Abstract
We study the asymptotic profile, as
ℏ
→
0
{\hbar\rightarrow 0}
, of positive solutions to
-
ℏ
2
Δ
u
+
V
(
x
)
u
-
ℏ
2
+
γ
u
Δ
u
2
=
K
(
x
)
|
u
|
p
-
2
u
,
x
∈
ℝ
N
,
-\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2%
}u,\quad x\in\mathbb{R}^{N},
where
γ
⩾
0
{\gamma\geqslant 0}
is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite
L
2
{L^{2}}
-energy solutions. We investigate the concentrating behavior of solutions when
γ
>
0
{\gamma>0}
and, differently from the case
γ
=
0
{\gamma=0}
where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for
γ
>
0
{\gamma>0}
we find a different concentration behavior of solutions in the case
p
=
2
N
N
-
2
{p=\frac{2N}{N-2}}
and when
2
N
N
-
2
<
p
<
4
N
N
-
2
{\frac{2N}{N-2}<p<\frac{4N}{N-2}}
. This phenomenon does not occur when
γ
=
0
{\gamma=0}
.
Asymptotic profile of solutions to the heat equation on thin plate with boundary heating