In this paper, we prove the existence of normalized solutions to the following fractional Schrödinger–Poisson system:
(
-
Δ
)
s
u
+
ϕ
u
=
f
(
u
)
+
λ
u
,
x
∈
R
3
,
λ
∈
R
(
-
Δ
)
t
ϕ
=
u
2
,
where
0
<
...s
,
t
<
1
,
2
s
+
2
t
>
3
,
λ
∈
R
,
f
∈
C
1
(
R
,
R
)
satisfies some general conditions which contain the case
f
(
u
)
∼
|
u
|
q
-
2
u
with
q
∈
(
4
s
+
2
t
s
+
t
,
4
s
3
+
2
)
∪
(
4
s
3
+
2
,
2
s
∗
)
,
2
s
∗
=
6
3
-
2
s
.
In this paper, we study the existence of ground state solutions for the nonlinear fractional Schrödinger–Poisson system with critical Sobolev ...exponent{(−Δ)su+V(x)u+ϕu=μ|u|q−1u+|u|2s⁎−2u,in R3,(−Δ)tϕ=u2,in R3, where μ∈R+ is a parameter, 1<q<2s⁎−1=3+2s3−2s, s,t∈(0,1) and 2s+2t>3. Under certain assumptions on V(x), using the method of Pohozaev–Nehari manifold and the arguments of Brezis–Nirenberg, the monotonic trick and global compactness Lemma, we prove the existence of a nontrivial ground state solution.
In this paper, we consider a new kind of electromagnetostatic Schrödinger–Maxwell system with critical exponent and prove the existence of vortex ground state solutions with cylindrically symmetry ...using the Nehari manifold approach combining with the Brezis–Nirenberg technique.
In this paper, we show the existence of at least two nontrivial solutions of the nonlinear elliptic problem driven by nonlocal integro-differential operators
L
K
u
=
λ
f
(
x
,
u
)
,
in
Ω
,
u
=
0
,
on
...R
N
\
Ω
,
where
λ
∈
R
is a parameter and
L
K
u
(
x
)
=
2
P
.
V
.
∫
R
N
|
u
(
x
)
-
u
(
y
)
|
p
-
2
(
u
(
x
)
-
u
(
y
)
)
K
(
x
-
y
)
d
y
and
K
belongs to a class of singular symmetric kernels modeled on the case
K
(
x
,
y
)
=
|
x
-
y
|
-
(
N
+
s
p
)
,
P
.
V
.
is a commonly used abbreviation for in the principal value sense.
In this paper, we consider the Cauchy problem for the 2D incompressible Navier–Stokes–Allen–Cahn system (
1.1
), which describes two-phase patterns in a flowing liquid including phase ...transformations. In this model, a new function
χ
is introduced to distinguish two substances and to indicate the location of the interface. Among them, the Navier–Stokes equation in the coupled system mainly describes the dynamic characteristics of the fluid, such as velocity. The Allen–Cahn equation describes the interaction between two fluids near their interface, such as the change of concentration caused by diffusion. We prove that there exists a global smooth solution of (
1.1
) under the assumptions of small initial data. Moreover, we show the large time behavior of smooth solution to the Cauchy problem through using the Fourier-splitting method.
In this paper, we study a class of Schrödinger–Bopp–Podolsky system. Under some suitable assumptions for the potentials, by developing some calculations of sharp energy estimates and using a ...topological argument involving the barycenter function, we establish the existence of positive bound state solution.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper we complete two tasks. We first extend the critical point theorem obtained by Li and Willem S.J. Li, M. Willem, Applications of local linking to critical point theory, J. Math. Anal. ...Appl. 189 (1995), 6–32 to the nonsmooth case in which the energy functional is locally Lipschitz and satisfies the weaker nonsmooth Cerami condition. Then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results.
We study the concentration and uniqueness of standing waves associated with the constraint minimization problems for the nonlinear Hartree type equations with homogeneous potentials and fractional ...Laplacian. This class of equations is an effective model to describe the fractional quantum mechanics with a convolution perturbation. By making full use of the Bessel kernel and adopting the iterative process, we establish the
L
∞
-estimates and decay properties of the solutions to the fractional Hartree equations. Based on the above basic conclusions, we establish the concentration and uniqueness of the constraint minimizers by exploring some fine energy estimates and studying some uniform regularity, while establishing local Pohozăev identity and overcoming the blow-up estimates to the nonlocal operator
(
-
Δ
)
s
. Compared with the classical local elliptic problems, we encounter some new difficulties because of the nonlocal nature of the fractional Laplace. One of the main difficulties is that the decay estimates of the sequences of solutions to the nonlocal problems at infinity are different from those in the case of the classical local problems. Another difficulty is that we have to consider the corresponding harmonic extension problems to construct the Pohozăev identity, which will cause us to have to estimate several kinds of integrals that never appear in the classic local problems. In addition, the presence of the potential
|
x
|
2
and the Hartree term
|
x
|
2
s
-
3
∗
u
2
will affect the blow-up frequency of the minimizers, and we have to control
s
to achieve the optimal blow-up speed.
In this article, we study a class of critical fractional Schrodinger-Poisson system with two perturbation terms. By using variational methods and Lusternik-Schnirelman category theory, the existence ...of ground state and two nontrivial solutions are established.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
PDF
