In this paper, we study the following 3D stationary Q-tensor system of liquid crystal:

$ \left\{\begin{array}{lr} \,-\mu\Delta u+(u\cdot\nabla)u+\nabla P=-\nabla\cdot(\nabla Q\odot\nabla Q)-\lambda\nabla\cdot(|Q|H)+\nabla\cdot(Q\Delta Q-\Delta QQ), \\[1ex] \,(u\cdot\nabla) Q+Q\Omega-\Omega Q-\lambda|Q|D=\Gamma H, \\[1ex] \,\nabla\cdot u=0, \end{array} \right.$

(1.1)

with

$ H = \Delta Q -aQ+b\left(Q^{2}-\frac{\mbox{tr}(Q^{2})}{3}I_{3\times3}\right)-cQ\mbox{tr}(Q^{2}). $

Here $ u \in \mathbb{R}^{3} $, $ P \in \mathbb{R} $ and

$ Q\in S^{3}_{0}\triangleq \{A=(a_{ij})_{3\times 3}|a_{ij}=a_{ji},\; \mbox{tr}(A)=0\} $

stand for the flow velocity, the scalar pressure and the nematic tensor order parameter, respectively. The parameters $ \mu > 0 $, $ \Gamma^{-1}>0 $ and $ \lambda\in \mathbb{R} $ represent the viscosity coefficient, the rotational viscosity and the nematic alignment, respectively. The coefficients $ a, b, c\in \mathbb{R} $ with $ c> 0 $ are constants. $ (\nabla Q\odot\nabla Q)_{i,j}=\partial_{x_{i}}Q:\partial_{x_{j}}Q $ is the symmetric additional stress tensor. $ D\triangleq\frac{1}{2}(\nabla u + \nabla u^{\mathsf{T}}) $ and $ \Omega\triangleq\frac{1}{2}(\nabla u - \nabla u^{\mathsf{T}}) $ are the symmetric and skew symmetric, respectively, where the notation $ \mathsf{T} $ represents the transposition of a matrix.

When $ Q=0 $, system (1.1) reduces to the 3D stationary Navier-Stokes system:

$ \left\{\begin{array}{lr} \,-\Delta u+(u\cdot\nabla)u+\nabla P=0, \\[1ex] \,\nabla\cdot u=0. \\[1ex] \end{array} \right.$

(1.2)

In the past decades, there have been many results in the study of Liouville theorem for system (1.2). Specifically, Galdi [1] proved that when $ u\in L^{\frac{9}{2}}(\mathbb{R}^{3}) $, then $ u=0 $. Chae-Wölf [2] gave a logarithmic improvement, namely when the solution $ u $ satisfies

$ \begin{equation} \nonumber \displaystyle{\int}_{\mathbb{R}^{3}}|u|^{\frac{9}{2}}\bigg\{\log\left(2+\frac{1}{|u|}\right)\bigg\}^{-1} dx<\infty, \end{equation} $

then $ u=0 $. Kozono-Terasawa-Wakasugi [3] investigated Liouville theorem in Lorentz spaces $ L^{\frac{9}{2},\infty}(\mathbb{R}^{3}) $. Li-Niu [4] further obtained sufficient conditions in Lorentz spaces, that is, $ u\in L^{p,q}(\mathbb{R}^3) $ for $ 3<p<\frac{9}{2},\; \; 3\leq q\leq\infty $ or $ p=q=3 $. Jarrin [5] proved in Morrey spaces that when $ u\in L^{2}_{loc}(\mathbb{R}^{3})\cap \dot M^{p,r}(\mathbb{R}^{3}) $ with $ 3\leq p<r<\frac{9}{2} $, then $ u=0 $. Later, Chamorro-Jarrin-Lemari-Rieusset [6] extended the result in [5] to $ u\in \dot M^{2,3}(\mathbb{R}^{3})\cap \dot M^{2,p}(\mathbb{R}^{3}) $ with $ 3< p <\infty $. For more works on Liouville theorem for the 3D stationary Navier-Stokes system, one could refer to [7–12] and references therein.

In recent years, the Q-tensor system of liquid crystal (1.1) has been received much attention, however there are few results on its Liouville theorem (see [13, 14]). Gong-Liu-Zhang [13] proved that when

$ u\in L^{\frac{9}{2}}(\mathbb{R}^{3})\cap \dot{H}_1(\mathbb{R}^3), \; Q\in H^{2}(\mathbb{R}^{3}) \; \rm{and} \;\mathit{b} ^{2}-24\mathit{ac}\leq 0,$

(1.3)

then $ u=Q\equiv 0 $. Later, Lai and Wu [14] generalized the condition (1.3) to

$ u\in L^{\frac{9}{2},\infty}(\mathbb{R}^{3})\cap \dot{H}_1(\mathbb{R}^3), \; Q\in H^{2}(\mathbb{R}^{3}) \; \rm{and}\; \mathit{b}^{2}-24\mathit{ac}\leq 0.$

(1.4)

Concerning Liouville theorem for other systems, one can refer to [15–21] etc.

