## Lϕ-Type Estimates for Very Weak Solutions of A-Harmonic Equation in Orlicz Spaces

Tong Yuxia, Wang Xinru, Gu Jiantao,

Abstract

The paper deals with the gradient estimates in Orlicz spaces for very weak solutions of A-harmonic equations under the assumptions that A satisfies some proper conditions and the given function satisfies some moderate growth condition.

Keywords： Orlicz space ; Very weak solution ; A-Harmonic equation

Tong Yuxia, Wang Xinru, Gu Jiantao. Lϕ-Type Estimates for Very Weak Solutions of A-Harmonic Equation in Orlicz Spaces. Acta Mathematica Scientia[J], 2020, 40(6): 1461-1480 doi:

## 1 引言

$\Omega $${{\Bbb R}} ^{n} 中的有界正则区域, n\geq 2 .在正则区域中, Hodge分解及其估计是满足的.例如, Lipschitz域是正则的.本文考虑如下的 A -调和方程 $${\rm div} A(x, \nabla u) = {\rm div}(\left | \textbf{f} \right |^{p-2}\textbf{f}), \quad\; x\in \Omega ,$$ 其中 \textbf{f} = (f^{1}, f^{2}, \cdots , f^{n}) 是给定的向量, 算子 A = A(x, \xi ):\Omega\times {{\Bbb R}} ^{n}\rightarrow {{\Bbb R}} ^{n} .对于每一个 \xi , A$$ x$上可测, 并且对于几乎所有的$x$, $A $$\xi 上连续.而且当 p\in(1, \infty) 时, 算子 A = A(x, \xi ) 满足如下结构条件 ({\rm H} _{1})\quad\left \langle A(x, \xi )-A(x, \zeta ) , \xi -\zeta \right \rangle\geq C_{1}\left | \xi -\zeta \right |^{p} , ({\rm H}_{2})\quad | A(x, \xi ) |\leq C_{2}\left | \xi \right |^{p-1} , ({\rm H} _{3})\quad A(x, 0) = 0, ({\rm H}_{4})\quad\left | A(x, \xi )-A(y, \xi ) \right |\leq C_{3}\omega (\left | x-y \right |)\left | \xi \right |^{p-1} , 其中 \xi, \zeta\in{{\Bbb R}} ^{n}, x, y\in\Omega, C_{i}> 0, i = 1, 2, 3 为正常数.这里的连续模 \omega (x):{{\Bbb R}} ^{+}\rightarrow {{\Bbb R}} ^{+} 非减, 而且满足 $$\omega (r)\rightarrow 0 \qquad \mbox{当} \quad r\rightarrow 0.$$ 方程 (1.1) 的原型是 p -调和方程 $${\rm div} (\left |\nabla u\right |^{p-2}\nabla u) = 0.$$ 定义1.1[12] 函数 u \in W_{loc}^{1, p} (\Omega) 称为方程 (1.1) 的局部弱解, 若有 $$\int _{\Omega }\left \langle A(x, \nabla u), \nabla \varphi \right \rangle{\rm d}x = \int _{\Omega }\left \langle | \textbf{f} |^{p-2} \textbf{f}, \nabla \varphi \right \rangle{\rm d}x$$ 对于任意的具有紧支集的 \varphi \in W_{0}^{1, p} (\Omega ) 都成立. 本文考虑很弱解的概念, 降低了自然可积性假设. 定义1.2 函数 u \in W_{loc}^{1, r} (\Omega) , \max\{1, p-1\}\leq r < p , 称为方程 (1.1) 的局部很弱解, 若对所有具有紧支集的 \varphi \in W_{0}^{1, \frac{r}{r-p+1}} (\Omega ) , 方程 (1.4) 成立. 关于梯度估计, 有下述研究成果. Acerbi-Mingione[13]证明了 p(x) -Laplacian方程组 $${\rm div}(\left | D u \right |^{p(x)-2} D u) = \rm {div}(\left | F \right |^{p(x)-2}F)$$ 的梯度估计; DiBenedetto-Manfredi[14]获得了 $${\rm div}(\left | D u \right |^{p-2} D u) = \rm {div}(\left | F \right |^{p-2}F)$$ 的弱解的梯度的更高可积性; Byun-Wang[15]和Kinnunen-Zhou[16]获得了 $${\rm div}((A D u\cdot D u)^{(p-2)/2}A D u) = {\rm div}(\left | F \right |^{p-2}F)$$ 的弱解的 L^{q}(q\geq p) 梯度估计; Byun-Wang[17]获得了一般非线性椭圆方程 $${\rm div} A(x, \nabla u) = {\rm div}f$$ L^{p} 估计; Yao[12]在2010年获得了方程(1.1)的弱解在Orlicz空间中的梯度估计. 尽管关于弱解的梯度估计有很多出色的成果, 但Orlicz空间中的很弱解的梯度估计尚未研究.本文旨在研究方程(1.1)的很弱解在Orlicz空间中的 L^{\phi} 估计.下面是本文的主要结论. 定理1.1 假设 \phi \in\triangle _{2}\bigcap \bigtriangledown_{2}, | {\bf f} |^{r}\in L_{loc}^{\phi}(\Omega) , \max\{1, p-1\}\leq r < p .如果 u 是方程 (1.1) 的局部很弱解, 且算子 A 满足条件 (H_{1})-(H_{4}) , 则有 \left | \nabla u \right |^{r} \in L_{loc}^{\phi }\left ( \Omega \right ), 且有估计式 $$\int _{B_{R}}\phi \left ( \left | \nabla u \right | ^{r}\right ){\rm d}x\leq C\left \{ \int _{B_{2R}}\phi\left(| {\bf f}| ^{r}\right ){\rm d}x+\phi\left ( \frac{1}{R^{r}}\int_{B_{2R}}\left | u-u_{B_{2R}} \right |^{r}{\rm d}x\right) +1\right \},$$ 其中 B_{2R}\subset \Omega , 且 C 是独立于 u$$ {\bf f}$的常数.

