# Liouville type theorems for the 3D stationary MHD equations in Lebesgue spaces with variable exponent

Jianfeng Sun, Gastón Vergara-Hermosilla, Jihong Zhao

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Jianfeng Sun, Gastón Vergara-Hermosilla, Jihong Zhao. Liouville type theorems for the 3D stationary MHD equations in Lebesgue spaces with variable exponent. 2025. hal-05157007

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# Liouville type theorems for the 3D stationary MHD equations in Lebesgue spaces with variable exponent*

Jianfeng Sun¹, Gastón Vergara-Hermosilla², Jihong Zhao¹

¹School of Mathematics and Information Science, Baoji University of Arts and Sciences

Baoji, Shaanxi 721013, China

²Institute for Theoretical Sciences, Westlake University, Hangzhou, Zhejiang 310024, China

July 10, 2025

## Abstract

In this paper, we study the Liouville-type problem of the three dimensional stationary MHD equations and establish some Liouville-type results under some additional conditions imposed on the velocity field $u$ and the magnetic field $b$ in terms of the Lebesgue spaces with variable exponent $L^{p(\cdot)}$. These functional spaces present some particularities that can relax the $L^{p(\cdot)}$ norm imposed on $(u, b)$ beyond the usual range $3 \le p(\cdot) \le \frac{9}{2}$ in some non-negligible regions of $\mathbb{R}^3$, and thus provide us with some new interesting insights to study the Liouville-type problem of the stationary MHD equations.

**Keywords:** MHD equations; Liouville theorems; Lebesgue spaces of variable exponent.

**2020 AMS Subject Classification:** 35Q35, 76D03

## 1 Introduction

The purpose of this paper is to establish some Liouville type theorems for the three dimensional incompressible stationary magnetohydrodynamic (MHD) equations

$$ egin{cases} -\Delta u + (u \cdot \nabla)u + \nabla\pi = (b \cdot \nabla)b, \\ -\Delta b + (u \cdot \nabla)b = (b \cdot \nabla)u, \\ \nabla \cdot u = 0, \quad \nabla \cdot b = 0, \end{cases} \quad (1.1) $$

where $u$ is the velocity field, $b$ is the magnetic field and $\pi$ is the scalar pressure, $u_0$ and $b_0$ are the initial velocity field and the initial magnetic field satisfying $\text{div } u_0 = \text{div } b_0 = 0$, respectively. For simplicity, we have assumed that the viscosity coefficient and the resistivity coefficient to be equal to 1.

If we neglect the magnetic field, i.e., taking $b = 0$ in (1.1), then the equations (1.1) becomes to the following incompressible stationary Navier-Stokes equations

$$ egin{cases} -\Delta u + (u \cdot \nabla)u + \nabla\pi = 0, \\ \nabla \cdot u = 0. \end{cases} \quad (1.2) $$

Owing to the pioneering works of Leray [17], there is an outstanding open problem that whether or not the solution $u \in \dot{H}^1(\mathbb{R}^3)$ must be identically zero if $u$ decays uniformly to zero at infinity. This is known as the famous Liouville type problem for equations (1.2). A well-known result is given in the book [13], where Galdi proved that if $u \in L^{\frac{9}{2}}(\mathbb{R}^3)$, then $u \equiv 0$. This is shown that under some additional assumptions imposed on the velocity field $u$, the answer to the Liouville type problem is affirmative. Subsequently, Chae and Wolf [3] gave a logarithmic improvement of the result in [13], Seregin [20] proved if $u \in L^6(\mathbb{R}^3) \cap \text{BMO}^{-1}(\mathbb{R}^3)$, then $u \equiv 0$. Recently, Chamorro, Jarrín and Lemarié-Rieusset [6] proved that if the velocity field $u \in L^p(\mathbb{R}^3)$ with $3 \le p \le \frac{9}{2}$, then $u \equiv 0$. For more interesting Liouville type results of the equations (1.2), We refer readers to see [9, 14, 15, 23].

<footnote>*Email addresses: sunjianfeng0715@163.com, coibungo@gmail.com, jihzhao@163.com.</footnote>
The Liouville type problem of the MHD equations (1.1) is also studied, and several works have been established. For the 2D case, Wang and Wang [22] proved that if the energy $D(u, b) = \|\nabla u\|_{L^2}^2 + \|\nabla v\|_{L^2}^2$ is finite and

$$ \|b\|_{L^1} D(u, b)^{\frac{1}{2}} \le C \min\{\mu\nu, \mu^{\frac{1}{2}}\nu^{\frac{3}{2}}\}, $$

then the solution $(u, b)$ must be constants. Nitti, Hounkpe and Schulz [19] proved that if $D(u, b)$ is finite, and the magnetic field $b \in L^2(\mathbb{R}^2)$, or the velocity field $u \in L^p(\mathbb{R}^2)$ and the magnetic field $b \in L^q(\mathbb{R}^2)$ with $p, q \in (2, \infty)$ such that $\frac{2}{\max\{p,q\}} + \frac{1}{p} \ge \frac{1}{2}$, then the solution $(u, b)$ must be constants. For the 3D case, Chae, Degond and Liu [2] proved that if $(u, b) \in L^\infty \cap L^\frac{9}{2}$ and $(\nabla u, \nabla b) \in L^2(\mathbb{R}^3)$, then $u = b = 0$. Schulz [21] removed the finite Dirichlet energy condition and proved that if the solution $(u, b) \in L^p(\mathbb{R}^3) \cap BMO^{-1}$ with $p \in (2, 6]$, then $u = b = 0$. Yuan and Xiao [23] further obtained the Liouville type result under the condition that

$$ u \in L^p(\mathbb{R}^3), b \in L^q(\mathbb{R}^3) \text{ for } 2 \le p \le \frac{9}{2} \text{ and } 4 \le q \le \frac{9}{2}. $$

Recently, Chen, Li and Wang [8] proved that if the solution $u \in L^3(\mathbb{R}^3)$ and $b \in L^6(\mathbb{R}^3)$, then $u = b = 0$. Chae and Lee [7] further extended this result provided that

$$ u \in L^p(\mathbb{R}^3), b \in L^q(\mathbb{R}^3) \text{ for } \frac{1}{p} + \frac{2}{q} \ge \frac{2}{3} \text{ and } 3 \le p \le \frac{9}{2}. $$

For more interesting Liouville type results of the equations (1.1), we refer the readers to see [8, 12, 18].

Notice that by the classical Sobolev embeddings we have the space inclusion $\dot{H}^1(\mathbb{R}^3) \subset L^6(\mathbb{R}^3)$, which combining with the Liouville type results obtained for equations (1.1), we know if $u \in \dot{H}^1(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)$ with $3 \le p \le \frac{9}{2}$, then $u = 0$. However, the case $\frac{9}{2} < p \le 6$ remains unsolved. Recently, Chamorro and the second author of this paper in [5] established several Liouville type results of equations (1.2) in the framework of the Lebesgue spaces with variable exponent. These results may allow the variable exponent $p(\cdot)$ beyond the range of $[3, \frac{9}{2}]$ in some non-negligible regions in $\mathbb{R}^3$, which provide us some new insights for the Liouville type problem of equations (1.1). Motivated by [5], in this paper we study the Liouville type problem of the stationary MHD equations (1.1) in the framework of the Lebesgue spaces with variable exponent.

The Lebesgue spaces with variable exponent $L^p(\cdot)$ is quite different from the usual Lebesgue spaces $L^p$. Indeed, given a measurable set $\Omega \subset \mathbb{R}^n$, the norm for the classical Lebesgue space $L^p(\Omega)$ ($1 \le p < \infty$) can be directly defined by

$$ \|f\|_{L^p} := \left( \int_{\Omega} |f(x)|^p dx \right)^{\frac{1}{p}}. $$

Such a definition obviously fails in the context of variable Lebesgue spaces since we cannot replace the constant exponent $\frac{1}{p}$ outside the integral by the exponent function $\frac{1}{p(\cdot)}$. The solution is a more subtle approach which is similar to that used to define the Luxemburg norm on the Orlicz spaces (cf. [10, 11]). Let $P(\Omega)$ be the set of all Lebesgue measurable functions $p(\cdot) : \Omega \to [1, +\infty]$. Then for $p(\cdot) \in P(\Omega)$, we define

$$ \|f\|_{L^{p(\cdot)}} := \inf \left\{ \lambda > 0 : \rho_{p(\cdot)} \left( \frac{f}{\lambda} \right) \le 1 \right\}, \quad (1.3) $$

where the modular function $\rho_{p(\cdot)}$ associated with $p(\cdot)$ is given by the expression

$$ \rho_{p(\cdot)}(f) := \int_{\Omega} |f(x)|^{p(x)} dx. $$

Moreover, if the set on the right-hand side of (2.1) is empty then we define $\|f\|_{L^{p(\cdot)}} = \infty$. With the Luxemburg norm (2.1), we define the Lebesgue space of variable exponent $L^{p(\cdot)}(\Omega)$ to be the set of Lebesgue measurable functions $f$ such that $\|f\|_{L^{p(\cdot)}} < +\infty$.

