有界平滑區域上擬線性橢圓型問題強解的存在性

Abstract

我們討論在三維（含）以上空間有界平滑區域Ω上之擬線性橢圓方程： $$\left\{ \begin{array}{lrc} Lu=\displaystyle\sum_{i,j=1}^{N}a_{ij}(x,u)\frac{{\partial}^{2}u}{\partial x_i\partial x_j}+\sum_{i=1}^{N}b_{i}(x,u)\frac{{\partial}u}{\partial x_i}+c(x,u)u=f(x,u,\nabla u)& \mbox{in $\Omega$,}\\\\ u=0 & \mbox{on $\partial \Omega$,} \end{array} \right.$$ 其中 $a_{ij}, b_{i},$ $c$ , $f$ 和 $\frac{\partial a_{ij}}{\partial x_{i}}, \frac{\partial a_{ij}}{\partial r}$ 都是 $Carath\acute{e}dory$ 函數。假設 $a_{ij} = a_{ij}(x,r)$ 屬於$C^{0,1}(\Omega \times R)$ ，且對於所有的 $a_{ij} , \frac{\partial a_{ij}}{\partial x_{i}} , \frac{\partial a_{ij}}{\partial r} , b_{i} , c \in L^{infty}(\Omega \times R)$ 以及 $c(x,r) < 0$ 。我們在$a_{ij}$ 適當的振當範圍內，假設滿足 $|f(x,r,\xi)| \le b(|r|)(1 + |\xi|^{\theta}),$ ,$0 \leq \theta < \frac{32}{(N+2)^{2}}$ ,這裡 $b$ 函數是近似於 $ o(|r|)$，則我們能證得方程式 $L u = f(x,u,\nabla u)$具有屬於 $u\in W^{2,p}(\Omega)\cap W^{1,p}_{0}(\Omega)$ 之解。 $$L_{0}u= - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i} }(a_{ij}(x,u)\frac{\partial u}{\partial x_j}) = -g(x,u,\nabla u) \quad \mbox{in $\Omega$,}$$ 我們在$a_{ij}$ 適當振當範圍內，滿足 $ |g(x,r,\xi)| \leq h(|r|) + k|r|^{\mu}|\xi|^{\nu}, \quad \mu + \nu = \theta, 0 \leq \theta < \frac{4}{N} $ 這裡 $h$ 函數是近似於$o (|r|^{\gamma)), $ \gamma < \frac{N+2}{N-2}$。則我們也能證得方程式 $L_{0} u + g(x,u,\nabla u)=0$具有屬於 $u\in W^{2,p}(\Omega)\cap W^{1,p}_{0}(\Omega)$之解。

We consider the following quasilinear elliptic problem in a bounded smooth domain $\Omega$ of $R^N$, $N\ge 3$:$$\left\{ \begin{array}{lrc} Lu=\displaystyle\sum_{i,j=1}^{N}a_{ij}(x,u)\frac{{\partial}^{2}u}{\partial x_i\partial x_j}+\sum_{i=1}^{N}b_{i}(x,u)\frac{{\partial}u}{\partial x_i}+c(x,u)u=f(x,u,\nabla u)& \mbox{in $\Omega$,}\\\\ u=0 & \mbox{on $\partial \Omega$,} \end{array} \right.$$where $|f(x,r,\xi)| \le b(|r|)(1 + |\xi|^{\theta}),$\,\,\,\,\,\,\,\,\,\,\,$0 \leq \theta < \frac{32}{(N+2)^{2}}$ ,\,\,\,\,\, here $\lim\limits_{|r| \rightarrow +\infty} \frac{b(| r|)}{| r|^{\gamma}} = 0 $ for $\gamma = 1$. The oscillations of $a_{ij}=a_{ij}(x,r)$ with respect to $r$ are sufficiently small. Then there exists a strong solution $u\in W^{2,p}(\Omega)\cap W^{1,p}_{0}(\Omega)$ for $L u = f(x,u,\nabla u)$.\\ \hspace*{\parindent}Next, we study the following quasilinear elliptic problem : $$L_{0}u= - \sum_{i,j=1}^{N}\frac{\partial}{\partial x_{i} }(a_{ij}(x,u)\frac{\partial u}{\partial x_j}) = -g(x,u,\nabla u) \quad \mbox{in $\Omega$,}$$where $g$ satisfies one-sided condition and the growth condition $$ |g(x,r,\xi)| \leq h(|r|) + k|r|^{\mu}|\xi|^{\nu}, \quad \mu + \nu = \theta, 0 \leq \theta < \frac{4}{N} $$ where $\lim\limits_{|r|\rightarrow +\infty} \frac{h(|r|)}{|r|^{\gamma}} = 0$ for some $ \gamma < \frac{N+2}{N-2}$. Then there exists a strong solution $u\in W^{2,p}(\Omega)\cap W^{1,p}_{0}(\Omega)$ for $L_{0} u + g(x,u,\nabla u)=0$.