The obstacle problem for nonlinear noncoercive elliptic equations with \(L^{1}\)-data

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R E S E A R C H

Open Access

The obstacle problem for nonlinear

noncoercive elliptic equations with

L

1

-data

Jun Zheng

1*

_{and Leandro S. Tavares}

2

*_{Correspondence:}

[email protected] 1_{School of Mathematics, Southwest}

Jiaotong University, Chengdu, China Full list of author information is available at the end of the article

Abstract

In this paper, we study the obstacle problem governed by nonlinear noncoercive elliptic equations withL1_{-data. We prove the existence of an entropy solution and}

show its continuous dependence on theL1_{-data in}_W1,q_{(Ω) with}_q_{> 1.}

Keywords: Obstacle problem; Noncoercive elliptic equation;L1_{-data; Entropy}

solution

1 Introduction

LetΩ⊂RN_(N_≥_{2) be a bounded domain,}_p_{> 1,}_θ_≥_0,_f_∈_L1₍_Ω_{), and}_g,_ψ_∈_W1,p₍_Ω₎_∩ L∞(Ω). We consider the obstacle problem governed by the noncoercive operator

Au=div a(x,∇u)

(1 +b(x)|u|)θ(p–1) (1)

associated with (f,ψ,g), wherea:Ω×RN →RN is a Carathéodory function satisfying:

a(x,ξ)·ξ≥α|ξ|p, (2)

a(x,ξ)≤βj(x) +|ξ|p–1, (3)

a(x,ζ) –a(x,η)(ζ–η) > 0, (4)

a(x,ξ) –a(x,η)≤γ

⎧ ⎨ ⎩

|ξ–η|p–1 if 1 <p< 2,

(1 +|ξ|+|η|)p–2_|_ξ_–_η_| _if_p_≥_2, (5) for almost everyx∈Ω and everyξ,ζ,η∈RN withζ =η, whereα,β,γ are positive con-stants,jis a nonnegative function inLpp–1₍_Ω_{), and}_b _{is an}_L∞_{-function satisfying, with}

someB≥0,

0≤b(x)≤B (6)

for almost everyx∈Ω.

If f ∈W–1,p₍_Ω_{), then the obstacle problem associated with (f}_,_ψ_,_{g) is formulated in} terms of the inequality

Ω

a(x,∇u)

(1 +b(x)|u|)θ(p–1)· ∇(v–u)dx+

Ω

(2)

wheneverKg,ψ={v∈W1,p(Ω);v–g∈W01,p(Ω),v≥ψa.e. inΩ} =∅. However, the second integration in (7) is not well deﬁned forf ∈L1₍_Ω_{). Following [}₁_,₃_,₅_{], and so on, we are} led to a more general deﬁnition of a solution to the obstacle problem with dataf∈L1₍_Ω_), using the truncation function

Ts(r) =max –s,min{s,r}

, s,r∈R.

Deﬁnition 1 An entropy solution to the obstacle problem associated with (f,ψ,g) is a measurable functionusuch thatu≥ψ a.e. inΩ,Ts(u) –Ts(g)∈W01,p(Ω) for everys> 0, and

Ω

a(x,∇u)

(1 +b(x)|u|)θ(p–1)· ∇Ts(v–u)dx+

Ω

fTs(v–u)dx≥0 ∀v∈Kg,ψ∩L∞(Ω). (8)

Observe that in the deﬁnition a global integrability condition is required neither onu nor on its gradient. As pointed out in [8], ifTs(u)∈W1,p(Ω) for alls> 0, then there exists a unique measurable vector ﬁeld U:Ω→RN such that ∇(Ts(u)) =χ{|u|<s}U a.e. inΩ, s> 0, which, in fact, coincides with the standard distributional gradient of∇uwhenever u∈W1,1₍_Ω_).

