# Existence ofW01,1Ω Solutions to Non-Coercivity Quasilinear Elliptic Problem

Authors: Jimao Xiawu; Shuibo Huang; Yingyuan Mi; Maoji Ri
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper we consider the existence ofW01,1Ω solutions to following kind of problems −div∇up−2∇u/1+uθp−1=fx,x∈Ω;ux=0,x∈∂Ω where Ω is an open bounded subset of RNN>2, maxp−2N+1/p−1N−1,0<θ<1 and 1<p⩽1+N−1/N1−θ+θ, f is a function which belongs to a suitable integrable space.


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## Body

## 1. Introduction

In this paper, we study the following kind of problems(1)−divax,u,∇u=fx,x∈Ω,ux=0,x∈∂Ω,whereΩ is an open bounded subset of RNN>2, f is a function which belongs to a suitable integrable space and a:Ω×R×RN→RN is a Carathéodory function, which satisfies the following assumptions: for almost every x∈Ω and every s∈R(2)α1+sθp−1ξp⩽ax,s,ξξ,(3)ax,s,ξ⩽βξp−1,(4)ax,s,ξ−ax,s,η·ξ−η>0,for everyξ,η∈RN with ξ≠η, where α,β are positive constants and 0⩽θ<1.We note that there is a difficulty in dealing with (2) under the assumption (3). Indeed the differential operator(5)Au=−divax,u,∇u,is well defined inW01,pΩ, but is not coercive on the same space when u is unbounded. Therefore, the classical Leray—Lions theorem cannot be apply even in the case the datum f belongs to W−1,p′Ω for p′=p/p−1. In order to overcome this difficulty, we will proceed by approximation by means of truncatures in ax,s,ξ to get a coercive differential operator on W01,p. In particular in [1–3], the authors analyzed the problems(6)−divax,u∇u=f,x∈Ω,ux=0,x∈∂Ω,wheref∈LmΩ with m⩾1, ax,s:Ω×R→R is a Carathéodory function such that(7)α1+sγ⩽ax,s⩽β,for alls∈R and 0<γ⩽1. Denote(8)r=Nm1−γN−2m,q=Nm1−γN−m1+γ.Main results of [1–3] summarized as:(1)
Iff∈LmΩ with 1⩽m⩽maxN/N+1−γN−1,1, then there exists a solution u to problem (6) such that(9)
u
s
∈
L
1
Ω
,
s
<
r
and
∇
u
s
∈
L
1
Ω
,
s
<
q
.(2) IfN/N+1−γN−1<m⩽2N/N1−γ+2γ+1, then u∈W01,qΩ;
(3) If2N/N1−γ+2γ+1⩽m<N/2, then u∈H01Ω∩LrΩ;
(4) Ifm>N/2, then u∈H01Ω∩L∞Ω.In the case whereγ>1, problem (6) may has no solution at all, even if the data f is a constant (see [1, 3]). Thus the non-coerciveness of operator A has great effect on the existence and regularity of the solution. Some other results of non-coercivity quasilinear elliptic problem, see [4–24].Furthermore, in the literature [21], they proved the L∞-regularity for solutions to the following nonlinear elliptic equations with degenerate coercivity(10)−divax,u,∇u=f,x∈Ω,ux=0,x∈∂Ω,whereΩ is an open bounded subset of RNN>2,1<p<N, f satisfies(11)f∈LN/plogαLwithα>Np−1/p, and a:Ω×R×RN→RN is a Carathéodory function which satisfies the following assumptions(12)ax,s,ξ⩾hp−1sξp,(13)ax,s,ξ⩽a0x+sp−1+ξp−1,(14)ax,s,ξ−ax,s,ξ′·ξ−ξ′>0,for almost everyx∈Ω, for every s∈R, where a0 is a nonnegative function in Lp′ with p′=p/p−1, and(15)ht=11+uθ,whereθ is a real such that 0⩽θ⩽1 and h:R+→0,∞ is a decreasing continuous function such that its primitive(16)Hs=∫0shtdt,is unbounded. Under the hypothesis (11)–(14), the authors proved that (10) has at least a weak solution u∈W01,pΩ∩L∞Ω in the sense that(17)∫Ωax,u,∇u·∇vdx=∫Ωfvdxfor allv in C0∞Ω.On the other hand, as in [22], it has been shown the existence results of solutions for nonlinear elliptic boundary value problem(18)−div∇up−2∇u=f,x∈Ω,ux=0,x∈∂Ω,wheref∈Lmm⩾1,1<p<N. We recall the following results contained in [22].(1) Letf∈LmΩ,m=N/Np−1+1,1<p<2−1/N. Then there exists a distributional solution u∈W01,1Ω of (18).(2) Letp=2−1/N, and if flog1+|f|∈L1Ω, then there exists a distributional solution u∈W01,1Ω of (18).The aim of this article is to study the existence ofW01,1Ω solutions to the problem (1). Our main results are following:Theorem 1.
Assume (2)–(4) hold and
f
∈
L
m
Ω
with
(19)
m
=
N
N
p
−
1
+
1
−
θ
p
−
1
N
−
1
,
max
p
−
2
N
+
1
p
−
1
N
−
1
,
0
<
θ
<
1
.Then problem (1) exists a distributional solution u∈W01,1Ω provided(20)1<p<1+N−1N1−θ+θ.Remark 2.
We point out that the existence inW01,1Ω is not so usual in the study of elliptic problems. The main difficulty of study is due to the a priori estimate of the sequence of the approximate solutions {un} of  (18) in the non-reflexive space W01,1Ω. For more results about existence of W01,1Ω to elliptic equations, see [23–25].Remark 3.
Observe that,m>1 if (20) holds.Remark 4.
It can be easily seen that, (20) implies p−2N+1/p−1N−1<θ, while 1+N−1/N1−θ+θ>2−1/N. Therefore, in (20), we should require maxp−2N+1/p−1N−1,0<θ, instead of p−2N+1/p−1N−1<θ.Remark 5.
It is worth pointing out that, the existence ofW01,1Ω solutions to (18) was established provided 1<p<2−1/N and f∈LmΩ with m=N/Np−1+1. According to Remark 4, we know that problem (1) has u∈W01,1Ω solution if p satisfies.(21)
2
−
1
N
≤
p
<
1
+
N
−
1
N
1
−
θ
+
θ
.This fact shows that the lack of coercivity of the differential operatorax,u,∇u has the great influence on the existence of W01,1Ω solutions to (1). Especially, for p=2−1/N, we obtain the existence of W01,1Ω solutions for f∈LmΩ with m=N2/N2−θN−12>1, rather than flog1+|f|∈L1Ω.Forflog1+|f|∈L1Ω, we have the following theorem.Theorem 6.
Assume (2)–(4) hold. Then problem (1) has a distributional solution
u
∈
W
0
1
,
1
Ω
if(22)
∫
Ω
f
log
1
+
f
<
∞
and
p
=
1
+
N
−
1
N
1
−
θ
+
θ
.Remark 7.
Note that ifθ=0, then p=2−1/N.The paper is organized as follows. In Section2 we collect some definitions and useful tools. The proof of Theorems 1 and 6 is given in Section 3.


