We study the regularity problem with non-homogeneous terms of p-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.
Citation: Qing Zhao, Shuhong Chen. Regularity for very weak solutions to elliptic equations of p-Laplacian type[J]. Electronic Research Archive, 2025, 33(6): 3482-3495. doi: 10.3934/era.2025154
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We study the regularity problem with non-homogeneous terms of p-Laplacian type, which is a still unsolved problem for nonlinear elliptic equations. The main results of this work are obtained by three steps. First, we use the Hodge decomposition theorem to construct a suitable test function that satisfies the solution definition. Second, by combining the solution definition with the Hodge decomposition theorem, we establish a properly formulated inverse Hölder inequality to enhance the integrability of the very weak solutions. Finally, through an iterative process, we show that the considered very weak solutions can be improved to classical weak solutions.
In this paper, we investigate the regularity theory of very weak solutions to the elliptic equations with p-Laplacian nonhomogeneous terms of the following type:
{divS(x,u,∇u)=div(|f|p−2f),in Ω, u=0,on ∂Ω, | (1.1) |
where Ω a Lipschitz domain in Rn with n⩾2, p∈(1,+∞), u is a vector-valued function taking values in Rn with N∈N, ∇u stands for the gradient matrix of u, and f:Ω→RnN is a given vector field.
To introduce solutions under the very weak formulation for the elliptic system (1.1), it is necessary to impose appropriate structural conditions on the operator S(x,u,⋅). To this end, we assume that the function S(x,u,∇u):Ω×RN×RnN→RnN satisfies the following structural assumptions (H1)–(H3) for a.e. x∈Ω and every u,u1,u2∈RN, z1,z2∈RnN. Here, u,u1 and u2 are arbitrary functions, and z,z1 and z2 are the gradient matrices corresponding to u,u1, and u2, respectively.
H1 (Coercivity assumption) There exists a positive constant ν such that
S(x,u,z)z⩾ν|z|p. |
H2 (Monotonicity assumption) There exists a positive constant β such that
(S(x,u1,z1)−S(x,u2,z2))(z1−z2)⩾β(|z1|+|z2|)p−2|z1−z2|2. |
H3 (Boundedness assumption) A positive constant γ can be chosen so that
|S(x,u,z)|⩽γ(|z|p−1+|u|μ(p−1)+ϕ(x)), |
where γ∈[β,+∞),μ∈(0,nn−p+1),ϕ(x)∈Lpp−1(Ω).
With these structural conditions, we are now in a position to define very weak solutions to the system (1.1).
Definition 1. A vector-valued function u∈W1,q0(Ω,RN), max{1,p−1}<q<p is called a very weak solution of (1.1) under the structural assumptions (H1)–(H3) if
∫ΩS(x,u,∇u)⋅∇φdx=∫Ω|f|p−2f⋅∇φdx | (1.2) |
holds for all test functions φ∈C∞0(Ω,RN).
The study of regularity theory is typically predicated on the existence of weak solutions. However, the existence of weak solutions remains unresolved for certain classes of equations within the current analytical framework. Prominent examples include elliptic equations with a p-Laplacian operator and a singular convection term.
Currently, for the p-Laplacian, the existence of solutions has been extensively studied [1,2,3,4]. However, under general structural assumptions, little is known about the existence of solutions when f∈Lq(Ω,RnN) with q<p and p≠2. Even for the simplest degenerate p-Laplacian system,
div(|∇u|p−2∇u)=div(|f|p−2f) |
subject to u=0 on Ω, the existence of solutions remains an open question when q<p and p=2.
Iwaniec [2] observed that the integral identity for weak solutions can still hold by weakening the integrability of these solutions. This observation led to the introduction of very weak solutions with exponents below the natural exponent. He further established the existence of such solutions for homogeneous p-harmonic equations. Bulíček and Schwarzacher [5] demonstrated the existence of very weak solutions to the system of p-Laplacian type
divS(x,∇u)=div|f|p−2f |
in Ω, subject to the boundary condition u=0 on ∂Ω. Chen and Guo [6] extended this result to Eq (1.1) for f∈Lq with q∈[p−λ,p], using the method of [5].
