Abstract
AbstractThe purpose of this paper is to study the existence of sign-changing solution to the following fourth-order equation: $$ \Delta ^{2}u- \biggl(a+ b \int _{\mathbb{R}^{N}} \vert \nabla u \vert ^{2}\,dx \biggr) \Delta u+V(x)u=K(x)f(u) \quad\text{in } \mathbb{R}^{N}, $$
Δ
2
u
−
(
a
+
b
∫
R
N
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
K
(
x
)
f
(
u
)
in
R
N
,
where $5\leq N\leq 7$
5
≤
N
≤
7
, $\Delta ^{2}$
Δ
2
denotes the biharmonic operator, $K(x), V(x)$
K
(
x
)
,
V
(
x
)
are positive continuous functions which vanish at infinity, and $f(u)$
f
(
u
)
is only a continuous function. We prove that the equation has a least energy sign-changing solution by the minimization argument on the sign-changing Nehari manifold. If, additionally, f is an odd function, we obtain that equation has infinitely many nontrivial solutions.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis