EXISTENCE OF STRONG SOLUTIONS TO SOME QUASILINEAR ELLIPTIC PROBLEMS ON BOUNDED SMOOTH DOMAINS

Abstract

We consider the following quasilinear elliptic problems in a bounded smooth domain $Z$ of ${\Bbb R}^N$, $N\ge 3$: $$\left\{\begin{array}{lr} Lu= \sum_{i,j=1}^{N}a_{ij}(x,u) \frac{\partial^2u} {\partial x_i\partial x_j}+ \sum_{i=1}^{N}b_{i}(x,u) \frac{\partial u} {\partial x_i}+c(x,u)u=f(x) &\mbox{in } Z, \\ u=0 &\mbox{on } \partial Z,\end{array}\right . $$ where $f(x)\in L^{p}(Z)$ and all the coefficients $a_{ij}, b_{i}, c$ are Carath\'{e}dory functions. Suppose that $a_{ij}\in C^{0,1}(\bar{Z}\times \Bbb R)$, $a_{ij}$, $\partial a_{ij}/\partial x_i$, $\partial a_{ij}/\partial r$, $b_{i}$, $c\in L^{\infty}(Z\times \Bbb R)$, $c\le 0$ for $i,j=1,...N$ and the oscillations of $a_{ij}=a_{ij}(x,r)$ with respect to $r$ are sufficiently small. A global estimate for a solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ is established and the existence of a strong solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ is proved for $p\gt N$. Furthermore, we replace $f(x)$ by $f(x,r,\xi)$ which is defined on $Z\times {\Bbb R}\times {\Bbb R}^N$ and is a $Carath\acute{e}dory$ function. Assume that $$ |f(x,r,\xi)|\le C_{0}+h(|r|)|\xi|^{\theta},\qquad 0\le\theta \lt 2,$$ where $C_0$ is a nonnegative constant, $h(|r|)$ is a locally bounded function, and $-c\ge \alpha_{0}\gt 0$ for some constant $\alpha_{0}$. We prove the existence of solution $u\in W^{2,p}(Z)\cap W^{1,p}_{0}(Z)$ for the equation $Lu=f(x,u,\nabla u)$.