The motivation of this paper comes from the study on the Dirichlet boundary value problem

⎧ ⎨ ⎩

div |∇u|p–2∇u

(1+|u|)θ(p–1)=f inΩ,

u= 0 on∂Ω. (9)

Indeed, for the p-Laplacian equation, that is,θ = 0 in (9), the existence and regularity of solutions whenf has a ﬁne regularity have been well studied. However, under weaker summability assumptions onf, for example,f ∈L1₍_Ω_{), the gradient of}_u_{(and even}_u_{itself )} may not be inL1₍_Ω_{). In this case, it is possible to give a meaning to solutions of problem (}₉₎ by using the concept of entropy solutions. The works on the theory of entropy solutions forp-Laplacian equations have been applied to unilateral problems in [5,7,17], and so on and have been extended in [8,20] to the obstacle problems withL1_{-data in Sobolev} spaces with variable exponents and Orlicz–Sobolev spaces, respectively. We remark that the classical obstacle problem for elliptic operators with nonlinear variational energies was considered in [12] and linear elliptic systems involving Radon measures were considered in [19]. Parabolic problems with irregular obstacles and nonstandardp(x,t)-growth were considered in [10] and references therein.

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and references therein). Particularly, a whole range of existence results have been proven in [6] forp= 2 andf regular enough, showing that solutions are in some Sobolev space W₀1,q(Ω) (1 <q≤2) (see also [13,14,18]). Nevertheless, little literature has considered the obstacle problem for noncoercive elliptic equations, particularly, for noncoercive elliptic equations withL1-data.

Motivated by this, we study the obstacle problem governed by (1) and (f,ψ,g) withL1 -data. The main ideas in this paper originate from [1,8], which can be also applied to the study on a large class of elliptic/parabolic equations [9,15,21], potential theory [22], and Schrödinger equations [11,16]. Throughout this paper, without special statements, we always assume that

2 – 1

N <p<N, 0≤θ(p– 1) <

N(p– 1)

N– 1 – 1, (10)

f∈L1₍_Ω_),_ψ_,_g_∈_W1,p₍_Ω₎_∩_L∞₍_Ω_{) satisfy (}_ψ_–_g)+_∈_W1,p

0 (Ω), andKg,ψ=∅. Note that (10) implies that

0≤θ< N N– 1–

1

p– 1, and

N(p– 1)(1 –θ) N– 1 –θ(p– 1)> 1. The main result in this paper is the following:

Theorem 1 There exists at least one entropy solution u to the obstacle problem associated with(f,ψ,g).In addition,u depends continuously on f,that is,if fn→f in L1(Ω)and un is a solution to the obstacle problem associated with(fn,ψ,g),then

un→u in W1,q(Ω)for all q∈

1, N(p– 1)(1 –θ) N– 1 –θ(p– 1)

.

Notations

up=uLp_(Ω)is the norm ofLp(_Ω), where1≤p≤ ∞.

u1,p=uW1,p_(Ω)is the norm ofW1,p(Ω), where1 <p<∞.

p=_pp_–1 with1 <p<∞.

u+₌_max_{_{u, 0}_}_,_u–_{= (–u)}+_{for a real-valued function}_u_. Cis a constant, which may be diﬀerent from line to line.

{u>s}={x∈Ω;u(x) >s}.

{u≤s}=Ω\ {u>s}.

{u<s}={x∈Ω;u(x) <s}.

{u≥s}=Ω\ {u<s}.

{u=s}={x∈Ω;u(x) =s}.

{t≤u<s}={u≥t} ∩ {u<s}.

LN _{is the Lebesgue measure in}_RN_.

|E|=LN_(E)_{for a measurable set}_E_.

2 Preliminaries on entropy solutions

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obtainW1,p_{-solutions due to assumptions (}₂_)–(₆_{), which, particularly, are also entropy} so-lutions. In this section, we establish several auxiliary results on convergence of sequences of entropy solutions asfn→f inL1(Ω). The main techniques used in this section come from [1,8]. We start with a priori estimate.

Lemma 2 Let v0∈Kg,ψ∩L∞(Ω),and let u be an entropy solution to the obstacle problem associated with(f,ψ,g).Then,we have

{|u|<t}

|∇u|p

(1 +b(x)|u|)θ(p–1)dx≤C

jp_p+∇v0pp+f1

t+v0∞ ∀t> 0,

where C is a positive constant depending only onα,β,and p. Proof Fort> 0, takingv0as a test function in (8), we compute

{|v0–u|<t}

a(x,∇u)· ∇u (1 +b(x)|u|)θ(p–1)dx≤

{|v0–u|<t}

a(x,∇u)· ∇v0 (1 +b(x)|u|)θ(p–1)dx

+

Ω

fTt(v0–u)

dx.

