ON THE EXISTENCE OF STRONG SOLUTIONS TO SOME SEMILINEAR ELLIPTIC PROBLEMS

Abstract

We study the following semilinear elliptic problem: $$ \left\{\begin{array}{l} \displaystyle\sum_{i,j=1}^N a_{ij}(x, u)\displaystyle\frac{\partial^2 u}{\partial x_i\partial x_j} +\displaystyle\sum_{i=1}^N b_i(x, u)\displaystyle\frac{\partial u}{\partial x_i}+c(x, u)u=f(x)\quad\mbox{ in }B, \\ u=0 \qquad\mbox{ on }{\partial B},\end{array} \right. $$ where $B$ is a ball in ${\Bbb R}^N$, $N\geq 3$, $a_{ij}=a_{ij}(x,r)\in C^{0,1} (\bar{B}\times{\Bbb R})$, $a_{ij}$, $\partial a_{ij}/\partial x_i$, $\partial a_{ij}/\partial r$, $b_i$, $c\in L^\infty(B\times{\Bbb R})$, with $i, j=1, 2, \cdots, N$ and $c\leq 0$, and $f\in L^p(B)$. For each $p$, $p\geq N$, there exists a strong solution $u\in W^{2,p}(B)\cap W_0^{1,p}(B)$ provided the oscillations of $a_{ij}$ with respect to $r$ are sufficiently small. Moreover, for $N/2