# Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

Authors: Xiaohua He; Shuibo Huang; Qiaoyu Tian; Yonglin Xu
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem−divMx∇u+up−1u=−divuEx+fx,x∈Ω,ux=0,x∈∂Ω, where Ω⊂ℝNN>2 is a bounded smooth domain with 0∈Ω, f belongs to the Lebesgue space LmΩ with m≥1,p>0. The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.


---

## Body

## 1. Introduction

The main purpose of this paper is to prove the existence of solutions to the following elliptic boundary value problem(1)−divMx∇u+up−1u=−divuEx+fx,x∈Ω,ux=0,x∈∂Ω,where Ω is a bounded smooth subset of ℝNN>2 with 0∈Ω, and M:Ω⟶ℝN2 is a bounded measurable matrix, which satisfies the following conditions: there are two positive constants α and β, such that, for a.e. x∈Ω and every ξ∈ℝN,
(2)αξ2≤Mxξ⋅ξ,Mx≤β.Moreover,Ex is a measurable vector field, which satisfies
(3)E≤Ax,A∈ℝ+,fx is a measurable function which satisfies
(4)f∈LmΩm≥1,where LmΩ represents the Lebesgue space.When the problem (1) without the lower-order terms, it becomes the following elliptic Dirichlet problem
(5)−divMx∇u=−divuEx+fx,x∈Ω,ux=0,x∈∂Ω,Problem (5) was studied by Boccardo in a series of papers. More precisely, in [1], the existence of solutions u to problem (5) was proved provided E∈LNΩN:(i)
Iff∈LmΩ with 2N/N+2≤m<N/2, then u∈W01,2Ω∩Lm∗∗Ω, where m∗∗=Nm/N−2m(ii)
Iff∈LmΩ with 1<m<2N/N+2, then u∈W01,m∗Ω, where m∗=Nm/N−m(iii)
Iff∈LmΩ with m>N/2, then u∈W01,2Ω∩L∞Ω(iv)
Iff∈L1Ω, then u∈W01,qΩ, where q<N/N−1Whenf∈LmΩ and vector field E satisfies (3), which does not belong to LNΩN, Boccardo [2] shown that
(i)
IfA<αN−2m/m and 2N/N+2≤m<N/2, then u∈W01,2Ω∩Lm∗∗Ω(ii)
IfA<αN−2m/m and 1<m<2N/N+2, then u∈W01,m∗Ω(iii)
IfA<αN−2 and m=1, then ∇u∈MN/N−1ΩN and u∈W01,qΩ, where q<N/N−1Furthermore, in the same paper, the existence of entropy solutions to problem (5) also be considered provided E∈L2ΩN and f∈L1Ω.Recently, continuation of [2], Boccardo and Orsina [3] studied the existence of distributional solution u∈W01,qΩ to problem (5) with q<Nα/A+α provided αN−2≤A<αN−1 and f∈L1Ω. Moreover, u verifies a prior estimation:
(6)∫Ω∇uq1/q≤CEfL1Ω.The constantCE lies on E, α, and Ω. Some other results about noncoercivity elliptic problems see [4–16] and reference therein.It is well known that the presence of lower-order term will improve the regularity properties of the solutions. Whenp=1 and Mx satisfying (2), Boccardo [2] shown that the existence of entropy solution u of (1) provided E∈L2ΩN and f∈L1Ω.Moreover,u verifies the following estimations:
(7)∫Ωu≤∫Ωf,∫Ω∇log1+u2≤1α2∫ΩE2+2α∫Ωf,α2∫Ω∇Tku2≤k22α∫ΩE2+k∫Ωf.Furthermore, there is a weak solutionu∈W01.2Ω∩Lp+1Ω of (1) provided (2) holds, p>N+2/N−2, f∈Lp+1/pΩ, and E∈L2p+1/p−1ΩN.For some other results about elliptic problems with lower-order terms, see [17–26] and reference therein.With motivation from the results of the above-cited papers, the main goal of this paper is to further study the regularity of solutions to problem (1) with f∈LmΩm≥1. The main features of this paper are the presence of the convection term divuEx, which leads to the noncoercivity of ‐divMx∇u+divuEx in W01,2Ω. Therefore, in order to overcome the coercivity difficulty, we use truncation technique and consider the corresponding approximate Dirichlet problem, see (19) for more details.The main results are the following:Theorem 1.
Suppose thatΩ is a bounded smooth domain of ℝNN>2 with 0∈Ω and (2)–(4) hold.
(a)
There is a weak solutionu∈W01.2Ω∩Lp+1Ω to problem (1) provided A<αN−2/2 and f∈LmΩ with 1≤m≤1+1/p(b)
There is a weak solutionu∈LmpΩ to problem (1) provided A<αN−2/pm−1+1 and f∈LmΩ with m>p+1/p. Furthermore,(8)uLmpΩ≤CfLmΩ,where
(9)mp=pm−1+1NN−2.Remark 2.
A point worth emphasizing is that our results further refine the conclusions of [2]. More precisely, under different assumptions on E, we give the existence of solutions to problem (1) with f∈LmΩ for 1≤m≤p+1/p and m>p+1/p, respectively, rather than f∈Lp+1/pΩ.Remark 3.
Obviously,(10)mp=pm−1+1NN−2⟶+∞asp⟶+∞,which shows the regularizing effect of the lower-order term for the regularity properties of the solutions to problem (1).Remark 4.
It is clear that(11)A<αN−2pm−1+1⟶0asp⟶+∞,which shows that the lower-order term intensifies the requirement on E.
The paper is organized as follows. In Section2, we give some definitions and lemmas. In Section 3, the Proof of Theorem 1 is given.


