Our concern is to solve the nonlinear perturbation problem for the semilinear elliptic equation
Δ
u
+
p
(
x
)
u
+
ϕ
(
x
,
u
)
=
0
\Delta u + p(x) u + \phi (x,u) = 0
in an exterior domain of
R
N
\mathbb {R}^N
with
N
≥
3
N \ge 3
. The lower limit of the nonlinear perturbed term
ϕ
(
x
,
u
)
\phi (x,u)
is given for all nontrivial solutions to be oscillatory. The tools for obtaining our theorems are the so-called “supersolution-subsolution” method and some results concerning the oscillation and nonoscillation of solutions of the ordinary differential equation associated with the elliptic equation. A simple example is given to illustrate the main results.