In this paper we study the existence of multiple asymmetric positive solutions for the following symmetric problem:
\[
{
−
Δ
u
+
(
λ
−
h
(
x
)
)
u
=
(
1
−
f
(
x
)
)
u
p
,
a
m
p
;
x
∈
R
N
,
u
(
x
)
>
0
,
a
m
p
;
x
∈
R
N
,
u
∈
H
1
(
R
N
)
,
\begin {cases} -\Delta u+(\lambda -h(x))u=(1-f(x))u^p, & x\in \mathbb {R}^N,\\ u(x)>0, &\quad x\in \mathbb {R}^N,\\ u\in H^1(\mathbb {R}^N), \end {cases}
\]
where
λ
>
0
\lambda >0
is a parameter,
h
(
x
)
h(x)
and
f
(
x
)
f(x)
are nonnegative radially symmetric functions in
L
∞
(
R
N
)
L^\infty (\mathbb {R}^N)
,
h
(
x
)
h(x)
and
f
(
x
)
f(x)
have compact support in
R
N
\mathbb {R}^N
,
f
(
x
)
≤
1
f(x)\leq 1
for all
x
∈
R
N
x\in \mathbb {R}^N
,
1
>
p
>
+
∞
1>p>+\infty
for
N
=
1
,
2
N=1,2
,
1
>
p
>
N
+
2
N
−
2
1>p>\frac {N+2}{N-2}
for
N
≥
3
N\geq 3
. We prove that for any
k
=
1
,
2
,
…
k=1,2,\,\ldots \,
, if
λ
\lambda
is large enough the above problem has positive solutions
u
λ
u_\lambda
concentrating at
k
k
distinct points away from the origin as
λ
\lambda
goes to
∞
\infty
.