## 2 Orlicz空间

$\Phi$表示所有函数$\phi :\left [ 0, + \infty \right )\rightarrow \left [ 0, + \infty \right )$组成的函数类, 其中$\phi$是递增的凸函数.

$$$\phi (2t)\leq K\phi (t),$$$

$$$\phi (t)\leq \frac{\phi (at)}{2a},$$$

$(2)$ 事实上, 如果$\phi \in \triangle _{2}\bigcap \bigtriangledown _{2}$, 则对于$0< \theta _{2}\leq 1\leq \theta _{1}< \infty$, $\phi$满足

$$$\phi (\theta _{1}t)\leq K\theta _{1}^{\alpha _{1}}\phi (t), \qquad \phi (\theta _{2}t)\leq 2a\theta _{2}^{\alpha _{2}}\phi (t),$$$

$(3)$ 在条件(2.3)下, 很容易得到$\phi \in \Phi$满足$\phi(0) = 0$以及

$$$\lim\limits_{t\rightarrow 0^{+}}\frac{\phi (t)}{t} = \lim\limits_{t\rightarrow +\infty }\frac{t}{\phi (t)} = 0.$$$

(1) $K^{\phi } = L^{\phi }, $$C_{0}^{\infty }$$ L^{\phi }$中稠密;

(2) $L^{\alpha _{1}}(\Omega )\subset L^{\phi }(\Omega )\subset L^{\alpha _{2}}(\Omega )\subset L^{1}(\Omega ),$其中$\alpha _{1} = \log _{2}K, \alpha _{2} = \log _{\alpha }2+1;$

(3) $\int _{\Omega }\phi (\left | g \right |){\rm d}x = \int_{0}^{\infty }\left | \left \{ x \in \Omega :\left | g \right |> \lambda \right \} \right |{\rm d}\left [ \phi (\lambda ) \right ];$

(4) $\int_{0}^{\infty }\frac{1}{\mu }\int _{\left \{ x \in \Omega :\left | g \right | > a\mu \right \}}\left | g \right |{\rm d}x{\rm d}\left [ \phi (b\mu ) \right ]\leq C\int _{\Omega }\phi (\left | g \right |){\rm d}x,$其中$a, b>0, C = C(a, b, \phi )$.

