On substructures of semigroups of inductive terms

1.   Introduction

This article deals with the nonlinear elliptic equation with measure data of the type

Here $\Omega\subset \mathbb{R}^n$ ($n\geq2$) is a bounded domain, the unknown $u\in W^{1, P}(\Omega)$ with an $N$-function $P(\cdot)$ which will be introduced in Section 2, and $\mathcal{A}(x, \nabla u)\in L_{\operatorname{loc}}^{1}(\Omega, \mathbb{R}^{n})$. In (1.1), $\mu$ is a Radon measure defined on $\Omega$ with finite total mass $|\mu|(\Omega) < \infty$. Moreover, if the measure $\mu$ is actually an $L^1$-function, then

for a measurable subset $Q\subset\Omega$. We assume that measure $\mu$ satisfies the following condition:

(M) There exists some $\theta_0\in(0, n-1)$ such that

holds for all $x_{0}\in\Omega$ and every $R > 0$.

In this article, we assume that the Carathéodory vector field $\mathcal{A}:\Omega\times\mathbb{R}^n\rightarrow \mathbb{R}^n$ is $C^1$-regular in the gradient variable $\xi$, and satisfies $\mathcal{A}(x, 0) = 0$, and the growth, ellipticity, and continuity conditions, i.e., there are constants $0\leq s\leq 1$, $0 < \nu\leq L$, and $K\geq 1$ such that

for any $x, \, y\in\Omega$ and $\xi, \, \xi_{1}, \, \xi_{2}\in \mathbb{R}^n$. In (A1), the function $\omega:[0, \infty)\rightarrow [0, \infty)$ has the following hypothesis.

Moreover, $\omega$ is assumed to satisfy the Dini-continuous condition:

for every $0 < R\leq 1$.

A significant example of (1.1) is the $p$-Laplacian type equation, for which $s = 0$, $p\in (1, \infty)$, $P(x) = x^p$, and $\mathcal{A}(\nabla u) = |\nabla u|^{p-2}\nabla u$. Then Eq (1.1) can be expressed as

The relevant research on the regularities of solutions to elliptic equations starts with Kilpeläinen and Malý ^[1,2], and extends with a different technique by Trudinger and Wang ^[3]. Later, Duzaar and Mingione make a further study in ^[4]. Those results show a standard fact that solutions to non-homogeneous $p$-Laplacian-type equations with measure data can be pointwise estimated in a natural way by involving the classical nonlinear Wolff potential $\mathbf{W}_{\beta, \, p} ^\mu(x, R)$ ^[5], that is,

Based on the relationship between Wolff potential and Riesz potential, Mingione et al. ^[6,7] find out pointwise gradient estimates hold for general quasilinear degenerate equations by applying the Riesz potential

Baroni proves pointwise gradient bounds for solutions in terms of linear Riesz potentials in ^[8]. In addition, the caloric Riesz potential serves as a means for pointwise estimation of the spatial gradient of solutions to nonlinear degenerate parabolic equations ^[9].

Further, pointwise gradient estimates via the nonlinear Wolff potentials for weak solutions to various quasilinear elliptic equations with measure data are obtained by Mingione ^[10] and Yao ^[11]. More generally, pointwise potential estimates for elliptic equations and systems with Orlicz growth are studied in ^[12,13,14], respectively.

In recent years, a great deal of effort has gone into investigating nonlinear elliptic equations and systems involving measure data. Chilebicka et al. ^[15] study estimates including precise continuity and Hölder continuity criteria by the means of potential of a Wolff type; they also provide regularity estimates of the solutions and their gradients in the generalized Marcinkiewicz scale ^[16]. The existence of solutions in the framework of renormalized solutions is introduced in ^[17]. There are many interesting results in ^[18,19,20].

In this article, a weak solution to (1.1) is a function $u\in W^{1, P}(\Omega)$ such that

whenever $\varphi\in C_0^\infty(\Omega)$. Inspired by Mingione et al. ^[4,10,21], the main objective is to present pointwise potential estimates and interior Hölder continuity of weak solutions to (1.1) by using (1.4) in the Sobolev-Orlicz spaces.

We state our pointwise estimates in Theorems 1.1 and 1.2. It is important to note that $\theta_0$ in the following theorems will be introduced in (1.3). The first main result is the gradient pointwise estimates of $u$ as follows.

Theorem 1.1. Let $u\in C^1(\Omega)\cap W^{1, P}(\Omega)$ be a weak solution to (1.1) under the assumptions (A1), (A2), (A3), Radon measure $\mu$ satisfy $\mu\in L^1(\Omega)$ and ${\text{(M)}}$, and $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$ satisfying Assumption 2.4 and (2.12). There exists a constant $C\equiv C(n, \, \nu, \, L, \, K, \, s, \, \theta_0, \, C_{\Delta_2}, \, C_1) > 0$ and a positive radius $\widetilde{R} < 1$ such that the pointwise estimate

holds whenever $B(x_0, R)\subset\Omega$ and $0 < R\leq\widetilde{R}$. In (1.6), $\rlap{-} {\int}$ denotes integral average, $\Delta_2$-condition, $C_{\Delta_2}$, and $C_1$ will be described in Definition 2.3 and Lemma 2.5, respectively.

On the basis of Theorem 1.1, we demonstrate the pointwise estimate of $u$.

Theorem 1.2. Let $u\in C^0(\Omega)\cap W^{1, P}(\Omega)$ be a weak solution to (1.1) under the assumptions ${\text{(A1)}}_1$, ${\text{(A1)}}_2$, Radon measure $\mu$ satisfy $\mu\in L^1(\Omega)$ and ${\text{(M)}}$, and $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$ satisfying Assumption 2.4, and (2.12). There exists a constant $C\equiv C(n, \, \nu, \, L, \, \theta_0, \, C_{\Delta_2}, \, C_1, \, \mathrm{diam}(\Omega)) > 0$ and a positive radius $R < 1$ such that for every $r\leq R$ the pointwise estimate

holds, where $\varepsilon_0$ and $\alpha$ will be introduced in Lemmas 2.5 and 3.3, respectively.

Inspired by Mingione's result ^[10], the following theorem expounds the Hölder continuity of the solution $u$.

Theorem 1.3 (Interior Hölder type estimate). Let $u\in C^0(\Omega)\cap W^{1, P}(\Omega)$ be a weak solution to (1.1) under the assumptions (A1), (A2), let $\mu$ be a Radon measure satisfying $\mu\in L^1(\Omega)$ and ${\text{(M)}}$, and let $P$ be an $N$-function satisfying $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4, and (2.12). Then there exists constants $\alpha\in[0, 1)$ and $0 < R < 1$ such that for every $x, \, y\in B\left(x_0, 2R\right)\subset\Omega$, there holds

where the constant $C$ depends on $n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1$, and $\mathrm{diam}(\Omega)$.

Remark 1.4. Drawing upon the Riesz potential, our results present pointwise estimates and Hölder continuity within the more generalized framework of Sobolev-Orlicz spaces. Notably, (1.6) provides an estimate of $\nabla u$, whereas (1.7) estimates $u$ itself. We shall leverage Lemma 2.11 to relate $u$ to $\nabla u$.

In this article, we adopt several technical tools and methods in Sobolev-Orlicz spaces, and explore the properties of solutions of homogeneous equations to those of inhomogeneous equations with measure data. We first establish Proposition 3.2, which reveals the density of the Riesz potential and serves as a crucial conclusion in facilitating the proof of subsequent comparison lemmas. The proof of the comparison estimate is divided into two intricate steps; the first step requires Sobolev-type embedding, while the second primarily employs scaling changes. Our primary objective is to prove pointwise potential estimates and Hölder continuity, which is shown essentially by oscillation estimates of solutions. By utilizing summation methods and involving the Riesz potential, we proceed from deriving the estimates of the gradient of $u$ to those of $u$ itself. Ultimately, we employ the sharp maximal functions, and achieve the interior Hölder estimates of solutions.

This article is organized as follows: In Section 2, we state fundamental tools and definitions such as $N$-functions and maximal functions. Section 3 is devoted to the proof of Lemmas 3.4 and 3.9, and Section 4 presents supporting results, to gather Caccioppoli-type inequality and maximal estimate towards the proof of the main theorems. In the last section, we present the proofs of Theorems 1.1–1.3, respectively.

2.   Preliminaries

In this section, we give the definitions and tools of $N$-functions, function spaces; classical inequalities, and maximal functions.

2.1.   $N$-functions

The following definitions and results are standard in the context of $N$-function; see ^[22].

Definition 2.1. A function $P:[0, \infty)\rightarrow [0, \infty)$ is said to be an $N$-function, if $P$ is convex, differentiable, and its derivative $P'$ is a right continuous, non-decreasing function satisfying that $P'(0) = 0$ and $P'(t) > 0$ for $t > 0$.

Definition 2.2. The complementary function $\widetilde{P}:[0, \infty)\rightarrow [0, \infty)$ is defined by

Definition 2.3. We say that $P$ satisfied the $\Delta_2$-condition, if there exists $C_{\Delta_2} > 0$ such that

for all $t\geq 0$. By $\Delta_2(P)$ we denote the smallest constant $C_{\Delta_2}$. The function $P$ is said to satisfy the $\nabla_2$-condition, if $\widetilde{P}\in\Delta_2$. Then we define

By the $\Delta_2$-condition of Definition 2.3, we can easily obtain

for every $x, \, y\geq 0$.

Assumption 2.4. Let $P$ be a convex function that satisfies $\Delta_2({P, \widetilde{P}}) < \infty$ as (2.1), and $P$ is $C^2$ on $(0, \infty)$. Moreover, let $P'(0) = 0$, $\lim_{t\rightarrow \infty}P'(t) = \infty$ and uniformly in $t\geq 0$

Assumption 2.4 assures that $P$ is an $N$-function.