## 2. Preliminaries

In order to prove our main results, we need to introduce some basic definitions and lemmas. We shall used the following notion of distributional solution for problem (1).Definition 8.
We say thatu∈W01,1Ω is a distributional solutions to problem (1) if(23)
∫
Ω
a
x
,
u
,
∇
u
·
∇
φ
=
∫
Ω
f
φ
,for everyφ∈C0∞Ω.Definition 9.
For allk⩾0, the truncation function is defined by(24)
T
k
s
=
max
−
k
,
min
k
,
s
.We recall the following lemma (see [26]).Lemma 10.
Letu:Ω→R be measurable function such that Tku∈H01Ω for k>0. Then there exists a unique measurable υ:Ω→RN such that (25)∇Tku=υχu<ka.e.inΩ,∀k>0,
the mapυis called the weak gradient ofu. Moreover ifu∈W01,1Ω, then υ coincides with the standard distributional gradient of u.Definition 11.
Letp⩾1 and α>0, the Zygmund space LplogαL, is made up of all measurable functions g such that(26)∫Ωgplogα1+gdx<∞,
forg on Ω.Remark 12.
Observe that ifp=1 and α=1, then the (26) becomes(27)
∫
Ω
f
log
1
+
f
<
∞
.We obtain the existence solutions to (2) by considering the following approximation problems(28)−divax,Tnun,∇un=f,x∈Ω,unx=0,x∈∂Ω.Lemma 13.
Let
u
n
be solutions to (28) and
f
∈
L
1
Ω
,
p
>
1
. Then the sequence
log
1
+
|
u
n
|
sgn
u
n
is bounded in
W
0
1
,
p
Ω.Proof.
Let us take1−1+|un|−1−θp−1sgnun as test function in (28), we get(29)C∫Ω∇log1+unp⩽C∫Ω∇unp1+unp⩽∫Ωf1−1+un−1−θp−1⩽fL1Ω.Then we arrive to the conclusion.In order to prove the existence part of Theorems1 and 6 we need the following preliminary result. This convergence of the gradients is a consequence of a classical result (see [22]).Lemma 14.
Let
u
n
be the sequence of solutions to (28), which are bounded in
W
0
1
,
p
Ω
, Assume
1
<
p
<
N
,
f
∈
L
m
Ω
m
⩾
1
and (2)–(4) hold. Suppose that
(1)
u
n
is such that
u
n
W
0
1
,
1
Ω
⩽
M
for constant
M;
(2)
u
n
converges to
u
almost everywhere in
Ω;
(3)
∇
T
k
u
n
converges weakly to
∇
T
k
u
in
W
0
1
,
p
Ω.
Then, up to a subsequence,
∇
u
n
converges almost everywhere in
Ω
to
∇
u.


