Asymptotic behavior of positive solutions of some elliptic problems. (Q1880056)

The authors deal with the asymptotic behavior of positive solutions of the problem \[ \begin{cases} -\Delta u=au-b(x)u^p,\quad & x\in\Omega\\ u=0, \quad &\text{on }\partial\Omega,\end{cases}\tag{1} \] for \(p\) near 1 and near \(\infty\), respectively. Here \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) \((N\geq 1)\) and \(b(x)\) is a nonnegative function in \(C (\overline\Omega)\), \(a\) and \(p\) are constants but the exponent \(p\) is always greater than 1. The authors show that, for each case, the behavior is determined by a limiting problem. Moreover, the limiting problem is of free boundary nature when \(p\to\infty\).