The existence results for very weak solutions naturally lead to the problem of characterizing their relationship to weak solutions. Iwaniec and Sbordone [7] showed that very weak solutions are in fact weak solutions for A-harmonic equations
divA(x,∇u)=0. |
Later, Kinnunen and Zhou [8] obtained the same result for the system
div(|∇u|p−2∇u)=0 |
when the exponent p is close to two. Greco and Verde [9] extended this result to the p-Laplacian system
div((G(x)∇u,∇u)p−22G(x)∇u)=0. |
Subsequently, similar results have been obtained for various systems, including p-Laplacian systems [10,11,12], elliptic systems [13,14,15,16,17], and parabolic systems [18,19,20].
We are now interested in whether very weak solutions to Eq (1.1) can be promoted to weak solutions when f∈Lq with q<p, given that the existence of these very weak solutions has been established in [6]. To address this issue, we aim to demonstrate the self-enhancement property of the gradients corresponding to the very weak solutions of (1.1).
The key challenge is to construct an appropriate test function, since conventional methods involving truncations and powers of u fail when the exponent is below the natural exponent p. Iwaniec and Sbordone [7] ingeniously applied the Hodge decomposition to address this challenge. Lewis [21] similarly overcame this difficulty by appealing to the Whitney extension theorem. This study adopts the method of Iwaniec and Sbordone, leveraging the stability of the Hodge-type decomposition to design an auxiliary function.
Compared to previous works, a key new aspect of our work is that an operator of A-harmonic S(x,u,∇u) in Eq (1.1) appears to be more general, as it depends not only on the gradient ∇u and additionally on u considered in the very weak framework. In this context, we refer to the operator S(x,u,∇u) in the system represented in (1.1) that satisfies the framework conditions (H1)–(H3) as an A-harmonic operator. Consequently, we must address the estimate issues arising from u under the structural assumption (H3). By applying regularity proof techniques, we establish a weak reverse Hölder inequality. Subsequently, we enhance the integrability exponent for very weak solutions of Eq (1.1) to the natural exponent p. These results extend the findings of [7]. And obtain the following main results.
Theorem 1. Let u∈W1,q0(Ω,RN) be a solution in the very weak sense for the system (1.1), subject to the structural conditions (H1)–(H3), where f∈Lq(Ω,RnN), q∈[p−λ,p], 1<p<+∞, 0<λ<1. This ensures the existence of an exponent q1 that satisfies p−λ⩽q1=q1(n,N,p,μ,β,γ)<p so that all very weak solutions u∈W1,q20(Ω) with q1⩽q2<p.
Corollary 1 Assuming that the hypotheses of Theorem 1 are fulfilled, one can find a constant q1=q1(n,N,p,μ,β,γ)<p such that for any solution u∈W1,q0(Ω) within the very weak solution framework corresponding to the elliptic model (1.1), if q1<q<p, then u∈W1,p0(Ω).
We now outline the structure of the remainder of this paper. In Section 2, we introduce the existence of very weak solutions to elliptic equations of p-Laplacian type (1.1), the theory of Hodge decomposition, a useful inequality, and a reverse Hölder inequality. In Section 3, we show that these very weak solutions to Eq (1.1) are, in fact, classical weak solutions.
In this section, we review several basic results and inequalities that will facilitate the proof of Theorem 1.
The first result, which forms the basis of our work, establishes the existence of very weak solutions to the p-Laplacian equations (1.1) when f∈Lq with q∈[p−λ,p]. This existence theorem was established through the selection of appropriately formed weight functions, in conjunction with methods such as the relative covering decomposition theorem, weighted techniques, and the divergence-curl lemma.
Lemma 1 ([6]). Let Ω⊂Rn be a bounded Lipschitz domain; the operator S(x,u,∇u) satisfy (H1)–(H3). Then, there exists a constant λ(ν,β,γ,n,N,p,Ω) such that for all q∈[p−λ,p](1<p<∞), the following results hold.
