Affiliation:
1. School of Mathematics and Physics, Yancheng Institute of Technology , Yancheng , 224051 , China
Abstract
Abstract
This article shows the existence and multiplicity of solutions for the following
p
p
-Kirchhoff-type equation:
a
+
b
∫
R
N
(
∣
∇
u
∣
p
+
V
(
x
)
∣
u
∣
p
)
d
x
(
−
△
p
u
+
V
(
x
)
∣
u
∣
p
−
2
u
)
=
λ
g
(
x
)
∣
u
∣
r
−
2
u
−
h
(
x
)
∣
u
∣
q
−
2
u
,
in
R
N
.
\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\left({| \nabla u| }^{p}+V\left(x){| u| }^{p}){\rm{d}}x\right)\left(-{\bigtriangleup }_{p}u+V\left(x){| u| }^{p-2}u)=\lambda g\left(x){| u| }^{r-2}u-h\left(x){| u| }^{q-2}u,\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}.
where
λ
\lambda
is a real parameter and
1
<
p
<
q
<
∞
1\lt p\lt q\lt \infty
,
a
a
and
b
b
are positive constants. Depending on the relationship of
p
,
q
p,q
, and
r
r
, we obtain the existence, multiplicity, and nonexistence of solutions to the abovementioned equation using variational methods.