We let $P^{-1}:[0, \infty)\rightarrow [0, \infty)$ be the right-continuous inverse function of $P$, and $(P')^{-1}:[0, \infty)\rightarrow [0, \infty)$ the inverse function of $P'$. Then $\widetilde{\widetilde{P}(t)} = P(t)$ and $(\widetilde{P})'(t) = (P')^{-1}(t)$ hold. By ^[23], one has

hold uniformly in $t\geq 0$. By (2.2) and (2.3), we have

Lemma 2.5. ^[23] Let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$. Then there exist $\varepsilon_0 > 0$, $C_1 > 0$ which only depend on $\Delta_2({P, \widetilde{P}})$ such that for all $t\geq 0$ and all $\lambda\in[0, 1]$, one has

From Definition 2.2 of $N$-function $\widetilde{P}(\cdot)$, it is easy to obtain that (2.5) holds for $\widetilde{P}$, i.e.,

Using (2.3) and (2.5), it is not difficult to obtain

In this article, we denote $a\sim b$ by $C\, b\leq a\leq C'\, b$ for two constants $C$ and $C'$.

Lemma 2.6. ^[23] Let $P$ be an $N$-function under Assumption 2.4, and $\mathcal{A}$ satisfies the continuity and growth condition of ${\text{(A1)}}_1$, ${\text{(A1)}}_2$. Then

Moreover,

uniformly in $\xi\in \mathbb{R}^n$ and $x\in\Omega$.

Example 2.7. ^[24] Assume that $P$ is a Young function such that

where $p_1 > 1$ and $p_2\in\mathbb{R}$. The derivative $P'$ of $P$ is as follow:

The complementary function $\widetilde{P}$ satisfies

It is not difficult to verify that $P(\cdot)$ is an $N$-function satisfying Definitions 2.1–2.3, and Assumption 2.4.

2.2.   Function spaces

In this article, we need the following definitions of function spaces. The classical Orlicz spaces $L^{P}(\mathbb{R}^n)$ with its norm are given via ^[22]

and

If $\|f(x)\|_{L^{P}(\mathbb{R}^n)}$ is finite, then $f(x)\in L^{P}(\mathbb{R}^n)$. The Sobolev-Orlicz spaces $W^{1, P}(\mathbb{R}^n)$ and its norm are given by ^[25]

and

If $\|f(x)\|_{W^{1, P}(\mathbb{R}^n)}$ is finite, then $f(x)\in W^{1, P}(\mathbb{R}^n)$. Both $L^{P}(\mathbb{R}^n)$ and $W^{1, P}(\mathbb{R}^n)$ are Banach spaces.

2.3.   Classical inequalities

In this subsection, we recall several classical inequalities.

Lemma 2.8. (Young's inequality ^[26]) For all $\varepsilon > 0$, there exist $C_\varepsilon$, $\widetilde{C}_\varepsilon$ depending on $\Delta_2({P, \widetilde{P}})$, such that for all $\zeta_1, \, \zeta_2\geq 0$, there holds

and

Lemma 2.9. Let $P$ be an $N$-function with the $\Delta_2$-condition. Then for all $\varepsilon > 0$, there exists a constant $C$ such that

for $x, \, y > 0$, where the constant $C$ depends on $C_{\Delta_2}$ and $C_1$.

Proof. By the mean value theorem, for $x > y > 0$, there exists $\lambda_0\in(0, 1)$ such that

We use (2.4) and (2.7) to get

Then applying (2.3) and (2.10), we obtain that

We complete the proof of Lemma 2.9. □

Let $B$ be a measurable set with positive measure, and $f:\, B\rightarrow \mathbb{R}^n$ a measurable function. We denote the integral average of $f$ by

Lemma 2.10. (Jesen's inequality ^[27]) Let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$. If $f\in W^{1, P}(B(x, R))$, then there exists $C\equiv C(n)$ for $B(x, R)\subset\Omega$ such that

Lemma 2.11. (Sobolev-Poincaré's inequality ^[27]) Let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$ and satisfy Assumption 2.4. If $f\in W^{1, P}(B(x, R))$, then there exist $0 < \theta_1 < 1$ and $C > 0$ such that

holds whenever $B(x, R)\subset\Omega$.

The following lemma describes an embedding into a space of continuous functions; see Theorem 8.39 in ^[28] and (2.22) in ^[24].

Lemma 2.12. If an $N$-function $P$ satisfies that

then $W^{1, P}_{0}(\Omega)\hookrightarrow C^{0}(\Omega)\cap L^{\infty}(\Omega)$, that is, there exists a constant $C\equiv C(n)$ such that

for all $f\in W^{1, P}(\Omega)$.

The following iteration lemma plays an essential role in proving the Caccioppoli-type inequality (4.1).

