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Research article

Regularity for very weak solutions to elliptic equations of p-Laplacian type

  • Received: 10 March 2025 Revised: 28 April 2025 Accepted: 21 May 2025 Published: 05 June 2025
  • 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|p2f),in Ω,         u=0,on Ω, (1.1)

    where Ω a Lipschitz domain in Rn with n2, p(1,+), u is a vector-valued function taking values in Rn with NN, u stands for the gradient matrix of u, and f:ΩRnN 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×RnNRnN satisfies the following structural assumptions (H1)–(H3) for a.e. xΩ and every u,u1,u2RN, z1,z2RnN. 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)zν|z|p.

    H2 (Monotonicity assumption) There exists a positive constant β such that

    (S(x,u1,z1)S(x,u2,z2))(z1z2)β(|z1|+|z2|)p2|z1z2|2.

    H3 (Boundedness assumption) A positive constant γ can be chosen so that

    |S(x,u,z)|γ(|z|p1+|u|μ(p1)+ϕ(x)),

    where γ[β,+),μ(0,nnp+1),ϕ(x)Lpp1(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 uW1,q0(Ω,RN), max{1,p1}<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|p2fφdx (1.2)

    holds for all test functions φC0(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 fLq(Ω,RnN) with q<p and p2. Even for the simplest degenerate p-Laplacian system,

    div(|u|p2u)=div(|f|p2f)

    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|p2f

    in Ω, subject to the boundary condition u=0 on Ω. Chen and Guo [6] extended this result to Eq (1.1) for fLq with q[pλ,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|p2u)=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)p22G(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 fLq 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 uW1,q0(Ω,RN) be a solution in the very weak sense for the system (1.1), subject to the structural conditions (H1)–(H3), where fLq(Ω,RnN), q[pλ,p], 1<p<+, 0<λ<1. This ensures the existence of an exponent q1 that satisfies pλq1=q1(n,N,p,μ,β,γ)<p so that all very weak solutions uW1,q20(Ω) with q1q2<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 uW1,q0(Ω) within the very weak solution framework corresponding to the elliptic model (1.1), if q1<q<p, then uW1,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 fLq with q[pλ,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 ΩRn 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[pλ,p](1<p<), the following results hold.

    If fLq(Ω,RnN), then there exists a very weak solution uW1,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

    uLq(Ω,RnN)CfLq(Ω,RnN)+ˉC.

    The Hodge decomposition theorem serves as the key tool in constructing a suitable test function.

    Lemma 2 ([7]). Let ΩRn be a domain with a regular boundary, where wW1,r0(Ω,RN),r>1 with λ satisfying 1<λ<r1. This yields the existence of a function φW1,r1+λ0(Ω,RN) and a matrix field with zero divergence HLr1+λ(Ω,RnN) such that

    |w|λw=φ+H. (2.1)

    Moreover,

    Hr1+λCr(Ω,N)|λ|w1+λr. (2.2)

    In Lemma 2, the most valuable case for the construction of a test function is when λ can be negative. Let p1<r<p and uW1,r0(Ω,RN) is considered a solution in the very weak formulation to the p-Laplacian type equations (1.1), by letting λ=rp, it follows that 1<λ<0. Hence φ in (2.1) may be employed as a test function within (1.2) since φLr1+rp(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,YRn, where X,Y are nonzero vectors and λ lies in [0,1), then, we have

    ||X|λX|Y|λY|2λ1+λ1λ|XY|1λ.

    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 XRn is a Lebesgue measurable set and |X| denotes the Lebesgue measure of X. In particular,

    MR(x0)g(x)dx=1αnRnMR(x0)g(x)dx,

    where, αn denotes the volume of the unit ball.

    Lemma 4 ([23]). Let 0<R<R0dist(x0,Ω), x0Ω. 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)|pdxθMR(x0)|u(x)|pdx+C(MR(x0)|u(x)|sdx)p/s+MR(x0)|f(x)|pdx

    with 1s<p, 0θ<1. Then there exists a constant p=p(θ,p,n,C) with tp>p such that

    uLploc(Ω),

    and

    (MR/2(x0)|u(x)|pdx)1/pC(MR(x0)|u(x)|pdx)1/p+C(MR(x0)|f(x)|pdx)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 uW1,q0(Ω,RN) to the elliptic equations (1.1) under the structural assumptions (H1)–(H3), where fLq(Ω,RnN), q[pλ,p], λ=λ(ν,β,γ,n,N,p,Ω). To prove Theorem 1, the key is to enhance the integrability exponent of uW1,q0(Ω,RN). To this end, we need to construct an appropriate test function. For convenience, we set q=pλ with 0<λ<12. Consequently, uW1,pλ0(Ω,RN)(0<λ<12) is a very weak solution to (1.1). Let η(x)C0(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 φW1,pλ1λ0(Ω) and HLpλ1λ(Ω) such that

    |(η(x)u(x))|λ(η(x)u(x))=φ+H (3.1)

    and

    Hpλ1λC(n,p)λ(ηu)1λpλ, (3.2)
    φpλ1λC(n,p)(ηu)1λpλ. (3.3)