It follows from (2), (3), and Young’s inequality withε> 0 that

{|v0–u|<t}

α|∇u|p

(1 +b(x)|u|)θ(p–1)dx≤

{|v0–u|<t}

β(|j|+|∇u|p–1₎_{· |∇}_v0_|

(1 +b(x)|u|)θ(p–1) dx+tf1

≤

{|v0–u|<t}

βε(|j|p₊_|∇_u_|p₎ (1 +b(x)|u|)θ(p–1)dx

+

{|v0–u|<t}

βC(ε)|∇v0|p

(1 +b(x)|u|)θ(p–1)dx+tf1

≤ε

{|v0–u|<t}

|∇u|p

(1 +b(x)|u|)θ(p–1)dx

+Cj_pp+∇v0pp

+tf1.

Thus we have

{|v0–u|<t}

|∇u|p

(1 +b(x)|u|)θ(p–1)dx≤C

jp_p+∇v0pp+tf1

. (11)

Replacingtwitht+v0∞ in (11) and noting that{|u|<t} ⊂ {|v0–u|<t+v0∞}, we

obtain the desired result.

In the rest of this section, let{un}be a sequence of entropy solutions to the obstacle problem associated with (fn,ψ,g) and assume that

fn→f inL1(Ω) and fn1≤ f1+ 1.

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Proof Lets,t,ε> 0. We can verify that, for allm,n≥1, LN _|_u

n–um|>s

≤LN _|_u n|>t

+LN _|_u m|>t

+LN Tt(un) –Tt(um)>s

(12)

and

LN _|_u n|>t

= 1 tp

{|un|>t}

tpdx≤ 1 tp

Ω

Tt(un)pdx. (13)

Sincev0=g+ (ψ–g)+∈Kg,ψ∩L∞(Ω), by Lemma2we have

Ω

_∇Tt(un) p

dx=

{|un|<t}

|∇un|pdx

≤C(1 +Bt)θ(p–1)_jp

p+∇v0 p p+f1

t+v0∞

. (14)

Note thatTt(un) –Tt(g)∈W01,p(Ω). By (13), (14), and Poincaré’s inequality, for everyt>

g∞and for some positive constantCindependent ofnandt, we have

LN _|_u n|>t

≤ 1

tp

Ω

Tt(un) p

dx

≤2p–1

tp

Ω

Tt(un) –Tt(g)pdx+ 2p–1

tp g p p

≤C

tp

Ω

_∇Tt(un) –∇Tt(g) p

dx+2 p–1 tp g

p p

≤C

tp

Ω

_∇Tt(un) p

dx+C tpg

p 1,p

≤C(1 +t)1+θ(p–1)

tp .

Since 0≤θ< 1, there existstε> 0 such that LN _|_u

n|>t

<ε

3 ∀t≥tε,∀n≥1. (15)

Now, as in (13), we have

LN _T

tε(un) –Ttε(um)>s

= 1 sp

{|Ttε(un)–Ttε(um)|>s} spdx

≤ 1

sp

Ω

Ttε(un) –Ttε(um)

p

dx. (16)

Using (14) and the fact thatTt(un)–Tt(g)∈W01,p(Ω)∩L∞(Ω) again, we see that{Ttε(un)}is

a bounded sequence inW1,p₍_Ω_{). Thus, up to a subsequence,}_{_T

tε(un)}converges strongly inLp(Ω). By (16) there existsn0=n0(tε,s)≥1 such that

LN _T

tε(un) –Ttε(um)>s

<ε

(6)

Combining (12), (15), and (17), we obtain

LN _|_u

n–um|>s

<ε, ∀n,m≥n0.

Hence {un} is a Cauchy sequence in measure, and therefore there exists a measurable function usuch thatun→uin measure. Note thatTk(un) –Tk(g)∈W01,p(Ω). By (14) and Poincaré’s inequality we conclude that, for ﬁxedk,{Tk(un)}is a bounded sequence in W1,p₍_Ω_{). Therefore,} _T

k(un)→Tk(u) strongly in Lp(Ω), and, up to a subsequence,

Tk(un)→Tk(u) a.e. inΩ.

Proposition 4 There exist a subsequence of{un}and a measurable function u such that, for each q∈(1,N_N(_–1–θ(p–1)(1–θ)_p_–1)),we have

un→u strongly in W1,q(Ω).

Furthermore,if0≤θ<min{_N_–1_p₊₁,_NN_–1–_p1_–1},then a(x,∇un)

(1 +b(x)|un|)θ(p–1)

→ a(x,∇u)

(1 +b(x)|u|)θ(p–1) strongly in

L1(Ω)N.