Motivated by the works [4–6], in this paper we will consider Liouville theorem for system (1.1) in Morrey and Lorentz spaces, respectively. First of all, we recall the definition of Lorentz spaces.

Let $ f: \mathbb{R}^3\rightarrow \mathbb{R} $ be a measurable function. The distribution function $ d_f(\alpha) $ is defined as

$ d_f(\alpha)=dx(\{x\in\mathbb{R}^3:|f(x)|>\alpha\}), $

where dx denotes the Lebesgue measure. For $ 1\leq p<+\infty $ and $ 1\leq q\leq+\infty $, the Lortenz spaces $ L^{p,q}(\mathbb{R}^3) $ are the spaces of measurable function $ f: \mathbb{R}^3\rightarrow \mathbb{R} $ such that

$ \|f\|_{L^{p,q}(\mathbb{R}^3)}<+\infty, $

where

$ \begin{eqnarray} \|f\|_{L^{p,q}(\mathbb{R}^3)}= \begin{cases} r^{\frac{1}{q}}\left(\int_{0}^{+\infty}\big(\alpha d_{f}^{\frac{1}{p}}(\alpha)\big)^q\frac{d\alpha}{\alpha}\right)^{\frac{1}{q}},\; \; \; \; \; &{\rm if}\; q<+\infty,\\ \sup\limits_{\alpha>0}\left\{\alpha d_{f}^{\frac{1}{p}}(\alpha)\right\},\; \; \; \; \; &{\rm if}\; q=+\infty.\\ \end{cases} \end{eqnarray} $

This space is a homogeneous space of degree $ -\frac{3}{r} $ and we have the following continuous embedding $ L^p(\mathbb{R}^3)=L^{p,p}(\mathbb{R}^3)\subset L^{p,q}(\mathbb{R}^3)\; {\rm for}\; 1\leq p<q\leq+\infty. $

In the framework of Lorentz spaces, our result is stated as follows.

Theorem 1.1  $ \rm Let $ $ (u,Q)\in L^2_{loc}(\mathbb{R}^3) $ be a smooth solution to system (1.1). If one of the following holds,

(1) $ u\in L^{p,q}(\mathbb{R}^3),\; Q\in H^2(\mathbb{R}^3),\; b^2-24ac\leq 0 ,\; $with$ \; 3< p\leq\frac{9}{2},\; 3\leq q<\infty,\; $or$ \; p=q=3, $

(2) $ u\in L^{p,q}(\mathbb{R}^3)\cap \dot{H}_1(\mathbb{R}^3),\; Q\in H^2(\mathbb{R}^3),\; b^2-24ac\leq 0 ,\; $with$ \; 3< p\leq\frac{9}{2},\; 3\leq q\leq\infty,\; $or$ \; p=q=3 $, then $ u=Q\equiv0. $

Remark 1  The difierence between (1) and (2) in Theorem 1.1 is that when $q=\infty$, the conditions for the velocity fleld need $ u\in\dot{H}_1(\mathbb{R}^3). $

Remark 2  $ \rm When $ $ (p,q)=(\frac{9}{2}, \frac{9}{2}), $ we get the condition (1.3) in [13]; while $ (p,q)=(\frac{9}{2}, \infty), $ that is the condition (1.4) in [13]. Therefore, Theorem 1.1 improves the results in [13] and [14].

We also establish the results in the framework of Morrey spaces. To this end, we first recall the definition of the homogeneous Morrey spaces $ \dot{M}^{p,q}(\mathbb{R}^{3}). $

For $ 1 < p < q < +\infty $, the homogeneous Morrey spaces $ \dot{M}^{p,q}(\mathbb{R}^{3}) $ are the set of all $ f\in L^{p}_{loc}(\mathbb{R}^{3}) $ such that

$ \begin{equation} \nonumber \|f\|_{\dot{M}^{p,q}(\mathbb{R}^{3})}=\sup\limits_{R>0,x_{0}\in\mathbb{R}^{3}}R^{\frac{3}{q}}\left(R^{-3}\displaystyle{\int}_{B(R(x_{0}))}|f(x)|^{p} dx\right)^{\frac{1}{p}}<+\infty, \end{equation} $

where $ B(R(x_{0})) $ is a ball of radius $ R $ centered at $ x_{0} $. This space satisfies the following embedding relation

$ \begin{equation} \nonumber L^{q}(\mathbb{R}^{3})\subset L^{q,r}(\mathbb{R}^{3})\subset L^{q,\infty}(\mathbb{R}^{3}) \subset \dot{M}^{p,q}(\mathbb{R}^{3}),\; 1< p < q \leq r\leq\infty. \end{equation} $

In the framework of Morrey spaces, our two results are stated as follows.

