τG(1) ^G

r⁻ⁿ Z

Br

|∇u−(∇u)_r|dx

≤ ^Cr

n+₁₊^τ_g

0

r^τg(₁) ^G

r⁻ⁿ Z

Br

|∇u−(∇u)_r|dx

. (3.3)

Note that convexity ofGand (3.1) imply that G

1

|B_r|

Z

Br

|∇u−(∇u)_r|dx

≤ ¹

|B_r|

Z

Br

G(|∇u−(∇u)_r|)dx≤Cr^τ. (3.4) By (G3), (3.3) and (3.4), one may get

Z

Br

|∇u−(∇u)_r|dx ≤Cr^n+1+τg0,

where Cdepends only ong(₁)_,g₀ and the volume of the unit ball. Now we have proven that (3.2) holds for anyr ≤ R. Thusu ∈ C^1,

τ 1+g0

loc (_Ω)by Campanato’s embedding theorem. Due to the arbitrariness ofλ∈(0,n),τ>0 can be arbitrary withτ<min{σ,^m−₂ⁿ}, which guarantees that Theorem1.1(i) holds true.

(ii) Ifm∈[n−1,n], we only prove form= n−1 due to the fact thatµ(B_r)≤ Cr^m ≤Crⁿ⁻¹ with smallr. By (G₄), Lemma2.4and Lemma2.6, we get

Z

Br

G(|∇u|)dx≤ C Z

Br

G(|∇u− ∇h|)dx+C Z

Br

G(|∇h|)dx

≤ Cr^m+Cr^m+2λ +Cr^λ

≤ Cr^m,

where in the last inequality we letn> λ> n−₁=m.

We claim that for anyr≤ R<1 withB_R ⊂⊂_Ωand some positive constantCindependent ofr, there holds

Z

Br

|∇u|dx ≤Cr^n−1+α0, (3.5)

with someα₀ ∈(0, 1).

Indeed, forr≤ Rsatisfying

r^−n+1−α0 Z

Br

|∇u|dx≤1, (3.6)

(3.5) holds withC =1. Forr≤ Rsatisfying r^−n+1−α0

Z

Br

|∇u|dx≥1,

due to the increasing monotonicity of F(t) =G(t)−G(1)t int≥1, it follows G

r^−n+1−α0 Z

Br

|∇u|dx

≥ G(1)·r^−n+1−α0 Z

Br

|∇u|dx.

Then we have

Z

Br

|∇u|dx≤Cr^n−1+α0(r^1−α0)^1+δG

r⁻ⁿ Z

Br

|∇u|dx

≤Cr^n−1+α0·(r^1−α0)^{1+δ 1}

|B_r|

Z

Br

G(|∇u|)dx

≤Cr^{n−1+α0+(1−α0)(1+δ)}·r⁻ⁿ·r^m

=Cr^{n−1+α0+(1−α0)(1+δ)+m−n}. (3.7) Combining (3.6) and (3.7), we may chooseα₀ =α₀+ (1−α₀)(1+δ) +m−n, i.e.,α₀ =1−ⁿ₁⁻₊^m

δ

such that (3.5) holds for allr ≤R.

Form=n−1, we conclude thatu∈ C_loc^0,α0(_Ω)by Morrey Theorem (see page 30, [12]) with α₀= ₁₊^δ

δ.

Note that infBru ≤ _infBrh (see the proof of Theorem 3.3 in [3]). Then by (2.1) and Lemma2.4, forλlarger thanm+α₀, we have

Z

Br

G(|∇u|)dx≤

Z

Br

(u−h)dµ+

Z

Br

G(|∇h|)dx

≤(sup

Br

u−inf

Br

h)µ(B_r) +

Z

Br

G(|∇h|)dx

≤(sup

Br

u−inf

Br

u)µ(Br) +

Z

Br

G(|∇h|)dx

≤Cosc(u,B_r)r^m+Cr^λ

≤Cr^α0+m+Cr^λ

≤Cr^m+α0, where osc(u,B_r) =sup_B

ru−inf_Bru. Arguing as (3.5), we getu∈ C_loc^0,α1(_Ω)with α₁ =1− ⁿ−(m+α₀)

1+δ = ^δ

1+δ + ^α0 1+δ. Repeating this process, we getu∈C^0,α_loc^k(_Ω)with

α_k = ^δ

1+δ + ^αk−1 1+δ. Finally, we haveα_k = ^α0

(1+δ)^k +δ∑^kj=1 1

(1+δ)^j, which leads to lim_k→_∞α_k=1, and the result follows.

(iii) If n−1−δ < m < n−1, checking the proof and repeating the process as above, we may get α₀ = 1− ⁿ₁⁻₊^m

δ, α₁ = ^1+δ₁⁺₊^m−n

δ + ₁^α₊⁰

δ, . . . , α_k = ^1+δ₁⁺₊^m−n

δ + ^α₁^k₊⁻¹

δ. Finally, one has u∈C^0,α_loc(_Ω)for anyα∈(0,^1+δ+_δ^m−n).

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