If f∈Lq(Ω,RnN), then there exists a very weak solution u∈W1,q0(Ω,RN) to the Eq (1.1).
Furthermore, there exist constants C(ν,β,γ,p,q,n,N,Ω) and ˉC(ν,β,γ,p,q,n,N,Ω) such that
‖∇u‖Lq(Ω,RnN)⩽C‖f‖Lq(Ω,RnN)+ˉC. |
The Hodge decomposition theorem serves as the key tool in constructing a suitable test function.
Lemma 2 ([7]). Let Ω⊂Rn be a domain with a regular boundary, where w∈W1,r0(Ω,RN),r>1 with λ satisfying −1<λ<r−1. This yields the existence of a function φ∈W1,r1+λ0(Ω,RN) and a matrix field with zero divergence H∈Lr1+λ(Ω,RnN) such that
|∇w|λ∇w=∇φ+H. | (2.1) |
Moreover,
‖H‖r1+λ⩽Cr(Ω,N)|λ|‖∇w‖1+λr. | (2.2) |
In Lemma 2, the most valuable case for the construction of a test function is when λ can be negative. Let p−1<r<p and u∈W1,r0(Ω,RN) is considered a solution in the very weak formulation to the p-Laplacian type equations (1.1), by letting λ=r−p, it follows that −1<λ<0. Hence φ in (2.1) may be employed as a test function within (1.2) since ∇φ∈Lr1+r−p(Ω,RnN).
The following inequality will be useful for estimating the left-hand side term in the Hodge-type decomposition.
Lemma 3 ([22]). Assume that X,Y∈Rn, where X,Y are nonzero vectors and λ lies in [0,1), then, we have
||X|−λX−|Y|−λY|≤2λ1+λ1−λ|X−Y|1−λ. |
Finally, we introduce a reverse Hölder inequality that implies the self-improvement of the integrability exponent of u(x). To clarify the notation used in the upcoming lemma, we recall that the average value of g(x) over a set X is defined as
−∫Xg(x)dx=1|X|∫Xg(x)dx, |
where X⊂Rn is a Lebesgue measurable set and |X| denotes the Lebesgue measure of X. In particular,
−∫MR(x0)g(x)dx=1αnRn∫MR(x0)g(x)dx, |
where, αn denotes the volume of the unit ball.
Lemma 4 ([23]). Let 0<R<R0⩽dist(x0,∂Ω), x0∈Ω. Suppose that u(x)∈Lp(MR(x0)), f(x)∈Lt(MR(x0)), t>p, 1<p<∞ satisfies the reverse Hölder inequality
−∫MR/2(x0)|u(x)|pdx⩽θ−∫MR(x0)|u(x)|pdx+C∗(−∫MR(x0)|u(x)|sdx)p/s+−∫MR(x0)|f(x)|pdx |
with 1⩽s<p, 0⩽θ<1. Then there exists a constant p′=p′(θ,p,n,C∗) with t⩾p′>p such that
u∈Lp′loc(Ω), |
and
(−∫MR/2(x0)|u(x)|p′dx)1/p′⩽C∗(−∫MR(x0)|u(x)|pdx)1/p+C∗(−∫MR(x0)|f(x)|p′dx)1/p′ |
where C∗=C∗(n,C∗,p,θ,R0).
In the present section, we first construct an appropriate test function using the Hodge decomposition and establish a reverse-form Hölder inequality for solutions within the very weak formulation associated with (1.1). We then explore the relationship between the aforementioned very weak solutions and standard weak solutions.