## 3 预备引理

$$$u_\lambda(x) = \frac{u(x)}{\lambda}, \textbf{f}_\lambda(x) = \frac{\textbf{f}(x)}{\lambda}, A_\lambda(x, \xi) = \frac{A(x, \lambda\xi)}{\lambda^{p-1}}.$$$

(1) $A_\lambda$满足条件(H$_1)$–(H$_4)$, 即有

$$$\left \langle A_\lambda(x, \xi )-A_\lambda(x, \zeta ) , \xi -\zeta \right \rangle\geq C_{1}\left | \xi -\zeta \right |^{p},$$$

$$$|A_\lambda(x, \xi ) |\leq C_{2}\left | \xi \right |^{p-1},$$$

$$$A_\lambda(x, 0) = 0,$$$

$$$\left | A_\lambda(x, \xi )-A_\lambda(y, \xi ) \right |\leq C_{3}\omega (\left | x-y \right |)\left | \xi \right |^{p-1},$$$

(2) $u_\lambda$是方程

$$${{\rm div}} A_\lambda(x, \nabla u) = {\rm div}(\left | \textbf{f}_\lambda \right |^{p-2}\textbf{f}_\lambda), \quad x\in \Omega$$$

使用文献[12, 引理2.1]中的类似方法来证明.

$$$\lambda _{0} = \left [-\!\!\!\!\!\!\int_{B_{1}} \left | \nabla u \right |^{r}{\rm d}x+\frac{1}{\varepsilon}-\!\!\!\!\!\!\int_{B_{1}}|\textbf{f}|^{r}{\rm d}x\right ]^{1/r},$$$

$$$0\leq \eta \leq 1;\qquad \eta \equiv 1, \ x\in B_{R};\qquad \eta \equiv 0, \ x\in {{\Bbb R}}^n \setminus B_{2R}, \qquad \left | \nabla \eta \right | \leq \frac{C}{R}.$$$

$|\nabla [\eta( u-u_{B_{2R}}\;)]|^{r-p}\nabla[\eta( u-u_{B_{2R}}\;)] \in L^{\frac{r}{r-p+1}}\;(\Omega)$进行Hodge分解, 参见文献[18]:

$$$|\nabla [\eta( u-u_{B_{2R}})]|^{r-p}\nabla[\eta( u-u_{B_{2R}})] = \nabla\varphi+H,$$$

$$$||\nabla \varphi||_{\frac{r}{r-p+1}}\leq C ||\nabla [\eta( u-u_{B_{2R}})]||_{r}^{r-p+1},$$$

$$$||H||_{\frac{r}{r-p+1}}\leq C(p-r) ||\nabla [\eta( u-u_{B_{2R}})]||_{r}^{r-p+1},$$$

$$$\nabla\varphi = |\eta\nabla(u-u_{B_{2R}})|^{r-p}\eta\nabla(u-u_{B_{2R}})+E-H.$$$

$\begin{eqnarray} && \int _{B_{2R}}\left \langle A(x, \nabla u), |\eta\nabla(u-u_{B_{2R}})|^{r-p}\eta\nabla(u-u_{B_{2R}})+E-H\right \rangle{\rm d}x \\ & = & \int _{B_{2R}}\left \langle |\textbf{f}|^{p-2}\textbf{f}, |\eta\nabla(u-u_{B_{2R}})|^{r-p}\eta\nabla(u-u_{B_{2R}})+E-H\right \rangle{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} && \int _{B_{2R}}\left \langle A(x, \nabla u), |\eta\nabla(u-u_{B_{2R}})|^{r-p}\eta\nabla(u-u_{B_{2R}})\right \rangle{\rm d}x \\ & = & -\int _{B_{2R}}\left \langle A(x, \nabla u), E\right \rangle{\rm d}x+\int _{B_{2R}}\left \langle A(x, \nabla u), H\right \rangle{\rm d}x \\ && +\int _{B_{2R}}\left \langle |\textbf{f}|^{p-2}\textbf{f}, |\eta\nabla(u-u_{B_{2R}})|^{r-p}\eta\nabla(u-u_{B_{2R}})\right \rangle{\rm d}x \\ && +\int _{B_{2R}}\left \langle |\textbf{f}|^{p-2}\textbf{f}, E \right \rangle{\rm d}x - \int _{B_{2R}}\left \langle |\textbf{f}|^{p-2}\textbf{f}, H \right \rangle{\rm d}x . \end{eqnarray}$