To prove Proposition4, we need two preliminary lemmas.

Lemma 5 There exists a subsequence of{un}and a measurable function u such that for each q∈(1,N_N(_–1–θ(p–1)(1–θ)_p_–1)),we have that unu weakly in W1,q(Ω)and un→u strongly in Lq₍_Ω_).

Proof Let k> 0 and n≥1. DeﬁneDk={|un| ≤k} andBk ={k≤ |un|<k+ 1}. Using Lemma2withv0=g+ (ψ–g)+_{, we get}

Dk

|∇un|p (1 +b(x)|un|)θ(p–1)

dx≤C(1 +k), (18)

whereCis a positive constant depending only onα,β,p,jp,f1,∇v0p, andv0∞.

Taking the functionTk(un) withk>{g∞,ψ∞}as a test function for the problem associated with (fn,ψ,g), we obtain

Ω

a(x,∇un)· ∇T1(un–Tk(un)) (1 +b(x)|un|)θ(p–1)

dx≤

Ω –fnT1

un–Tk(un)

dx,

which, together with (2), gives

Bk

α|∇un|p (1 +b(x)|un|)θ(p–1)

dx≤ fn1≤ f1+ 1. (19)

Let q ∈(1,N_N(_–1–θ(p–1)(1–θ)_p_–1)) and r= qθ(_pp–1). Noting that N_N(_–1–θ(p–1)(1–θ)_p_–1) <p, it follows q<p. Let

Θ =θ(p– 1). Since B–1 A–1 ≤

B

A for all A≥B> 1, we have p–Θ–1 N–Θ–1 ≤

p–Θ

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qΘ p–q<

Nq

N–q:=q∗. For allk> 0, we estimate

Bk|∇un|

q_dx:

Bk

|∇un|qdx=

Bk

|∇un|q (1 +b(x)|un|)r·

1 +b(x)|un|

r dx

≤

Bk

|∇un|p (1 +b(x)|un|)θ(p–1)

dx

q

p

Bk

1 +b(x)|un|

pr p–q_dx

p–q p

≤C|Bk|

p–q

p ₊_C

Bk

|un|

pr p–q_dx

p–q p

by (19) and (6)

≤C|Bk|

p–q

p ₊_C

Bk

|un|q ∗

dx

r

q∗

· |Bk|

p–q p –qr∗_.

Since|Bk| ≤_k1q∗

Bk|un|

q∗_{dx, for}_k_≥_k0_≥_{1, we have}

Bk

|∇un|qdx≤C

1 kq∗

Bk

|un|q ∗

dx

p–q p

+C 1 kq∗(

p–q p –qr∗)

Bk

|un|q ∗

dx

p–q p

≤ 2C

kq∗(

p–q p –qr∗)

Bk

|un|q ∗

dx

p–q p

.

Summing up fromk=k0tok=Kand using Hölder’s inequality, we have

K

k=k0

Bk

|∇un|qdx≤C

_K

k=k0

1

kq∗(

p–q p –qr∗)

p q q p · _K

k=k0

Bk

|un|q ∗

dx

p–q p

. (20)

Note that

{|un|≤K}

|∇un|qdx=

Dk₀

|∇un|qdx+ K

k=k0

Bk

|∇un|qdx. (21)

To estimate the ﬁrst integral in the right-hand side of (21), using Hölder’s inequality, (18), and (6), we obtain

Dk₀

|∇un|qdx≤

Dk₀

|∇un|p (1 +b(x)|un|)θ(p–1)

dx

q

p

Dk₀

1 +b(x)|un|

pr p–q_dx

p–q p

≤C, (22)

whereCdepends only onα,β,B,p,θ,jp,f1,∇v0p,v0∞, andk0. Note thatK_k₌_k

0

1 kq

∗(p–_pq– r

q∗) p

q converges sinceq

∗₍p–q p –

r q∗)

p

q > 1. Combining (20)–(22), we get, fork0large enough,

{|un|≤K}

|∇un|qdx≤C+C

{|un|≤K}

|un|q ∗

dx

p–q p

. (23)

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obtain

TK(un)q_q∗≤2q–1TK(un) –gq_q∗+gqq∗

≤C∇TK(un) –g q q+g

q q∗

≤C∇TK(un) q q+∇g

q q+g

q q∗

≤C

1 +

{|un|≤K}

|∇un|qdx

. (24)

