Abstract
In this paper, we study the following generalized Kadomtsev-Petviashvili equation
u
t
+
u
x
x
x
+
(
h
(
u
)
)
x
=
D
x
−
1
Δ
y
u
,
{u}_{t}+{u}_{xxx}+{\left(h\left(u))}_{x}={D}_{x}^{-1}{\Delta }_{y}u,
where
(
t
,
x
,
y
)
∈
R
+
×
R
×
R
N
−
1
\left(t,x,y)\in {{\mathbb{R}}}^{+}\times {\mathbb{R}}\times {{\mathbb{R}}}^{N-1}
,
N
≥
2
N\ge 2
,
D
x
−
1
f
(
x
,
y
)
=
∫
−
∞
x
f
(
s
,
y
)
d
s
{D}_{x}^{-1}f\left(x,y)={\int }_{-\infty }^{x}f\left(s,y){\rm{d}}s
,
f
t
=
∂
f
∂
t
{f}_{t}=\frac{\partial f}{\partial t}
,
f
x
=
∂
f
∂
x
{f}_{x}=\frac{\partial f}{\partial x}
and
Δ
y
=
∑
i
=
1
N
−
1
∂
2
∂
y
i
2
{\Delta }_{y}={\sum }_{i=1}^{N-1}\frac{{\partial }^{2}}{{\partial }_{{y}_{i}}^{2}}
. We get the existence of infinitely many nontrivial solutions under certain assumptions in bounded domain without Ambrosetti-Rabinowitz condition. Moreover, by using the method developed by Jeanjean [13], we establish the existence of ground state solutions in
R
N
{{\mathbb{R}}}^{N}
.
Asymptotic behaviour of ground state solutions for the fractional Hénon equation