## 2. Useful Tools and Function Setting

In order to prove Theorem1, the following basic definitions and lemmas are needed. First of all, we give the definitions of weak solution to (1).Definition 5.
We say thatu∈W01,2Ω is a weak solution to problem (1), if ∫Ωup−1udx<∞ and
(12)∫ΩMx∇u∇v+∫Ωup−1uv=∫ΩuEx∇v+∫Ωfv,for every v∈W01,2Ω.The following is the definition of the truncation function.Definition 6.
For∀k≥0,s∈ℝ, the truncation function defined by
(13)Tks=max−k,mink,s,Gks=s−Tks.
Now, let us briefly recall the Sobolev’s embedding theorem.Lemma 7.
Assume thatp=2, then there is a normal number S, such that, for ∀u∈C0∞ℝN satisfies
(14)u2∗≤S∇u2,where
(15)2∗=2NN−2.
The following Hölder inequality plays an important role in this paper.Lemma 8 (Hölder inequality).
Assume that1<p,q<∞,1/p+1/q=1. Then, if u∈LpΩ,v∈LqΩ, we have
(16)∫Ωuvdx≤uLpΩvLqΩ.
The results of the Hardy inequality and its generalization can be founded in [27–29]. In this paper, we use the following Hardy inequality repeatedly.Lemma 9 (see [3]).
For∀vx∈W01,2Ω, we have
(17)H∫Ωv2x21/2≤∫Ω∇v21/2,where
(18)H=N−22.


