This paper is concerned with necessary conditions for the existence of positive solutions of the semilinear problem
Δ
u
+
f
(
u
)
=
0
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
\Delta u + f(u) = 0,x \in \Omega ,u = 0,x \in \partial \Omega
, whose supremum norm bears a certain relationship to zeros of the nonlinearity
f
f
. We first discuss the smooth case (i.e.,
f
f
and
∂
Ω
\partial \Omega
smooth) and then show how to obtain similar results in the nonsmooth case.