Theorem 1.2  $ \rm Let $ $ (u,Q)\in L^2_{loc}(\mathbb{R}^3) $ be a smooth solution to system (1.1). Assume $ u\in\dot{M}^{2,3}(\mathbb{R}^3)\cap\dot{M}^{2,q}(\mathbb{R}^3) $, $ Q\in H^2(\mathbb{R}^3),\; b^2-24ac\leq 0, $ with $ 3< q <\infty, $ then $ u=Q\equiv0. $

Theorem 1.3  $ \rm Let $ $ (u,Q)\in L^2_{loc}(\mathbb{R}^3) $ be a smooth solution to system (1.1). Suppose $ u\in\dot{M}^{p,q}(\mathbb{R}^3) $, $ Q\in H^2(\mathbb{R}^3),\; b^2-24ac\leq 0, $ with $ 3\leq p< q \leq \frac{9}{2}, $ then $ u=Q\equiv0. $

Remark 3  $ \rm When $ $ q=\frac{9}{2} $, compared to the conditions (1.3) and (1.4), due to the embedding relationship $ L^{\frac{9}{2}}(\mathbb{R}^3)\subset L^{\frac{9}{2},\infty}(\mathbb{R}^3)\subset\dot{M}^{p,\frac{9}{2}}(\mathbb{R}^3), $ we know that Theorem 1.3 extends the results in [13, 14].

The rest of this paper is organized as follows. In Section 2 we will prove Theorem 1.1 in the Lorentz spaces, while Section 3 is devoted to the proofs of Theorem 1.2 and Theorem 1.3 in the Morrey spaces.

In this section, we will prove Theorem 1.1. Firstly, we provide two known results as follows.

Lemma 2.1  (see [22]) Let $ \beta(Q)=1-6\frac{(\mbox{tr}Q^{3})^{2}}{|Q|^{6}}. $ Assume $ Q\in S^{3}_{0} $, then $ 0\leq\beta(Q)\leq 1 $.

Lemma 2.2  (see [6]) Let $ U\in L^{\infty}(\mathbb{R}^3)\cap\dot{M}^{2,3}(\mathbb{R}^3). $ Then $ \|U\|_{\dot{M}^{3,\frac{9}{2}}(\mathbb{R}^3)}\leq C\|U\|^{\frac{1}{3}}_{L^{\infty}(\mathbb{R}^3)}\|U\|^{\frac{2}{3}}_{{\dot{M}^{2,3}(\mathbb{R}^3)}} $.

We consider a smooth function $ \phi\in C^{\infty}_{c}(\mathbb{R}^{3}) $ as

$ \begin{equation} \nonumber \phi(x)= \left\{ \begin{aligned} & 1,\; \; \; \rm{if}\; |\mathit{x}|<1 \\ &0,\; \; \; \rm{if}\; |\mathit{x}|>2 \end{aligned} \right. \end{equation} $

and define $ \phi_{R}(x)=\phi(\frac{x}{R}), \; x\in \mathbb{R}^3, $ so we have $ {\rm supp}\; (\nabla\phi_{R})\subset\{x\in\mathbb{R}^3; R<|x|<2R\}=B(2R/R) $.

Proof  Multiplying the first equation and the second equation in (1.1) by $ u\phi_{R} $ and $ -H\phi_{R} $, respectively, integrating them over $ \mathbb{R}^3 $, and integration by parts, we have