Lemma 2.13. (Iteration lemma ^[29]) Let $f:[\gamma R, R]\rightarrow [0, \infty)$ be a bounded function such that the inequality

holds for fixed constants $C_2, \, \kappa\geq 0$, and $\gamma R\leq\varrho\leq r\leq R$ with $0 < \gamma < 1$. Then we have

for a constant $C$ depending only on $\kappa$.

2.4.   Maximal functions and sharp maximal functions

Throughout this subsection, we provide powerful tools for Hölder estimate of Theorem 1.3. Analogously to the definition of the classical maximal operator in ^[30], we define the generalized maximal operator and sharp maximal function in Sobolev-Orlicz spaces.

Definition 2.14. ^[31] Let $-1 < \tau < n$, $R < \mathrm{dist}(x, \partial\Omega), \, x\in\Omega$, where $\partial\Omega$ is the boundary of $\Omega$. Let $f$ be a function in Orlicz space $L^P(\Omega)$ or a measure with finite mass, and $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$. The function defined by

is called the restricted fractional $\tau$ generalized maximal function of $f$.

Definition 2.15. ^[31] Let $\beta\in(0, 1)$, $x\in\Omega$, and $R < \mathrm{dist}(x, \partial\Omega)$, let $f\in L^P(\Omega)$ and $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$. The function defined by

is called the restricted fractional $\beta$ generalized sharp maximal function of $f$. For $\theta > 0$, we also denote

The following note gives us a connection between maximal functions and sharp maximal functions.

Remark 2.16. Combining the generalized sharp maximal functions and Lemma 2.11, we see that if Assumption 2.4 holds, then it follows that

3.   A comparison argument

In this section, we collect the relevant difference estimates and decay estimates, and consider the density of the Riesz potential $\mathbf{I}_{\beta}^\mu$ explicitly. We define $v\in u+W_{0}^{1, P}(B(x_{0}, 2R))$ as the unique solution to the homogeneous Dirichlet problem

The existence of $v$ is guaranteed by a standard monotonicity argument; see ^[13]. One obtains the following control estimate.

Lemma 3.1. Let $u\in W^{1, P}(\Omega)$ be as in Theorem 1.1 satisfying the continuity and growth condition of (A1), $v\in u+W_0^{1, P}(B(x_{0}, 2R))$ be a solution to $(3.13)_1$, let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$ and satisfy Assumption 2.4. Then the following estimate

holds, where $C$ depends on $n, \, \nu, \, s, \, L$.

We consider the estimate of the difference of a solution to (1.1), and that of the corresponding solution to the Dirichlet problem (3.1).

Proposition 3.2. Let $u\in W^{1, P}(\Omega)$ be as in Theorem 1.1 satisfying ${\text{(A1)}}_2$, $v\in u+W_0^{1, P}(B(x_{0}, 2R))$ be a solution to (3.1), let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$, and satisfy Assumption 2.4, (2.12). Radon measure $\mu$ satisfies ${\text{(M)}}$. There exists a constant $C\equiv C(n, \, \nu, \, C_1)$ such that

Proof. Without loss of generality, we first assume that $B(x_{0}, 2R)\equiv B(0, 1)$ and $|\mu|(B(0, 1)) = 1$. Then we shall remove these two conditions for general situations.

Step 1. We assume that $B(x_{0}, 2R)\equiv B(0, 1)$ with $|\mu|(B(0, 1)) = 1$. By choosing $\varphi\equiv u-v$ as a test function in (1.5), and using (1.1) and $(3.1)_1$, we obtain

By using the ellipticity assumption ${\text{(A1)}}_2$, we deduce that

By (3.4) and Lemma 2.12, it follows that

That is, there exists a constant $C_{3}$ such that

We claim that there exists a positive constant $C\equiv C(n, \, \nu)$ such that

with $2R = 1$ and $|\mu|(B(0, 1)) = 1$. According to (2.8), we write $\|\nabla u-\nabla v\|_{L^{P}(B(0, 1))} = k$. We shall prove (3.6) in the following two scenarios. On the one hand, if $0 < k\leq 1$, then it follows from (3.5) that

On the other hand, if $k > 1$, then we set $k = \left(C_{1}C_3\right)^{-(2+\varepsilon_0)}$, where $C_1$ and $C_3$ are given in (2.5) and (3.5), respectively. It is obvious that $\frac{1}{C_{1}C_3} > 1$, and so

By involving $\lambda = C_{1}C_3 < 1$ in (2.5), we obtain that

By (3.7) and (2.8), one finds that the norm $k\leq\frac{1}{C_{1}C_3}$, that is, $\frac{1}{C_{1}C_3}\leq 1$, which is contradictory to the condition $\frac{1}{C_{1}C_3} > 1$. Therefore, the inequality (3.6) holds.

Step 2. Scaling procedures.

