AbstractIn this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$
−
div
(
a
(
x
,
∇
u
)
)
+
V
(
x
)
|
x
|
−
α
p
∗
|
u
|
p
−
2
u
=
K
(
x
)
|
x
|
−
α
p
∗
f
(
x
,
u
)
in
R
N
,
where $N\geq 3$
N
≥
3
, $1< p< N$
1
<
p
<
N
, $-\infty <\alpha <\frac{N-p}{p}$
−
∞
<
α
<
N
−
p
p
, $\alpha \leq e\leq \alpha +1$
α
≤
e
≤
α
+
1
, $d=1+\alpha -e$
d
=
1
+
α
−
e
, $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$
p
∗
:
=
p
∗
(
α
,
e
)
=
N
p
N
−
d
p
(critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.
Infinitely many solutions for a quasilinear Schrödinger equation with Hardy potentials