We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity
(
-
Δ
)
s
u
(
x
)
=
K
(
x
)
u
-
α
(
x
)
+
μ
u
p
-
1
(
x
)
in
Ω
,
u
>
0
in
Ω
,
u
...=
0
in
Ω
c
:
=
R
N
\
Ω
,
where
s
∈
(
0
,
1
)
,
α
>
0
and
Ω
⊂
R
N
is a bounded domain with smooth boundary
∂
Ω
and
N
>
2
s
.
Under some appropriate assumptions of
α
,
p
,
μ
and
K
, we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of
C
loc
1
,
1
∩
L
∞
solutions are also established for subcritical exponent
p
when the domain is a ball. Nonexistence of
C
1
,
1
∩
L
∞
solutions are obtained for star-shaped domain under a condition of
K
.
We consider the following semi-linear equations
(
-
Δ
)
p
u
=
u
+
γ
in
R
n
,
where
γ
∈
(
1
,
n
+
2
p
n
-
2
p
)
,
n
>
2
p
>
0
,
u
+
=
max
{
u
,
0
}
, and
2
≤
p
∈
N
or
p
∈
(
0
,
1
)
. Subject to the ...integral constraint
u
+
γ
∈
L
1
(
R
n
)
,
we obtain the classification of solutions to the above polyharmonic equation for any
γ
<
n
+
2
p
n
-
2
p
and
γ
≤
n
n
-
2
p
, according to the two different assumptions:
Δ
u
(
x
)
→
0
and
u
(
x
)
=
o
(
|
x
|
2
)
at infinity, respectively. Under the other integral constraint
u
+
q
∈
L
1
(
R
n
)
,
q
=
n
(
γ
-
1
)
2
p
,
γ
<
n
+
2
p
n
-
2
p
,
which is scaling invariant, the classification of solutions with the decay assumption
Δ
u
(
x
)
→
0
at infinity is established for any integer
p
≥
2
, and the classification of solutions with the growth assumption
u
(
x
)
=
o
(
|
x
|
2
)
at infinity is proved for integers
p
=
2
,
3
as well. In the fractional equation case, namely
p
∈
(
0
,
1
)
, under either of the above two integral constraints, we also complete the classification of solutions with certain growth assumption at infinity.
We consider periodic solutions of the following problem associated with the fractional Laplacian:
in
. The smooth function
is periodic about
and is a double-well potential with respect to
with wells ...at
and -1 for any
. We prove the existence of periodic solutions whose periods are large integer multiples of the period of
about the variable
by using variational methods. An estimate of the energy functional, Hamiltonian identity and Modica-type inequality for periodic solutions are also established.
$ \begin{align*} \begin{cases} (-\Delta)^s u+V(x)u = f(x,u), \; \; \; x\in \mathbb{R}^N ,\\ u\in H^{s}(\mathbb {R}^N) , \\ \end{cases} \end{align*} $ where both $ V(x) $ and $ f(x, u) $ are periodic ...in $ x $, $ 0 $ belongs to a spectral gap of the operator $ (-\Delta)^s+V $ and $ f(x, u) $ is subcritical in $ u $. We obtain the existence of nontrivial solutions by using a generalized linking theorem, and based on this existence we further establish infinitely many geometrically distinct solutions. We weaken the super-quadratic condition of $ f $, which is usually assumed even in the standard Laplacian case so as to obtain the existence of solutions.
