AbstractThe paper deals with the equation $$-\Delta u+a(x) u =|u|^{p-1}u $$
-
Δ
u
+
a
(
x
)
u
=
|
u
|
p
-
1
u
, $$u \in H^1({\mathbb {R}}^N)$$
u
∈
H
1
(
R
N
)
, with $$N\ge 2$$
N
≥
2
, $$p> 1,\ p< {N+2\over N-2}$$
p
>
1
,
p
<
N
+
2
N
-
2
if $$N\ge 3$$
N
≥
3
, $$a\in L^{N/2}_{loc}({\mathbb {R}}^N)$$
a
∈
L
loc
N
/
2
(
R
N
)
, $$\inf a> 0$$
inf
a
>
0
, $$\lim _{|x| \rightarrow \infty } a(x)= a_\infty $$
lim
|
x
|
→
∞
a
(
x
)
=
a
∞
. Assuming that the potential a(x) satisfies $$\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta > 0$$
lim
|
x
|
→
∞
[
a
(
x
)
-
a
∞
]
e
η
|
x
|
=
∞
∀
η
>
0
, $$ \lim _{\rho \rightarrow \infty } \sup \left\{ a(\rho \theta _1) - a(\rho \theta _2) \ :\ \theta _1, \theta _2 \in {\mathbb {R}}^N,\ |\theta _1|= |\theta _2|=1 \right\} e^{\tilde{\eta }\rho } = 0 \quad \text{ for } \text{ some } \ \tilde{\eta }> 0$$
lim
ρ
→
∞
sup
a
(
ρ
θ
1
)
-
a
(
ρ
θ
2
)
:
θ
1
,
θ
2
∈
R
N
,
|
θ
1
|
=
|
θ
2
|
=
1
e
η
~
ρ
=
0
for
some
η
~
>
0
and other technical conditions, but not requiring any symmetry, the existence of infinitely many positive multi-bump solutions is proved. This result considerably improves those of previous papers because we do not require that a(x) has radial symmetry, or that $$N=2$$
N
=
2
, or that $$|a(x)-a_\infty |$$
|
a
(
x
)
-
a
∞
|
is uniformly small in $${\mathbb {R}}^N$$
R
N
, etc. ....
Qualitative aspects for the cubic nonlinear Schrödinger equations with localized damping: Exponential and polynomial stabilization
