Abstract
This paper studies the non-linear biharmonic Schödinger equation
i
u
˙
+
Δ
2
u
±
F
(
x
,
|
u
|
)
u
=
0
,
where
F
(
x
,
|
u
|
)
∈
{
|
x
|
-
2
b
|
u
|
2
(
q
-
1
)
,
|
x
|
-
b
|
u
|
p
-
2
(
I
α
∗
|
·
|
-
b
|
u
|
p
)
}
, where
b
>
0
and the source terms are inter-critical. First one develops a local theory in
L
2
and in the energy space
H
2
, by use of a sharp Gagliardo–Nirenberg type inequality. Then, one considers the global theory. Indeed, a sharp dichotomy of global versus non-global existence of solutions is obtained by use of the existence of ground states. Moreover, the strong instability of standing waves is proved. This note is a natural extension of Saanouni (Commun Pure Appl Anal 19(10): 5033–5057, 2020) to the inhomogeneous regime and gives some essential tools for the scattering of the focusing global solutions proved by Saanouni (Calc Var 60(113), 2021).