We assume that $|\mu|(B(x_0, 2R)) = 1$, and we shall reduce to the case $B(x_{0}, 2R)\equiv B(0, 1)$ by a standard scaling argument. By letting

for $y \in B(0, 1)$, one has the following equations:

With the definition of Radon measure (1.2), we find the following relation between $|\tilde{\mu}|(B(0, 1))$ and $|\mu|(B(x_0, 2R))$, namely,

Next, we shall reduce the general case to the special case $|\mu|(B(0, 1)) = 1$. We define

By (1.3), one has $M\leq 1$. Hence the new solution, coefficient, and datum become

Then we find that

hold in the weak sense and $|\bar{\mu}|(B(0, 1)) = 1$. Then by applying the result (3.6) in Step 1, one has

Considering (3.9) on $B(x_0, 2R)$, then

we apply Lemma 2.5 with $\lambda$ replaced by $M$ to obtain

where $C$ depends on $n, \, \nu$, and $C_1$. From (3.10), we complete the proof of Proposition 3.2. □

Via a classical approach, we have the following estimate inspired by ^[10].

Lemma 3.3. Let $v\in W^{1, P}(\Omega)$ be a weak solution to the Dirichlet problem (3.1) under the assumptions (A1), let $P$ be an $N$-function satisfying $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4 and (2.12). There exist constants $\alpha\in (0, 1]$ and $C_4\equiv C_4(n, \nu, L)\geq 1$ such that the estimate

holds whenever $B(x_0, \rho)\subset B(x_0, R)\subset\Omega$.

We note that the estimate (3.11) is tenable by imitating the proof for the $p$-Laplacian equations.

Lemma 3.4. Let $u\in W^{1, P}(\Omega)$ be a weak solution to (1.1) under the assumptions ${\text{(A1)}}_2$. Let $P$ be an $N$-function satisfying $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4, and (2.12). Then there exist constants $\ell\equiv \ell(n, \, \nu, \, L, \, C_{\Delta_2})\geq 1$ and $C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1)\geq 1$ such that

holds whenever $B(x_0, \rho)\subset B(x_0, R)\subset\Omega$.

Proof. From the triangle inequality and (2.2), we see that

By applying (3.11), one carries out

Using the triangle inequality $|\nabla v|\leq|\nabla u-\nabla v|+|\nabla u|$ again, the inequality (3.12) leads to

with $\ell = C_{\Delta_2}^{2}C_4$. It follows from (3.3) that

Notice that $\rho\leq R$ with

we complete the proof of Lemma 3.4. □

We also define $w\in v+W_{0}^{1, P}(B(x_0, R))$ as the unique solution to the homogeneous Dirichlet problem with frozen coefficients

We have the following decay estimate.

Lemma 3.5. Let $w\in W^{1, P}(\Omega)$ be a weak solution to (3.13) under the assumption (A1). Then there exist constants $\tilde{\alpha}\in(0, 1]$ and $C\geq 1$, both depending on $n, \, \nu, \, L$, such that

holds whenever $B(x_0, \rho)\subset B(x_0, R)\subset\Omega$.

Notice that the conclusion (3.14) is inspired by ^[32].

Lemma 3.6. Under the assumptions (A1) and (A2) of Theorem 1.1, with $v$ as in (3.1) and $w$ as in (3.13), there exists a constant $C\equiv C(n, \, \nu, \, L)$ such that

for $B(x_{0}, R)\subset\Omega$, where $K$ and $\omega(R)$ are given in the assumption (A1).

Proof. We test Eq $(3.13)_1$ with $v-w$. Since both $v$ and $w$ are weak solutions, then the assumption ${\text{(A1)}}_2$ gives us that

By using ${\text{(A1)}}_3$ with $|x-x_{0}|\leq R$ and Young's inequality (2.9), we derive that

Finally, Lemma 3.6 is proved by using the assumptions (A2), (3.16), and (2.3). □

Lemma 3.7. Assume that $u\in W^{1, P}(\Omega)$ is a weak solution to (1.1) satisfying (A1), (A2), and $P$ is an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$, and satisfies Assumption 2.4, (2.12). Let $w$ be defined in (3.13), and $\mu$ be a Radon measure that satisfies ${\text{(M)}}$. There exists a constant $C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1)$ such that

The key to the proof of Lemma 3.7 is the triangle inequality as follows:

with (3.3), (3.15), and (3.2).

Corollary 3.8. Let $u\in W^{1, P}(\Omega)$ be a weak solution to (1.1) under the assumptions ${\text{(A1), (A2)}}$. Let $P$ be an $N$-function satisfying $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4, and (2.12). Then there exists a constant $C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1)\geq 1$ such that

holds whenever $B(x_0, \rho)\subset B(x_0, R)\subset B(x_0, 2R)\subset\Omega$.

Our goal is to derive an oscillation decay estimate of $\nabla u$. Based on Lemmas 3.5 and 3.7, we first involve the corresponding oscillation decay estimate (3.14) of $\nabla w$, and then compare $\nabla u$ and $\nabla w$ by (3.17). We note that Lemma 2.10 and the triangle inequality play an essential role in the following lemma.

