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 $1 < p < {n+2}/{n-1}$. Inspired by their result, in this
article, we use the method of moving planes to fill the gap between
${n+2}/{n-1}$ and the critical Sobolev exponent ${n+2}/{n-2}$. We
obtain a priori estimates for the solutions for all $1 < p < {n+2}/{n-2}$.
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