Now we state the main results. For any $p(\cdot) \in P(\Omega)$, we denote

$$ p_{\Omega} := \operatorname{essinf}_{x \in \Omega} p(x), \quad p_{\Omega}^{+} := \operatorname{esssup}_{x \in \Omega} p(x). $$
**Theorem 1.1** Let E be a cylinder in $\mathbb{R}^3$ defined by

$$E = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le 1, x_1 \in \mathbb{R}\}.$$

The variable exponents $p(\cdot), q(\cdot) \in \mathcal{P}(\mathbb{R}^3)$ satisfy the condition $\frac{1}{p(\mathbb{R}^3 \setminus E)} + \frac{2}{q(\mathbb{R}^3 \setminus E)} > \frac{2}{3}$, as well as the following conditions:

$$3 < p_{(\mathbb{R}^3 \setminus E)} \le p_{(\mathbb{R}^3 \setminus E)}(x) \le p_{(\mathbb{R}^3 \setminus E)}^{+} < \frac{9}{2}, \quad \frac{9}{2} < p_{\bar{E}} \le p_{\bar{E}}(x) \le p_{\bar{E}}^{+} < +\infty, \quad (1.4)$$

$$3 < q_{(\mathbb{R}^3 \setminus E)} \le q_{(\mathbb{R}^3 \setminus E)}(x) \le q_{(\mathbb{R}^3 \setminus E)}^{+} < 6, \quad 6 < q_{\bar{E}} \le q_{\bar{E}}(x) \le q_{\bar{E}}^{+} < +\infty. \quad (1.5)$$

For any weak solution $(u, b) \in L_{\text{loc}}^2(\mathbb{R}^3)$ and $\pi \in D'(\mathbb{R}^3)$ of the stationary MHD equations (1.1), if we further assume that $u \in L^{p(\cdot)}(\mathbb{R}^3)$, $b \in L^{q(\cdot)}(\mathbb{R}^3)$ and $\pi \in L^{\frac{p(\cdot)}{2}}(\mathbb{R}^3)$, then we have $(u, b) = (0, 0)$.

**Remark 1.1** The main novelty of Theorem 1.1 lies in we can take the values of variable exponents $p(\cdot)$ and $q(\cdot)$ beyond $\frac{9}{2}$. Indeed, by the conditions (1.4) and (1.5), when $x \in E$, the variable exponents $p(\cdot)$ and $q(\cdot)$ can be taken values beyond $\frac{9}{2}$, even larger than the Sobolev embedding integrability index 6. This gives us some new insights for the Liouville type problem of the equations (1.1). However, when $x \in \mathbb{R}^3 \setminus E$, i.e., outside the cylinder $E$, we still need to impose to the variable exponent $p(\cdot)$ to be in the open interval $(3, \frac{9}{2})$ and this is the known restrictions from which we can deduce the uniqueness of the trivial solution of equations (1.2) (cf. [6]). The range of variable exponent $q(\cdot)$ can be determined by $p(\cdot)$ and the condition $\frac{1}{p(\cdot)} + \frac{2}{q(\cdot)} > \frac{2}{3}$.

**Remark 1.2** Recall that, using the divergence-free conditions $\nabla \cdot u = 0$, $\nabla \cdot b = 0$, we have the following good relationship

$$\pi = \sum_{i,j=1}^{3} \mathcal{R}_i \mathcal{R}_j (u_i u_j + b_i b_j),$$

where $\mathcal{R}_i$ ($i = 1, 2, 3$) are Riesz operators, which is bounded in the usual Lebesgue spaces $L^p$ for $1 < p < +\infty$. Thus, if $(u, b) \in L^p$, we have $\pi \in L^{\frac{p}{2}}$. However, since the Riesz operators are not bounded in the Lebesgue spaces with variable exponent $L^{p(\cdot)}$, we can not deduce the desired control of the pressure $\pi$ by the solution $(u, b)$, therefore in Theorem 1.1, we ask the constraint $\pi \in L^{\frac{p(\cdot)}{2}}(\mathbb{R}^3)$ which does not interfere with the general purposes of our theorems.

**Remark 1.3** In Theorem 1.1, we assume that the velocity field $u \in L^{p(\cdot)}$ and the magnetic field $b \in L^{q(\cdot)}$, where the variable exponents $p(\cdot)$ and $q(\cdot)$ can be take larger than $\frac{9}{2}$ in some regions in $\mathbb{R}^3$, thus it can be regarded as the generalization of the results in Chae and Lee [7].

In the following theorem, we seek more regions of $\mathbb{R}^3$, where the values of the variable exponents $p(\cdot)$ and $q(\cdot)$ can be taken larger than $\frac{9}{2}$, and the Liouville type theorem still holds. For simplicity, we consider more details about the behavior of the variable exponent $p(\cdot)$ in the case of $p(\cdot) > \frac{9}{2}$ over larger subsets of $\mathbb{R}^3$, but the variable exponent $q(\cdot)$ satisfies the same condition as (1.5).

**Theorem 1.2** Let E be a cylinder in $\mathbb{R}^3$ defined by

$$E = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le 1, x_1 \in \mathbb{R}\},$$

and G a subset in $\mathbb{R}^3$ defined by

$$G = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le x_1^\gamma, x_1 > 0\}, \quad 0 < \gamma < 1.$$

The variable exponents $p(\cdot), q(\cdot) \in \mathcal{P}(\mathbb{R}^3)$ satisfy the condition $\frac{1}{p(\mathbb{R}^3 \setminus (E \cup G))} + \frac{2}{q(\mathbb{R}^3 \setminus (E \cup G))} > \frac{2}{3}$, as well as the following conditions:

$$3 < \mathfrak{p}_{(\mathbb{R}^3 \setminus G)} \le \mathfrak{p}_{(\mathbb{R}^3 \setminus G)}(x) \le \mathfrak{p}_{\bar{G}}^{+} < \frac{9}{2}, \quad \frac{9}{2} < \mathfrak{p}_{\bar{G}} \le \mathfrak{p}_{\bar{G}}(x) \le \mathfrak{p}_{\bar{G}}^{+} < \frac{3\gamma + 3}{\gamma}, \quad (1.6)$$

$$3 < q_{(\mathbb{R}^3 \setminus E)} \le q_{(\mathbb{R}^3 \setminus E)}(x) \le q_{(\mathbb{R}^3 \setminus E)}^{+} < 6, \quad 6 < q_{\bar{E}} \le q_{\bar{E}}(x) \le q_{\bar{E}}^{+} < +\infty. \quad (1.7)$$

For any weak solution $(u, b) \in L_{\text{loc}}^2(\mathbb{R}^3)$ and $\pi \in D'(\mathbb{R}^3)$ of the stationary MHD equations (1.1), if we further assume that $u \in L^{p(\cdot)}(\mathbb{R}^3)$, $b \in L^{q(\cdot)}(\mathbb{R}^3)$ and $\pi \in L^{\frac{p(\cdot)}{2}}(\mathbb{R}^3)$, then we have $(u, b) = (0, 0)$.
**Remark 1.4** Notice that, if we let $\gamma \to 0^+$, then the shape of the region $G$ tend to $E$, which we have considered in Theorem 1.1, and the upper bound $\frac{3\gamma+3}{\gamma} \to +\infty$, which is the second condition stated in (1.4). On the other hand, in the case $\gamma \to 1^-$ (the set $G$ becomes larger), the upper bound $\frac{3\gamma+3}{\gamma} \to 6^+$, which is exactly the Sobolev embedding integrability index 6. Thus Theorem 1.2 shows that if the measure of this set is “reasonable big" then we can consider a mild decay at infinity and we can still obtain the uniqueness of the trivial solution. However, if the measure of this set is “too big" then we shall need to impose strong decay which allows us to solve the uniqueness problem considered here.

Notice that, in above two theorems, we can take values of the variable exponents $p(\cdot)$ and $q(\cdot)$ larger than $\frac{9}{2}$ in some regions in $\mathbb{R}^3$, but we have always assumed that the upper bounds $p_E^+, q_E^+, p_G^+ < +\infty$. In the following theorem, we consider the case where the upper bounds can be taken $+\infty$ over a specific subset of $\mathbb{R}^3$. For simplicity, let us consider $q^+ = +\infty$ over a specific subset of $\mathbb{R}^3$, but the variable exponent $p(\cdot)$ satisfies the same condition as (1.6).

