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

Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic Dirichlet problems in bounded smooth domains $\Omega$ with a large parameter $\lambda$. It is shown that the large positive solution has flat core. The distance of its flat core to the boundary $\partial \Omega$ is exactly measured as $\lambda \to \infty$