    Since φLpλ1λ(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|p2fφdx. (3.4)

    Let

    E(η,u)=|(ηu)|λ(ηu)|ηu|ληu. (3.5)

    By Lemma 3, we obtain that

    |E(η,u)|2λ1+λ1λ|uη|1λ. (3.6)

    By combining Eq (3.1) with Eq (3.5), it follows that

    φ=E(η,u)+|ηu|ληuH.

    Substituting this equation into Eq (3.4), we obtain

    MR(x0)S(x,u,u)|ηu|ληudx=MR(x0)S(x,u,u)E(η,u)dx+MR(x0)S(x,u,u)Hdx+MR(x0)|f|p2fφdx.

    Combining (H1), (H3), and (3.6), we can conclude that

    νMR(x0)η1λ|u|pλdxMR(x0)S(x,u,u)|ηu|ληudxMR(x0)|S(x,u,u)||E(η,u)|dx+MR(x0)|S(x,u,u)||H|dx+MR(x0)|f|p1|φ|dx.C1MR(x0)|u|p1|u|1λdx+C1MR(x0)|u|μ(p1)|u|1λdx+C1MR(x0)|ϕ(x)||u|1λdx+γMR(x0)|u|p1|H|dx+γMR(x0)|u|μ(p1)|H|dx+γMR(x0)|ϕ(x)||H|dx+MR(x0)|f|p1|φ|dxI1+I2+I3+I4+I5+I6+I7, (3.7)

    where

    I1=C1MR(x0)|u|p1|u|1λdx;I2=C1MR(x0)|u|μ(p1)|u|1λdx;I3=C1MR(x0)|ϕ(x)||u|1λdx;I4=γMR(x0)|u|p1|H|dx;I5=γMR(x0)|u|μ(p1)|H|dx;I6=γMR(x0)|ϕ(x)||H|dx;I7=MR(x0)|f|p1|φ|dx;

    with

    C1=2λ1+λ1λ(4R)1λγ.

    To derive a weak reverse Hölder inequality for |u|pλ of the form presented in Lemma 4, we need to estimate I1I7 appropriately.

    We begin by estimating I1.

    In view of Hölder's inequality with exponents

    p=n(pλ)(n+1λ)(p1),q=n(pλ)(np+1)(1λ),

    where 1<p<, 1<q<, and 1p+1q=1, and the Sobolev embedding theorem with exponent

    p=n(pλ)n+1λ,

    such that npnp=n(pλ)np+1, we find that

    I1C1(MR(x0)|u|n(pλ)n+1λdx)(n+1λ)(p1)n(pλ)(MR(x0)|u|n(pλ)np+1dx)(np+1)(1λ)n(pλ)C1C1λs(MR(x0)|u|n(pλ)n+1λdx)(n+1λ)(p1)n(pλ)(MR(x0)|u|n(pλ)n+1λdx)(n+1λ)(1λ)n(pλ)=C1C1λs(MR(x0)|u|n(pλ)n+1λdx)n+1λn.

    Using the same techniques as for I1, we can estimate I2.

    I2C1MR(x0)|u|μ(p1)+1λdxC1(MR(x0)|u|n(pλ)np+1dx)(np+1)[μ(p1)+1λ]n(pλ)(MR(x0)dx)(p1)[n(1μ)+μ(p1)+1λ]n(pλ)C1C(|MR(x0)|,Cs)(MR(x0)|u|n(pλ)n+1λdx)n+1λn(pλ)[μ(p1)+1λ].

    We now discuss the estimate of I2 for different values of μ.

    If 0<μ1, it is evident that

    I2C1C(|MR(x0)|,Cs)(MR(x0)|u|n(pλ)n+1λdx)n+1λn.

    If 1<μ<nnp+1, we obtain

    I2C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)(MR(x0)|u|n(pλ)n+1λdx)n+1λn.

    Thus, we can conclude that

    I2C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)(MR(x0)|u|n(pλ)n+1λdx)n+1λn.