## 3. Proof of Main Results

In this section, we prove the existence results of Theorems1 and 6 by a classical approach. Consider the following approximate problems(30)−divax,Tnun,∇un=Tnf,x∈Ω,unx=0,x∈∂Ω.Proof of Theorem1.
Step 1: Let(31)σ=pm′−p∗θp′pm′−p∗,
and we know the function(32)ν=1+un1−p1−σsgnun
is bounded since1−p1−σ>0. Thus, choose ν as a test function in the approximate problem (28), we find(33)C∫Ω∇unp1+unθp−1+p1−σ⩽fLmΩ∫Ω1+un1−p1−σm′1/m′,
whereC denotes a strictly positive constant, depending only on α,p,N,σ and fLmΩ. By the Sobolev embedding theorem, we have(34)C∫Ω1+unσ−θ/p′p∗p/p∗⩽∫Ω∇unp1+unθp−1+p1−σ.Combining (34), (35), then(35)C∫Ω1+unσ−θ/p′p∗p/p∗⩽C∫Ω∇unp1+unθp−1+p1−σ⩽fLmΩ∫Ω1+un1−p1−σm′1/m′.Note thatp/p∗>1/m′ since m<N/p. Moreover the choice of σ implies(36)σ−θp′p∗=1−p1−σm′=NN−1.Thus we prove that(37)C∫ΩunN/N−11/m−p/N⩽fLmΩ.This estimate implies that(38)∫Ω∇unp1+unθp−1+p1−σ<∞.Now we can estimate∫Ω∇un. Indeed, apply the Hölder inequality and we obtain(39)∫Ω∇un=∫Ω∇un1+unθ/pp−1+1−σ1+unθ/pp−1+1−σ⩽∫Ω∇unp1+unθp−1+p1−σ1/p∫Ω1+unθ+p′1−σ1/p′.Note thatθ+p′1−σ=N/N−1, so the right hand side is bounded. Then we get that the sequence {un} is uniformly bounded in W01,1Ω, subsequently there exist M>0 such that(40)unW01,1Ω⩽M.As a consequence there exists a subsequence{un} converging to u a.e. in LrΩ,1⩽r<N/N−1.Step 2: Let us chooseTkun as test function in (28)(41)∫Ωax,Tnun,∇un·∇Tkun=∫ΩfnTkun.Using assumption (2) on a, we get(42)C∫Ω∇unp1+Tkunθp−1⩽kfL1Ω,and so(43)∫Ω∇Tkunp⩽Ckθp−1+1,forallk>0.This implies thatTkun→Tku weakly in W01,pΩ.Now we are also going to estimate∫{k⩽|un|}|∇un|. Choosing 1+|un|1−p1−σ+sgnun as a test function in (28), we have(44)∫k⩽un∇unp1+unθp−1+p1−σ⩽∫k⩽unfm1/m∫k⩽un1+un1−p1−σm′1/m′⩽C∫k⩽unfm1/m,where we have use (37) in the last passage. By the Hölder inequality we get(45)∫k⩽un∇un=∫k⩽un∇un1+unθ/pp−1+1−σ1+unθ/pp−1+1−σ⩽∫k⩽un∇unp1+unθp−1+p1−σ1/p∫k⩽un1+unθ+p′1−σ1/p′⩽C∫k⩽unfm1/m.LetE be a measurable subset of Ω, and let i=1,…,N. Then by (43), (45) and Hölder inequality, we have(46)∫E∂un∂xi⩽∫E∇un⩽∫E∇Tkun+∫k⩽un∇un⩽measE1/p′∫E∇Tkunp1/p+∫k⩽un∇un⩽CmeasE1/p′k1/pθp−1+1+C∫k⩽unfm1/m.The estimates (37) and (46) imply that the sequence {∂un/∂xi} is equiintegrable. Thus, by the Dunford—Pettis theorem, and up to subsequences, there exists Yi in L1Ω such that ∂un/∂xi weakly converges to Yi in L1Ω. Since ∂un/∂xi is the distributional partial derivative of un, then for every n in N, we have(47)∫Ω∂iunφ=−∫Ωun∂iφ,whereφ∈C0∞Ω. We now pass to the limit in the above identities, using that ∂iun weakly