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.
Harmonic Suppression Strategy of Photovoltaic Grid Connected Inverter Based on Repetitive and PI Control
