DOI: 10.1515/ans-2023-0136 ISSN: 1536-1365
Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials
Caifeng Zhang, Maochun Zhu Abstract
The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint:
(0.1)
sup
u
∈
H
1
(
R
2
)
,
∫
R
2
(
|
∇
u
|
2
+
V
(
x
)
u
2
)
d
x
≤
1
∫
R
2
e
4
π
1
−
β
2
u
2
−
1
|
x
|
β
d
x
<
+
∞
,
$$\underset{u\in {H}^{1}({\mathbb{R}}^{2}),\underset{{\mathbb{R}}^{2}}{\int }(\vert \nabla u{\vert }^{2}+V(x){u}^{2})\mathrm{d}x\le 1}{\mathrm{sup}}\underset{{\mathbb{R}}^{2}}{\int }\frac{{e}^{4\pi \left(1-\tfrac{\beta }{2}\right){u}^{2}}-1}{\vert x{\vert }^{\beta }}\mathrm{d}x< +\infty ,$$
where
0
<
β
<
2
$0< \beta < 2$
,
V
(
x
)
≥
0
$V(x)\ge 0$
and may vanish on an open set in
R
2
${\mathbb{R}}^{2}$
. Second, we consider the existence of ground states to the following Schrödinger equations with critical exponential growth in
R
2
${\mathbb{R}}^{2}$
:
(0.2)
−
Δ
u
+
γ
u
=
f
(
u
)
|
x
|
β
,
$${-}{\Delta }u+\gamma u=\frac{f(u)}{\vert x{\vert }^{\beta }},$$
where the nonlinearity