**Theorem 1.3** Let $H$ be a subset in $\mathbb{R}^3$ defined by

$$H = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le x_1^{-\sigma}, x_1 > 0\}, \quad 0 < \sigma < 1,$$

and $G$ a subset in $\mathbb{R}^3$ defined by

$$G = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le x_1^{\gamma}, x_1 > 0\}, \quad 0 < \gamma < 1.$$

The variable exponents $\mathbf{p}(\cdot), \mathbf{q}(\cdot) \in \mathcal{P}(\mathbb{R}^3)$ satisfy the condition $\frac{1}{\mathbf{p}(\mathbb{R}^3 \setminus (H \cup G))}(\cdot) + \frac{2}{\mathbf{q}(\mathbb{R}^3 \setminus (H \cup G))}(\cdot) > \frac{2}{3}$, as well as the following conditions:

$$3 < \bar{\mathbf{p}}_{(\mathbb{R}^3 \setminus G)} \le \mathbf{p}_{(\mathbb{R}^3 \setminus G)}(x) \le \bar{\mathbf{p}}_{(\mathbb{R}^3 \setminus G)}^{+} < \frac{9}{2}, \quad \frac{9}{2} < \bar{\mathbf{p}}_{G} \le \mathbf{p}_{G}(x) \le \bar{\mathbf{p}}_{G}^{+} < \frac{3\gamma + 3}{\gamma}, \quad (1.8)$$

$$3 < \bar{\mathbf{q}}_{(\mathbb{R}^3 \setminus H)} \le \mathbf{q}_{(\mathbb{R}^3 \setminus H)}(x) \le \bar{\mathbf{q}}_{(\mathbb{R}^3 \setminus H)}^{+} < 6, \quad \mathbf{q}_{H}(x) = +\infty. \quad (1.9)$$

For any weak solution $(u, b) \in L_{\text{loc}}^2(\mathbb{R}^3)$ and $\pi \in \mathcal{D}'(\mathbb{R}^3)$ of the stationary MHD equations (1.1), if we further assume that $u \in L^{\mathbf{p}(\cdot)}(\mathbb{R}^3)$, $b \in L^{\mathbf{q}(\cdot)}(\mathbb{R}^3)$ and $\pi \in L^{\frac{\mathbf{p}(\cdot)}{2}}(\mathbb{R}^3)$, then we have $(u, b) = (0, 0)$.

**Remark 1.5** Notice that the Lebesgue measure of $H$ is infinite, and the volume of $H$ will decrease as $\sigma \to 1^-$, thus in Theorem 1.3, we assumed that the magnetic field $b$ have a suitable decay at infinity in a large portion $\mathbb{R}^3$ but that can be constant at infinity over a non-negligible subset of $\mathbb{R}^3$. This implies that the decay conditions imposed on solutions at infinity may not absolutely necessary to solve the Liouville problem of equations (1.1) and (1.2).

Before ending this section, let us give a conclusion about our main results. In Theorems 1.1–1.3, we study the Liouville type problem of the MHD equations (1.1) in the framework of Lebesgue spaces with variable exponents. We proved the uniqueness of trivial solution under the assumptions that the solution $u$ and $b$ belong to the Lebesgue spaces with different variable integrability exponents, and these variable exponents can take values, inside some subsets of $\mathbb{R}^3$ (i.e., $E, G, H \subset \mathbb{R}^3$), beyond the known range $[3, \frac{9}{2}]$. Notice that the Lebesgue measure of these subsets are all infinity, thus they are non-negligible, and we can extend our main theorems to many different subsets of $\mathbb{R}^3$ (for example a finite union of cylinders like $E$) as long as in these regions, some restrictions for the integrability indices should be satisfied. In the future, we hope to seek more subsets of $\mathbb{R}^3$ that the Liouville type theorem still holds.

The rest of this paper is organized as follows. Section 2 is devoted to proving Theorem 1.1, and in Sections 3 and 4 we prove Theorems 1.2 and 1.3, respectively. Throughout this paper, we will use $C$ to denote a finite inessential constant which may be different from line to line, but do not depend on particular solutions or functions.

## 2 Proof of Theorem 1.1

Let us first emphasize that under the assumptions of Theorem 1.1, the solution $(u, b)$ and the pressure $P$ are actually regular functions. Indeed, let $(u, b) \in L_{\text{loc}}^2(\mathbb{R}^3)$ satisfy in the weak sense of equations (1.1), by Theorem 2.51 in [10], we know the space inclusion relation

$$L^{\mathbf{p}(\cdot)}(\mathbb{R}^3) \subset L^{\mathbf{p}^{-}}(\mathbb{R}^3) + L^{\mathbf{p}^{+}}(\mathbb{R}^3),$$
and by the definition of the variable exponent $p(\cdot)$ in Theorem 1.1, we have the lower and upper bounds $3 < p^{-} \le p^{+} \le +\infty$, which leads to the following embedding results

$$L^{p(\cdot)}(\mathbb{R}^3) \subset L^{-p}(\mathbb{R}^3) + L^{p^+}(\mathbb{R}^3) \subset L_{\text{loc}}^{-p}(\mathbb{R}^3) + L_{\text{loc}}^{p^+}(\mathbb{R}^3) \subset L_{\text{loc}}^{3}(\mathbb{R}^3).$$

Thus we know $(u, b) \in L_{\text{loc}}^{3}(\mathbb{R}^3)$, and we can follow the same arguments used in the proof of Theorem X.1.1 of [16] to prove that the solution $(u, b)$ and the pressure $P$ are actually regular functions.

Next, we present some useful lemmas used in the proofs of our main results.

**Lemma 2.1** Consider the variable exponent functions $p(\cdot), p_1(\cdot), p_2(\cdot) \in \mathcal{P}(\Omega)$ such that the following pointwise relationship holds $\frac{1}{p(x)} = \frac{1}{p_1(x)} + \frac{1}{p_2(x)}$, $x \in \Omega$. Then there exists a constant $C$ such that for all $f \in L^{p_1(\cdot)}(\Omega)$ and $g \in L^{p_2(\cdot)}(\Omega)$, we have $fg \in L^{p(\cdot)}(\Omega)$ and

$$\|fg\|_{L^{p(\cdot)}(\Omega)} \le C \|f\|_{L^{p_1(\cdot)}(\Omega)} \|g\|_{L^{p_2(\cdot)}(\Omega)} \quad (2.1)$$

Clearly, (2.1) is a generalization of the Hölder's inequality, see Corollary 2.28 in [10], or Lemma 3.2.20 in [11] for its proof, moreover, the Hölder's inequality is still holds for the case $1 \le p(\cdot) \le \infty$, see Theorem 2.26 in [10] for more details.

For any $f \in L^{p(\cdot)}(\Omega)$, we can use the characteristic function to rewrite the Luxemburg norm as

$$\|f\|_{L^{p(\cdot)}(\Omega)} = \|f1_{\Omega}\|_{L^{p(\cdot)}(\mathbb{R}^3)}, \quad (2.2)$$

and we have the following result ([11], Lemma 3.2.12).

**Lemma 2.2** Given a measurable set $\Omega \subset \mathbb{R}^3$ such that $|\Omega| < +\infty$. Then for any $p(\cdot) \in \mathcal{P}(\Omega)$, we have

$$\|1\|_{L^{p(\cdot)}(\Omega)} \le 2 \max \left\{ |\Omega|^{\frac{1}{p}}, |\Omega|^{\frac{1}{p^+}} \right\}.$$

We also need to use the following embedding result ([10], Proposition 2.43).

**Lemma 2.3** Assume that a measurable set $\Omega \subset \mathbb{R}^3$ and $p(\cdot) \in \mathcal{P}(\Omega)$. Then the embedding $L^{\infty}(\Omega) \subset L^{p(\cdot)}(\Omega)$ holds if and only if $1 \in L^{p(\cdot)}(\Omega)$ and we have the estimate

$$\|f\|_{L^{p(\cdot)}(\Omega)} \le \|f\|_{L^{\infty}(\Omega)} \|1\|_{L^{p(\cdot)}(\Omega)}.$$

In particular, the embedding holds if $|\Omega| < +\infty$.

Now we present the proof of Theorem 1.1. The main idea is to localize (1.1) by a smooth cut-off function supported in a ball $B(0, R)$, then we carefully analyze the behavior of all localized terms as $R \to +\infty$, and in this step we will exploit the hypotheses stated in Theorem 1.1 to deduce the uniqueness of the trivial solution. For this purpose, let $\varphi \in C_0^\infty(\mathbb{R}^3)$ be a cut-off function such that $0 < \varphi \le 1$, $\varphi(x) = 1$ if $|x| \le \frac{1}{2}$, $\varphi(x) = 0$ if $|x| \ge 1$. Given $R > 1$, we define the function $\varphi_R$ by $\varphi_R(x) = \varphi(\frac{x}{R})$, thus $\varphi_R(x) = 1$ if $|x| \le \frac{R}{2}$, $\varphi_R(x) = 0$ if $|x| \ge R$, $|\nabla \varphi_R| \le CR^{-1}$ and $|\Delta \varphi_R| \le CR^{-2}$. Multiplying the first equations of (1.1) by $\varphi_R u$ and integrating over $\mathbb{R}^3$, by using the fact that $\text{supp}(\varphi_R u) = B_R = B(0, R)$, we get

$$\int_{B_R} [-\Delta u \cdot (\varphi_R u) + (u \cdot \nabla) u \cdot (\varphi_R u) + \nabla \pi \cdot (\varphi_R u) - (b \cdot \nabla)b \cdot (\varphi_R u)] dx = 0. \quad (2.3)$$

