Abstract
We consider the following p-harmonic problem
Δ
(
|
Δ
u
|
p
−
2
Δ
u
)
+
m
|
u
|
p
−
2
u
=
f
(
x
,
u
)
,
x
∈
R
N
,
u
∈
W
2
,
p
(
R
N
)
,
$$\begin{array}{}
\displaystyle
\left\{ \displaystyle\begin{array}{ll} \displaystyle {\it\Delta} (|{\it\Delta} u|^{p-2}{\it\Delta} u)+m|u|^{p-2}u=f(x,u), \ \ x\in {\mathbb R}^N, \\
u \in W^{2,p}({\mathbb R}^N),
\end{array}
\right.
\end{array}$$
where m > 0 is a constant, N > 2p ≥ 4 and
lim
t
→
∞
f
(
x
,
t
)
|
t
|
p
−
2
t
=
l
$\begin{array}{}
\displaystyle
\lim\limits_{t\rightarrow \infty}\frac{f(x,t)}{|t|^{p-2}t}=l
\end{array}$
uniformly in x, which implies that f(x, t) does not satisfy the Ambrosetti-Rabinowitz type condition. By showing the Pohozaev identity for weak solutions to the limited problem of the above p-harmonic equation and using a variant version of Mountain Pass Theorem, we prove the existence and nonexistence of nontrivial solutions to the above equation. Moreover, if f(x, u) ≡ f(u), the existence of a ground state solution and the nonexistence of nontrivial solutions to the above problem is also proved by using artificial constraint method and the Pohozaev identity.