Structure of large positive solutions of some semilinear elliptic problems where the nonlinearity changes sign (Q1599785)

The present paper deals with the study of the eigenvalue problem \[ \begin{cases} -\Delta u=\lambda f(u)\quad & \text{in }\Omega\\ u=0\quad & \text{on } \partial \Omega,\end{cases} \tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) \((N\geq 2)\) with smooth boundary \(\partial\Omega\), \(\lambda>0\). The author is mainly interested in the structure of positive solutions of (1) for large positive \(\lambda\) in the case \(f(0)=0\), \(f'(0)=0\), \(f(a)=f(b)=0\), \(0<a <b\), \(f\) changes sign on \([0,\infty)\). Existence and uniqueness of the solutions are presented. It is shown that the large positive solution has flat core and the distance of its flat core to the boundary \(\partial\Omega\) is exactly measured as \(\lambda\to\infty\).