Using the fact that

{|un|≤K}

|un|q ∗

dx≤

{|un|≤K}

TK(un) q∗

dx≤TK(un) q∗

q∗, (25)

from (23)–(24) we obtain

{|un|≤K}

|∇un|qdx≤C+C

1 +

{|un|≤K}

|∇un|qdx

q∗ q

p–q p

. (26)

Note that p < N ⇔ q_q∗p–_pq < 1. It follows from (26) that, for k0 large enough,

{|un|≤K}|∇un|

q_dx _{is bounded independently of} _n _and_K_{. Using (}₂₄_{) and (}₂₅_{), we}

de-duce that_{|_u

n|≤K}|un|

q∗_dx_{is also bounded independently of}_n_and_{K. Letting}_K_{→ ∞}_,

we deduce that∇unq andunq∗ are uniformly bounded independently ofn. Partic-ularly, un is bounded inW1,q(Ω). Therefore, there exist a subsequence of {un} and a functionv∈W1,q₍_Ω_{) such that}_u

nvweakly inW1,q(Ω) andun→vstrongly inLq(Ω) and a.e. inΩ. By Lemma3, un→uin measure in Ω, and we conclude thatu=vand

u∈W1,q(Ω).

Lemma 6 There exist a subsequence of{un}and a measurable function u such that∇un converges to∇u almost everywhere inΩ.

Proof The proof is similar to that of [1, Thm. 4.1] and can be also found in [4]. Here we sketch only the main steps due to slight modiﬁcations. Forr2> 1, letλ= _prq

2 < 1, where

qis the same as in Lemma5. DeﬁneA(x,u,ξ) = a(x,ξ)

(1+b(x)|u|)θ(p–1) (for simplicity, we omit the

dependence ofA(x,u,ξ) onx) and

I(n) =

Ω

A(un,∇un) –A(un,∇u)

· ∇(un–u)

λ dx.

We ﬁxk> 0 and split the integral inI(n) on the sets{|u|>k}and{|u| ≤k}, obtaining

I1(n,k) =

{|u|>k}

A(un,∇un) –A(un,∇u)

· ∇(un–u)

λ dx

and

I2(n,k) =

{|u|≤k}

A(un,∇un) –A(un,∇u)

· ∇(un–u)

(9)

ForI2(n,k), we have

I2(n,k)≤I3(n,k) =

Ω

An(un,∇un) –An

un,∇Tk(u)

· ∇un–Tk(u)

λ

dx.

Fixh> 0 and splitI3(n,k) on the sets{|un–Tk(u)|>h}and{|un–Tk(u)| ≤h}, obtaining

I4(n,k,h) =

{|un–Tk(u)|>h}

An(un,∇un) –An

un,∇Tk(u)

· ∇un–Tk(u)

λ

dx

and

I5(n,k,h) =

{|un–Tk(u)|≤h}

An(un,∇un) –An

un,∇Tk(u)

· ∇un–Tk(u)

λ

dx

=

Ω

An(un,∇un) –An

un,∇Tk(u)

· ∇Th

un–Tk(u)

λ

dx

≤ |Ω|1–λ

Ω

An(un,∇un) –An

un,∇Tk(u)

· ∇Th

un–Tk(u)

dx

λ

=|Ω|1–λI6(n,k,h)λ.

ForI6(n,k,h), we can split it as the diﬀerenceI7(n,k,h) –I8(n,k,h), where

I7(n,k,h) =

Ω

A(un,∇un)· ∇Th

un–Tk(u)

dx

and

I8(n,k,h) =

Ω

Aun,∇Tk(u)

· ∇Th

un–Tk(u)

dx.

Note that|∇un|is bounded inLq(Ω) andλpr2=q. Due to Lemmas3and5, in the same way as Theorem 4.1 in [1], we can get that

lim

k→∞lim supn→∞ I1(n,k) = 0, hlim→∞lim sup_k_→∞ lim supn→∞ I4(n,k,h) = 0,

lim

n→∞I8(n,k,h) = 0.

ForI7(n,k,h), letk>max{g∞,ψ∞}andn≥h+k. TakeTk(u) as a test function for (8), obtaining

I7(n,k,h)≤

Ω –fnTh

un–Tk(u)

dx.