    By applying Hölder's inequality, Poincaré's inequality, and Young's inequality, we obtain

    I3C1(MR(x0)|ϕ(x)|pλp1dx)p1pλ(MR(x0)|u|pλdx)1λpλC1C1λpλp(MR(x0)|ϕ(x)|pλp1dx)p1pλ(MR(x0)|u|pλdx)1λpλC1C1λpλpεMR(x0)|u|pλdx+C1C1λpλpC(ε)MR(x0)|ϕ(x)|pλp1dx.

    By Hölder's inequality and (3.2), we have

    I4γ(MR(x0)|u|pλdx)p1pλ(MR(x0)|H|pλ1λdx)1λpλC(n,p)γλ(MR(x0)|u|pλdx)p1pλ(MR(x0)|(ηu)|pλdx)1λpλ.

    By employing Minkowski's inequality and Poincaré's inequality, we derive

    (MR(x0)|(ηu)|pλdx)1pλ=(MR(x0)|uη+ηu|pλdx)1pλ(MR(x0)|uη|pλdx)1pλ+(MR(x0)|ηu|pλdx)1pλ4RC1pλp(MR(x0)|u|pλdx)1pλ+(MR(x0)|u|pλdx)1pλ=(4RC1pλp+1)(MR(x0)|u|pλdx)1pλ. (3.8)

    Substituting (3.8) into the preceding estimate of I4, we further obtain

    I4C(n,p)(4RC1pλp+1)1λγλ(MR(x0)|u|pλdx)p1pλ(MR(x0)|u|pλdx)1λpλC(n,p)(4RC1pλp+1)1λγλMR(x0)|u|pλdx.

    By Hölder's inequality, (3.2) and (3.8), we can estimate

    I5γ(MR(x0)|u|μ(pλ)dx)p1pλ(MR(x0)|H|pλ1λdx)1λpλC(n,p)γλ(MR(x0)|u|μ(pλ)dx)p1pλ(MR(x0)|(ηu)|pλdx)1λpλC(n,p)(4RC1pλp+1)1λγλ(MR(x0)|u|μ(pλ)dx)p1pλ(MR(x0)|u|pλdx)1λpλ.

    As before, we consider μ in two cases.

    Assuming 0<μ1, and utilizing Hölder's, Young's, and Poincaré's inequalities, the following equation is obtained.

    (MR(x0)|u|μ(pλ)dx)p1pλ(MR(x0)|u|pλdx)μ(p1)pλ(MR(x0)dx)(1μ)(p1)pλ(MR(x0)|u|pλdx)p1pλ+(MR(x0)dx)p1pλCp1pλp(MR(x0)|u|pλdx)p1pλ+(MR(x0)dx)p1pλ.

    If 1<μ<nnp+1, using the Sobolev embedding theorem, we can find

    (MR(x0)|u|μ(pλ)dx)p1pλCμ(p1)s(MR(x0)|u|pλdx)μ(p1)pλC(Cs,uW1,pλ0)(MR(x0)|u|pλdx)p1pλ.

    Combining the above two inequalities, we can deduce that

    (MR(x0)|u|μ(pλ)dx)p1pλC(Cp,Cs,uW1,pλ0)(MR(x0)|u|pλdx)p1pλ+(MR(x0)dx)p1pλ.

    Substituting the above inequality into the estimate for I5, we obtain the following bound:

    I5C(n,p,γ,CP,Cs,uW1,pλ0)λ[MR(x0)|u|pλdx+MR(x0)dx].

    To estimate I6, we apply Hölder's inequality, (3.2), (3.8), and Young's inequality to obtain

    I6γ(MR(x0)|ϕ(x)|pλp1dx)p1pλ(MR(x0)|H|pλ1λdx)1λpλC(n,p)γλ(MR(x0)|ϕ(x)|pλp1dx)p1pλ(MR(x0)|(ηu)|pλdx)1λpλC(n,p)(4RC1pλP+1)1λγλ(MR(x0)|ϕ(x)|pλp1dx)p1pλ(MR(x0)|u|pλdx)1λpλC(n,p)(4RC1pλP+1)1λγλ(MR(x0)|u|pλdx+MR(x0)|ϕ(x)|pλp1dx).

    Finally, by applying Hölder's inequality, (3.3), (3.8), and Young's inequality, we arrive at

    I7(MR(x0)|f|pλdx)p1pλ(MR(x0)|φ|pλ1λdx)1λpλC(n,p)(MR(x0)|f|pλdx)p1pλ(MR(x0)|(ηu)|pλdx)1λpλC(n,p)(4RC1pλP+1)1λ(MR(x0)|f|pλdx)p1pλ(MR(x0)|u|pλdx)1λpλC(n,p)(4RC1pλP+1)1λ(εMR(x0)|u|pλdx+C(ε)MR(x0)|f|pλdx).

