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

Abstract

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem where is a bounded smooth domain with , belongs to the Lebesgue space with The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

1. Introduction

The main purpose of this paper is to prove the existence of solutions to the following elliptic boundary value problem where is a bounded smooth subset of with , and is a bounded measurable matrix, which satisfies the following conditions: there are two positive constants and , such that, for a.e. and every ,

Moreover, is a measurable vector field, which satisfies is a measurable function which satisfies where represents the Lebesgue space.

When the problem (1) without the lower-order terms, it becomes the following elliptic Dirichlet problem

Problem (5) was studied by Boccardo in a series of papers. More precisely, in [1], the existence of solutions to problem (5) was proved provided (i)If with , then , where (ii)If with , then , where (iii)If with , then (iv)If , then , where

When and vector field satisfies (3), which does not belong to , Boccardo [2] shown that (i)If and , then (ii)If and , then (iii)If and , then and , where

Furthermore, in the same paper, the existence of entropy solutions to problem (5) also be considered provided and .

Recently, continuation of [2], Boccardo and Orsina [3] studied the existence of distributional solution to problem (5) with provided and . Moreover, verifies a prior estimation:

The constant lies on , , and . 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. When and satisfying (2), Boccardo [2] shown that the existence of entropy solution of (1) provided and .

Moreover, verifies the following estimations:

Furthermore, there is a weak solution of (1) provided (2) holds, , , and .

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 . The main features of this paper are the presence of the convection term , which leads to the noncoercivity of in . 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 that is a bounded smooth domain of with and (2)–(4) hold. (a)There is a weak solution to problem (1) provided and with (b)There is a weak solution to problem (1) provided and with . Furthermore,where

Remark 2. A point worth emphasizing is that our results further refine the conclusions of [2]. More precisely, under different assumptions on , we give the existence of solutions to problem (1) with for and , respectively, rather than .

Remark 3. Obviously, 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 which shows that the lower-order term intensifies the requirement on .
The paper is organized as follows. In Section 2, 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 Theorem 1, the following basic definitions and lemmas are needed. First of all, we give the definitions of weak solution to (1).

Definition 5. We say that is a weak solution to problem (1), if and for every .

The following is the definition of the truncation function.

Definition 6. For the truncation function defined by Now, let us briefly recall the Sobolev’s embedding theorem.

Lemma 7. Assume that , then there is a normal number , such that, for satisfies where The following Hölder inequality plays an important role in this paper.

Lemma 8 (Hölder inequality). Assume that Then, if we have 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 , we have where

3. Proof of Main Theorem

In this part, we are going to give the Proof of Theorem 1 in a similar way as [2, 17–21, 23, 26]. In order to do this, first of all, we consider the following approximate problem: where

Let us start with the following conclusions.

First of all, the following lemma gives an information on the summability of .

Lemma 10. Let Then, for every there exists a solution to (19) such that

Proof. In order to get the estimates (21), we will consider the following two cases separately.
Case . Select as a test function in (19), by (3), we have According to (2) and (22), we get Using the Hölder inequality and the Hardy inequality for the first term on the right of (23), we obtain Since Thus, taking into account (23)–(25), we arrive at This fact leads to provided where .
Applying the Hölder inequality on the right-hand side of (27), we get where which together with (27), implies that (21) holds.
When , taking as a test function in (19), according to (3), we have For the first term on the left-hand side of (30), we have For the first term on the right-hand side of (30), using the Hölder inequality and the Hardy inequality, we get Combining (30)–(31) with (2), we have Since , we obtain Fatou lemma implies, for , the expression of (21) holds.

Next, we will prove the following existence result.

Lemma 11. Assume (2), (3) hold with , with Then, there is a weak solution to problem (1).

Proof. Set obviously, in as . Let as a test function in (19), using the Hölder inequality and the Hardy inequality, we get Moreover, by (2) and a direct calculation, we get Since , by Lemma 10, we get . Together (35) with (36), it follows that We deduce that is uniformly bounded in Then, we pass to the limit in the approximation problem (19); up to a subsequence, there is a function
Now, we want to prove that . Let be defined by Selecting as a test function in (19), we have which implies that Let is measurable. For any we get The above fact and allow us to say that, for any given , there is satisfies Thus, Therefore, Thus, we prove that Vitali theorem implies that in . In other words, Finally, let us show the second part of Theorem 1, that is, the existence of solution to problem (1) in the case where with

Lemma 12. Assume that with , and Then, there is a weak solution to problem (1) that such that where

Proof. Define , it satisfies Applying as a test function in (19), by (3), we get By (2), (49) can become Since Applying the Hölder inequality and the Hardy inequality for the first term on the right-hand side of (50), it can be rewritten as We conclude that With this choice of , by Lemma 10, we arrive at According to the Sobolev embedding theorem, we obtain Moreover, where .
As a consequence, there exists a function

Similarly, we can prove Then, we have proved the existence result.

Proof of Theorem 1. We can combine Lemmas 11–12 to get Theorem 1.

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to this work.

Acknowledgments

This research was partially supported by the Program for Yong Talent of State Ethnic Affairs Commission of China (No. XBMU-2019-AB-34), Fundamental Research Funds for the Central Universities (No.31920200036), Innovation Team Project of Northwest Minzu University (No.1110130131), General Topics of National Education and Scientific Research (No. ZXYB18019), and First-rate Discipline of Northwest Minzu University.