Using the strong convergence offninL1(Ω), we have

lim

n→∞I7(n,k,h)≤

Ω –fTh

u–Tk(u)

dx.

Note thath> 0 andlimk→∞Th(u–Tk(u)) = 0. It follows that

lim

(10)

Putting together all the limitations and noting thatI(n)≥0, we have

lim

n→∞I(n) = 0.

The same arguments as in [1] give that, up to a subsequence,∇un(x)→ ∇u(x)a.e.

Proof of Proposition4 We shall prove that∇unconverges strongly to∇uninLq(Ω) for eachq∈(1,N_N(_–1–θ(p–1)(1–θ)_p_–1)). To this end, we will apply Vitalli’s theorem, using the fact that by Lemma5,∇unis bounded inLq(Ω) for eachq∈(1,N_N(_–1–θ(p–1)(1–θ)_p_–1)). Letr∈(q,N_N(_–1–θ(p–1)(1–θ)_p_–1)), and letE⊂Ωbe a measurable set. We have, by Hölder’s inequality,

E

|∇un|qdx≤

E

|∇un|rdx

q

r

· |E|r–rq ≤_C|_E| r–q

r →₀

uniformly innas|E| →0. From this and from Lemma6we deduce that∇unconverges strongly to∇uinLq(Ω).

Now assume that 0≤θ <min{_N_–1_p₊₁,_NN_–1–_p_–11 }. Note that since∇unconverges to∇u a.e. inΩ, to prove the convergence

a(x,∇un) (1 +b(x)|un|)θ(p–1)

→ a(x,∇u)

(1 +b(x)|u|)θ(p–1) strongly in

L1(Ω)N,

it suﬃces, thanks to Vitalli’s theorem, to show that, for every measurable subsetE⊂Ω,

E|

a(x,∇un)

(1+b(x)|un|)θ(p–1)|dxconverges to 0 uniformly innas|E| →0. Note thatp– 1 <

N(p–1)(1–θ) N–1–θ(p–1) by the assumptions. For anyq∈(p– 1,N_N(_–1–θ(p–1)(1–θ)_p_–1)), we deduce by Hölder’s inequality that

E

a(x,∇un)

(1 +b(x)|un|)θ(p–1)

dx≤β

E

j+|∇un|p–1

dx

≤βjp|E|

1

p₊_β

E

|∇un|qdx

p–1

q

|E|

q–p+1

q

→0 uniformly innas|E| →0.

Lemma 7 There exists a subsequence of{un}such that,for all k> 0, a(x,∇Tk(un))

(1 +b(x)|Tk(un)|)θ(p–1)→

a(x,∇Tk(u)) (1 +b(x)|Tk(u)|)θ(p–1)

strongly inL1(Ω)N.

Proof Letkbe a positive number. It is well known that if a sequence of measurable func-tions{un}with uniform boundedness inLp(Ω) (p> 1) converges in measure tou, then unconverges strongly tou inL1(Ω). First note that the sequence{₍₁₊_ba₍_x(x₎_|,_T∇Tk(un))

k(un)|)θ(p–1)}is

bounded inLp₍_Ω_{). Indeed, we have by (}₃_{) and Lemma}₂_,

Ω

a(x,∇Tk(un)) (1 +b(x)|Tk(un)|)θ(p–1)

p

dx≤βjp_p+β

Ω

|∇Tk(un)|p (1 +b(x)|Tk(un)|)θp

dx

≤βjp_p+β

Ω

|∇Tk(un)|p (1 +b(x)|Tk(un)|)θ(p–1)

dx

(11)

Next, it suﬃces to show that there exists a subsequence of{un}such that

a(x,∇Tk(un)) (1 +b(x)|Tk(un)|)θ(p–1)→

a(x,∇Tk(u)) (1 +b(x)|Tk(u)|)θ(p–1)

in measure.

Note thatun,u∈W1,q(Ω), whereqis the same as in Proposition4. The a.e. convergence ofuntouand the fact that∇un→ ∇uin measure imply that

∇Tk(un)→ ∇Tk(u) in measure.

Lets,tbe positive numbers and write∇Au=₍₁₊_ba₍_x(x₎_|,_u∇_|u₎θ)(p–1). Deﬁne

En= ∇ATk(un) –∇ATk(u)>s

,

E1_n= ∇Tk(un)>t

,

E2_n= ∇Tk(u)>t

,

E3_n=En∩ ∇Tk(un)≤t

∩ ∇Tk(u)≤t

.