## 3. Proof of Main Theorem

In this part, we are going to give the Proof of Theorem1 in a similar way as [2, 17–21, 23, 26]. In order to do this, first of all, we consider the following approximate problem:
(19)−divMx∇un+unp−1un=−divun1+1/nunEx1+1/nEx+fnx,x∈Ω,unx=0,x∈∂Ω,where
(20)fnx=fx1+1/nfx.Let us start with the following conclusions.First of all, the following lemma gives an information on the summability ofup−1u.Lemma 10.
Letf∈LmΩ,m≥1. Then, for every n∈ℕ, there exists a solution un∈W01,2Ω to (19) such that
(21)∫Ωunpm≤∫Ωfm.Proof.
In order to get the estimates (21), we will consider the following two cases separately.
Casem>1. Select ϕ=unpm−1−1un as a test function in (19), by (3), we have
(22)∫ΩMx∇un⋅∇unpm−1−1un+∫Ωunpm=∫Ωun1+1/nunEx1+1/nEx⋅∇unpm−1−1un+∫Ωfnxunpm−1−1un≤A∫Ωunx∇unpm−1−1un+∫Ωfnxunpm−1−1un.
According to (2) and (22), we get
(23)αpm−1∫Ω∇un2unpm−1−1+∫Ωunpm≤Apm−1∫Ωunx∇ununpm−1−1+∫Ωfnxunpm−1−1un=2Apm−1pm−1+1∫Ωunpm−1+1/2x∇unpm−1+1/2+∫Ωfnxunpm−1−1un.
Using the Hölder inequality and the Hardy inequality for the first term on the right of (23), we obtain
(24)2Apm−1pm−1+1∫Ωunpm−1+1/2x∇unpm−1+1/2≤2Apm−1pm−1+1H∫Ω∇unpm−1+1/22.
Since(25)αpm−1∫Ω∇un2unpm−1−1=4αpm−1pm−1+12∫Ω∇unpm−1+1/22.
Thus, taking into account (23)–(25), we arrive at
(26)4αpm−1pm−1+12−2Apm−1pm−1+1H∫Ω∇unpm−1+1/22+∫Ωunpm≤∫Ωfnxunpm−1−1un.
This fact leads to(27)∫Ωunpm≤∫Ωfnxunpm−1−1un,provided
(28)A<αN−2pm−1+1,where H=N−2/2.
Applying the Hölder inequality on the right-hand side of (27), we get
(29)∫Ωfnxunpm−1−1un≤fLmΩ∫Ωunpm1/m′,where m′=m/m−1, which together with (27), implies that (21) holds.
Whenm=1, taking Tkun/k as a test function in (19), according to (3), we have
(30)∫ΩMx∇un⋅∇Tkunk+∫Ωunp−1unTkunk=∫Ωun1+1/nunEx1+1/nEx⋅∇Tkunk+∫ΩfnxTkunk≤A∫Ωunx∇Tkunk+∫ΩfnxTkunk.
For the first term on the left-hand side of (30), we have
(31)∫ΩMx∇un⋅∇Tkunk=1k∫un≤kMx∇un⋅∇un.
For the first term on the right-hand side of (30), using the Hölder inequality and the Hardy inequality, we get
(32)A∫Ωunx∇Tkunk=Ak∫un≤kunx∇un≤AHk∫un≤k∇un2.
Combining (30)–(31) with (2), we have
(33)αk−AHk∫∣un∣≤k∇un2+∫ΩunpTkunk≤∫ΩfnxTkunk.
SinceA<αN−2/2, we obtain
(34)∫un≤kunpTkunk≤∫un≤kfnxTkunk≤∫Ωfnx≤∫Ωf.
Fatou lemma implies, fork⟶∞, the expression of (21) holds.Next, we will prove the following existence result.Lemma 11.
Assume (2), (3) hold with A<αN−2/2, f∈LmΩ with 1≤m≤1+1/p. Then, there is a weak solution u∈W01.2Ω∩Lp+1Ω to problem (1).Proof.
Setfnx=fx/1+1/nfx obviously, fnx⟶fx in L1Ω as n⟶∞. Let ϕ=Tkun as a test function in (19), using the Hölder inequality and the Hardy inequality, we get