By using the divergence free conditions $\nabla \cdot u = 0$ and $\nabla \cdot b = 0$, after integration by parts, all terms on the left-hand side of (2.3) can be rewritten as:

$$\begin{aligned} - \int_{B_R} \Delta u \cdot (\varphi_R u) dx &= - \sum_{i,j=1}^3 \int_{B_R} \partial_j^2 u_i \varphi_R u_i dx \\ &= \sum_{i,j=1}^3 \int_{B_R} \partial_j u_i \partial_j (\varphi_R u_i) dx \end{aligned}$$
$$
\begin{align*}
&= \sum_{i,j=1}^{3} \int_{B_R} \partial_j \left( \frac{u_i^2}{2} \right) \partial_j \varphi_R dx + \sum_{i,j=1}^{3} \int_{B_R} (\partial_j u_i)^2 \varphi_R dx \\
&= - \int_{B_R} \Delta \varphi_R \left( \frac{|u|^2}{2} \right) dx + \int_{B_R} \varphi_R |\nabla \otimes u|^2 dx;
\end{align*}
$$

$$
\begin{align*}
\int_{B_R} (u \cdot \nabla) u \cdot (\varphi_R u) \, dx &= \sum_{i,j=1}^{3} \int_{B_R} u_i \partial_i u_j \varphi_R u_j \, dx \\
&= \frac{1}{2} \sum_{i,j=1}^{3} \int_{B_R} \partial_i (u_i u_j^2) \varphi_R \, dx \\
&= -\frac{1}{2} \int_{B_R} \nabla \varphi_R \cdot (|u|^2 u) \, dx;
\end{align*}
$$

$$
\int_{B_R} \nabla \pi \cdot (\varphi_R u) \, dx = - \int_{B_R} \nabla \varphi_R \cdot (\pi u) \, dx;
$$

and

$$
\begin{align*}
- \int_{B_R} (b \cdot \nabla) b \cdot (\varphi_R u) \, dx &= - \sum_{i,j=1}^{3} \int_{B_R} \partial_i (b_i b_j) \varphi_R u_j \, dx \\
&= \sum_{i,j=1}^{3} \int_{B_R} \varphi_R b_i \partial_i u_j b_j \, dx + \sum_{i,j=1}^{3} \int_{B_R} b_i \partial_i \varphi_R b_j u_j \, dx \\
&= \int_{B_R} \varphi_R (b \cdot \nabla) u \cdot b \, dx + \int_{B_R} (b \cdot \nabla) \varphi_R b \cdot u \, dx.
\end{align*}
$$

Taking all above identities into (2.3), we obtain

$$
\int_{B_R} \varphi_R |\nabla \otimes u|^2 dx = \int_{B_R} \Delta \varphi_R \left(\frac{|u|^2}{2}\right) + \nabla \varphi_R \cdot \left(\frac{|u|^2}{2} u\right) + \nabla \varphi_R \cdot (\pi u) - (b \cdot \nabla) \varphi_R b \cdot u \, dx \\
- \int_{B_R} \varphi_R (b \cdot \nabla) u \cdot b \, dx. \tag{2.4}
$$

Similarly, by testing the second equations of (1.1) with $\varphi_R b$, we obtain

$$
\int_{B_R} [-\Delta b \cdot (\varphi_R b) + (u \cdot \nabla) b \cdot (\varphi_R b) - (b \cdot \nabla) u \cdot (\varphi_R b)] dx = 0. \quad (2.5)
$$

We can also rewritten the first two terms on the left-hand side of (2.5) as:

$$
\begin{align*}
- \int_{B_R} \Delta b \cdot (\varphi_R b) \, dx &= \sum_{i,j=1}^{3} \int_{B_R} \partial_j b_i \partial_j (\varphi_R b_i) \, dx \\
&= \sum_{i,j=1}^{3} \int_{B_R} \partial_j \left( \frac{b_i^2}{2} \right) \partial_j \varphi_R \, dx + \sum_{i,j=1}^{3} \int_{B_R} (\partial_j b_i)^2 \varphi_R \, dx \\
&= - \int_{B_R} \Delta \varphi_R \left( \frac{|b|^2}{2} \right) \, dx + \int_{B_R} \varphi_R |\nabla \otimes b|^2 \, dx;
\end{align*}
$$

and

$$
\int_{B_R} (u \cdot \nabla) b \cdot (\varphi_R b) \, dx = \sum_{i,j=1}^{3} \int_{B_R} u_i \partial_i b_j \varphi_R b_j \, dx
$$
$$
\begin{align*}
&= \frac{1}{2} \sum_{i,j=1}^{3} \int_{B_R} \partial_i (u_i b_j^2) \varphi_R dx \\
&= -\frac{1}{2} \int_{B_R} \nabla \varphi_R \cdot (|b|^2 u) dx,
\end{align*}
$$

and taking above two identities to (2.5), we see that

$$
\int_{B_R} \varphi_R |\nabla \otimes b|^2 dx = \int_{B_R} \Delta \varphi_R \left(\frac{|b|^2}{2}\right) + \nabla \varphi_R \cdot \left(\frac{|b|^2}{2} u\right) dx + \int_{B_R} \varphi_R (b \cdot \nabla) u \cdot b dx. \quad (2.6)
$$

Adding (2.4) and (2.6) together, since the last terms of (2.4) and (2.6) can be cancelled, we obtain

$$
\begin{align*}
& \int_{B_R} \varphi_R |\nabla \otimes u|^2 dx + \int_{B_R} \varphi_R |\nabla \otimes b|^2 dx \\
&= \int_{B_R} \Delta \varphi_R \left(\frac{|u|^2}{2}\right) + \Delta \varphi_R \left(\frac{|b|^2}{2}\right) + \nabla \varphi_R \cdot \left(\frac{|u|^2}{2} u\right) dx \\
& \quad + \int_{B_R} \nabla \varphi_R \cdot (\pi u) + \nabla \varphi_R \cdot \left(\frac{|b|^2}{2} u\right) - (b \cdot \nabla) \varphi_R b \cdot u dx \\
&:= I_1 + I_2 + I_3 + I_4 + I_5 + I_6. \tag{2.7}
\end{align*}
$$

Since $\varphi_R(x) = 1$ over the set $|x| \le \frac{R}{2}$, one can easily see that

$$
\int_{B_{\frac{R}{2}}} |\nabla \otimes u|^2 dx + \int_{B_{\frac{R}{2}}} |\nabla \otimes b|^2 dx \leq \sum_{i=1}^{6} |I_i|.
$$

Thus if we can prove all the limits

$$
\lim_{R \to +\infty} I_i = 0, \text{ for } i = 1, 2, \dots, 6, \tag{2.8}
$$

then we obtain

$$
\lim_{R \to +\infty} \left( \int_{B_{\frac{R}{2}}} |\nabla \otimes u|^2 dx + \int_{B_{\frac{R}{2}}} |\nabla \otimes b|^2 dx \right) = \|(u, b)\|_{\dot{H}^1}^2 = 0,
$$

which implies that $\|(u, b)\|_{L^6} = 0$ by the Sobolev embedding $\dot{H}^1(\mathbb{R}^3) \hookrightarrow L^6(\mathbb{R}^3)$, and we get $(u, b) = (0, 0)$.
Now we estimate the terms $I_i$ ($i = 1, 2, \dots, 6$) one by one to complete the proof of (2.8).

Estimate of $I_1$. Applying the Hölder's inequality with $\frac{2}{p(\cdot)} + \frac{1}{r(\cdot)} = 1$, one sees that

$$
I_1 \le \int_{\mathbb{R}^3} |\Delta \varphi_R| \frac{|u|^2}{2} dx \le C \|\Delta \varphi_R\|_{L^{p(\cdot)}(\mathbb{R}^3)} \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}^2. \quad (2.9)
$$

Under the assumptions of Theorem 1.1, we know that $\|u\|_{L^{p(\cdot)}(\mathbb{R}^3)} < +\infty$, thus it suffices to estimate the quantity
$\|\Delta\varphi_R\|_{L^{r(\cdot)}(\mathbb{R}^3)}$ in (2.9). Since the variable exponent $p(\cdot)$ satisfies the condition (1.4), it is easy to verify that the
variable exponent $r(\cdot) = \frac{p(\cdot)}{p(\cdot)-2}$ satisfies

$$
\frac{9}{5} < r_{(\mathbb{R}^3 \setminus E)}^{-} \le r_{(\mathbb{R}^3 \setminus E)}(x) \le r_{(\mathbb{R}^3 \setminus E)}^{+} < 3 \quad \text{and} \quad 1 < r_{E}^{-} \le r_{E}(x) \le r_{E}^{+} < \frac{9}{5}, \qquad (2.10)
$$

where $E = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le 1, x_1 \in \mathbb{R}\}$. Since $\text{supp}(\Delta\varphi_R) \subset C(\frac{R}{2}, R) = \{x \in \mathbb{R}^3 : \frac{R}{2} \le |x| \le R\}$, we have

$$
\|\Delta\varphi_R\|_{L^{r(\cdot)}(\mathbb{R}^3)} = \|\Delta\varphi_R\|_{L^{r(\cdot)}(C(\frac{R}{2}, R))}.
$$

To continue, we denote by $E_1$ and $E_2$ the following sets defined by

$$
E_1 := C(\frac{R}{2}, R) \cap E \quad \text{and} \quad E_2 := C(\frac{R}{2}, R) \setminus E, \qquad (2.11)
$$
and using the fact (2.2), one has

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{r(\cdot)}(C(\frac{R}{2},R))} &= \|\Delta\varphi_R(1_{E_1} + 1_{E_2})\|_{L^{r(\cdot)}(C(\frac{R}{2},R))} \\ &\le \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_1)} + \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_2)}. \end{aligned} \quad (2.12) $$

To estimate the first term on the right-hand side of (2.12), one can apply Lemma 2.3 to get