Proof of Theorem 1 By Lemma 1, there exists a very weak solution u∈W1,q0(Ω,RN) to the elliptic equations (1.1) under the structural assumptions (H1)–(H3), where f∈Lq(Ω,RnN), q∈[p−λ,p], λ=λ(ν,β,γ,n,N,p,Ω). To prove Theorem 1, the key is to enhance the integrability exponent of u∈W1,q0(Ω,RN). To this end, we need to construct an appropriate test function. For convenience, we set q=p−λ with 0<λ<12. Consequently, u∈W1,p−λ0(Ω,RN)(0<λ<12) is a very weak solution to (1.1). Let η(x)∈C∞0(MR(x0)) with 0<R<min{1,dist(x0,∂Ω)} be a truncation function with η(x) taking values in [0,1] and satisfying |∇η(x)|⩽4R and identically equal to 1 on MR/2(x0)
Based on the Hodge decomposition theorem in Lemma 2, for any 0<λ<12, there exists φ∈W1,p−λ1−λ0(Ω) and H∈Lp−λ1−λ(Ω) such that
|∇(η(x)u(x))|−λ∇(η(x)u(x))=∇φ+H | (3.1) |
and
‖H‖p−λ1−λ⩽C(n,p)λ‖∇(ηu)‖1−λp−λ, | (3.2) |
‖∇φ‖p−λ1−λ⩽C(n,p)‖∇(ηu)‖1−λp−λ. | (3.3) |
Since ∇φ∈Lp−λ1−λ(Ω), φ is appropriate to act as a test function within the framework of very weak solutions. Consequently,
∫MR(x0)S(x,u,∇u)⋅∇φdx=∫MR(x0)|f|p−2f⋅∇φdx. | (3.4) |
Let
E(η,u)=|∇(ηu)|−λ∇(ηu)−|η∇u|−λη∇u. | (3.5) |
By Lemma 3, we obtain that
|E(η,u)|⩽2λ1+λ1−λ|u∇η|1−λ. | (3.6) |
By combining Eq (3.1) with Eq (3.5), it follows that
∇φ=E(η,u)+|η∇u|−λη∇u−H. |
Substituting this equation into Eq (3.4), we obtain
∫MR(x0)S(x,u,∇u)⋅|η∇u|−λη∇udx=−∫MR(x0)S(x,u,∇u)⋅E(η,u)dx+∫MR(x0)S(x,u,∇u)⋅Hdx+∫MR(x0)|f|p−2f⋅∇φdx. |
Combining (H1), (H3), and (3.6), we can conclude that
ν∫MR(x0)η1−λ|∇u|p−λdx⩽∫MR(x0)S(x,u,∇u)|η∇u|−λη∇udx⩽∫MR(x0)|S(x,u,∇u)||E(η,u)|dx+∫MR(x0)|S(x,u,∇u)||H|dx+∫MR(x0)|f|p−1|∇φ|dx.⩽C1∫MR(x0)|∇u|p−1|u|1−λdx+C1∫MR(x0)|u|μ(p−1)|u|1−λdx+C1∫MR(x0)|ϕ(x)||u|1−λdx+γ∫MR(x0)|∇u|p−1|H|dx+γ∫MR(x0)|u|μ(p−1)|H|dx+γ∫MR(x0)|ϕ(x)||H|dx+∫MR(x0)|f|p−1|∇φ|dx⩽I1+I2+I3+I4+I5+I6+I7, | (3.7) |
where
I1=C1∫MR(x0)|∇u|p−1|u|1−λdx;I2=C1∫MR(x0)|u|μ(p−1)|u|1−λdx;I3=C1∫MR(x0)|ϕ(x)||u|1−λdx;I4=γ∫MR(x0)|∇u|p−1|H|dx;I5=γ∫MR(x0)|u|μ(p−1)|H|dx;I6=γ∫MR(x0)|ϕ(x)||H|dx;I7=∫MR(x0)|f|p−1|∇φ|dx; |
with
C1=2λ1+λ1−λ(4R)1−λγ. |
To derive a weak reverse Hölder inequality for |∇u|p−λ of the form presented in Lemma 4, we need to estimate I1−I7 appropriately.
We begin by estimating I1.