$\begin{eqnarray} I_1 \geq \int _{B_{R}}\left \langle A(x, \nabla u), |\nabla u|^{r-p}\nabla u \right \rangle{\rm d}x \geq C_1\int _{B_{R}}|\nabla u|^{r}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} I_2 &\leq& \int _{B_{2R}} |A(x, \nabla u)||E|{\rm d}x \\ &\leq& C_2C(n, p, r)\int _{B_{2R}}|\nabla u|^{p-1}|(u-u_{B_{2R}})\nabla \eta|^{r-p+1}{\rm d}x \\ &\leq& C(n, p, r, C_2)\Big(\tau\int _{B_{2R}}|\nabla u|^{r}{\rm d}x + C{(\tau)}\int _{B_{2R}}|(u-u_{B_{2R}})\nabla \eta|^{r}{\rm d}x\Big) \\ &\leq& C(n, p, r, C_2)\Big(\tau\int _{B_{2R}}|\nabla u|^{r}{\rm d}x + \frac{C(\tau)}{R^{r}}\int _{B_{2R}}|(u-u_{B_{2R}}|^{r}{\rm d}x\Big). \end{eqnarray}$

$\begin{eqnarray} I_3 &\leq & \int _{B_{2R}} |A(x, \nabla u)||H|{\rm d}x \leq C_2\int _{B_{2R}}|\nabla u|^{p-1}|H|{\rm d}x {} \\ &\leq& C_2\left(\int _{B_{2R}}|\nabla u|^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{2R}}|H|^{\frac{r}{r-p+1}}\right)^{\frac{r-p+1}{r}} {} \\ &\leq & C_2C(n, p)(p-r)\left(\int _{B_{2R}}|\nabla u|^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{2R}}|\nabla[\eta(u-u_{B_{2R}})]|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}}. \end{eqnarray}$

$\begin{eqnarray} I_3 &\leq& C_2C(n, p)(p-r)\left(\int _{B_{2R}}|\nabla u|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{2R}}|\eta\nabla u|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ && +C_2C(n, p)(p-r)\left(\int _{B_{2R}}|\nabla u|^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{2R}}|(u-u_{B_{2R}})\nabla\eta|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, C_2)(p-r)\int _{B_{2R}}|\nabla u|^{r}{\rm d}x +C(n, p, C_2)(p-r)\tau\int _{B_{2R}}|\nabla u|^{r}{\rm d}x \\ && +C(n, p, C_2, \tau)(p-r)\int _{B_{2R}}|(u-u_{B_{2R}})\nabla\eta|^{r}{\rm d}x \\ &\leq& C(n, p, C_2)(p-r)(1+\tau)\int _{B_{2R}}|\nabla u|^{r}{\rm d}x +\frac{C(n, p, C_2, \tau)(p-r)}{R^{r}}\int _{B_{2R}}|(u-u_{B_{2R}})|^{r}{\rm d}x. {}\\ \end{eqnarray}$

$$$I_4\leq \int _{B_{2R}} |\textbf{f}|^{p-1}|\nabla u|^{r-p+1}{\rm d}x \leq \tau\int _{B_{2R}}|\nabla u|^{r}{\rm d}x+C(\tau)\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x.$$$

$\begin{eqnarray} I_5 &\leq& \int _{B_{2R}} |\textbf{f}|^{p-1}|E|{\rm d}x \leq C(n, p, r)\int _{B_{2R}}|\textbf{f}|^{p-1}|(u-u_{B_{2R}})\nabla\eta|^{r-p+1}{\rm d}x \\ &\leq& C(n, p, r)\left(\tau\int _{B_{2R}}|(u-u_{B_{2R}})\nabla\eta|^{r}{\rm d}x+C(\tau)\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right) \\ &\leq& C(n, p, r)\left(\frac{\tau}{R^{r}}\int _{B_{2R}}|(u-u_{B_{2R}})|^{r}{\rm d}x+C(\tau)\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right). \end{eqnarray}$

$\begin{eqnarray} I_6 &\leq& \int _{B_{2R}} |\textbf{f}|^{p-1}|H|{\rm d}x \leq C(n, p)\left(\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{2R}}|H|^{\frac{r}{r-p+1}}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p)(p-r)\left(\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{2R}}|\nabla\eta(u-u_{B_{2R}})|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, r)(p-r)\left(\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{2R}}|\eta\nabla u|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ && +C(n, p, r)(p-r)\left(\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{2R}}|(u-u_{B_{2R}})\nabla \eta|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, r)(p-r)\tau \int _{B_{2R}}|\nabla u|^{r}{\rm d}x+C(n, p, r, \tau)(p-r)\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x \\ && +C(n, p, r)(p-r)\tau \int _{B_{2R}}|(u-u_{B_{2R}})\nabla \eta|^{r}{\rm d}x \\ &\leq& C(n, p, r)(p-r)\tau \int _{B_{2R}}|\nabla u|^{r}{\rm d}x+C(n, p, r, \tau)(p-r)\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x \\ && +\frac{C(n, p, r)(p-r)\tau}{R^{r}}\int _{B_{2R}}|(u-u_{B_{2R}})|^{r}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} C_1\int _{B_{R}}|\nabla u|^{r}{\rm d}x &\leq& [C\tau +C(p-r)(1+\tau)+\tau+C(p-r)\tau]\int _{B_{2R}}|\nabla u|^{r}{\rm d}x \\ && +\frac{C\tau +C(p-r)+C\tau+C(p-r)\tau}{R^{r}}\int _{B_{2R}}|u-u_{B_{2R}}|^{r}{\rm d}x \\ && +[C\tau +C(p-r)+C(\tau)]\int _{B_{2R}}|\textbf{f}|^{r}{\rm d}x, \end{eqnarray}$