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_1)} &\le \|\Delta\varphi_R\|_{L^{\infty}(E_1)} \|1\|_{L^{r(\cdot)}(E_1)} \\ &\le \|\Delta\varphi_R\|_{L^{\infty}(C(\frac{R}{2},R))} \|1\|_{L^{r(\cdot)}(E_1)} \\ &\le CR^{-2} \|1\|_{L^{r(\cdot)}(E_1)}, \end{aligned} \quad (2.13) $$

where we have used the fact $E_1 \subset C(\frac{R}{2}, R)$ and the definition of the function $\varphi_R$. On the other hand, as $E_1 \subset E$, we know $r_{\bar{E}} \le r_{\bar{E}_1} \le r_{\bar{E}_1}^{+} \le r_{\bar{E}}^{+}$, thus applying the Lemma 2.2 leads to

$$ \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_1)} \le CR^{-2} \max \left\{ |E_1|^{\frac{1}{r_E}}, |E_1|^{\frac{1}{r_E^+}} \right\}. \quad (2.14) $$

Notice that $E_1$ is actually included in a cylinder with diameter 1 and height $2R$, i.e.,

$$ E_1 \subset \mathcal{A} = \left\{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le 1, -R < x_1 < R \right\}. $$

Then we have $|E_1| \le |\mathcal{A}| = CR$ and

$$ \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_1)} \le C \max \left\{ R^{-2+\frac{1}{r_E}}, R^{-2+\frac{1}{r_E^+}} \right\}. \quad (2.15) $$

By (2.10), we know $\frac{5}{9} < \frac{1}{r_E^+} \le \frac{1}{r_E} < 1$, which yields that $-2 + \frac{1}{r_E^+} \le -2 + \frac{1}{r_E} < -1$, thus we obtain

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_1)} \to 0. \quad (2.16) $$

For the second term on the right-hand side of (2.12), applying the same arguments as above, one sees that

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_2)} &\le \|\Delta\varphi_R\|_{L^{\infty}(E_2)} \|1\|_{L^{r(\cdot)}(E_2)} \\ &\le CR^{-2} \|1\|_{L^{r(\cdot)}(E_2)} \\ &\le CR^{-2} \max \left\{ |E_2|^{\frac{1}{r_E}}, |E_2|^{\frac{1}{r_E^+}} \right\}, \end{aligned} \quad (2.17) $$

where we have used the facts $E_2 \subset C(\frac{R}{2}, R)$ and $\|\Delta\varphi_R\|_{L^{\infty}(E_2)} \le \|\Delta\varphi_R\|_{L^{\infty}(C(\frac{R}{2},R))} \le CR^{-2}$. By (2.10) again, we know that

$$ \frac{1}{3} < \frac{1}{r_{\mathbb{R}^3 \setminus E}^+} < \frac{1}{r_{E_2}^+} \le \frac{1}{r_{\bar{E}_2}^+} \le \frac{1}{r_{\mathbb{R}^3 \setminus E}^+} < \frac{5}{9}. $$

Moreover, since $|E_2| \le |C(\frac{R}{2}, R)| \le CR^3$, we have

$$ \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_2)} \le C \max \left\{ R^{-2+\frac{3}{r_{\mathbb{R}^3 \setminus E}^+}}, R^{-2+\frac{3}{r_{\mathbb{R}^3 \setminus E}^+}} \right\}. \quad (2.18) $$

Observing that $-2 + \frac{3}{r_{\mathbb{R}^3 \setminus E}^+} \le -2 + \frac{3}{r_{\mathbb{R}^3 \setminus E}^+} < -\frac{1}{3}$, which yields that

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(E_2)} \to 0. \quad (2.19) $$

Taking (2.16) and (2.19) back to (2.12), we obtain

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(\mathbb{R}^3)} = \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(C(\frac{R}{2},R))} = 0, $$

which directly yields that $\lim_{R \to +\infty} |I_1| = 0$.
Estimate of $I_2$. Applying the Hölder's inequality and using the fact $\text{supp}(\nabla \varphi_R) \subset C(\frac{R}{2}, R)$, we see that

$$|I_2| = \left| \int_{B_R} \Delta \varphi_R \frac{|b|^2}{2} dx \right| \le C \|\Delta \varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \|b\|_{L^{q(\cdot)}(\mathbb{R}^3)}^2, \quad (2.20)$$

where $\frac{2}{q(\cdot)} + \frac{1}{s(\cdot)} = 1$. Since the variable exponent $q(\cdot)$ satisfies (1.5), we can deduce that the variable exponent $s(\cdot) = \frac{q(\cdot)}{q(\cdot)-2}$ satisfies

$$\frac{9}{5} < s_{(\mathbb{R}^3 \setminus E)}^{-} \le s_{(\mathbb{R}^3 \setminus E)}(x) \le s_{(\mathbb{R}^3 \setminus E)}^{+} < 3 \quad \text{and} \quad 1 < s_{E}^{-} \le s_{E}(x) \le s_{E}^{+} < \frac{9}{5}. \quad (2.21)$$

Using the sets $E_1$ and $E_2$ as defined in (2.11) and proceeding just as the estimate (2.12) above, we have

$$\|\Delta\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \le \|\Delta\varphi_R\|_{L^{s(\cdot)}(E_1)} + \|\Delta\varphi_R\|_{L^{s(\cdot)}(E_2)}.$$

Following the same ideas that leads us to estimates (2.15) and (2.18), one gets

$$\|\Delta\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \le C \max \left\{ R^{-2+\frac{1}{s_E}}, R^{-2+\frac{1}{s_E}} \right\} + C \max \left\{ R^{-2+\frac{3}{s_{\mathbb{R}^3 \setminus E}}}, R^{-2+\frac{3}{s_{\mathbb{R}^3 \setminus E}}} \right\}. \quad (2.22)$$

By (2.21), one can easily deduce that

$$-2 + \frac{1}{s_E^+} \le -2 + \frac{1}{s_E^-} < -1, \quad -2 + \frac{3}{s_{\mathbb{R}^3 \setminus E}^+} \le -2 + \frac{3}{s_{\mathbb{R}^3 \setminus E}^-} < 0,$$

which yields that $\lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \to 0$ by (2.22). Since $\|b\|_{L^{q(\cdot)}(\mathbb{R}^3)} < +\infty$, we conclude from (2.20) that $\lim_{R \to +\infty} |I_2| = 0$.

Estimate of $I_3$. Using the Hölder's inequality yields that

$$|I_3| \le \frac{1}{2} \int_{C(\frac{R}{2}, R)} |u|^3 |\nabla \varphi_R| dx \le C \|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}^3, \quad (2.23)$$

where $\frac{3}{p(\cdot)} + \frac{1}{l(\cdot)} = 1$. Similarly, by (1.4), we deduce that the variable exponent $l(\cdot) = \frac{p(\cdot)}{p(\cdot)-3}$ satisfies

$$3 < l_{(\mathbb{R}^3 \setminus E)}^{-} \le l_{(\mathbb{R}^3 \setminus E)}(x) \le l_{(\mathbb{R}^3 \setminus E)}^{+} < +\infty \quad \text{and} \quad 1 < l_{E}^{-} \le l_{E}(x) \le l_{E}^{+} < 3, \quad (2.24)$$

and

$$\|\nabla\varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \le \|\nabla\varphi_R\|_{L^{l(\cdot)}(E_1)} + \|\nabla\varphi_R\|_{L^{l(\cdot)}(E_2)}.$$

Recalling that $\|\nabla\varphi_R\|_{L^\infty} \le CR^{-1}$, we can follow the same ideas that leads to estimates (2.15) and (2.18) to deduce that

$$\|\nabla\varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \le C \max \left\{ R^{-1+\frac{1}{l_E}}, R^{-1+\frac{1}{l_E}} \right\} + C \max \left\{ R^{-1+\frac{3}{l_{\mathbb{R}^3 \setminus E}^{-}}}, R^{-1+\frac{3}{l_{\mathbb{R}^3 \setminus E}^{+}}} \right\}, \quad (2.25)$$

which by the values of the function $l(\cdot)$ in (2.24), one sees that

$$-1 + \frac{1}{l_E^+} \le -1 + \frac{1}{l_E^-} < 0, \quad -1 + \frac{3}{l_{\mathbb{R}^3 \setminus E}^+} \le -1 + \frac{3}{l_{\mathbb{R}^3 \setminus E}^-} < 0.$$

This readily implies that $\lim_{R \to +\infty} \|\nabla\varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} = 0$. Since $\|u\|_{L^{p(\cdot)}(\mathbb{R}^3)} < +\infty$, we obtain from (2.23) that $\lim_{R \to +\infty} |I_3| = 0$.
*Estimate of $I_4$ Since $u \in L^{p(\cdot)}(\mathbb{R}^3)$ and $\pi \in L^{\frac{p(\cdot)}{2}}(\mathbb{R}^3)$, we can get the limit $\lim_{R \to +\infty} |I_4| = 0$ by (2.25) and the following inequality $(\frac{3}{p(\cdot)} + \frac{1}{l(\cdot)} = 1)$:

$$|I_4| \le \int_{C(\frac{R}{2}, R)} |\pi| |u| |\nabla \varphi_R| dx \le C \|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \|\pi\|_{L^{\frac{p(\cdot)}{2}}(\mathbb{R}^3)} \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)} \quad (2.26)$$