Lemma 3.9. Let $u$ be a weak solution to (1.1) under the assumptions (A1), (A2), (M), $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4, and (2.12). Then there exists $C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) > 0$ such that

for $B(x_{0}, \rho)\subset B(x_{0}, 2R)\subset\Omega$. Here the constant $\tilde{\alpha}$ is introduced in Lemma 3.5.

4.   Caccioppoli-type inequality and maximal estimate

In this section, we use the estimate established in Proposition 3.2 to derive the Caccioppoli-type inequality and the maximal estimate. First, the following Caccioppoli-type inequality gives a connection between $\nabla u$ and $u$.

Proposition 4.1. (Caccioppoli-type inequality) Let $u\in W^{1, P}(\Omega)$ be a weak solution to (1.1) with measurable coefficients and satisfy ${\text{(A1)}}_1$, ${\text{(A1)}}_2$. Let $\mu$ be a Radon measure with (M). Suppose $P$ is an $N$-function satisfying Assumption 2.4. Then there exists a constant $C\equiv C(n, \, \nu, \, L, \, C_1, \, C_{\Delta_2})$ such that

where $B(x_0, 2R)\subset\Omega$.

Proof. We may assume that $(u)_{B(x_0, 2R)} = 0$ as if $u$ solves (1.1) also $u-(u)_{B(x_0, 2R)}$ dose. Let $\eta\in C_{0}^{\infty}(B(x_0, 2R))$ such that $0\leq \eta\leq 1$, and

Let $v\in u+W_0^{1, P}(B(x_{0}, 2R))$ be the weak solution to (3.1). We choose a test function $\varphi: = v\eta$ to (3.1), and obtain

It is clear that

By Lemma 2.6 and (4.2), we deduce that

By (4.3), (4.4), Cauchy-Schwartz inequality, and Lemma 2.6, one has

According to Young's inequality (2.10), we derive that for $\varepsilon > 0$, there exists $\widetilde{C}_{\varepsilon}$ such that

By (2.3) and Lemma 2.13, we deduce that

Dividing by $|B(x_0, R)|$, one gives

Applying the triangle inequality with $P(|\nabla u|)\leq C_{\Delta_2}\left(P(|\nabla u-\nabla v|)+P(|\nabla v|)\right)$ and (4.5), one has

In order to estimate the last term of (4.6), we use the triangle inequality again to obtain

Lemma 2.11 and the classical Hölder's inequality give us that

Thus we combine (3.3) and (4.6)–(4.8), and conclude that

This establishes the Caccioppoli-type inequality (4.1). □

Based on the definition of maximal functions in Section 2.4 and the control estimate in Lemma 3.4, we present the following pointwise estimate involving the maximal functions.

Proposition 4.2. (Maximal estimate) Let $u\in W^{1, P}(\Omega)$ be a weak solution to (1.1) under (A1), (A2). Let $P$ be an $N$-function with $\Delta_2({P, \widetilde{P}}) < \infty$, Assumption 2.4, and (2.12). Let Radon measure $\mu$ satisfy (M). Then there exists a constant $C\equiv C(n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1)$ such that

Here $\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R)$ is a Riesz potential that is introduced in (1.4). In (4.9), the constants $\varepsilon_{0}$, $\alpha\in(0, 1]$, $\theta_1$ and $\theta_{0}$ are given in Lemmas 2.5, 3.3, 2.11, and (1.3), respectively.

Proof. The key of the proof is to consider the radii $R$ satisfying that $R\leq R_0$, where the quantity $R_0 > 0$ is in dependence of the data $n, \, \nu, \, L, \, \alpha$, and $\omega(\cdot)$. More precisely, by (A2), we shall choose $R_0$ so that

where $\delta$ will be a small quantity that will be reduced at several stages, as a decreasing function of the quantities $n, \, \nu, \, L$, and also $\alpha$. The proof of Proposition 4.2 is accomplished through two steps, to which the following content is devoted.

By (2.15) and (2.13) with $\theta = 1+\varepsilon_0-\alpha$ and $\tau = (1+\varepsilon_0-\alpha)\theta_1$, there holds

By using Hölder's inequality, we obtain

Then the inequality (4.9) will follow if we are able to show that

Step 1. The case for small radii $R\leq R_0$.

We take $0 < \rho\leq r/2\leq r \leq R$, and adopt the estimate in Lemma 3.4 with two radii $\rho$ and $r/2$. There exists a constant $C_5\equiv C_5(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1)$ such that

Multiplying both sides of (4.13) by $\rho^{1+\varepsilon_0-\alpha}$, and taking $S = r/\rho$, it follows that

for $\rho\leq r/2\leq R/2$. We choose the constant $S\geq2$ large enough, which satisfies that

and take the supremum with $0 < r\leq R$ such that the following estimate holds

where $0 < r\leq R$ is equivalent to $0 < \rho\leq R/S$. By (2.13) and (4.14), we obtain

with a constant $C$ depending on $n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, S$, and $\alpha$.