$$$\int _{B_{\widetilde{R}}}\left \langle A(x^{*}, \nabla v), \nabla \varphi \right \rangle{\rm d}x = 0$$$

$$$E(u, v) = |\nabla(v-u)|^{r-p}\nabla(v-u)-|\nabla v|^{r-p}\nabla v,$$$

$$$|E(u, v)|\leq2^{p-r}\frac{p-r+1}{r-p+1}|\nabla u|^{r-p+1},$$$

$$$\nabla\varphi = |\nabla\widetilde{\varphi}|^{r-p}\nabla\widetilde{\varphi}-H = E-H+|\nabla v|^{r-p}\nabla v,$$$

$\begin{eqnarray} \int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x^{*}, \nabla v), |\nabla v|^{r-p}\nabla v\right \rangle{\rm d}x & = & -\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x^{*}, \nabla v), E-H\right \rangle{\rm d}x. \end{eqnarray}$

$$$\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x^{*}, \nabla v), |\nabla v|^{r-p}\nabla v\right \rangle{\rm d}x\geq C_1\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x,$$$

$$$\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x \leq C\int _{B_{10\rho _{i}}(x_{i})}|\nabla u|^{r}{\rm d}x,$$$

$$$\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla u) , \nabla \varphi\right \rangle{\rm d}x = \int _{B_{10\rho _{i}}(x_{i})}\left \langle |\textbf{f} |^{p-2}\textbf{f} , \nabla \varphi\right \rangle{\rm d}x.$$$

$$$\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x^{*}, \nabla v) , \nabla \varphi)\right \rangle{\rm d}x = 0.$$$

$\begin{eqnarray} && \int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla u)-A(x, \nabla v), \nabla \varphi\right \rangle{\rm d}x \\ & = & -\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla v)-A(x^{*}, \nabla v), \nabla \varphi\right \rangle{\rm d}x +\int _{B_{10\rho _{i}}(x_{i})}\left \langle |\textbf{f} |^{p-2}\textbf{f} , \nabla \varphi\right \rangle{\rm d}x. \end{eqnarray}$

$$$K_1 = K_2+K_3+K_4+K_5+K_6,$$$

$\begin{eqnarray} &&K_1 = \int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla u)-A(x, \nabla v), |\nabla(u-v)|^{r-p}\nabla(u-v)\right \rangle{\rm d}x, \\ && K_2 = \int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla u)-A(x, \nabla v), H\right \rangle{\rm d}x, \\ && K_3 = -\int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla v)-A(x^{*}, \nabla v), |\nabla(u-v)|^{r-p}\nabla(u-v)\right \rangle{\rm d}x, \\ && K_4 = \int _{B_{10\rho _{i}}(x_{i})}\left \langle A(x, \nabla v)-A(x^{*}, \nabla v), H\right \rangle{\rm d}x, \\ && K_5 = \int _{B_{10\rho _{i}}(x_{i})}\left \langle |\textbf{f} |^{p-2}\textbf{f}, |\nabla(u-v)|^{r-p}\nabla(u-v)\right \rangle{\rm d}x, \\ && K_6 = -\int _{B_{10\rho _{i}}(x_{i})}\left \langle |\textbf{f} |^{p-2}\textbf{f}, H\right \rangle{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} K_1 &\geq& C_1\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} K_2 &\leq& \int _{B_{10\rho _{i}}(x_{i})}(|A(x, \nabla u)|+|A(x, \nabla v)|)|H|{\rm d}x \\ &\leq& C_2\int _{B_{10\rho _{i}}(x_{i})}\Big(|\nabla u|^{p-1}+|\nabla v|^{p-1}\Big)|H|{\rm d}x \\ &\leq& C_2\left(\int _{B_{10\rho _{i}}(x_{i})}(|\nabla u|^{r}+|\nabla v|^{r}){\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{10\rho _{i}}(x_{i})}|H|^{\frac{r}{r-p+1}}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, C_2)(p-r)\left(\int _{B_{10\rho _{i}}(x_{i})}(|\nabla u|^{r}+|\nabla v|^{r}){\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, C_2)(p-r)\left(\int _{B_{10\rho _{i}}(x_{i})}(|\nabla (u-v)|^{r}+|\nabla v|^{r}){\rm d}x\right)^{\frac{p-1}{r}} \\ && \cdot\left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p, \tau , C_2)(p-r)\left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla (u-v)|^{r}{\rm d}x+\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x\right) \\ && +C(n, p)(p-r)\tau\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x. \end{eqnarray}$