Macrophage therapy for liver fibrosis is on the cusp of meaningful clinical utility. Due to the heterogeneities of macrophages, it is urgent to develop safer macrophages with a more stable and ...defined phenotype for the treatment of liver fibrosis. Herein, a new macrophage‐based immunotherapy using macrophages stably expressing a pivotal cytokine from Toxoplasma gondii, a parasite that infects ≈ 2 billion people is developed. It is found that Toxoplasma gondii macrophage migration inhibitory factor‐transgenic macrophage (Mφtgmif) shows stable fibrinolysis and strong chemotactic capacity. Mφtgmif effectively ameliorates liver fibrosis and deactivates aHSCs by recruiting Ly6Chi macrophages via paracrine CCL2 and polarizing them into the restorative Ly6Clo macrophage through the secretion of CX3CL1. Remarkably, Mφtgmif exhibits even higher chemotactic potential, lower grade of inflammation, and better therapeutic effects than LPS/IFN‐γ‐treated macrophages, making macrophage‐based immune therapy more efficient and safer. Mechanistically, TgMIF promotes CCL2 expression by activating the ERK/HMGB1/NF‐κB pathway, and this event is associated with recruiting endogenous macrophages into the fibrosis liver. The findings do not merely identify viable immunotherapy for liver fibrosis but also suggest a therapeutic strategy based on the evolutionarily designed immunomodulator to treat human diseases by modifying the immune microenvironment.
TgMIF‐reprogrammed macrophages with stable pro‐resolution and strong chemotactic capacity effectively reverse liver fibrosis by modulating the immune microenvironment and promoting fibrinolysis. These findings not only identify viable immunotherapy for liver fibrosis but also suggest a therapeutic strategy based on the evolutionarily designed immunomodulator to treat human diseases.
We obtain some existence theorems for periodic solutions to several linear and nonlinear equations involving fractional Laplacian. We also prove that the lower bound of all periods for semilinear ...elliptic equations involving fractional Laplacian is not larger than some exact positive constant. Hamiltonian identity, Modica-type inequalities and an estimate of the energy for periodic solutions are also established.
We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian (−∂xx)su(x)+∇F(u(x))=0in R,where u(x)=(u(x),v(x)). The function F:R2→R is a smooth ...double-well potential. We prove the existence of periodic solutions with large period T by using variational methods. Moreover, we draw a conclusion that the second component of periodic solution is identical to zero if the origin is a saddle point of F, whereas the second component is not identical to zero if the origin is a local maximum point of F. A Hamiltonian identity for periodic solutions is also established.
We consider the problem
ε
2
s
(
-
∂
x
x
)
s
u
~
(
x
~
)
-
V
(
x
~
)
u
~
(
x
~
)
(
1
-
u
~
2
(
x
~
)
)
=
0
in
R
,
where
(
-
∂
x
x
)
s
denotes the usual fractional Laplace operator,
ε
>
0
is a small ...parameter and the smooth bounded function
V
satisfies
inf
x
~
∈
R
V
(
x
~
)
>
0
. For
s
∈
(
1
2
,
1
)
, we prove the existence of separate multi-layered solutions for any small
ε
, where the layers are located near any non-degenerate local maximal points and non-degenerate local minimal points of function
V
. We also prove the existence of clustering-layered solutions, and these clustering layers appear within a very small neighborhood of a local maximum point of
V
.
Melanomas most commonly localized in the skin can arise anywhere in the body, and approximately 5% of all melanomas appear in other sites of mucosal surfaces out of skin. Primary melanoma from nasal ...mucosa is quite rare. We present this case: a 46-year-old man had complained a pain in the left upper abdomen for 2 months when he was admitted to the Northern Jiangsu People’s Hospital. The pain was paroxysmal and enhanced when eating. There was no nausea, vomiting, or anorexia. There had been no change in weight in previous months. This patient had a past history of surgery for nasal mucosal malignant melanoma 2 years ago. Abdominal enhanced computed tomography (CT) indicated that a mass originated from small bowel and occupied the left upper abdomen. The patient underwent a laparotomy during which a black lesion measuring about 5 cm × 5 cm × 4 cm was found at the jejunum and resected totally together with partial jejunum. The patient was eventually diagnosed as secondary jejunal malignant melanoma from nasal mucosal melanoma. For patients with a history of melanoma, gastrointestinal metastasis should be considered when patients develop gastrointestinal symptoms. In addition, we recommend positive anti-tumor therapy after surgery.
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