$ \begin{align} &\mu\int_{\mathbb{R}^3}|\nabla u|^2\phi_R(x){\rm d}x+\Gamma\int_{\mathbb{R}^3}|H|^2\phi_R(x){\rm d}x\\ =&\left(\int_{\mathbb{R}^3}(u\cdot\nabla Q):\Delta Q\phi_R(x){\rm d}x-\int_{\mathbb{R}^3}(\Omega Q-Q\Omega):\Delta Q\phi_R(x){\rm d}x\right)\\ &-\int_{\mathbb{R}^3}(u\cdot \nabla Q):\left[aQ+b\left(Q^2-\frac{{\rm tr}(Q^2)}{3} I_{3\times3}\right)+cQ|Q|^2\right]\phi_R(x){\rm d}x\\ &+\int_{\mathbb{R}^3}(\Omega Q-Q\Omega):\left[aQ+b\left(Q^2-\frac{{\rm tr}(Q^2)}{3} I_{3\times3}\right)+cQ|Q|^2\right]\phi_R(x){\rm d}x\\ &-\left(\lambda\int_{\mathbb{R}^3}|Q|D:H\phi_R(x){\rm d}x+\int_{\mathbb{R}^3}\nabla\cdot(\nabla Q\odot\nabla Q)\cdot u\phi_R(x){\rm d}x\right)\\ &+\left(\lambda\int_{\mathbb{R}^3}|Q|H:\nabla(u\phi_R(x)){\rm d}x-\int_{\mathbb{R}^3}(Q\Delta Q-\Delta QQ):\nabla(u\phi_R(x)){\rm d}x\right)\\ &+\int_{\mathbb{R}^3}\frac{1}{2}|u|^2u\cdot\nabla\phi_R(x){\rm d}x+\left(\int_{\mathbb{R}^3}Pu\cdot\nabla\phi_R(x)+\frac{\mu}{2}|u|^2\Delta\phi_R(x){\rm d}x\right)\\ \triangleq & \sum\limits_{i=1}^{7}I_i. \end{align} $

(2.1)

Firstly, we deduce that $ I_{4} = 0 $ due to $ Q $ is symme tr ic and $ \Omega $ is skew symmetric. In what follows, we estimate the remaining terms on the right-hand side of (2.1) one by one.

For $ I_{1} $, applying Hölder inequality in Lorentz spaces, we see that

$ \begin{align} I_{1}&=\int_{\mathbb{R}^3}(u\cdot\nabla Q):\Delta Q \phi_{R} dx-\int_{\mathbb{R}^3}\nabla\cdot(\nabla Q\odot\nabla Q)\cdot u\phi_{R} {\rm d}x\\ &=\int_{\mathbb{R}^3}(u\cdot\nabla Q):\Delta Q \phi_{R} dx-\int_{\mathbb{R}^3}(u\cdot\nabla Q):\Delta Q \phi_{R} dx-\frac{1}{2}\int_{\mathbb{R}^3}u\cdot\nabla|\nabla Q|^{2}\phi_{R} {\rm d}x\\ &=\frac{1}{2}\int_{\mathbb{R}^3}u|\nabla Q|^{2}\nabla\phi_{R} {\rm d}x\\ &\leq \frac{C}{R}\||\nabla Q|^2\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{2p}{p-2},\frac{2q}{q-2}}(B(2R\setminus R))}\\ &\leq CR^{\frac{1}{2}-\frac{3}{p}}\||\nabla Q|^2\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.2)

By Sobolev imbedding theorem, we have

$ \begin{aligned} I_2 & \left.=-\int_{\mathbb{R}^3}(u \cdot \nabla Q):\left[a Q-b\left(Q^2-\frac{\operatorname{tr}\left(Q^2\right)}{3} I_{3 \times 3}\right)+c Q|Q|^2\right)\right] \phi_R \mathrm{d} x \\ & =\int_{\mathbb{R}^3}\left(\frac{a}{2}|Q|^2-\frac{b}{3} \operatorname{tr}\left(Q^3\right)+\frac{c}{4}|Q|^4\right) u \cdot \nabla \phi_R \mathrm{d} x \\ & \leq \frac{C}{R}\left(1+\|Q\|_{L^{\infty}(B(2 R \backslash R))}+\|Q\|_{L^{\infty}(B(2 R \backslash R))}^2\right)\left\||Q|^2\right\|_{L^2(B(2 R \backslash R))}\|u\|_{L^{p, q}(B(2 R \backslash R))}\|1\|_{L^{\frac{2 p}{p-2}, \frac{2 q}{q-2}(B(2 R \backslash R))}} \\ & \leq C R^{\frac{1}{2}-\frac{3}{p}}\left\||Q|^2\right\|_{L^2(B(2 R \backslash R))}\|u\|_{L^{p, q}(B(2 R \backslash R))} \rightarrow 0 \quad(\text { as } \quad R \rightarrow \infty) . \end{aligned}$

(2.3)

