AbstractIn this paper, we focus on the existence of solutions for the Choquard equation $$\begin{aligned} \textstyle\begin{cases} {-}\Delta {u}+V(x)u=(I_{\alpha }* \vert u \vert ^{\frac{\alpha }{N}+1}) \vert u \vert ^{ \frac{\alpha }{N}-1}u+\lambda \vert u \vert ^{p-2}u,\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases}\displaystyle \end{aligned}$$
{
−
Δ
u
+
V
(
x
)
u
=
(
I
α
∗
|
u
|
α
N
+
1
)
|
u
|
α
N
−
1
u
+
λ
|
u
|
p
−
2
u
,
x
∈
R
N
;
u
∈
H
1
(
R
N
)
,
where $\lambda >0$
λ
>
0
is a parameter, $\alpha \in (0,N)$
α
∈
(
0
,
N
)
, $N\ge 3$
N
≥
3
, $I_{\alpha }: \mathbb{R}^{N}\to \mathbb{R}$
I
α
:
R
N
→
R
is the Riesz potential. As usual, $\alpha /N+1$
α
/
N
+
1
is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if $\lambda >\lambda _{*}$
λ
>
λ
∗
for some given number $\lambda _{*}$
λ
∗
in three cases: (i) $2< p<\frac{4}{N}+2$
2
<
p
<
4
N
+
2
, (ii) $p=\frac{4}{N}+2$
p
=
4
N
+
2
, and (iii) $\frac{4}{N}+2< p<2^{*}$
4
N
+
2
<
p
<
2
∗
. Our result improves the previous related ones in the literature.
Nontrivial solutions for the choquard equation with indefinite linear part and upper critical exponent