The Dirichlet problem Δ u = λ f ( u ) \Delta u = \lambda \,f(u) in a domain Ω , u = 1 \Omega ,\,u = 1 on ∂ Ω \partial \Omega is considered with f ( t ) = 0 f(t) = 0 if t ≤ 0 , f ( t ) > 0 t \leq 0,\,f(t) > 0 if t > 0 , f ( t ) ∼ t p t > 0,\,f(t) \sim {t^p} if t ↓ 0 , 0 > p > 1 ; f ( t ) t \downarrow 0,0 > p > 1;f(t) is not monotone in general. The set { u = 0 } \{ u = 0\} and the “free boundary” ∂ { u = 0 } \partial \{ u = 0\} are studied. Sharp asymptotic estimates are established as λ → ∞ \lambda \to \infty . For suitable f f , under the assumption that Ω \Omega is a two-dimensional convex domain, it is shown that { u = 0 } \{ u = 0\} is a convex set. Analogous results are established also in the case where ∂ u / ∂ v + μ ( u − 1 ) = 0 \partial u/\partial v + \mu (u - 1) = 0 on ∂ Ω \partial \Omega .