We consider a semilinear boundary value problem
−
Δ
u
+
f
(
u
,
x
)
=
0
- \Delta u + f(u,x) = 0
in
Ω
⊂
R
N
\Omega \subset {\mathbb {R}^N}
and
u
=
0
u = 0
on
∂
Ω
\partial \Omega
. We assume that
f
f
is a
C
∞
{C^\infty }
-smooth function and
Ω
\Omega
is a bounded domain with a smooth boundary. For any
C
α
{C^\alpha }
-smooth perturbation
h
(
x
)
h(x)
of the right-hand side of the equation we consider the function
N
h
(
S
)
{N_h}(S)
defined as the number of
C
2
+
α
{C^{2 + \alpha }}
-smooth solutions
u
u
such that
‖
u
‖
C
0
(
Ω
)
≤
S
\left \| u\right \| _{{C^0}(\Omega )} \leq S
of the perturbed problem. How "small"
N
h
(
S
)
{N_h}(S)
can be made by a perturbation
h
(
x
)
h(x)
such that
‖
h
‖
C
0
(
Ω
)
≤
ε
?
\left \| h\right \| _{{C^0}(\Omega )} \leq \varepsilon ?
We present here an explicit upper bound in terms of
ε
\varepsilon
,
S
S
and
\[
max
|
u
|
≤
S
,
x
∈
Ω
¯
‖
D
u
i
f
(
u
,
x
)
‖
(
i
∈
{
0
,
1
,
2
}
)
.
\max \limits _{|u| \leq S,x \in \bar \Omega } \left \| D_u^i f(u,x)\right \| \quad (i \in \{ 0,1,2\} ).
\]
If
S
S
is fixed then
h
h
can be chosen by such a way that the upper bound persists under small in
C
0
{C^0}
-topology perturbations of
h
h
. We present an explicit lower bound for the radius of the ball of such admissible perturbations.