Abstract
We study the following Kirchhoff type equation:
−
a
+
b
∫
R
N
|
∇
u
|
2
d
x
Δ
u
+
u
=
k
(
x
)
|
u
|
p
−
2
u
+
m
(
x
)
|
u
|
q
−
2
u
in
R
N
,
$$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array}
\end{equation*}$$
where N=3,
a
,
b
>
0
$ a,b \gt 0 $
,
1
<
q
<
2
<
p
<
min
{
4
,
2
∗
}
$ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $
, 2≤=2N/(N − 2), k ∈ C (ℝ
N
) is bounded and m ∈ L
p/(p−q)(ℝ
N
). By imposing some suitable conditions on functions k(x) and m(x), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding
H
1
(
R
N
)
↪
L
r
(
R
N
)
(
2
≤
r
<
2
∗
)
$ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $
; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.
The Nehari manifold for elliptic equation involving the square root of the Laplacian