Note thatEn⊂E1n∪En2∪E3nfor eachn≥1. Since by Lemma5the sequence{un}and the functionuare uniformly bounded inW1,q₍_Ω_{), we obtain}

LN_E1 n

≤ 1

tq

Ω

_∇Tk(un)qdx≤ 1 tq

Ω

|∇un|qdx≤ C tq,

LN_E2 n

≤ 1

tq

Ω

_∇Tk(u) q

dx≤ 1 tq

Ω

|∇u|qdx≤C tq.

We deduce that, for anyε> 0, there existstε> 0 such that LN_E1

n

+LNE_n2<ε

3 ∀t≥tε,∀n≥1. (27)

Note that, fora≥b≥0 andτ ≥0, we have the inequality

_{(1 +}1_a)τ – 1 (1 +b)τ

=τ(b–a) (1 +c)1+τ

≤τ|b–a| for somec∈[b,a].

From (3), (5), and (6) we deduce that, inE3 n,

s<∇ATk(un) –∇ATk(u)

=a(x,∇Tk(un)) –a(x,∇Tk(u)) (1 +b(x)|Tk(un)|)θ(p–1)

+

1

(1 +b(x)|Tk(un)|)θ(p–1)

– 1

(1 +b(x)|Tk(u)|)θ(p–1)

ax,∇Tk(u)

≤θ(p– 1)BTk(un) –Tk(u)·β

j+∇Tk(u) p–1

+γ

⎧ ⎨ ⎩

|∇Tk(un) –∇Tk(u)|p–1 if 1 <p< 2,

(12)

≤C0jTk(un) –Tk(u)

+C01 +tp–1+tp–2Tk(un) –Tk(u)+∇Tk(un) –∇Tk(u),

which leads toE3

n⊂F1∪F2with

F1=

jTk(un) –Tk(u)> s 2C0

,

F2=

Tk(un) –Tk(u)+∇Tk(un) –∇Tk(u)>

s

2C0(1 +tp–1₊_tp–2₎

.

InF1, we have

LN_(F 1) =

2C0 s

F1

s 2C0dx<

2C0 s

F1

jTk(un) –Tk(u)dx.

By Lemma3we deduce that there existsn0=n0(S,C0,ε) such that LN_(F1)_≤ε

3 ∀n≥n0. (28)

Note thatF2⊂F3∪F4with

F3=Tk(un) –Tk(u)>

s

4C0(1 +tp–1₊_tp–2₎

,

F4=∇Tk(un) –∇Tk(u)>

s

4C0(1 +tp–1₊_tp–2₎

.

Using the convergence in measure of∇Tk(un) to∇Tk(u) and ofTk(un) toTk(u), fort=tε, we obtain the existence ofn1=n1(s,ε)≥1 such that

LN_(F2)_≤_LN_{(F3) +}_LN_{(F4) <}ε

3 ∀n≥n1. (29)

Combining (27), (28), and (29), we obtain

LN _∇

ATk(un) –∇ATk(u)>s

<ε ∀n≥max{n0,n1}.

Hence the sequence{∇ATk(un)} converges in measure to∇ATk(u), and the lemma

fol-lows.

3 Proof of the main result

Now we have gathered all the lemmas needed to prove the existence of an entropy solu-tion to the obstacle problem associated with (f,ψ,g). In this section, letfnbe a sequence of smooth functions converging strongly tof inL1₍_Ω_{) with}_f

n1≤ f1+ 1. We consider the sequence of approximated obstacle problems associated with (fn,ψ,g). The proof orig-inates from [8]. We provide details for readers’ convenience.

Proof of Theorem1 Letv∈Kg,ψ∩L∞(Ω). Takingvas a test function in (8) associated with (fn,ψ,g), we get

Ω

a(x,∇un) (1 +b(x)|un|)θ(p–1)

· ∇Tt(v–un)dx≥

Ω

(13)

Since{|v–un|<t} ⊂ {|un|<s}withs=t+v∞, the previous inequality can be written as

Ω

χn∇ATs(un)· ∇v dx≥

Ω

–fnTt(v–un)dx+

Ω

χn∇ATs(un)· ∇Ts(un)dx, (30)

whereχn=χ{|v–un|<t}and∇Au=

a(x,∇u)

(1+b(x)|u|)θ(p–1). It is clear thatχn χweakly* inL∞(Ω). Moreover,χnconverges a.e. toχ{|v–u|<t}inΩ\ {|v–u|=t}. It follows that

χ=

⎧ ⎨ ⎩

1 in{|v–u|<t},

0 in{|v–u|>t}.

