IJPAM: Volume 55, No. 1 (2009)

EXISTENCE OF STRONG SOLUTIONS TO SOME
QUASILINEAR ELLIPTIC EQUATIONS

Tsang-Hai Kuo

, Chu-Ching Huang

, Yi-Jung Chen

$^{1,2}$ Faculty of Mathematics
Center for General Education
Chang Gung University
256, Wen Hwa, 1-st Road, Kwei-Shan, Tao-Yuan, 333, TAIWAN

e-mail: thkuo@mail.cgu.edu.tw

e-mail: cchuang@mail.cgu.edu.tw

Department of Mathematics
Tamkang University
Tamsui, TAIWAN
e-mail: yjchen373@hotmail.com

Abstract.Let $L u = - \sum_{i, j = 1}^N a_{i j} (x, u) D_{i j} u$ . Consider the quasilinear elliptic equation $L u + f ( x, u, \nabla u ) = 0$ on a bounded smooth domain $\Omega$ in ${\mathbb R}^N$ . It is shown that if the oscillation of $a_{i j} ( x, r )$ with respect to is sufficiently small and $f ( x, r, \xi )$ has a sub-linear growth in and $\xi$ , then there exists a solution $u \in W^{2, p} ( \Omega ) \cap W^{1, p}_0 ( \Omega )$ . The existence of $W^{2, p} ( \Omega ) \cap W^{1, p}_0 ( \Omega )$ solutions to the equation $L u + c (x, u) u + f (x, u, \nabla u) = 0$ , where $\beta \geq c (x, r) \geq \alpha > 0$ , remains valid if has a sub-quadratic growth in $\xi$ .