*Estimates of $I_5 + I_6$. Notice that*

$$\begin{aligned} |I_5 + I_6| & \le C \int_{C(\frac{R}{2}, R)} |b|^2 |u| |\nabla \varphi_R| dx \\ & \le C \|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \|b|^2 \| L^{\frac{q(\cdot)}{2}}(C(\frac{R}{2}, R)) \| u \|_{L^{p(\cdot)}(C(\frac{R}{2}, R))} \\ & \le C \|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \|b\|_{L^{q(\cdot)}(\mathbb{R}^3)}^2 \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}, \end{aligned} \quad (2.27)$$

where $\frac{1}{m(\cdot)} + \frac{2}{q(\cdot)} + \frac{1}{p(\cdot)} = 1$. Then we can use (1.4), (1.5) and the condition $\frac{2}{3} < \frac{1}{p(\mathbb{R}^3 \setminus E)(\cdot)} + \frac{2}{q(\mathbb{R}^3 \setminus E)(\cdot)}$ to deduce that the variable exponent $m(\cdot) = \frac{p(\cdot)q(\cdot)}{p(\cdot)q(\cdot) - q(\cdot) - 2p(\cdot)}$ satisfies

$$3 < m_{(\mathbb{R}^3 \setminus E)} \le m_{(\mathbb{R}^3 \setminus E)}(x) \le m_{(\mathbb{R}^3 \setminus E)}^{+} < \infty \quad \text{and} \quad 1 < m_{\bar{E}} \le m_{\bar{E}}(x) \le m_{\bar{E}}^{+} < 3. \quad (2.28)$$

Recalling the definition of the sets $E_1$ and $E_2$ in (2.14), one gets

$$\|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \le \|\nabla \varphi_R\|_{L^{m(\cdot)}(E_1)} + \|\nabla \varphi_R\|_{L^{m(\cdot)}(E_2)}. \quad (2.29)$$

Following the same ideas that leads us to estimates (2.15) and (2.18) (with the difference that $\|\nabla \varphi_R\|_{L^\infty} \le CR^{-1}$), we deduce that

$$\|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \le C \max \left\{ R^{-1+\frac{1}{m_E}}, R^{-1+\frac{1}{m_E^+}} \right\} + C \max \left\{ R^{-1+\frac{3}{m_{\mathbb{R}^3 \setminus E}^+}}, R^{-1+\frac{3}{m_{\mathbb{R}^3 \setminus E}^+}} \right\}. \quad (2.30)$$

Now, by using the values of the function $m(\cdot)$ given in (2.28), one obtains

$$-1 + \frac{1}{m_E^+} \le -1 + \frac{1}{m_E} < 0, -1 + \frac{3}{m_{\mathbb{R}^3 \setminus E}^+} \le -1 + \frac{3}{m_{\mathbb{R}^3 \setminus E}^-} < 0,$$

i.e., all powers in (2.30) are negative, from which we deduce that $\lim_{R \to +\infty} \|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} = 0$, and thus, by the assumptions $u \in L^{p(\cdot)}(\mathbb{R}^3)$ and $b \in L^{q(\cdot)}(\mathbb{R}^3)$, we conclude that $\lim_{R \to +\infty} |I_5 + I_6| = 0$.

We have thus proven that, under the assumptions of Theorem 1.1, all the terms $|I_i|$ ($i = 1, 2, \dots, 6$) given in (2.7) tend to 0 as $R \to +\infty$, the proof of Theorem 1.1 is completed.

# 3 Proof of Theorem 1.2

Following the main ideas used in the proof of Theorem 1.1, it suffices to prove that, under the assumptions of Theorem 1.2, all the terms $|I_i|$ ($i = 1, 2, \dots, 6$) given in (2.7) tend to 0 as $R \to +\infty$. Moreover, the terms $I_2$ and $I_4$ can be exactly estimated by the same argument used in the proof of Theorem 1.1, thus we need only to focus on the terms $I_1$, $I_3$ and $I_5 + I_6$.

*Estimate of $I_1$. We first see that*

$$|I_1| \le C \|\Delta \varphi_R\|_{L^{\varphi(\cdot)}(\mathbb{R}^3)} \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}^2, \quad (3.1)$$

where $\frac{2}{p(\cdot)} + \frac{1}{q(\cdot)} = 1$. By (1.6), we know the variable exponent $\varphi(\cdot) = \frac{p(\cdot)}{p(\cdot)-2}$ satisfies

$$\frac{9}{5} < \bar{\mathbf{r}}_{(\mathbb{R}^3 \setminus G)} \le \mathbf{r}_{(\mathbb{R}^3 \setminus G)}(x) \le \mathbf{r}_{(\mathbb{R}^3 \setminus G)}^{+} < 3 \quad \text{and} \quad \frac{\gamma+1}{2} < \bar{\mathbf{r}}_G \le \mathbf{r}_G(x) \le \mathbf{r}_G^{+} < \frac{9}{5}, \quad (3.2)$$
where $G = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le x_1^\gamma, x_1 > 0\}$. By using the support properties of the localized function $\theta_R$, we have

$$ \|\Delta\varphi_R\|_{L^{\gamma}(\mathbb{R}^3)} = \|\Delta\varphi_R\|_{L^{\gamma}(C(\frac{R}{2}, R))}. $$

Let $G_1$ and $G_2$ be the following sets defined by

$$ G_1 = C(\frac{R}{2}, R) \cap G \quad \text{and} \quad G_2 = C(\frac{R}{2}, R) \setminus G, \qquad (3.3) $$

and using the fact (2.2), we have

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{\gamma}(C(\frac{R}{2}, R))} &= \|\Delta\varphi_R(1_{G_1} + 1_{G_2})\|_{L^{\gamma}(C(\frac{R}{2}, R))} \\ &\le \|\Delta\varphi_R\|_{L^{\gamma}(G_1)} + \|\Delta\varphi_R\|_{L^{\gamma}(G_2)}. \end{aligned} \qquad (3.4) $$

For the first term of the right-hand side of (3.4), we can bound it as

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{\gamma}(G_1)} &\le \|\Delta\varphi_R\|_{L^{\infty}(G_1)} \|1\|_{L^{\gamma}(G_1)} \\ &\le C R^{-2} \|1\|_{L^{\gamma}(G_1)} \\ &\le C R^{-2} \max \left\{ |G_1|^{\frac{1}{\gamma_G}}, |G_1|^{\frac{1}{\gamma_G}} \right\}. \end{aligned} $$

Notice that

$$ G_1 = C(\frac{R}{2}, R) \cap G \subset \mathcal{B} = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \le x_1^\gamma, 0 < x_1 < R\}, $$

which yields that the volume of $G_1$ can be controlled by the volume of $\mathcal{B}$, i.e.,

$$ |G_1| \le |\mathcal{B}| = C \int_0^R x_1^\gamma dx_1 = C R^{\gamma+1}, \qquad (3.5) $$

and we obtain

$$ \|\Delta\varphi_R\|_{L^{\gamma}(G_1)} \le C \max \left\{ R^{-2+\frac{\gamma+1}{\gamma_G}}, R^{-2+\frac{\gamma+1}{\gamma_G}} \right\}. \qquad (3.6) $$

By (3.2), we know $\frac{5}{9} < \frac{1}{\tau_G^+} \le \frac{1}{\tau_G(x)} \le \frac{1}{\tau_G^-} < \frac{\gamma+3}{3\gamma+3}$, which combining with the range of $0 < \gamma < 1$, we have

$$ -2 + \frac{\gamma+1}{\tau_G^+} \le -2 + \frac{\gamma+1}{\tau_G^-} < -2 + (\gamma+1) \frac{\gamma+3}{3\gamma+3} < 0. $$

This tell us that all powers in (3.6) are negative, thus we obtain

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{\gamma}(G_1)} = 0. \qquad (3.7) $$

For the second term on the right-hand side of (3.4), we can also bound it as

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{\gamma}(G_2)} &\le \|\Delta\varphi_R\|_{L^{\infty}(G_2)} \|1\|_{L^{\gamma}(G_2)} \\ &\le C R^{-2} \|1\|_{L^{\gamma}(G_2)} \\ &\le C \max \left\{ R^{-2+\frac{3}{\tau^3 \setminus G}}, R^{-2+\frac{3}{\tau^3 \setminus G}} \right\}. \end{aligned} \qquad (3.8) $$

By (3.2), we know $-2 + \frac{3}{\tau^3 \setminus G} < -2 + \frac{3}{\tau^3 \setminus G} < 0$, which directly yields that

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{\gamma}(G_2)} = 0. \qquad (3.9) $$

Putting (3.7) and (3.9) together, we conclude from (3.4) that $\lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{\gamma}(C(\frac{R}{2}, R))} = 0$. Now by the assumption $\|u\|_{L^{p}(\mathbb{R}^3)} < +\infty$, we obtain from (3.1) that $\lim_{R \to +\infty} |I_1| = 0$.
Estimate of $I_3$. We first use the Hölder's inequality to bound $I_3$ as

$$|I_3| \le C \|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}^3 \quad (3.10)$$

where $\frac{3}{p(\cdot)} + \frac{1}{l(\cdot)} = 1$. Since the variable exponent $p(\cdot)$ satisfies (1.6), the variable exponent $l(\cdot) = \frac{p(\cdot)}{p(\cdot)-3}$ satisfies

$$3 < \overline{l_{(\mathbb{R}^3 \setminus G)}} \le l_{(\mathbb{R}^3 \setminus G)}(x) \le l_{(\mathbb{R}^3 \setminus G)}^{+} < +\infty \quad \text{and} \quad \gamma + 1 < \overline{l_G} \le l_G(x) \le l_G^{+} < 3. \quad (3.11)$$