converges to Yi in L1Ω, and that un strongly converges to u in L1Ω, we find(48)∫ΩYiφ=−∫Ωu∂iφ.This implies thatYi=∂un/∂xi, and this result holds for every i. Since Yi belongs to L1Ω for every i, u belongs to W01,1Ω, as desired.Step 3: We can apply Lemma14 and conclude that ∇un→∇u a.e., that is ax,Tnun,∇un almost everywhere converges to ax,u,∇u. Since un is bounded in W01,pΩ, the inequality(49)ax,un,∇un⩽β∇unp−1and the Vitali theorem imply thatax,un,∇un converges to ax,u,∇u in L1/p−1ΩN. Hence we are able to pass to the limit in (28). Thus we prove that u∈W01,1Ω is a distributional solution of (2) and yields the conclusion of the proof of Theorem 1.Proof of Theorem6.
Step 1: Let1−1+un1−λ. Choosing 1−1+un1−λsgnun as a test function in (28) and by (3), we get.(50)
C
∫
Ω
∇
u
n
p
1
+
u
n
θ
p
−
1
+
λ
⩽
∫
Ω
f
n
1
−
1
+
u
n
1
−
λ
.Then using the Sobolev embedding theorem we have(51)C∫Ω1+un1−θp−1+λ/p−1p∗p/p∗⩽∫Ω∇unp1+unθp−1+λ.So combining (50) and (51),(52)C∫Ω1+un1−θp−1+λ/p−1p∗p/p∗⩽fL1Ω.Note that(53)1−θp−1+λpp∗<NN−1sinceλ>1, then we prove that the sequence {un} is bounded in LrΩ,1⩽r<N/N−1, that is(54)unLrΩ⩽C.Step 2: We use the following inequality: there existsar>0 only depending on r such that(55)tlog1+s⩽tlog1+t+sr+ar,for alls,t∈R+.We takelog1+|un|sgnun as a test function in the approximated problem (28) and by inequality (55), we obtain(56)α∫Ω∇unp1+unθp−1+1⩽∫Ωflog1+un⩽flog1+fL1Ω+∫Ωunr+armeasΩ.Now we can apply Hölder and Sobolev inequalities, since(57)p′pθp−1+1=NN−1and1−1p=N−1N−11−θ+N,then(58)SN,pα1/p∫ΩunN/N−1N−1/N⩽α1/p∫Ω∇un=∫Ω∇un1+un1/pθp−1+1.1+un1/pθp−1+1⩽∫Ω∇unp1+unθp−1+11/p⋅∫Ω1+unp′/pθp−1+11/p′⩽flog1+fL1Ω+Cr+armeasΩ1/p⋅∫Ω1+unN/N−1N−1/N−11−θ+N.SinceN−1/N−11−θ+N<N−1/N, we prove that the sequence {un} is bounded in W01,1Ω and so it is compact in LrΩ,1⩽r<N/N−1.Step 3: Choosinglog1+|un|−log1+ksgnun as a test function in (28), we get(59)α∫k⩽un∇unp1+unθp−1+1⩽∫k⩽unflog1+un⩽∫k⩽unflog1+f+∫k⩽ununr+armeask⩽un.In addition,(60)∫k⩽un∇un=∫k⩽un∇un1+un1/pθp−1+1⋅1+un1/pθp−1+1⩽∫k⩽un∇unp1+unθp−1+11/p⋅∫k⩽un1+unp′/pθp−1+11/p′⩽C∫k⩽unflog1+f+∫k⩽ununr+armeask⩽un1/p.Therefore, for every measurable subsetE, following (46) and we have (61)∫E∂un∂xi⩽∫E∇un⩽∫E∇Tkun+∫k⩽un∇un⩽CmeasE1/p′k1/pθp−1+1+C∫k⩽unflog1+f+∫k⩽ununr+armeask⩽un1/p.Thus we prove thatun weakly converges to u in W01,1Ω. Furthermore, the last part of the proof we follow the approach of the Theorem 1 and we can proof that u∈W01,1Ω is a distributional solution of (1).



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*Source: 5017818-2020-02-26.xml*