Note that we haveLN₍_{|_v_–_u_|₌_t_}_{) = 0 for a.e.}_t_∈_(0,_∞_{). So there exists a measurable} setO⊂(0,∞) such thatLN₍_{|_v_–_u_|₌_t_}_{) = 0 for all}_t_∈_(0,_∞₎_\_O_{. Assume that}_t_∈_(0,

∞)\O. Thenχnconverges weakly* inL∞(Ω) and a.e. inΩtoχ=χ{|v–u|<t}. Since∇Ts(un) converges a.e. to∇Ts(u) inΩ(Proposition4), by Fatou’s lemma we obtain

lim inf

n→∞

Ω

χn∇ATs(un)· ∇Ts(un)dx≥

Ω

χ∇ATs(u)· ∇Ts(u)dx. (31)

Using the strong convergence of∇ATs(un) to∇ATs(u) inL1(Ω) (Lemma7) and the weak* convergence ofχntoχinL∞(Ω), we obtain

lim

n→∞

Ω

χn∇ATs(un)· ∇v dx=

Ω

χ∇ATs(u)· ∇v dx. (32)

Moreover, sincefnconverges tof inL1(Ω) andTt(v–un) converges toTt(v–u) weakly* inL∞(Ω), by passing to the limit in (30) and taking into account (31)–(32) we obtain

Ω

χ∇ATs(u)· ∇v dx–

Ω

χ∇ATs(u)· ∇Ts(u)dx≥

Ω

–fTt(v–u)dx,

which can be written as

{|v–u|≤t}

χ∇ATs(u)·(∇v–∇u)dx≥

Ω

–fTt(v–u)dx

or, sinceχ=χ{|v–u|<t}and∇(Tt(v–u)) =χ{|v–u|<t}∇(v–u),

Ω

∇Au· ∇Tt(v–u)dx≥

Ω

–fTt(v–u)dx, ∀t∈(0,∞)\O.

Fort∈O, we know that there exists a sequence{tk}in (0,∞)\Osuch thattk→tdue to

|O|= 0. Therefore we have

Ω

∇Au· ∇Ttk(v–u)dx≥

Ω

–fTtk(v–u)dx ∀k≥1. (33)

Since∇(v–u) = 0 a.e. in{|v–u|=t}, the left-hand side of (33) can be written as

Ω

∇Au· ∇Ttk(v–u)dx=

Ω\{|v–u|=t}

(14)

The sequenceχ_{|v–u|<tk}converges toχ{|v–u|<t}a.e. inΩ\ {|v–u|=t}and therefore

con-verges weakly* inL∞(Ω\ {|v–u|=t}). We obtain

lim

k→∞

Ω

∇Au· ∇Ttk(v–u)dx=

Ω\{|v–u|=t}

χ_{|v–u|<t}∇Au· ∇(v–u)dx

=

Ω

χ{|v–u|<t}∇Au· ∇(v–u)dx

=

Ω

∇Au· ∇Tt(v–u)dx. (34)

For the right-hand side of (33), we have

_Ω–fTtk(v–u)dx–

Ω

–fTt(v–u)dx

≤ |tk–t|f1→0 ask→ ∞. (35)

It follows from (33)–(35) that we have the inequality

Ω

∇Au· ∇Tt(v–u)dx≥

Ω

–fTt(v–u)dx ∀t∈(0,∞).

Hence,uis an entropy solution of the obstacle problem associated with (f,ψ,g). The reg-ularity of the entropy solutionuis guaranteed by Proposition4.

Acknowledgements

The authors would like to thank the reviewers for his/her value comments on this paper.

Funding

This research was partially supported by NSFC: 51708459.

Availability of data and materials

Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors prepared the organizing and writing of the paper and have the same contribution to the paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Mathematics, Southwest Jiaotong University, Chengdu, China.}2_{Centro de Ciências e Tecnologia, Universidade}

Federal do Cariri, Juazeiro do Norte, Brazil.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 21 December 2018 Accepted: 11 March 2019

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