Abstract
In this paper, we study the following nonlinear magnetic Kirchhoff equation:
{
-
(
a
ϵ
2
+
b
ϵ
[
u
]
A
/
ϵ
2
)
Δ
A
/
ϵ
u
+
V
(
x
)
u
=
f
(
|
u
|
2
)
u
in
ℝ
3
,
u
∈
H
1
(
ℝ
3
,
ℂ
)
,
\left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/%
\epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\displaystyle%
\phantom{}\text{in }\mathbb{R}^{3},\\
&\displaystyle u\in H^{1}(\mathbb{R}^{3},\mathbb{C}),\end{aligned}\right.
where
ϵ
>
0
{\epsilon>0}
,
a
,
b
>
0
{a,b>0}
are constants,
V
:
ℝ
3
→
ℝ
{V:\mathbb{R}^{3}\rightarrow\mathbb{R}}
and
A
:
ℝ
3
→
ℝ
3
{A:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}}
are continuous
potentials, and
Δ
A
u
{\Delta_{A}u}
is the magnetic Laplace operator. Under a local assumption on the potential V, by combining variational methods, a penalization technique and the Ljusternik–Schnirelmann theory, we prove multiplicity properties of solutions and concentration phenomena for ϵ small. In this problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.
Concentration of solutions for fractional double-phase problems: critical and supercritical cases
