IJPAM: Volume 55, No. 1 (2009)

EXISTENCE OF STRONG SOLUTIONS TO SOME
QUASILINEAR ELLIPTIC EQUATIONS

Tsang-Hai Kuo$^1$, Chu-Ching Huang$^2$, Yi-Jung Chen$^3$
$^{1,2}$Faculty of Mathematics
Center for General Education
Chang Gung University
256, Wen Hwa, 1-st Road, Kwei-Shan, Tao-Yuan, 333, TAIWAN
$^1$e-mail: [email protected]
$^2$e-mail: [email protected]
$^3$Department of Mathematics
Tamkang University
Tamsui, TAIWAN
e-mail: [email protected]


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 $r$ is sufficiently small and $f ( x, r, \xi )$ has a sub-linear growth in $r$ 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 $f$ has a sub-quadratic growth in $\xi$.

Received: April 25, 2009

AMS Subject Classification: 35D05, 35J25, 46E35

Key Words and Phrases: quasilinear elliptic equation, $W^{2,p}$-estimate, $W^{2, p} ( \Omega ) \cap W^{1, p}_0 ( \Omega )$ solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 1