In view of Hölder's inequality with exponents
p′=n(p−λ)(n+1−λ)(p−1),q′=n(p−λ)(n−p+1)(1−λ), |
where 1<p′<∞, 1<q′<∞, and 1p′+1q′=1, and the Sobolev embedding theorem with exponent
p″=n(p−λ)n+1−λ, |
such that np″n−p″=n(p−λ)n−p+1, we find that
I1⩽C1(∫MR(x0)|∇u|n(p−λ)n+1−λdx)(n+1−λ)(p−1)n(p−λ)(∫MR(x0)|u|n(p−λ)n−p+1dx)(n−p+1)(1−λ)n(p−λ)⩽C1C1−λs(∫MR(x0)|∇u|n(p−λ)n+1−λdx)(n+1−λ)(p−1)n(p−λ)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)(n+1−λ)(1−λ)n(p−λ)=C1C1−λs(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn. |
Using the same techniques as for I1, we can estimate I2.
I2⩽C1∫MR(x0)|u|μ(p−1)+1−λdx⩽C1(∫MR(x0)|u|n(p−λ)n−p+1dx)(n−p+1)[μ(p−1)+1−λ]n(p−λ)(∫MR(x0)dx)(p−1)[n(1−μ)+μ(p−1)+1−λ]n(p−λ)⩽C1C(|MR(x0)|,Cs)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn(p−λ)[μ(p−1)+1−λ]. |
We now discuss the estimate of I2 for different values of μ.
If 0<μ⩽1, it is evident that
I2⩽C1C(|MR(x0)|,Cs)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn. |
If 1<μ<nn−p+1, we obtain
I2⩽C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn. |
Thus, we can conclude that
I2⩽C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn. |
By applying Hölder's inequality, Poincaré's inequality, and Young's inequality, we obtain
I3⩽C1(∫MR(x0)|ϕ(x)|p−λp−1dx)p−1p−λ(∫MR(x0)|u|p−λdx)1−λp−λ⩽C1C1−λp−λp(∫MR(x0)|ϕ(x)|p−λp−1dx)p−1p−λ(∫MR(x0)|∇u|p−λdx)1−λp−λ⩽C1C1−λp−λpε∫MR(x0)|∇u|p−λdx+C1C1−λp−λpC(ε)∫MR(x0)|ϕ(x)|p−λp−1dx. |
By Hölder's inequality and (3.2), we have
I4⩽γ(∫MR(x0)|∇u|p−λdx)p−1p−λ(∫MR(x0)|H|p−λ1−λdx)1−λp−λ⩽C(n,p)γλ(∫MR(x0)|∇u|p−λdx)p−1p−λ(∫MR(x0)|∇(ηu)|p−λdx)1−λp−λ. |
By employing Minkowski's inequality and Poincaré's inequality, we derive
(∫MR(x0)|∇(ηu)|p−λdx)1p−λ=(∫MR(x0)|u∇η+η∇u|p−λdx)1p−λ⩽(∫MR(x0)|u∇η|p−λdx)1p−λ+(∫MR(x0)|η∇u|p−λdx)1p−λ⩽4RC1p−λp(∫MR(x0)|∇u|p−λdx)1p−λ+(∫MR(x0)|∇u|p−λdx)1p−λ=(4RC1p−λp+1)(∫MR(x0)|∇u|p−λdx)1p−λ. | (3.8) |
Substituting (3.8) into the preceding estimate of I4, we further obtain
I4⩽C(n,p)(4RC1p−λp+1)1−λγλ(∫MR(x0)|∇u|p−λdx)p−1p−λ(∫MR(x0)|∇u|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λp+1)1−λγλ∫MR(x0)|∇u|p−λdx. |
By Hölder's inequality, (3.2) and (3.8), we can estimate
I5⩽γ(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ(∫MR(x0)|H|p−λ1−λdx)1−λp−λ⩽C(n,p)γλ(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ(∫MR(x0)|∇(ηu)|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λp+1)1−λγλ(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ(∫MR(x0)|∇u|p−λdx)1−λp−λ. |
As before, we consider μ in two cases.
