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

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem −div ðMðxÞ∇uÞ + jujp−1u = − div ðuEðxÞÞ + f ðxÞ, x ∈Ω, uðxÞ = 0, x ∈ ∂Ω, ( where Ω ⊂RNðN > 2Þ is a bounded smooth domain with 0 ∈Ω, f belongs to the Lebesgue space LmðΩÞ 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.


Introduction
The main purpose of this paper is to prove the existence of solutions to the following elliptic boundary value problem where Ω is a bounded smooth subset of ℝ N ðN > 2Þ with 0 ∈ Ω, and M : Ω ⟶ ℝ N 2 is a bounded measurable matrix, which satisfies the following conditions: there are two positive constants α and β, such that, for a.e.
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 ∈ L m ðΩÞðm ≥ 1Þ. The main features of this paper are the presence of the convection term div ðuEðxÞÞ, which leads to the noncoercivity of -div ðMðxÞ∇uÞ + div ðuEðxÞÞ in W 1,2 0 ðΩÞ. 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: where 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 ∈ L m ðΩÞ for 1 ≤ m ≤ p + 1/p and m > p + 1/p, respectively, rather than f ∈ L p+1/p ðΩÞ.

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 E. 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. Journal of Function Spaces

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).
The following is the definition of the truncation function.
Definition 6. For ∀k ≥ 0, s ∈ ℝ, the truncation function defined by Now, let us briefly recall the Sobolev's embedding theorem.
Lemma 7. Assume that p = 2, then there is a normal number S , such that, for ∀u ∈ C ∞ 0 ðℝ N Þ satisfies where The following Hölder inequality plays an important role in this paper.

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 juj p−1 u.

Lemma 10.
Let f ∈ L m ðΩÞ, m ≥ 1: Then, for every n ∈ ℕ, there exists a solution u n ∈ W 1,2 0 ðΩÞ to (19) Proof. In order to get the estimates (21), we will consider the following two cases separately. Case m > 1. Select ϕ = ju n j pðm−1Þ−1 u n as a test function in (19) 3 Journal of Function Spaces 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 ð where H = N − 2/2. Applying the Hölder inequality on the right-hand side of (27), we get where m ′ = m/m − 1, which together with (27), implies that (21) holds.
When m = 1, taking T k ðu n Þ/k 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 A < αðN − 2Þ/2, we obtain ð u n j j≤k u n j j p T k u n ð Þ k ≤ ð u n j j≤k Fatou lemma implies, for k ⟶ ∞, the expression of (21) holds.
Next, we will prove the following existence result. Proof. Set f n ðxÞ = f ðxÞ/1 + ð1/nÞjf ðxÞj obviously, f n ðxÞ ⟶ f ðxÞ in L 1 ðΩÞ as n ⟶ ∞. Let ϕ = T k ðu n Þ as a test function in (19), using the Hölder inequality and the Hardy inequality, we get