On the other hand, we notice that

Recalling the constant $S$, and putting (4.15) and (4.16) together, we obtain the following:

The definition of the supremum shows that for any $\varepsilon > 0$, there is $r\in (0, R]$ such that

This leads to

Since $\varepsilon$ is arbitrary, and $0 < r < 2r\leq 2R$, the preceding estimates (4.18) and (4.19) show that there exists a constant $C\equiv C(n, \, \theta_0)$ such that

Combining (4.17) and (4.20), we deduce the desired estimate (4.12) with $R\leq R_0$.

Step 2. Removing the condition $R\leq R_0$.

Our goal is to prove (4.12) without the restriction $R\leq R_0$. Taking $R > R_0$ and recalling Definition (2.13), it is clear that

We apply (4.12) with radius $R_0$, i.e.,

By the definition of Riesz potential (1.4), one has

It is apparent to enlarge the integral by

By using (4.21)–(4.24), we derive

Since $\Omega$ is bounded, then (4.12) holds.

Combining (4.10)–(4.12), we obtain (4.9), which means the proof of Proposition 4.2 is completed. □

5.   Proofs of Theorems 1.1–1.3

By establishing the preceding technical tools and lemmas, we are in a position to present the proofs of the main theorems. We first have the following proof.

Proof of Theorem 1.1. We set a sequence of balls $\{B_{i}\}_{i = 0}^{\infty}$ by

where $2\Lambda > 1$ will be chosen later. It is clear that $B_{i+1}\subset B_i$ for every $i\geq 0$. We set two sequences $\{K_{i}\}_{i = 0}^{\infty}$ and $\{k_{i}\}_{i = 0}^{\infty}$ by

We also introduce $\widetilde{k}_{0}$ by

By (2.10), it is obvious that

as well as

Step 1. An estimate of the summation of $K_i$.

An application of Lemma 3.9 with $B(x_0, \rho)\equiv B\left(x_0, \frac{R}{2\Lambda}\right)\subset B(x_0, R)$ shows that

where the constants $C$ depend on $n, \, \nu, \, L, \, s, \, C_{\Delta_2}$, and $C_1$. Using (5.1), we choose $\Lambda\equiv \Lambda(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) > 1$ large enough such that

where $\tilde{\alpha}\in(0, 1]$ is given in Lemma 3.5. By (2.2), it is clear that

Hence there exists a constant $C_{6} > 0$ depending on $n, \, \nu, \, L, \, K, \, s, \, C_{\Delta_2}$, and $C_1$ such that

Using (A2), we take $\widetilde{R}$ small enough to obtain

It follows that if $R\leq\widetilde{R}$, then all $R_{i}\leq\widetilde{R}$. Applying the estimate (5.6) with $R\equiv R_{i-1}$, and noting that $\omega(\cdot)$ is non-decreasing, it yields that

which can be simplified as

Via a summation, one deduces that

for $C\equiv C(n, \, \nu, \, L, \, K, \, s, \, C_{\Delta_2}, \, C_1)$ and for every integer $m$. This implies that

holds for every $m\in\mathbb{N}$.

Step 2. An estimate of $k_{m+1}$.

Using (5.2), we have

By using Lemma 2.9 with $x = \left|(\nabla u)_{B_{i+1}}\right|$, $y = \left|(\nabla u)_{B_{i}}\right|$, we estimate the difference as

Considering the triangle inequality $\left|\, |x|-|y|\, \right|\leq |x-y|$ and Lemma 2.10, we have

Hence there exists a constant $C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1)$ such that

For each $i$, we choose $\varepsilon = \varepsilon(i, n, C_{\Delta_2}, \Lambda)$ small enough such that

Then we have

Here we use the fact that the sum of geometric series is finite, i.e.,

By combining (5.9)–(5.12), and (5.4), there exists a constant $C$ such that

Making use of (5.13), (5.8), and (5.4), one derives that for every integer $m\geq 1$, there holds

For the second term on the right side of (5.14), one has

Using $\Lambda > 1$, the method adopted in (4.20), and (1.4), we obtain that

holds with a constant $C$ depending on $n$ and $\theta_0$. Inserting (5.15) in (5.14), we obtain the following inequality:

Step 3. An induction approach.

By setting

we shall use the mathematical induction to prove

Initial Step. If $m = -1$, then by (5.4), we see that (5.17) is trivial. For the case $m = 0$, (5.17) holds by using (5.13).

Inductive Step. Assuming that (5.17) is valid for any $\tilde{m} < m$, we shall prove it for $m+1$. By (5.16), we have

Due to the fact that $\omega(\cdot)$ is non-decreasing, we estimate

Considering the fact that $\Lambda > 1$ and the definition of $d(\cdot)$ in (A3), we have

By applying (5.18), we complete the proof of the inequality (5.17).

