In this paper, we consider the existence of nontrivial solutions for a fractional
p
-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and ...multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.
In this paper, our main purpose is to establish the existence of multiple solutions of a class of
p
–
q
-Laplacian equation involving concave–convex nonlinearities:
{
−
Δ
p
u
−
Δ
q
u
=
θ
V
(
x
)
|
u
...|
r
−
2
u
+
|
u
|
p
⁎
−
2
u
+
λ
f
(
x
,
u
)
,
x
∈
Ω
,
u
=
0
,
x
∈
∂
Ω
where
Ω is a bounded domain in
R
N
,
λ
,
θ
>
0
,
1
<
r
<
q
<
p
<
N
and
p
⁎
=
N
p
N
−
p
is the critical Sobolev exponent,
Δ
s
u
=
div
(
|
∇
u
|
s
−
2
∇
u
)
is the
s-Laplacian of
u. We prove that for any
λ
∈
(
0
,
λ
⁎
)
,
λ
⁎
>
0
is a constant, there is a
θ
⁎
>
0
, such that for every
θ
∈
(
0
,
θ
⁎
)
, the above problem possesses infinitely many weak solutions. We also obtain some results for the case
1
<
q
<
p
<
r
<
p
⁎
. The existence results of solutions are obtained by variational methods.
In this paper, we consider the existence of nontrivial solutions for a fractional
p
-
q
Laplacian equation with critical nonlinearity in a bounded domain. Our approach is based on variational methods ...and some analytical techniques.
This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity
u
log
|
u
|
−
⨏
Ω
u
log
|
u
|
d
x
in a bounded domain, subject to homogeneous Neumann boundary value ...condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.
In this paper, we deal with the following quasilinear attraction–repulsion model:
{
u
t
=
∇
⋅
(
D
(
u
)
∇
u
)
−
∇
⋅
(
S
(
u
)
χ
(
v
)
∇
v
)
+
∇
⋅
(
F
(
u
)
ξ
(
w
)
∇
w
)
+
f
(
u
)
,
x
∈
Ω
,
t
>
0
,
v
...t
=
Δ
v
+
β
u
−
α
v
,
x
∈
Ω
,
t
>
0
,
0
=
Δ
w
+
γ
u
−
δ
w
,
x
∈
Ω
,
t
>
0
,
u
(
x
,
0
)
=
u
0
(
x
)
,
v
(
x
,
0
)
=
v
0
(
x
)
,
x
∈
Ω
with homogeneous Neumann boundary conditions in a smooth bounded domain
Ω
⊂
R
n
(
n
≥
2
). Let the chemotactic sensitivity
χ
(
v
)
be a positive constant, and let the chemotactic sensitivity
ξ
(
w
)
be a nonlinear function. Under some assumptions, we prove that the system has a unique globally bounded classical solution.
This article concerns quenching properties of solutions for a semilinear parabolic system with multi-singular reaction terms. We obtain sufficient conditions for the existence of finite time ...quenching of global solutions. The blow up of time-derivatives at the quenching point is verified. In addition, we identify simultaneous and non-simultaneous quenching, and provide a classification of parameters for the simultaneous quenching rates.
In this paper, our main purpose is to consider the quasilinear equation
div
(
|
∇
u
|
p
-
2
∇
u
)
=
m
(
x
)
f
(
u
)
on a domain
Ω
⊆
R
N
,
N
⩾
3, where
f is a nonnegative, nondecreasing continuous ...function which vanishes at the origin, and
m is a nonnegative continuous function with the property that any zero of
m is contained in a bounded domain in
Ω such that
m is positive on its boundary. For
Ω bounded, we show that a nonnegative solution
u satisfying
u(
x)
→
∞ as
x
→
∂
Ω exists. For
Ω un-boundary (including
Ω
=
R
N
), we show that a similar result holds where
u(
x)
→
∞ as ∣
x∣
→
∞ within
Ω and
u(
x)
→
∞ as
x
→
∂
Ω.
In this article, we investigate how the coefficient $f(z)$ affects the number of positive solutions of the quasilinear elliptic system $$displaylines{ -Delta_{p}u =lambda ...g(z)|u|^{q-2}u+frac{alpha}{alpha+eta} f(z)|u|^{alpha-2}u|v|^{eta} quadhbox{in }Omega,cr -Delta_{p}v =mu h(z)|v|^{q-2}v +frac{eta}{alpha+eta}f(z)|u|^{alpha}|v|^{eta-2}v quadhbox{in }Omega,cr u=v=0quadhbox{on }partialOmega, }$$ where $0inOmegasubset mathbb{R}^{N}$ is a bounded domain, $alpha >1$, $eta>1$ and $1<p<q<alpha+eta=p^{*}=frac{Np}{N-p}$ for $N>2p$.
In this article, we explore the existence of multiple solutions for a p-Kirchhoff equation with the nonlinearity containing both singular and critical terms. By means of the concentration compactness ...principle and Ekeland's variational principle, we obtain two positive weak solutions.
This article deals with the parabolic-parabolic chemotaxis system{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇φ(v))+f(u),x∈Ω,t>0,vt=△v−v+u,x∈Ω,t>0 in a bounded domain Ω⊂Rn(n≥1) with smooth boundary conditions, ...D,S∈C2(0,+∞)) nonnegative, with D(u)=a0(u+1)−α for a0>0 and α<0, 0≤S(u)≤b0(u+1)β for b0>0,β∈R, and where the singular sensitivity satisfies 0<φ′(v)≤χvk for χ>0,k≥1. In addition, f:R→R is a smooth function satisfying f(s)≡0 or generalizing the logistic source f(s)=rs−μsm for all s≥0 with r∈R,μ>0, and m>1. It is shown that for the case without a growth source, if 2β−α<2, the corresponding system possesses a globally bounded classical solution. For the case with a logistic source, if 2β+α<2 and n=1 or n≥2 with m>2β+1, the corresponding system has a globally classical solution.
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