$\begin{eqnarray} K_3 &\leq& C_3\omega(|x-x^{*}|)\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{p-1}|\nabla(u-v)|^{r-p+1}{\rm d}x \\ &\leq& C_3\omega(R_0)\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{p-1}|\nabla(u-v)|^{r-p+1}{\rm d}x \\ &\leq& C_3\varepsilon\left\{C(\tau)\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x+\tau\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x\right\}. \end{eqnarray}$

$\begin{eqnarray} K_4 &\leq& C_3\omega(|x-x^{*}|)\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{p-1}|H|{\rm d}x \\ &\leq& C_3\omega(R_0)\left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x\right)^{\frac{p-1}{r}}\left(\int _{B_{10\rho _{i}}(x_{i})}|H|^{\frac{r}{r-p+1}}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C_3C(n, p)(p-r)\varepsilon\left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C_3C(n, p)(p-r)\varepsilon \Bigg\{C(\tau)\int _{B_{10\rho _{i}}(x_{i})}|\nabla v|^{r}{\rm d}x +\tau\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x \Bigg\}. \end{eqnarray}$

$\begin{eqnarray} K_5 &\leq& \int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{p-1}|\nabla(u-v)|^{r-p+1}{\rm d}x \\ &\leq& C(\tau)\int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{r}{\rm d}x+\tau\int _{B_{10\rho _{i}}(x_{i})}|\nabla(u-v)|^{r}{\rm d}x . \end{eqnarray}$

$\begin{eqnarray} K_6 &\leq& \int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{p-1}|H|{\rm d}x \\ &\leq& \left(\int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{10\rho _{i}}(x_{i})} |H|^{\frac{r}{r-p+1}}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p)(p-r)\left(\int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{r}{\rm d}x\right)^{\frac{p-1}{r}} \left(\int _{B_{10\rho _{i}}(x_{i})} |\nabla(u-v)|^{r}{\rm d}x\right)^{\frac{r-p+1}{r}} \\ &\leq& C(n, p)(p-r)\Bigg\{C(\tau)\int _{B_{10\rho _{i}}(x_{i})} |\textbf{f} |^{r}{\rm d}x +\tau\int _{B_{10\rho _{i}}(x_{i})} |\nabla(u-v)|^{r}{\rm d}x\Bigg\}. \end{eqnarray}$

令$v$是Dirichlet问题$(4.21)$的很弱解.由于$v-u \in W_{0}^{1, r}(B_{\widetilde{R}})$, 则在${\Omega} \backslash B_{\widetilde{R}} $$v = u . B_{2\widetilde R}\subset\subset\Omega, 0<\widetilde R\leq \widetilde\tau <t\leq2\widetilde R .考虑函数 v(x)$$ k>0$

$$${\rm supp}\eta \subset B_t, 0\leq \eta \leq 1, \eta \equiv 1 \; {\rm in} \;B_{\tilde{\tau}}, \left | \nabla \eta \right | \leq 2(t-\widetilde\tau)^{-1}.$$$

$$$\phi = \eta^{r}(v-t_k(v)),$$$

$$$t_k(v) = \min\left\{v, k\right\}, \quad k \geq 0.$$$

$$$|\nabla[\eta^{r}(v-t_k(v))]|^{r-p}\nabla[\eta^{r}(v-t_k(v))] = \nabla\varphi+H,$$$

$$$||H||_{\frac{r}{r-p+1}}\leq C(p-r)||\nabla[\eta^{r}(v-t_k(v))]||_r^{r-p+1},$$$

