Affiliation:
1. School of Mathematics and Computer Science, Yunnan Minzu University , Kunming , 650500 , P. R. China
Abstract
Abstract
In this article, we study the following quasilinear equation with nonlocal nonlinearity
−
Δ
u
−
κ
u
Δ
(
u
2
)
+
λ
u
=
(
∣
x
∣
−
μ
*
F
(
u
)
)
f
(
u
)
,
in
R
N
,
-\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},
where
κ
\kappa
is a parameter,
N
≥
3
N\ge 3
,
μ
∈
(
0
,
N
)
\mu \in \left(0,N)
,
F
(
t
)
=
∫
0
t
f
(
s
)
d
s
F\left(t)={\int }_{0}^{t}f\left(s){\rm{d}}s
, and
λ
\lambda
is a positive constant. We are going to analyze two cases: the
L
2
{L}_{2}
-norm of the solution is not confirmed and the
L
2
{L}_{2}
-norm of the solution is prescribed. Under the almost optimal assumptions on
f
f
, we obtain the existence of a sequence of radial solutions for two cases.