Indefinite elliptic problems in a domain
Wenxiong Chen  Department of Mathematics, Southwest Missouri State University, United States (email) Abstract: In this paper, we study the elliptic boundary value problem in a bounded domain $\Omega$ in $R^n$, with smooth boundary: $\Delta u = R(x) u^p \quad \quad u > 0 x \in \Omega$
$u(x) = 0 \quad \quad x \in \partial \Omega.$ where $R(x)$ is a smooth function that may change signs. In [2], using a blowing up argument, Berestycki, Dolcetta, and Nirenberg, obtained a priori estimates and hence the existence of solutions for the problem when the exponent $1 < p < {n+2}/{n1}$. Inspired by their result, in this article, we use the method of moving planes to fill the gap between ${n+2}/{n1}$ and the critical Sobolev exponent ${n+2}/{n2}$. We obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n2}$.
Keywords: Indefinite nonlinear elliptic equations,
a priori estimates, method of moving planes.
Received: January 1997; Available Online: April 1997. 
2016 Impact Factor1.099