For $ I_{3} $, we obtain

$ \begin{align} I_{3}&=-\int_{\mathbb{R}^3}(\Omega Q-Q\Omega):\Delta Q\phi_{R}-\int_{\mathbb{R}^3}(Q\Delta Q-\Delta QQ):\nabla(u\phi_{R}) {\rm d}x\\ &=-\int_{\mathbb{R}^3}(Q\Delta Q-\Delta QQ):u\otimes\nabla\phi_{R} {\rm d}x\\ &\leq \frac{C}{R}\|\nabla^2 Q\|_{L^2(B(2R\setminus R))}\|Q\|_{L^{\infty}(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{2p}{p-2},\frac{2q}{q-2}}(B(2R\setminus R))}\\ &\leq CR^{\frac{1}{2}-\frac{3}{p}}\|\nabla^2 Q\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.4)

With regard to $ I_{5} $, we observe

$ \begin{align} I_5&=-\lambda\int_{\mathbb{R}^3}|Q|D:H\phi_R(x){\rm d}x+\lambda\int_{\mathbb{R}^3}|Q|H:\nabla(u\phi_R(x)){\rm d}x \\ &=\lambda\int_{\mathbb{R}^3}|Q|H:(u\otimes\nabla\phi_R(x)){\rm d}x \\ &\leq \frac{C}{R}\|H\|_{L^2(B(2R\setminus R))}\|Q\|_{L^{\infty}(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{2p}{p-2},\frac{2q}{q-2}}(B(2R\setminus R))}\\ \label{2.5} &\leq CR^{\frac{1}{2}-\frac{3}{p}}\|H\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.5)

By Hölder inequality in Lorentz spaces, we see

$ \begin{align} I_{6}&=\frac{1}{2}\int_{\mathbb{R}^3}|u|^{2}u\cdot\nabla\phi_{R} {\rm d}x\\ &\leq \frac{C}{R}\|u\|^3_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{p}{p-3},\frac{q}{q-3}}(B(2R\setminus R))}\\ &\leq CR^{2-\frac{9}{p}}\|u\|^3_{L^{p,q}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.6)

To deal with $ I_{7} $, we first need to estimate the pressure. For that, taking the divergence to the first equation of (1.1) yields

$ \begin{align} \Delta P=-\mbox{divdiv}\left(u\otimes u+\nabla Q\odot\nabla Q+\lambda|Q|H+Q\Delta Q-\Delta QQ\right). \end{align} $

(2.7)

Let $ P=P_{1}+P_{2} $ such that $ \Delta P_{1}=-\mbox{divdiv}(f_{1}) $ and $ \Delta P_{2}=-\mbox{divdiv}(f_{2}) $, where

$ \begin{align} f_{1}=u\otimes u,\; \; \; \; \; \; \quad f_{2}=\nabla Q\odot\nabla Q+\lambda|Q|H+Q\Delta Q-\Delta QQ. \end{align} $

(2.8)

In view of the conditions $ u\in L^{p,q}(\mathbb{R}^3) $ and $ Q\in H^{2}(\mathbb{R}^3) $, it is not difficult to obtain that

$ f_{1}\in L^{\frac{p}{2},\frac{q}{2}}(\mathbb{R}^3),\; \; \; f_{2}\in L^{2}(\mathbb{R}^3). $

By Calderón-Zygmund theorem, we have $ P_1\in L^{\frac{p}{2},\frac{q}{2}}(\mathbb{R}^3),\; P_{2}\in L^{2}(\mathbb{R}^3) $, and then

$ \begin{align} I_{7}&=\int_{\mathbb{R}^3}\left(Pu\cdot\nabla\phi_{R}+\frac{\mu}{2}|u|^{2}\Delta\phi_{R}\right) {\rm d}x\\ &=\int_{\mathbb{R}^3}P_{1}u\cdot\nabla\phi_{R} {\rm d}x+\int_{\mathbb{R}^3}P_{2}u\cdot\nabla\phi_{R} {\rm d}x+\frac{\mu}{2}\int_{\mathbb{R}^3}|u|^{2}\Delta\phi_{R} {\rm d}x\\ &\leq \frac{C}{R}\|P_1\|_{L^{\frac{p}{2},\frac{q}{2}}(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{p}{p-3},\frac{q}{q-3}}(B(2R\setminus R))}\\ &\; \; \; \; \; \; \; +\frac{C}{R}\|P_2\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{2p}{p-2},\frac{2q}{q-2}}(B(2R\setminus R))}\\ &\; \; \; \; \; \; \; +\frac{C}{R^2}\|u\|^2_{L^{p,q}(B(2R\setminus R))}\|1\|_{L^{\frac{p}{p-2},\frac{q}{q-2}}(B(2R\setminus R))}\\ &\leq CR^{2-\frac{9}{p}}\|P_1\|_{L^{\frac{p}{2},\frac{q}{2}}(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\\ &\; \; \; \; \; \; \; +CR^{\frac{1}{2}-\frac{3}{p}}\|P_2\|_{L^2(B(2R\setminus R))}\|u\|_{L^{p,q}(B(2R\setminus R))}\\ &\; \; \; \; \; \; \; +CR^{1-\frac{6}{p}}\|u\|^2_{L^{p,q}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.9)

