By using the penalization method and the Ljusternik–Schnirelmann theory, we investigate the multiplicity of positive solutions of the following fractional Schrödinger equation
ε
2
s
(
-
Δ
)
s
u
+
V
(
...x
)
u
=
f
(
u
)
in
R
N
where
ε
>
0
is a parameter,
s
∈
(
0
,
1
)
,
N
>
2
s
,
(
-
Δ
)
s
is the fractional Laplacian,
V
is a positive continuous potential with local minimum, and
f
is a superlinear function with subcritical growth. We also obtain a multiplicity result when
f
(
u
)
=
|
u
|
q
-
2
u
+
λ
|
u
|
r
-
2
u
with
2
<
q
<
2
s
∗
≤
r
and
λ
>
0
, by combining a truncation argument and a Moser-type iteration.
We focus on the following fractional (
,
)-Choquard problem:
where
> 0 is a small parameter, 0 <
< 1,
, 0 <
<
,
, with
∈ {
,
}, is the fractional
-Laplacian operator,
is a positive continuous ...potential satisfying a local condition,
is a continuous nonlinearity with subcritical growth at infinity and
. Applying suitable variational and topological methods, we relate the number of solutions with the topology of the set where the potential
attains its minimum value.
In this paper, we are concerned with the following fractional relativistic Schrödinger equation with critical growth:
where
is a small parameter,
,
,
,
is the fractional critical exponent,
is the ...fractional relativistic Schrödinger operator,
is a continuous potential, and
is a superlinear continuous nonlinearity with subcritical growth at infinity. Under suitable assumptions on the potential
, we construct a family of positive solutions
, with exponential decay, which concentrates around a local minimum of
as
We deal with the following fractional Schrödinger-Poisson equation with magnetic field
ε
2
s
(
−
Δ
)
A
/
ε
s
u
+
V
(
x
)
u
+
ε
−
2
t
(
|
x
|
2
t
−
3
∗
|
u
|
2
)
u
=
f
(
|
u
|
2
)
u
+
|
u
|
2
s
∗
−
2
...u
in
ℝ
3
,
where
ε
> 0 is a small parameter,
s
∈
(
3
4
,
1
)
,
t
∈ (0, 1),
2
s
∗
=
6
3
−
2
s
is the fractional critical exponent,
(
−
Δ
)
A
s
is the fractional magnetic Laplacian,
V
:
ℝ
3
→
ℝ
is a positive continuous potential,
A
:
ℝ
3
→
ℝ
3
is a smooth magnetic potential and
f
:
ℝ
→
ℝ
is a subcritical nonlinearity. Under a local condition on the potential
V
, we study the multiplicity and concentration of nontrivial solutions as
ε
→
0
. In particular, we relate the number of nontrivial solutions with the topology of the set where the potential
V
attains its minimum.
In this paper, we deal with the following class of fractional (
p
,
q
)-Laplacian Kirchhoff type problem:
1
+
u
s
,
p
p
(
-
Δ
)
p
s
u
+
1
+
u
s
,
q
q
(
-
Δ
)
q
s
u
+
V
(
ε
x
)
(
|
u
|
p
-
2
u
+
...|
u
|
q
-
2
u
)
=
f
(
u
)
in
R
N
,
u
∈
W
s
,
p
(
R
N
)
∩
W
s
,
q
(
R
N
)
,
u
>
0
in
R
N
,
where
ε
>
0
,
s
∈
(
0
,
1
)
,
1
<
p
<
q
<
N
s
<
2
q
,
(
-
Δ
)
t
s
, with
t
∈
{
p
,
q
}
, is the fractional
t
-Laplacian operator,
V
:
R
N
→
R
is a positive continuous potential such that
inf
∂
Λ
V
>
inf
Λ
V
for some bounded open set
Λ
⊂
R
N
, and
f
:
R
→
R
is a superlinear continuous nonlinearity with subcritical growth at infinity. By combining the method of Nehari manifold, a penalization technique, and the Lusternik–Schnirelman category theory, we study the multiplicity and concentration properties of solutions for the above problem when
ε
→
0
.
In this paper, we obtain the boundedness of solutions for a class of fractional elliptic equations driven by the relativistic Schrödinger operator
(
−
Δ
+
I
)
s
, with
s
∈
(
0
,
1
)
. The proof ...relies on a distributional Kato’s inequality for
(
−
Δ
+
I
)
s
and on some properties of the Bessel kernel.
In this work we study the following fractional scalar field equation:
where
,
,
is the fractional Laplacian and the nonlinearity
is such that
.
By using variational methods, we prove the existence of ...a positive solution which is spherically symmetric and decreasing in
We use a variant of the fountain Theorem to prove the existence of infinitely many weak solutions for the fractional p-Laplace equation $$\displaylines{ (-\Delta)_p^s u + V(x) |u|^{p-2}u = f(x, u) ...\quad \text{in } \mathbb{R}^N, }$$ where $s\in (0,1)$, $p\geq 2$, $N\geq 2$, $(- \Delta)_{p}^s$ is the fractional p-Laplace operator, the nonlinearity f is p-superlinear at infinity and the potential V(x) is allowed to be sign-changing.
In this paper, we investigate the existence of multiple solutions for the following two fractional problems:
where
,
, Ω is a smooth bounded domain of
, and
is a superlinear continuous function which ...does not satisfy the well-known Ambrosetti–Rabinowitz condition. Here
is the spectral Laplacian and
is the fractional Laplacian in
. By applying variational theorems of mixed type due to Marino and Saccon and the Linking Theorem, we prove the existence of multiple solutions for the above problems.
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