    Substituting the estimates for I1 through I7 into (3.7) and using the definition of η, the final expression is derived.

    νMR/2(x0)|u|pλdxC1C1λs(MR(x0)|u|n(pλ)n+1λdx)n+1λn+C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)(MR(x0)|u|n(pλ)n+1λdx)n+1λn+C1C1λpλPεMR(x0)|u|pλdx+C1C1λpλPC(ε)MR(x0)|ϕ(x)|pλp1dx+C(n,p)(4RC1pλP+1)1λγλMR(x0)|u|pλdx+C(n,p,γ,CP,Cs,uW1,pλ0)λ(MR(x0)|u|pλdx+MR(x0)dx)+C(n,p)(4RC1pλP+1)1λγλMR(x0)|u|pλdx+C(n,p)(4RC1pλP+1)1λγλMR(x0)|ϕ(x)|pλp1dx+C(n,p)(4RC1pλP+1)1λεMR(x0)|u|pλdx+C(n,p)(4RC1pλP+1)1λC(ε)MR(x0)|f|pλdx.

    Rearranging this inequality yields

    νMR/2(x0)|u|pλdx[(2C(n,p)(4RC1pλP+1)1λγ+C(n,p,γ,CP,Cs,uW1,pλ0))λ+(C1C1λpλP+C(n,p)(4RC1pλP+1)1λ)ε]MR(x0)|u|pλdx+[C1C1λs+C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)](MR(x0)|u|n(pλ)n+1λdx)n+1λn+[C1C1λpλPC(ε)+C(n,p)(4RC1pλP+1)1λγλ]MR(x0)|ϕ(x)|pλp1dx+C(n,p)(4RC1pλP+1)1λC(ε)MR(x0)|f|pλdx+C(n,p,γ,CP,Cs,uW1,pλ0)λMR(x0)dx.

    Dividing both sides of the above inequality by |MR(x0)|=αnRn, where αn denotes the volume of the unit ball, we obtain

    νMR/2(x0)|u|pλdx2n[(2C(n,p)(4RC1pλP+1)1λγ+C(n,p,γ,CP,Cs,uW1,pλ0))λ+(C1C1λpλP+C(n,p)(4RC1pλP+1)1λ)ε]MR(x0)|u|pλdx+2n[C1C1λs+C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)](αnRn)1λn(MR(x0)|u|n(pλ)n+1λdx)n+1λn+2n[C1C1λpλPC(ε)+C(n,p)(4RC1pλP+1)1λγλ]MR(x0)|ϕ(x)|pλp1dx+2nC(n,p)(4RC1pλP+1)1λC(ε)MR(x0)|f|pλdx+C(n,p,γ,CP,Cs,uW1,pλ0)λ.

    Choosing λ, ε small enough such that

    2n[(2C(n,p)(4RC1pλP+1)1λγ+C(n,p,γ,CP,Cs,uW1,pλ0))λ+(C1C1λpλP+C(n,p)(4RC1pλP+1)1λ)ε]<ν,

    and setting τ=n(pλ)n+1λ, then we can deduce that

    MR/2(x0)|u|pλdxθMR(x0)|u|pλdx+M(MR(x0)|u|τdx)pλτ+MR(x0)|F|pλdx,

    with 0<θ<1, 1<τ<pλ and here

    M=2nν[C1C1λs+C1C(|MR(x0)|,Cs,uW1,n(pλ)n+1λ0)](αnRn)1λn,

    and

    F={2nν[C1C1λpλPC(ε)+C(n,p)(4RC1pλP+1)1λγλ]|ϕ(x)|pλp1+2nνC(n,p)(4RC1pλP+1)1λC(ε)|f|pλ+1νC(n,p,γ,CP,Cs,uW1,pλ0)λ}1pλ.

    Thus, by Lemma 4, one can find an exponent q>q=pλ for which uW1,q0(Ω) holds. Note that fLq(Ω,RnN) for q[pλ,p]. By similar reasoning, we derive an alternative estimate comparable to the reverse Hölder inequality for |u|pλ, where the exponents q and τ replace q=pλ and τ, respectively. Specifically,

    MR/2(x0)|u|qdxθMR(x0)|u|qdx+M(MR(x0)|u|τdx)qτ+MR(x0)|F|qdx.

    We then obtain uW1,q0(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 uW1,p0(Ω) for any uW1,q0(Ω) with q[pλ,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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