Based on the above estimates $ I_1-I_7 $, returning to (2.1), we find that

$ \begin{align} \lim\limits_{R\rightarrow +\infty}\left(\mu\int_{B(R)}|\nabla u|^{2} {\rm d}x+\Gamma\int_{B(R)}|H|^{2} {\rm d}x\right) = 0. \end{align} $

(2.10)

Thanks to the Sobolev embedding, we have

$ \begin{align} \|u\|_{L^{6}(\mathbb{R}^3)}\leq C\|\nabla u\|_{L^{2}(\mathbb{R}^3)}, \end{align} $

(2.11)

which implies $ u = H \equiv 0. $

In what follows, we give the estimate about $ |\nabla Q| $. By the definition of $ H $, we have

$ \begin{align} -\Delta Q = -aQ+b\left(Q^{2}-\frac{tr(Q^{2})}{3}I_{3\times3}\right)-cQ\mbox{tr}(Q^{2}). \end{align} $

(2.12)

Multiplying (2.12) by $ Q\phi_{R} $ and integrating over $ \mathbb{R}^3 $, one has from Lemma 2.1 that

$ \begin{align} \int_{\mathbb{R}^3}|\nabla Q|^{2}\phi_{R} {\rm d}x &=\int_{\mathbb{R}^3}|Q|^{2}\Delta \phi_{R} {\rm d}x-\int_{\mathbb{R}^3}\Big(a\mbox{tr}(Q^{2})-b\mbox{tr}(Q^{3})+c[\mbox{tr}(Q^{2})]^2\phi_{R}\Big) {\rm d}x\\ &\leq\frac{C}{R^{2}}\int_{B(2R\setminus R)}|Q|^{2} {\rm d}x+\int_{\mathbb{R}^3}\left(\frac{b^{2}-24ac}{24c}\right)|Q|^{2}{\rm d}x\\ &\leq\frac{C}{R^{2}}\int_{B(2R\setminus R)}|Q|^{2} {\rm d}x\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align} $

(2.13)

In view of $ Q\in H^{2}({\mathbb{R}^3}) $, we obtain $ Q = 0 $. The proof of Theorem 1.1 is finished.

In this section, we are going to prove Theorem 1.2 and Theorem 1.3. Theorem 1.2 is demonstrated through the essential application of Lemma 2.2. To this end, we first claim that $ u\in L^{\infty}(\mathbb{R}^3) $. In fact, according to the conditions of Theorem 1.2, we know that $ u\in \dot{M}^{2,q}(\mathbb{R}^3), \; Q\in H^2(\mathbb{R}^3) $, with $ 3<q<\infty $. By [23], and using the same method as in [6], we could obtain $ u\in L^{\infty}(\mathbb{R}^3) $.

Proof of Theorem 1.2  We reestimate the terms $ I_1-I_7 $ on the right-hand side of (2.1) in the framework of the Morrey spaces. For $ I_{1} $, making use of (2.2), and the Hölder inequality, we observe

$ \begin{align*} \nonumber I_{1}&=\frac{1}{2}\int_{\mathbb{R}^3}u|\nabla Q|^{2}\nabla\phi_{R} {\rm d}x\\\nonumber &\leq\frac{C}{R}\left(\int_{B(2R\setminus R)}|u|^{2} {\rm d}x\right)^{\frac{1}{2}}\left(\int_{B(2R\setminus R)}|\nabla Q|^{4} {\rm d}x\right)^{\frac{1}{2}}\\ &\leq CR^{-\frac{1}{2}}\|u\|_{{\dot{M}^{2,3}}(B(2R\setminus R))}\||\nabla Q|^{2}\|_{L^{2}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align*} $

(3.1)

