Let
p
,
φ
:
[
0
,
T
]
→
R
p,\varphi :[0,T] \to R
be bounded functions with
φ
>
0
\varphi > 0
. Let
g
:
R
→
R
g:{\mathbf {R}} \to {\mathbf {R}}
be a locally Lipschitzian function satisfying the superlinear jumping condition: (i)
lim
u
→
−
∞
(
g
(
u
)
/
u
)
∈
R
{\lim _{u \to - \infty }}(g(u)/u) \in {\mathbf {R}}
(ii)
lim
u
→
∞
(
g
(
u
)
/
u
1
+
ρ
)
=
∞
{\lim _{u \to \infty }}(g(u)/{u^{1 + \rho }}) = \infty
for some
ρ
>
0
\rho > 0
, and (iii)
lim
u
→
∞
(
u
/
g
(
u
)
)
N
/
2
(
N
G
(
κ
u
)
−
(
(
N
−
2
)
/
2
)
u
⋅
g
(
u
)
)
=
∞
{\lim _{u \to \infty }}{(u/g(u))^{N/2}}(NG(\kappa u) - ((N - 2)/2)u \cdot g(u)) = \infty
for some
κ
∈
(
0
,
1
]
\kappa \in (0,1]
where
G
G
is the primitive of
g
g
. Here we prove that the number of solutions of the boundary value problem
Δ
u
+
g
(
u
)
=
p
(
‖
x
‖
)
+
c
φ
(
‖
x
‖
)
\Delta u + g(u) = p(\left \| x\right \|) + c\varphi (\left \| x\right \|)
for
x
∈
R
N
x \in {{\mathbf {R}}^N}
with
‖
x
‖
>
T
,
u
(
x
)
=
0
\left \| x\right \| > T,u(x) = 0
for
‖
x
‖
=
T
\left \| x\right \| = T
tends to
+
∞
+ \infty
when
c
c
tends to
+
∞
+ \infty
. The proofs are based on the "energy" and "phase plane" analysis.