关键词: 非线性椭圆随机偏微分方程, 弱解, Leray-Schauder连续方法

Abstract: In this paper the authors study of following problem: Let $D$ be a bounded open set of $R^N(N>1)$ and
$(\Omega,F,P)$ is a probability space. The authors study the existence of weak solutions of the
following stochastic boundary value problem:
$$
\left\{
\begin{array}{ll}
-{\rm div} A(x,\omega,u, \nabla u)=f(x,\omega, u),\,\, &(x,\omega)\in D\times \Omega,\\
u=0, &(x,\omega)\in \partial D\times \Omega,
\end{array}\right.
$$
where by div and $\nabla$ the authors denote differentiation with respect to $x$ only. First, the authors
introduce the concept of the weak solution, then the authors transform the stochastic problem into a deterministic
one in high-dimensions. Finally, the authors prove the existence of weak solutions.

Key words: Nonlinear elliptic stochastic partial differential equations, Weak solutions,
Leray-Schauder continuation method