With respect to $ I_{2} $, it follows from (2.3) and the Sobolev imbedding that

$ \begin{align*} \nonumber I_{2} &=\int_{\mathbb{R}^3}\left(\frac{a}{2}|Q|^{2}-\frac{b}{3}{\mathtt{tr}(Q^{3})}+\frac{c}{4}|Q|^{4}\right)u\cdot\nabla\phi_{R} {\rm d}x\\\nonumber &\leq \frac{C}{R}\left(1+\|Q\|_{L^{\infty}(B(2R\setminus R))}+\|Q\|^{2}_{L^{\infty}(B(2R\setminus R))}\right)\|Q\|^{2}_{L^{4}(B(2R\setminus R))}\|u\|_{L^{2}(B(2R\setminus R))}\\ &\leq CR^{-\frac{1}{2}}\|u\|_{{\dot{M}^{2,3}}(B(2R\setminus R))}\|Q\|^2_{L^{4}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align*} $

(3.2)

One deduces from (2.4) and Lemma 2.2 that

$ \begin{align*} \nonumber I_{3} &=-\int_{\mathbb{R}^3}(Q\Delta Q-\Delta QQ):u\otimes\nabla\phi_{R} {\rm d}x\\\nonumber &\leq\frac{C}{R}\|\nabla^{2}Q\|_{L^{2}(B(2R\setminus R))}\|u\|_{L^{3}(B(2R\setminus R))}\|Q\|_{L^{6}(B(2R\setminus R))}\\ &\leq CR^{-\frac{2}{3}}\|u\|_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}\|\nabla^{2}Q\|_{L^{2}(B(2R\setminus R))}\|Q\|_{L^{6}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align*} $

(3.3)

We get from (2.5) that when $ R\rightarrow \infty $,

$ \begin{align*} \nonumber I_5&=\lambda\int_{\mathbb{R}^3}|Q|H:(u\otimes\nabla\phi_R(x)){\rm d}x \\\nonumber &\leq\frac{C}{R}\|H\|_{L^2(B(2R\setminus R))}\|u\|_{L^3(B(2R\setminus R))}\|Q\|_{L^6(B(2R\setminus R))}\\ &\leq CR^{-\frac{2}{3}}\|u\|_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}\|H\|_{L^2(B(2R\setminus R))}\|Q\|_{L^6(B(2R\setminus R))}\rightarrow 0. \end{align*} $

(3.4)

For $ I_{6} $, similar to $ I_1 $, we see by (2.6) that

$ \begin{align*} \nonumber I_{6}&=\frac{1}{2}\int_{\mathbb{R}^3}|u|^{2}u\cdot\nabla\phi_{R} {\rm d}x\\\nonumber &\leq\frac{C}{R}\int_{B(2R\setminus R)}|u|^{3} {\rm d}x \leq\frac{C}{R}\left(R^{9(\frac{1}{3}-\frac{2}{9})}\|u\|^3_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}\right)\\ &\leq C\|u\|^3_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align*} $

(3.5)

In view of (2.7)-(2.9), we can get

$ \begin{align*} \nonumber I_{7}&=\int_{\mathbb{R}^3}\left(Pu\cdot\nabla\phi_{R}+\frac{\mu}{2}|u|^{2}\Delta\phi_{R}\right) {\rm d}x\\\nonumber &=\int_{\mathbb{R}^3}P_{1}u\cdot\nabla\phi_{R} {\rm d}x+\int_{\mathbb{R}^3}P_{2}u\cdot\nabla\phi_{R} {\rm d}x+\frac{\mu}{2}\int_{\mathbb{R}^3}|u|^{2}\Delta\phi_{R} {\rm d}x\\\nonumber &\leq\frac{C}{R}\int_{B(2R\setminus R)}|u|^{3} {\rm d}x+\frac{C}{R}\|u\|_{L^{2}(B(2R\setminus R))}\|P_{2}\|_{L^{2}(B(2R\setminus R))} +\frac{C}{R^2}\int_{B(2R\setminus R)}|u|^{2} {\rm d}x\\\nonumber &\leq C\|u\|^3_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}+CR^{-\frac{1}{2}}\|u\|_{{\dot{M}^{2,3}}(B(2R\setminus R))}\|P_{2}\|_{L^{2}(B(2R\setminus R))}\\ &\; \; \; \; \; +CR^{-\frac{1}{3}}\|u\|^2_{{\dot{M}^{3,\frac{9}{2}}}(B(2R\setminus R))}\rightarrow 0\; \; \; (as\; \; \; R\rightarrow \infty). \end{align*} $

(3.6)

Setting $ R\rightarrow +\infty $, by (2.10) and (2.11), yields $ u = H \equiv 0. $ Similarly, from (2.12) and (2.13), we also obtain $ Q\equiv0. $ Therefore, we have completed the proof of Theorem 1.2.

