Let
Ω
\Omega
be a bounded domain in
R
n
,
\mathbf {R}^n,
n
≥
3
,
n \ge 3,
with a boundary
∂
Ω
∈
C
2
.
\partial \Omega \in C^2.
We consider the following singularly perturbed nonlinear elliptic problem on
Ω
\Omega
:
\[
ε
2
Δ
u
−
u
+
f
(
u
)
=
0
,
u
>
0
on
Ω
,
u
=
0
on
∂
Ω
,
\varepsilon ^2 \Delta u - u + f(u) = 0, \ \ u > 0 \textrm { on }\Omega , \quad u = 0 \textrm { on } \partial \Omega ,
\]
where the nonlinearity
f
f
is of subcritical growth. Under rather strong conditions on
f
,
f,
it has been known that for small
ε
>
0
,
\varepsilon > 0,
there exists a mountain pass solution
u
ε
u_\varepsilon
of above problem which exhibits a spike layer near a maximum point of the distance function
d
d
from
∂
Ω
\partial \Omega
as
ε
→
0.
\varepsilon \to 0.
In this paper, we construct a solution
u
ε
u_\varepsilon
of above problem which exhibits a spike layer near a maximum point of the distance function under certain conditions on
f
f
, which we believe to be almost optimal.