Asymptotic behavior of minimal solutions of $-\Delta u=\lambda f(u)$ as $\lambda \to -\infty$

Abstract

We consider the Dirichlet problem $-\Delta u=\lambda f(u)$ with $\lambda <0$ and $f$ non-negative and non-decreasing. We show existence and uniqueness of solutions $u_{\lambda }$ for any $\lambda$ and discuss their asymptotic behavior as $\lambda \to -\infty$. In the expansion of $u_{\lambda }$ large solutions naturally appear.

18 pages