For every Lebesgue point $x_{0}$ of $P(|(\nabla u)|)$, we let $m\rightarrow \infty$, and show that

where $C$ depends on $n, \, \nu, \, L, \, K, \, s, \, \theta_0, \, C_{\Delta_2}$ and $C_1$. Therefore, (1.6) has been proved. □

Via a similar approach to the previous proof, we are in a position to prove Theorem 1.2.

Proof of Theorem 1.2. We introduce a sequence of concentric balls $\{\widetilde{B}_{i}\}_{i = 0}^{\infty}$ by

where $H > 1$ is a constant determined later, and $r\leq R < 1$. Hence $\widetilde{B}_{i+1}\subset B\left(x_0, \frac{r_i}{2}\right)\subset \widetilde{B}_i$ for every $i\ge 0$. We define

We also introduce $\widetilde{a}_{0}$ by

By (2.10) and (2.5), it is obvious that

as well as

One applies Lemma 2.11 and Hölder's inequality to obtain

Applying Lemma 3.4 with $\rho\equiv r_{i+1}$, $R\equiv \frac{r_i}{2}$, one has

where the constant $C$ depends on $n, \, \nu, \, L, \, C_{\Delta_2}$, and $C_1$. By Caccioppoli-type inequality (4.1) and Definition 2.3, we obtain

Combining (5.21) and (5.22) with $r_{i+1}\leq r_{i}$, it follows that

where $C_{7}$ depends on $n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1$, and $R$. By choosing $H\equiv H(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R)$ large enough, one has

which implies immediately that

We consider a summation with respect to $i$ from $0$ to $m-1$, and deduce that

For every non-negative integer $m$, one writes

Adopting a similar approach as in the proof of (5.10), we have

and choose $\varepsilon = (2H)^{-i(n+1)}$ sufficiently small. With the help of (5.19), it follows that there exists a constant $C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R)$ such that

Analogous to (5.15), we obtain

Applying (5.23)–(5.25) and (5.20), one gets that for every integer $m\geq 1$, there holds

By the dominated convergence theorem, for every Lebesgue point $x_{0}\in P(|u|)$, there holds

where constant $C$ depends on $n, \, \nu, \, L, \, \theta_0, \, C_{\Delta_2}, \, C_1$, and $\mathrm{diam}(\Omega)$. □

By establishing (5.26), we are ready to prove the result of Theorem 1.3.

Proof of Theorem 1.3. For any real number $g$, we observe that if $u$ is a weak solution to (1.1), then $u-g$ is still a solution to (1.1). Let $B(x_{0}, 2R)\subset\Omega$. We consider $x, \, y\in B\left(x_{0}, \frac{R}{2}\right)$ satisfying that $r: = |x-y| < \frac{R}{4}$. By (5.26), it follows that

By (1.4), one has the following monotone property:

Via (2.2) and a direct computation, one has

We take $g: = (u)_{B(x, 2r)}$, and observe that $B(x, r)\cup B(y, r)\subset B(x, 2r)\subset B\left(x, \frac{R}{2}\right)$. Using Definition 2.3, we deduce that

Since $2r < \frac{R}{2}$, then (2.14) gives us that

Then it follows from Proposition 4.2 that

To estimate the last integral, we use Caccioppoli-type inequality (4.1) and (4.19) to obtain that

Substituting (5.29) into (5.28) and considering $0\leq\alpha < 1$, it yields that

Combining (5.27)–(5.30) together with $B(x, R)\subset B(x_0, 2R)$, there is a constant $C$ such that

where the constant $C$ depends on $n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1$, and $\mathrm{diam}(\Omega)$. Since $R < 1$ and $\alpha\in[0, 1)$, then the estimate (5.31) is the desired interior Hölder estimate of Theorem 1.3. □

6.   Conclusions

In this work, we establish pointwise potential estimates of weak solutions to a class of elliptic equations in divergence form with measure data. Our primary result is to employ the Riesz potential to prove the pointwise estimates of the solutions. The key innovation of this paper manifests in the proof of Proposition 3.2, which enables the relationship between measure data and the Riesz potential in the Sobolev-Orlicz spaces. Furthermore, we obtain Hölder continuity estimates for the solutions by establishing the Caccioppoli-type inequality and the maximal estimate. This systematic approach extends the potential estimates of regularity for nonlinear elliptic equations in the existing literature.

Author contributions

Zhaoyue Sui: Conceptualization, methodology, writing–original draft preparation; Feng Zhou: Supervision, funding acquisition, project administration, writing–review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the anonymous referee for their careful reading and valuable comments. The authors are supported by the National Natural Science Foundation of China (NNSF Grant No. 12001333) and the Shandong Provincial Natural Science Foundation (Grant No. ZR2020QA005).

Conflict of interest

The authors declare no conflicts of interest regarding the publication of this article.