Next, using the sets $G_1$ and $G_2$ defined in (3.3) and the fact(2.2), one has

$$\|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \le \|\nabla \varphi_R\|_{L^{l(\cdot)}(G_1)} + \|\nabla \varphi_R\|_{L^{l(\cdot)}(G_2)}. \quad (3.12)$$

Notice that $\|\nabla \varphi_R\|_{L^\infty} \le CR^{-1}$, we can follow the same ideas used in the estimates of (3.6) and (3.8) to obtain that

$$\|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} \le C \max \left\{ R^{-1+\frac{\gamma+1}{l_G}}, R^{-1+\frac{\gamma+1}{l_G^+}} \right\} + C \max \left\{ R^{-1+\frac{3}{l_{\mathbb{R}^3 \setminus G}}}, R^{-1+\frac{3}{l_{\mathbb{R}^3 \setminus G}^+}} \right\}. \quad (3.13)$$

By (3.11), one can readily calculate that

$$-1 + \frac{\gamma + 1}{l_G^+} \le -1 + \frac{\gamma + 1}{l_G^-} < 0, \quad -1 + \frac{3}{l_{\mathbb{R}^3 \setminus G}^+} \le -1 + \frac{3}{l_{\mathbb{R}^3 \setminus G}^-} < 0,$$

from which we know $\lim_{R \to +\infty} \|\nabla \varphi_R\|_{L^{l(\cdot)}(C(\frac{R}{2}, R))} = 0$, and by the assumption $\|u\|_{L^{p(\cdot)}(\mathbb{R}^3)} < +\infty$, we have $\lim_{R \to +\infty} |I_3| = 0$.

Estimate of $I_5 + I_6$. Using the Hölder's inequality yields that

$$|I_5 + I_6| \le C \|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \|b\|_{L^{q(\cdot)}(\mathbb{R}^3)}^2 \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}, \quad (3.14)$$

where $\frac{1}{m(\cdot)} + \frac{2}{q(\cdot)} + \frac{1}{p(\cdot)} = 1$. Since the variable exponents $p(\cdot)$ and $q(\cdot)$ satisfy (1.6), (1.7) and the condition $\frac{2}{3} < \frac{1}{p_{(\mathbb{R}^3 \setminus (E \cup G))}(\cdot)} + \frac{2}{q_{(\mathbb{R}^3 \setminus (E \cup G))}(\cdot)}$, the variable exponent $m(\cdot) = \frac{p(\cdot)q(\cdot)}{p(\cdot)q(\cdot)-q(\cdot)-2p(\cdot)}$ satisfies

$$3 < \bar{m}_{(\mathbb{R}^3 \setminus (E \cup G))} \le \bar{m}_{(\mathbb{R}^3 \setminus (E \cup G))}(x) \le \bar{m}_{(\mathbb{R}^3 \setminus (E \cup G))}^{+} < \infty, \\ \frac{3\gamma + 3}{\gamma + 2} < \bar{m}_{G \setminus E} \le \bar{m}_{G \setminus E}(x) \le \bar{m}_{G \setminus E}^{+} < 9, \quad \frac{3\gamma + 3}{2\gamma + 3} < \bar{m}_{E} \le \bar{s}_{E}(x) \le \bar{m}_{E}^{+} < 3. \quad (3.15)$$

Using the sets $G_1$ and $G_2$ defined in (3.2), and the sets $E_1$ and $E_2$ as defined in (2.14), we denote by $\mathcal{E}_1$, $\mathcal{E}_2$ and $\mathcal{E}_3$ the following sets defined by

$$\mathcal{E}_1 := C(\frac{R}{2}, R) \cap E, \quad \mathcal{E}_2 := (C(\frac{R}{2}, R) \cap G) \setminus E, \quad \mathcal{E}_3 := C(\frac{R}{2}, R) \setminus (E \cup G),$$

and by the fact (2.2), one has

$$\|\nabla \varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} \le \|\nabla \varphi_R\|_{L^{m(\cdot)}(\mathcal{E}_1)} + \|\nabla \varphi_R\|_{L^{m(\cdot)}(\mathcal{E}_2)} + \|\nabla \varphi_R\|_{L^{m(\cdot)}(\mathcal{E}_3)}. \quad (3.16)$$

For the three terms on the right-hand side of (3.16), using the fact $\|\nabla \varphi_R\|_{L^\infty} \le CR^{-1}$, we can bound them as

$$\|\nabla \varphi_R\|_{L^{m(\cdot)}(\mathcal{E}_1)} \le CR^{-1} \max \left\{ |\mathcal{E}_1|^{\frac{1}{m_E}}, |\mathcal{E}_1|^{\frac{1}{m_E^+}} \right\} \\ \le C \max \left\{ R^{-1+\frac{1}{m_E}}, R^{-1+\frac{1}{m_E^+}} \right\}; \quad (3.17)$$

$$\|\nabla \varphi_R\|_{L^{m(\cdot)}(\mathcal{E}_2)} \le CR^{-1} \max \left\{ |\mathcal{E}_2|^{\frac{1}{m_{\mathbb{G}\setminus E}^+}}, |\mathcal{E}_2|^{\frac{1}{m_{\mathbb{G}\setminus E}^+}} \right\}$$
$$ \leq C \max \left\{ R^{-1+\frac{\gamma+1}{m_{G\setminus E}^{+}}}, R^{-1+\frac{\gamma+1}{m_{G\setminus E}^{+}}} \right\}; \quad (3.18) $$

and

$$ \begin{aligned} \|\nabla\varphi_R\|_{L^{m(\cdot)}(\varepsilon_3)} &\leq C R^{-1} \max \left\{ |\varepsilon_3|^{\frac{1}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}}, |\varepsilon_3|^{\frac{1}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}} \right\} \\ &\leq C \max \left\{ R^{-1+\frac{3}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}}, R^{-1+\frac{3}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}} \right\}. \end{aligned} \quad (3.19) $$

Putting above estimates (3.17)–(3.19) together, we see that

$$ \begin{aligned} \|\nabla\varphi_R\|_{L^{m(\cdot)}(C(\frac{R}{2}, R))} &\leq C \max \left\{ R^{-1+\frac{1}{m_E}}, R^{-1+\frac{1}{s_E}} \right\} + C \max \left\{ R^{-1+\frac{\gamma+1}{m_{G\setminus E}^{+}}}, R^{-1+\frac{\gamma+1}{m_{G\setminus E}^{+}}} \right\} \\ &\quad + C \max \left\{ R^{-1+\frac{3}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}}, R^{-1+\frac{3}{m_{\mathbb{R}^3\setminus(E\cup G)}^{+}}} \right\}. \end{aligned} \quad (3.20) $$

By the values of the function $m(\cdot)$ given in (3.15), and using the condition $0 < \gamma < 1$, one can easily calculate that

$$ \begin{aligned} -1 + \frac{1}{m_E^+} &\leq -1 + \frac{1}{m_E^-} < 0, \quad -1 + \frac{\gamma+1}{m_{\mathbb{R}^3 \cap G}^+} \leq -1 + \frac{\gamma+1}{m_{\mathbb{R}^3 \cap G}^-} < 0, \\ -1 + \frac{3}{m_{\mathbb{R}^3 \setminus (E \cup G)}^+} &\leq -1 + \frac{3}{m_{\mathbb{R}^3 \setminus (E \cup G)}^-} < 0, \end{aligned} $$

from which directly implies the limit $\lim_{R \to +\infty} \|\nabla\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} = 0$. This fact combines with the assumptions $u \in L^{p(\cdot)}(\mathbb{R}^3)$ and $b \in L^{q(\cdot)}(\mathbb{R}^3)$, we know $\lim_{R \to +\infty} |I_5 + I_6| \leq 0$.

We have thus proven that, under the assumptions of Theorem 1.2, all the terms $|I_i|$ ($i = 1, 2, \dots, 6$) given in (2.7) tend to 0 as $R \to +\infty$, the proof of Theorem 1.1 is completed.

# 4 Proof of Theorem 1.3

Similarly, to prove Theorem 1.3, it suffices to prove that, under the assumptions of Theorem 1.3, all the terms $|I_i|$ ($i = 1, 2, \dots, 6$) given in (2.7) tend to 0 as $R \to +\infty$. Moreover, the terms $I_1, I_3$ and $I_4$ can be exactly estimated by the same argument used in the proofs of Theorem 1.1 and 1.2, thus we need only to focus on the terms $I_2$ and $I_5 + I_6$.