$\begin{eqnarray} \int_{A_{k, t}^{+}} \langle A(x^*, \nabla v), |\eta^{r}\nabla v|^{r-p} \eta^{r}\nabla v\rangle{\rm d}x &\geq& \int_{A_{k, \widetilde \tau}^{+}} \langle A(x^*, \nabla v), |\nabla v|^{r-p} \nabla v\rangle{\rm d}x \\ &\geq& C_1\int_{A_{k, \widetilde \tau}^{+}}|\nabla v|^{r}{\rm d}x, \end{eqnarray}$

$\begin{eqnarray} |I_1|&\leq& C\int_{A_{k, t}^{+}} |\nabla v|^{p-1}|r\eta^{r-1}\nabla \eta(v-t_k(v))|^{r-p+1}{\rm d}x \\ &\leq& C\left[\tau\int_{A_{k, t}^{+}} |\nabla v|^{r}{\rm d}x+C(\tau)\int_{A_{k, t}^{+}}|r\eta^{r-1}(v-t_k(v))\nabla \eta|^{r}{\rm d}x\right] , \end{eqnarray}$

$$$\sup\limits_{B_{5\rho _{i}}(x_{i})}|\nabla v| \leq N_0.$$$

令$v$是方程$(4.21)$的很弱解.由定理$4.4$可知$v$是局部有界的.令$g(t) = rt^{r-1}$.然后根据引理$4.3$, 有

$$$G(t) = \int_{0}^{ t }g(s){\rm d}s = \int_{0}^{ t }rs^{r-1}{\rm d}s = t^{r}.$$$

$$$\sup\limits_{B_{5\rho _{i}}(x_{i})}|\nabla v_{\lambda}| \leq N_0.$$$

$$$\int_{B_1}\phi (| \nabla u|^{r}){\rm d}x\leq C_4\varepsilon\int_{B_2}\phi (| \nabla u|^{r}){\rm d}x+C_6\int_{B_2}\phi (| \textbf{f}|^{r}){\rm d}x +C_7\phi \Big( \int_{B_{2}}\left | u-u_{B_{2R}} \right |^{r}{\rm d}x \Big)+1,$$$

$\begin{eqnarray} C_4\varepsilon = \frac{1}{2}. \end{eqnarray}$

$\begin{eqnarray} \int_{B_1}\phi (| \nabla u|^{r}){\rm d}x&\leq& C\left\{\int_{B_2}\phi (| \textbf{f}|^{r}){\rm d}x+ \phi \Big( \int_{B_{2}}\left | u-u_{B_{2R}} \right |^{r}{\rm d}x \Big)+1\right\}. \end{eqnarray}$

### 5.2 逼近

$\begin{eqnarray} \lim\limits_{k\rightarrow \infty} \int _{B_{R}(x_0)} \phi ( |\textbf{f}_k|^{r} ){\rm d}x = \int _{B_{R}(x_0)} \phi ( |\textbf{f}|^{r} ){\rm d}x. \end{eqnarray}$

$\begin{eqnarray} \left\{\begin{array}{ll} {\rm div}A(x, \nabla u_{k}) = {\rm div}(|\textbf{f}|^{p-2}\textbf{f}), \quad x\in B_{2R}, \\ u_{k}-u \in W_{0}^{1, r}(B_{2R}). \end{array}\right. \end{eqnarray}$

$$$\| u_{k}-u \| _{W^{1, r} (B_{2R})}\rightarrow 0 .$$$

$$$\nabla u_{k}\rightarrow \nabla u \quad a.e. \quad\ {\rm in} \quad B_{2R} .$$$

$\begin{eqnarray} \int _{B_{R}} \phi ( |\nabla u|^{r} ){\rm d}x &\leq& \liminf\limits_{k\rightarrow \infty} \int _{B_{R}} \phi ( |\nabla u_{k}|^{r} ){\rm d}x \\ &\leq& C\liminf\limits_{k\rightarrow \infty} \left \{ \phi\left ( \frac{1}{R^{r}}\int _{B_{2R}}|u_{k}-(u_{k})_{2R}|^{r}{\rm d}x \right )+\int _{B_{2R}}\phi(|\textbf{f}_k|^{r}){\rm d}x +1 \right \} \\ &\leq& C\left \{ \phi\left ( \frac{1}{R^{r}} \int _{B_{2R}}|u-u_{2R}|^{r}{\rm d}x \right )+\int _{B_{2R}}\phi(|\textbf{f}|^{r}){\rm d}x +1 \right \}. \end{eqnarray}$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Orlicz W .

