In this paper, we discuss the asymptotic behavior of the positive solutions of the problem $-\Delta u=au-bu^p,\, u|_{\partial \Omega}=0$ as $p\to 1+0$ and as $p\to\infty$. We show that, for each case, the behavior is determined by a limiting problem. Moreover, the limiting problem is of free boundary nature when $p\to\infty$.