*Estimate of $I_2$. Applying the Hölder's inequality yields that*

$$ |I_2| \leq C \|\Delta\varphi_R\|_{L^{r(\cdot)}(\mathbb{R}^3)} \|b\|_{L^{q(\cdot)}(\mathbb{R}^3)}^2, \quad (4.1) $$

where $\frac{2}{q(\cdot)} + \frac{1}{r(\cdot)} = 1$. As the variable exponent $q(\cdot)$ satisfies (1.9), we deduce that the variable exponent $r(\cdot) = \frac{q(\cdot)}{q(\cdot)-2}$ satisfies

$$ \frac{9}{5} < \bar{r}_{(\mathbb{R}^3 \setminus H)} \leq \mathbf{r}_{(\mathbb{R}^3 \setminus H)}(x) \leq \mathbf{r}_{(\mathbb{R}^3 \setminus H)}^{+} < 3 \quad \text{and} \quad \mathbf{r}_{H}(x) = \bar{r}_{H} = \mathbf{r}_{H}^{+} = 1, \quad (4.2) $$

where $H = \{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \leq x_1^{-\sigma}, x_1 > 0\}$. We denote by $H_1$ and $H_2$ the following sets defined by

$$ H_1 = C(\frac{R}{2}, R) \cap H \quad \text{and} \quad H_2 = C(\frac{R}{2}, R) \setminus H, \quad (4.3) $$

and using the fact (2.2), one has

$$ \|\Delta\varphi_R\|_{L^{r(\cdot)}(\mathbb{R}^3)} \leq \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_1)} + \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_2)}. \quad (4.4) $$
For the first term on the right-hand side of (4.4), we can estimate it as

$$ \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_1)} \leq CR^{-2} \max \left\{ |H_1|^{\frac{1}{r_H}}, |H_1|^{\frac{1}{r_H^+}} \right\}. \quad (4.5) $$

Notice that

$$ H_1 = C(\frac{R}{2}, R) \cap H \subset \mathcal{N} = \{ (x_1, x_2, x_3) \in \mathbb{R}^3 : x_2^2 + x_3^2 \leq x_1^{-\sigma}, 0 < x_1 < R \}, $$

which yields that the volume of $H_1$ can be controlled by the volume of $\mathcal{N}$, i.e.,

$$ |H_1| \leq |\mathcal{N}| = C \int_0^R x_1^{-\sigma} dx_1 = CR^{1-\sigma}. \quad (4.6) $$

Recalling that $0 < \sigma < 1$, by (4.5) and (4.6), we obtain

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_1)} = 0. \quad (4.7) $$

For the second term on the right-hand side of (4.4), we can directly bound it as

$$ \begin{aligned} \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_2)} &\leq |\Delta\varphi_R\|_{L^\infty(H_2)} \|1\|_{L^{r(\cdot)}(H_2)} \\ &\leq CR^{-2}\|1\|_{L^{r(\cdot)}(H_2)} \\ &\leq C \max \left\{ R^{-2+\frac{3}{r_{\mathbb{R}^3 \setminus H}}}, R^{-2+\frac{3}{r_{\mathbb{R}^3 \setminus H}}} \right\}. \end{aligned} \quad (4.8) $$

By (4.2), we know $-2 + \frac{3}{r_{\mathbb{R}^3 \setminus H}^+} < -2 + \frac{3}{r_{\mathbb{R}^3 \setminus H}^-} < 0$, which yields that

$$ \lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(H_2)} = 0. \quad (4.9) $$

Taking (4.7) and (4.9) into (4.4), we get $\lim_{R \to +\infty} \|\Delta\varphi_R\|_{L^{r(\cdot)}(C(\frac{R}{2}, R))} = 0$, and since $\|b\|_{L^{q(\cdot)}(\mathbb{R}^3)} < +\infty$, we obtain from (4.1) that $\lim_{R \to +\infty} |I_2| = 0$.

Estimate of $I_5 + I_6$. Using the Hölder's inequality, one has

$$ |I_5 + I_6| \leq C \|\nabla\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \|b\|_{L^{q(\cdot)}(\mathbb{R}^3)}^2 \|u\|_{L^{p(\cdot)}(\mathbb{R}^3)}, $$

where $\frac{1}{s(\cdot)} + \frac{2}{q(\cdot)} + \frac{1}{p(\cdot)} = 1$. As the variable exponents $p(\cdot)$ and $q(\cdot)$ satisfy (1.8), (1.9) and the condition $\frac{1}{p(\cdot)q(\cdot)} + \frac{2}{q(\cdot)p(\cdot)} > \frac{3}{2}$, we can deduce that the variable exponent $s(\cdot) = \frac{p(\cdot)q(\cdot)}{p(\cdot)q(\cdot)-q(\cdot)-2p(\cdot)}$ satisfies

$$ \begin{aligned} 3 < &s_{\mathbb{R}^3 \setminus (H \cup G)}^{-} \leq s_{\mathbb{R}^3 \setminus (H \cup G)}(x) \leq s_{\mathbb{R}^3 \setminus (H \cup G)}^{+} < \infty, \\ \frac{3\gamma + 3}{\gamma + 2} < &s_{G \setminus H}^{-} \leq s_{G \setminus H}(x) \leq s_{G \setminus H}^{+} < 9, \quad \frac{3\gamma + 3}{2\gamma + 3} \leq s_{H}^{-} \leq s_{H}(x) \leq s_{H}^{+} \leq \frac{3}{2}. \end{aligned} \quad (4.10) $$

Using the sets $G_1$ and $G_2$ as defined in (3.2), and the sets $H_1$ and $H_2$ as defined in (4.3), we denote by $\mathcal{H}_1$, $\mathcal{H}_2$ and $\mathcal{H}_3$ the following sets defined by

$$ \mathcal{H}_1 := C(\frac{R}{2}, R) \cap H, \quad \mathcal{H}_2 := (C(\frac{R}{2}, R) \cap G) \setminus H, \quad \mathcal{H}_3 := C(\frac{R}{2}, R) \setminus (H \cup G), $$

and by the fact (2.2), one has

$$ \|\nabla\varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} \leq \|\nabla\varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_1)} + \|\nabla\varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_2)} + \|\nabla\varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_3)}. \quad (4.11) $$

Following the same arguments used in above and using the fact $\|\nabla\varphi_R\|_{L^\infty} \leq CR^{-1}$, we can bound three terms on the right-hand side of (4.11) as

$$ \|\nabla\varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_1)} \leq CR^{-1} \max \left\{ |\mathcal{H}_1|^{\frac{1}{s_H}}, |\mathcal{H}_1|^{\frac{1}{s_H^+}} \right\} $$
$$ \begin{aligned} & \leq C \max \left\{ R^{-1+\frac{1-\sigma}{s_H}}, R^{-1+\frac{1-\sigma}{s_H^+}} \right\} \\ & \leq C R^{-1+(1-\sigma)\frac{2\gamma+3}{3\gamma+3}}; \end{aligned} \quad (4.12) $$

$$ \begin{aligned} \|\nabla \varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_2)} & \leq C R^{-1} \max \left\{ |\mathcal{H}_2|^{\frac{1}{s_G \setminus H}}, |\mathcal{H}_2|^{\frac{1}{s_{G \setminus H}^+}} \right\} \\ & \leq C \max \left\{ R^{-1+\frac{\gamma+1}{s_{\mathbb{R}^3 \cap G}^+}}, R^{-1+\frac{\gamma+1}{s_{\mathbb{R}^3 \cap G}^+}} \right\}; \end{aligned} \quad (4.13) $$

and

$$ \begin{aligned} \|\nabla \varphi_R\|_{L^{s(\cdot)}(\mathcal{H}_3)} & \leq C R^{-1} \max \left\{ |\mathcal{H}_3|^{\frac{1}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}}, |\mathcal{H}_3|^{\frac{1}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}} \right\} \\ & \leq C \max \left\{ R^{-1+\frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}}, R^{-1+\frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}} \right\}. \end{aligned} \quad (4.14) $$

Now taking (4.12)–(4.14) into (4.11), we obtain

$$ \begin{aligned} \|\nabla \varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} & \leq C R^{-1+(1-\sigma)\frac{2\gamma+3}{3\gamma+3}} + C \max \left\{ R^{-1+\frac{\gamma+1}{s_{G \setminus H}^+}}, R^{-1+\frac{\gamma+1}{s_{G \setminus H}^+}} \right\} \\ & \quad + C \max \left\{ R^{-1+\frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}}, R^{-1+\frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+}} \right\}. \end{aligned} \quad (4.15) $$

By the values of the function $s(\cdot)$ and $0 < \sigma, \gamma < 1$, we can deduce that

$$ -1 + (1 - \sigma) \frac{2\gamma + 3}{3\gamma + 3} < 0, \quad -1 + \frac{\gamma + 1}{s_{G \setminus H}^+} \leq -1 + \frac{\gamma + 1}{s_{G \setminus H}^-} < 0, \quad -1 + \frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^+} \leq -1 + \frac{3}{s_{\mathbb{R}^3 \setminus (H \cup G)}^-} < 0, $$

which implies that $\lim_{R \to +\infty} \|\nabla \varphi_R\|_{L^{s(\cdot)}(C(\frac{R}{2}, R))} = 0$. Thus by the assumptions $u \in L^{p(\cdot)}(\mathbb{R}^3)$ and $b \in L^{q(\cdot)}(\mathbb{R}^3)$, we conclude that $\lim_{R \to +\infty} |I_5 + I_6| = 0$. The proof of Theorem 1.3 is completed.

**Acknowledgements.** The authors declared that they have no conflict of interest. J. Zhao was partially supported by the National Natural Science Foundation of China (no. 12361034) and the Natural Science Foundation of Shaanxi Province (no. 2022JM-034).

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