(35)∫ΩMx∇un⋅∇Tkun+∫Ωunp−1unTkun=∫Ωun1+1/nunEx1+1/nEx⋅∇Tkun+∫ΩfnxTkun≤A∫Ωunx∇Tkun+k∫Ωfx=A∫ΩTkunx∇Tkun+k∫Ωfx≤AH∫Ω∇Tkun2+k∫Ωfx.
Moreover, by (2) and a direct calculation, we get
(36)α∫Ω∇Tkun2+∫ΩTkp+1un≤∫ΩMx∇Tkun⋅∇Tkun+∫ΩunpTkun.
SinceTkun∈W01,2Ω, by Lemma 10, we get m=1+1/p. Together (35) with (36), it follows that
(37)α−AH∫Ω∇Tkun2+∫ΩTkp+1un≤Cα,N,f,k.
We deduce thatTkun is uniformly bounded in W01,2Ω∩Lp+1Ω. Then, we pass to the limit in the approximation problem (19); up to a subsequence, there is a function u∈W01,2Ω∩Lp+1Ω.
Now, we want to prove thatunp⟶up. Let ψiσ be defined by
(38)ψiσ=1,σ≥t,0,σ<t,−1,σ≤−t.
Selectingϕ=ψiun as a test function in (19), we have
(39)∫Ωunpψiun≤∫Ωfψiun,which implies that
(40)∫un>t∩Ωunp≤∫un>t∩Ωf.
LetE⊂Ω is measurable. For any t>0, we get
(41)∫Eunp≤tpE+∫E∩un>tunp≤tpE+∫un>tf.
The above fact andf∈L1Ω allow us to say that, for any given ε>0, there is tε satisfies
(42)∫un>tεf≤ε.
Thus,(43)∫Eunp≤tεpE+ε.
Therefore,(44)limE→0∫Eunp≤ε.
Thus, we prove thatlimE→0∫Eunp=0. Vitali theorem implies that unp⟶up in L1Ω. In other words,
(45)limn→∞∫Ωunp=∫Ωup.
Finally, let us show the second part of Theorem1, that is, the existence of solution to problem (1) in the case where f∈LmΩ with m>p+1/p.Lemma 12.
Assume thatf∈LmΩ with m>p+1/p, and
(46)A≤αN−2pm−1+1.
Then, there is a weak solutionu to problem (1) that such that
(47)uLmpΩ≤CfLmΩ,where
(48)mp=pm−1+1NN−2.Proof.
Defineβ=pm−1>1, it satisfies p+β=βm′. Applying ϕ=unβ−1un as a test function in (19), by (3), we get
(49)∫ΩMx∇un⋅∇unβ−1un+∫Ωunp+β=∫Ωun1+1/nunEx1+1/nEx⋅∇unβ−1un+∫Ωfnxunβ−1un≤A∫Ωunx∇unβ−1un+∫Ωfnxunβ−1un=Aβ∫Ωunβx∇un+∫Ωfnxunβ−1un.
By (2), (49) can become
(50)αβ∫Ω∇un2unβ−1+∫Ωunp+β≤Aβ∫Ωunβx∇un+∫Ωfnxunβ−1un.
Since(51)αβ∫Ω∇un2unβ−1=4αββ+12∫Ω∇unβ+1/22.
Applying the Hölder inequality and the Hardy inequality for the first term on the right-hand side of (50), it can be rewritten as
(52)4αββ+12∫Ω∇unβ+1/22+∫Ωunp+β≤2Aββ+1H∫Ω∇unβ+1/22+∫Ωfnxunβ−1un.
We conclude that(53)4αββ+12−2Aββ+1H∫Ω∇unβ+1/22+∫Ωunp+β≤∫Ωfnxunβ−1un≤fLmΩ∫Ωunβm′1/m′.
With this choice ofβ, by Lemma 10, we arrive at
(54)∫Ω∇unβ+1/22≤CfLmΩ.
According to the Sobolev embedding theorem, we obtain(55)∫Ωunβ+12∗/21/2∗≤CfLmΩ.
Moreover,(56)β+12∗2=Npm−1+1N−2=mp,where 2∗=2N/N−2.
As a consequence, there exists a functionu∈Lβ+12∗/2Ω.Similarly, we can proveunp⟶up. Then, we have proved the existence result.Proof of Theorem 1.
We can combine Lemmas11–12 to get Theorem 1.



---

*Source: 8816696-2020-10-28.xml*