Proof of Theorem 1.3  To prove Theorem 1.3, we use different Hölder indicators to reestimate the terms $ I_1-I_7 $ in (2.1). For $ I_1 $, by Hölder inequality and the definition of Morrey spaces, we deduce from (3.1) that

$ \begin{align} I_{1}&=\frac{1}{2}\int_{\mathbb{R}^3}u|\nabla Q|^{2}\nabla\phi_{R} {\rm d}x\\ &\leq\frac{C}{R}\left(\int_{B(2R\setminus R)}|u|^{p} {\rm d}x\right)^{\frac{1}{p}}\left(\int_{B(2R\setminus R)}|\nabla Q|^{4} {\rm d}x\right)^{\frac{1}{2}}\left(\int_{B(2R\setminus R)}1 \; {\rm d}x\right)^{\frac{1}{2}-\frac{1}{p}}\\ &\leq CR^{-1}R^{(\frac{3}{2}-\frac{3}{p})}R^{3(\frac{1}{p}-\frac{1}{q})}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\||\nabla Q|^{2}\|_{L^{2}(B(2R\setminus R))}\\ &\leq CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\||\nabla Q|^{2}\|_{L^{2}(B(2R\setminus R))}. \end{align} $

Similarly, we could obtain the following new estimations of (3.2)-(3.6) as

$ \begin{align*} &I_2\leq CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|Q\|^{2}_{L^{4}(B(2R\setminus R))},\\ &I_3\leq CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|\nabla^{2}Q\|_{L^{2}(B(2R\setminus R))},\\ &I_5\leq CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|H|_{L^{2}(B(2R\setminus R))},\\ &I_6\leq CR^{2-\frac{9}{q}}\|u\|^3_{{\dot{M}^{p,q}}(B(2R\setminus R))},\\ &I_7\leq CR^{2-\frac{9}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|P_1\|_{{\dot{M}^{\frac{p}{2},\frac{q}{2}}}(B(2R\setminus R))}\\ &\; \; \; \; \; \; +CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|P_{2}\|_{L^{2}(B(2R\setminus R))}+CR^{1-\frac{6}{q}}\|u\|^2_{{\dot{M}^{p,q}}(B(2R\setminus R))}. \end{align*} $

Collecting all the above estimations implies

$ \begin{align*} \nonumber &\mu\int_{\mathbb{R}^3}|\nabla u|^2\phi_R(x){\rm d}x+\Gamma\int_{\mathbb{R}^3}|H|^2\phi_R(x){\rm d}x\\\nonumber \leq& CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\||\nabla Q|^{2}\|_{L^{2}(B(2R\setminus R))}+CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|Q\|^{2}_{L^{4}(B(2R\setminus R))}\\\nonumber &\; \; \; \; \; \; +CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|\nabla^{2}Q\|_{L^{2}(B(2R\setminus R))}+CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|H|_{L^{2}(B(2R\setminus R))}\\\nonumber &\; \; \; \; \; \; \; +CR^{2-\frac{9}{q}}\|u\|^3_{{\dot{M}^{p,q}}(B(2R\setminus R))}+CR^{2-\frac{9}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|P_1\|_{{\dot{M}^{\frac{p}{2},\frac{q}{2}}}(B(2R\setminus R))}\\ &\; \; \; \; \; \; +CR^{\frac{1}{2}-\frac{3}{q}}\|u\|_{{\dot{M}^{p,q}}(B(2R\setminus R))}\|P_{2}\|_{L^{2}(B(2R\setminus R))}+CR^{1-\frac{6}{q}}\|u\|^2_{{\dot{M}^{p,q}}(B(2R\setminus R))}. \end{align*} $

(3.7)

By the condition $ 3\leq p< q \leq \frac{9}{2} $, we have

$ \frac{1}{2}-\frac{3}{q}\leq 0, \; 2-\frac{9}{q}\leq 0, \; 1-\frac{6}{q}\leq 0. $

Thus, returning to (3.7), we get

$ \lim\limits_{R\rightarrow +\infty}\left(\mu\displaystyle{\int}_{B(R)}|\nabla u|^{2} {\rm d}x+\Gamma\displaystyle{\int}_{B(R)}|H|^{2} {\rm d}x\right) = 0. $

By virtue of the Sobolev embedding $ \|u\|_{L^{6}(\mathbb{R}^3)}\leq C\|\nabla u\|_{L^{2}(\mathbb{R}^3)}, $ we have $ u = H \equiv 0. $ And using the same method dealing with (2.12) and (2.13) one concludes $ Q\equiv0 $, which ends the proof of Theorem 1.3.