Assuming 0<μ⩽1, and utilizing Hölder's, Young's, and Poincaré's inequalities, the following equation is obtained.
(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ⩽(∫MR(x0)|u|p−λdx)μ(p−1)p−λ(∫MR(x0)dx)(1−μ)(p−1)p−λ⩽(∫MR(x0)|u|p−λdx)p−1p−λ+(∫MR(x0)dx)p−1p−λ⩽Cp−1p−λp(∫MR(x0)|∇u|p−λdx)p−1p−λ+(∫MR(x0)dx)p−1p−λ. |
If 1<μ<nn−p+1, using the Sobolev embedding theorem, we can find
(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ⩽Cμ(p−1)s(∫MR(x0)|∇u|p−λdx)μ(p−1)p−λ⩽C(Cs,‖u‖W1,p−λ0)(∫MR(x0)|∇u|p−λdx)p−1p−λ. |
Combining the above two inequalities, we can deduce that
(∫MR(x0)|u|μ(p−λ)dx)p−1p−λ⩽C(Cp,Cs,‖u‖W1,p−λ0)(∫MR(x0)|∇u|p−λdx)p−1p−λ+(∫MR(x0)dx)p−1p−λ. |
Substituting the above inequality into the estimate for I5, we obtain the following bound:
I5⩽C(n,p,γ,CP,Cs,‖u‖W1,p−λ0)λ[∫MR(x0)|∇u|p−λdx+∫MR(x0)dx]. |
To estimate I6, we apply Hölder's inequality, (3.2), (3.8), and Young's inequality to obtain
I6⩽γ(∫MR(x0)|ϕ(x)|p−λp−1dx)p−1p−λ(∫MR(x0)|H|p−λ1−λdx)1−λp−λ⩽C(n,p)γλ(∫MR(x0)|ϕ(x)|p−λp−1dx)p−1p−λ(∫MR(x0)|∇(ηu)|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λP+1)1−λγλ(∫MR(x0)|ϕ(x)|p−λp−1dx)p−1p−λ(∫MR(x0)|∇u|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λP+1)1−λγλ(∫MR(x0)|∇u|p−λdx+∫MR(x0)|ϕ(x)|p−λp−1dx). |
Finally, by applying Hölder's inequality, (3.3), (3.8), and Young's inequality, we arrive at
I7⩽(∫MR(x0)|f|p−λdx)p−1p−λ(∫MR(x0)|∇φ|p−λ1−λdx)1−λp−λ⩽C(n,p)(∫MR(x0)|f|p−λdx)p−1p−λ(∫MR(x0)|∇(ηu)|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λP+1)1−λ(∫MR(x0)|f|p−λdx)p−1p−λ(∫MR(x0)|∇u|p−λdx)1−λp−λ⩽C(n,p)(4RC1p−λP+1)1−λ(ε∫MR(x0)|∇u|p−λdx+C(ε)∫MR(x0)|f|p−λdx). |
Substituting the estimates for I1 through I7 into (3.7) and using the definition of η, the final expression is derived.
