Existence of Strong Solutions for Certain Quasilinear Elliptic Problem on Bounded Smooth Domains

• Main Authors: Fan-Hsuan Lai, 賴凡暄
• Other Authors: Tsang-Hai Kuo
• Format: Others
• Language: en_US
• Published: 2000
• Online Access: http://ndltd.ncl.edu.tw/handle/51401181772311998639

碩士 === 國立交通大學 === 應用數學系 === 88 === 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 $L_{0} u + g(x,u,\nabla u)=0$.