Üeber eine gewisse Klasse von Räumen vom Typus B

Bulletin International de l'Académie Polonaise Série A, 1932, 8: 207- 220

Rao M M , Ren Z D . Applications of Orlicz Spaces. New York: Marcel Dekker, 2002

Bulíček M , Diening L , Schwarzacher S .

Existence, uniqueness and optimal regularity results for very weak solutions to nonlinear elliptic systems

Analysis and PDE, 2016, 9 (5): 1115- 1151

Merker J , Rakotoson J M .

Very weak solutions of Poisson's equation with singular data under Neumann boundary conditions

Calculus of Variations and Partial Differential Equations, 2015, 52 (3/4): 705- 726

Bulíček M , Schwarzacher S .

Existence of very weak solutions to elliptic systems of $p$-Laplacian type

Calculus of Variations and Partial Differential Equations, 2016, 55 (3): 1- 14

Bao G , Wang T , Li G .

On very weak solutions to a class of double obstacle problems

Journal of Mathematical Analysis and Applications, 2013, 402 (2): 702- 709

Gao H , Liang S , Cui Y .

Integrability for very weak solutions to boundary value problems of $p$-harmonic equation

Czechoslovak Mathematical Journal, 2016, 66 (1): 101- 110

Tong Y , Liang S , Zheng S .

Integrability of very weak solution to the Dirichlet problem of nonlinear elliptic system

Electronic Journal of Differential Equations, 2019, (1): 1- 11

Zhou S Q , Hu Z H , Peng D Y .

Global regularity for very weak solutions to obstacle promlems corresponding to a class of $A$-harmonic equations

Acta Mathematica Scientia, 2014, 34A (1): 27- 38

Gao H Y , Jia M M .

Local regularity and local boundedness for solutions to obstacle problems

Acta Mathematica Scientia, 2017, 37A (4): 706- 713

Tong Y X , Zheng S Z , Cheng L N .

Removable singularities of weakly $A$-harmonic tensors

Acta Mathematica Scientia, 2017, 37A (6): 1001- 1011

Yao F. Gradient estimates for weak solutions of $A$-Harmonic equations. Journal of Inequalities and Applications, 2010, Article ID: 685046

Acerbi E , Mingione G .

Gradient estimates for the $p(x)$-Laplacean system

J Reine Angew Math, 2005, 584: 117- 148

Dibenedetto E , Manfredi J .

On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems

American Journal of Mathematics, 1993, 115 (5): 1107- 1134

Byun S S , Wang L .

Quasilinear elliptic equations with BMO coefficients in Lipschitz domains

Transactions of the American Mathematical Society, 2007, 359 (12): 5899- 5913

Kinnunen J , Zhou S .

A Local estimate for nonlinear equations with discontinuous coefficients

Communications in Partial Differential Equations, 1999, 24 (11/12): 2043- 2068

Byun S S , Wang L .

$L.{p}$-estimates for general nonlinear elliptic equations

Indiana University Mathematics Journal, 2007, 56 (6): 3193- 3221

Iwaniec T , Sbordone C .

Weak minima of variational integrals

J Reine Angew Math, 1994, 454: 143- 161

Iwaniec T , Migliaccio L , Nania L , Sbordone C .

Integrability and removability results for quasiregular mappings in high dimensions

Mathematica Scandinavica, 1994, 75 (2): 263- 279

Giaquinta M . Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton: Princeton University Press, 1983

Hong M C .

Some remarks on the minimizers of variational integrals with nonstandard growth conditions

Bollettino dell'Unione Matematica Italiana, 1992, 6 (1): 91- 101

Lieberman G M .

The natural generalization of the natural conditions of Ladyzhenskaya and Ural'tseva for elliptic equations

Communications in Partial Differential Equations, 1991, 16: 311- 361

Yao F , Sun Y , Zhou S .

Gradient estimates in Orlicz spaces for quasilinear elliptic equation

Nonlinear Analysis, 2008, 69: 2553- 2565

Zhang Y N , Yan S , Tong Y X .

Gradient estimates for weak solutions to non-homogeneous $A$-Harmonic equations under natural growth

Acta Mathematica Scientia, 2020, 40A (2): 379- 394

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