ν∫MR/2(x0)|∇u|p−λdx⩽C1C1−λs(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn+C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)(∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn+C1C1−λp−λPε∫MR(x0)|∇u|p−λdx+C1C1−λp−λPC(ε)∫MR(x0)|ϕ(x)|p−λp−1dx+C(n,p)(4RC1p−λP+1)1−λγλ∫MR(x0)|∇u|p−λdx+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0)λ(∫MR(x0)|∇u|p−λdx+∫MR(x0)dx)+C(n,p)(4RC1p−λP+1)1−λγλ∫MR(x0)|∇u|p−λdx+C(n,p)(4RC1p−λP+1)1−λγλ∫MR(x0)|ϕ(x)|p−λp−1dx+C(n,p)(4RC1p−λP+1)1−λε∫MR(x0)|∇u|p−λdx+C(n,p)(4RC1p−λP+1)1−λC(ε)∫MR(x0)|f|p−λdx. |
Rearranging this inequality yields
ν∫MR/2(x0)|∇u|p−λdx⩽[(2C(n,p)(4RC1p−λP+1)1−λγ+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0))λ+(C1C1−λp−λP+C(n,p)(4RC1p−λP+1)1−λ)ε]∫MR(x0)|∇u|p−λdx+[C1C1−λs+C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)](∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn+[C1C1−λp−λPC(ε)+C(n,p)(4RC1p−λP+1)1−λγλ]∫MR(x0)|ϕ(x)|p−λp−1dx+C(n,p)(4RC1p−λP+1)1−λC(ε)∫MR(x0)|f|p−λdx+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0)λ∫MR(x0)dx. |
Dividing both sides of the above inequality by |MR(x0)|=αnRn, where αn denotes the volume of the unit ball, we obtain
ν−∫MR/2(x0)|∇u|p−λdx⩽2n[(2C(n,p)(4RC1p−λP+1)1−λγ+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0))λ+(C1C1−λp−λP+C(n,p)(4RC1p−λP+1)1−λ)ε]−∫MR(x0)|∇u|p−λdx+2n[C1C1−λs+C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)](αnRn)1−λn(−∫MR(x0)|∇u|n(p−λ)n+1−λdx)n+1−λn+2n[C1C1−λp−λPC(ε)+C(n,p)(4RC1p−λP+1)1−λγλ]−∫MR(x0)|ϕ(x)|p−λp−1dx+2nC(n,p)(4RC1p−λP+1)1−λC(ε)−∫MR(x0)|f|p−λdx+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0)λ. |
Choosing λ, ε small enough such that
2n[(2C(n,p)(4RC1p−λP+1)1−λγ+C(n,p,γ,CP,Cs,‖u‖W1,p−λ0))λ+(C1C1−λp−λP+C(n,p)(4RC1p−λP+1)1−λ)ε]<ν, |
and setting τ=n(p−λ)n+1−λ, then we can deduce that
−∫MR/2(x0)|∇u|p−λdx⩽θ−∫MR(x0)|∇u|p−λdx+M(−∫MR(x0)|∇u|τdx)p−λτ+−∫MR(x0)|F|p−λdx, |
with 0<θ<1, 1<τ<p−λ and here
M=2nν[C1C1−λs+C1C(|MR(x0)|,Cs,‖u‖W1,n(p−λ)n+1−λ0)](αnRn)1−λn, |
and
F={2nν[C1C1−λp−λPC(ε)+C(n,p)(4RC1p−λP+1)1−λγλ]|ϕ(x)|p−λp−1+2nνC(n,p)(4RC1p−λP+1)1−λC(ε)|f|p−λ+1νC(n,p,γ,CP,Cs,‖u‖W1,p−λ0)λ}1p−λ. |
Thus, by Lemma 4, one can find an exponent q′>q=p−λ for which u∈W1,q′0(Ω) holds. Note that f∈Lq(Ω,RnN) for q∈[p−λ,p]. By similar reasoning, we derive an alternative estimate comparable to the reverse Hölder inequality for |∇u|p−λ, where the exponents q′ and τ′ replace q=p−λ and τ, respectively. Specifically,
−∫MR/2(x0)|∇u|q′dx⩽θ−∫MR(x0)|∇u|q′dx+M(−∫MR(x0)|∇u|τ′dx)q′τ′+−∫MR(x0)|F|q′dx. |
We then obtain u∈W1,q″0(Ω) for some q″>q′. Moreover, the reverse Hölder inequality remains valid with the new exponents q″ and τ″ in place of q′ and τ′, respectively. Therefore, by repeating the above process, we can continuously improve the integrability of ∇u. Consequently, we infer that u∈W1,p0(Ω) for any u∈W1,q0(Ω) with q∈[p−λ,p).
The proof of Theorem 1 is completed.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The paper is supported by the National Natural Science Foundation of China (No: 12301585); the Natural Science Foundation of Fujian Province (No: 2024J01321).
The authors declare there is no conflicts of interest.
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