DOI: 10.68381/jca24069 ISSN: 0944-6532
Existence of Solutions for a Nonlocal Variational Problem in ℝ
2
with Exponential Critical Growth
Claudianor O. Alves, Minbo Yang
We study the existence of nontrivial solutions for the following class of nonlocal problem,
-\Delta u +V(x)u =\Big( I_\mu\ast F(x,u)\Big)f(x,u) \quad \text{in} \quad \mathbb{R}^2,
−
Δ
u
+
V
(
x
)
u
=
(
I
μ
∗
F
(
x
,
u
)
)
f
(
x
,
u
)
in
R
2
,
where
V
V
is a positive periodic potential,
I_\mu=\frac{1}{|x|^\mu}
I
μ
=
1
∣
x
∣
μ
,
0<\mu<2
0
<
μ
<
2
and
F(x,s)
F
(
x
,
s
)
is the primitive function of
f(x,s)
f
(
x
,
s
)
in the variable
s
s
. By assuming that the nonlinearity
f(x,s)
f
(
x
,
s
)
has an exponential critical growth at infinity, we prove the existence of solutions by variational methods.