Let
N
≥
3
,
R
>
ρ
>
0
and
A
ρ
:
=
{
x
∈
R
N
;
ρ
<
|
x
|
<
R
}
. Let
U
n
,
ρ
±
,
n
≥
1
, be a radial solution with
n
nodal domains of
Δ
U
+
|
x
|
α
|
U
|
p
-
1
U
=
0
in
A
ρ
,
U
=
0
on
∂
A
ρ
.
We show ...that if
p
=
N
+
2
+
2
α
N
-
2
,
α
>
-
2
and
N
≥
3
, then
U
n
,
ρ
±
is nondegenerate for small
ρ
>
0
and the Morse index
m
(
U
n
,
ρ
±
)
satisfies
m
(
U
n
,
ρ
±
)
=
n
(
N
+
2
ℓ
-
1
)
(
N
+
ℓ
-
1
)
!
(
N
-
1
)
!
ℓ
!
for small
ρ
>
0
,
where
ℓ
=
α
2
+
1
. Using Jacobi elliptic functions, we show that if
(
p
,
α
)
=
(
3
,
N
-
4
)
and
N
≥
3
, then the Morse index of a positive and negative solutions
m
(
U
1
,
ρ
±
)
is completely determined by the ratio
ρ
/
R
∈
(
0
,
1
)
. Upper and lower bounds for
m
(
U
n
,
ρ
±
)
,
n
≥
1
, are also obtained when
(
p
,
α
)
=
(
3
,
N
-
4
)
and
N
≥
3
.
The paper focuses on a class of fractional elliptic system with critical Sobolev exponents, where there is no compact embedding under proper assumptions on potential functions. The proof of the ...existence results mainly relies on concentration-compactness principle of fractional Sobolev space and genus theory.
This paper is concerned with existence and bifurcation of nontrivial solutions for the following critical nonlocal problem −LKu=λf(x,u)+|u|2∗−2uinΩ,u=0inRn∖Ω,where α∈(0,1), Ω is an open bounded ...subset of Rn(n>2α) with continuous boundary, λ is a positive real parameter, 2∗=2nn−2α is the fractional critical Sobolev exponent and f satisfies suitable growth condition, LK is the integrodifferential operator defined as LKu(x)=∫Rn(u(x+y)+u(x−y)−2u(x))K(y)dy,x∈Rn.The single solution results extend the main results of Servadei et al. (2015) 3,18, and include Theorem 1.1 of Servadei (2013), the bifurcation result extends those got by Sang (1994) for classical elliptic equations, to the case of nonlocal fractional operators, and also generalizes the result got by Fiscella et al. (2016).
In this paper, we study the following two-component elliptic system: Δu−(λa(x)+a0)u+u3+βv2u=0inR4,Δv−(λb(x)+b0)v+v3+βu2v=0inR4,(u,v)∈H1(R4)×H1(R4),where a0,b0∈R are constants, λ>0 and β∈R are ...parameters, a(x),b(x)≥0 are potentials. By using variational methods, we prove the existence and nonexistence of general ground state solutions (maybe semi-trivial) and ground state solutions of the above system under some further conditions on the potentials a(x),b(x) and the parameters λ,β. It is worth pointing out that the cubic nonlinearities and the coupled terms of the above system are all of critical growth owing to the Sobolev embedding theorem. Furthermore, by introducing some ideas which are different from that in the literature, the phenomenon of phase separation of ground state solutions of the above system is also obtained without any symmetry conditions.
In this paper, we study the following Kirchhoff-type system:
<disp-formula> 0.1 <tex-math id="E0.1"> \begin{document}$ \begin{equation} \left\{ \begin{array}{ll} ...-(a_{1}+b_{1}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx)\Delta u = \frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta}+\varepsilon f(x), \\ -(a_{2}+b_{2}\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx)\Delta v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v+\varepsilon g(x), \\ (u, v)\in D^{1, 2}(\mathbb{R}^{3})\times D^{1, 2}(\mathbb{R}^{3}), \end{array} \right. \end{equation} $\end{document} </tex-math></disp-formula>
where $ a_{1}, a_{2}\geq0, \; b_{1}, b_{2} > 0, \; \alpha, \beta > 1, \; \alpha+\beta = 6 $ and $ f(x), g(x)\geq0, \; f(x), g(x)\in L^{\frac{6}{5}}(\mathbb{R}^3). $ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.
We prove some existence and nonexistence results for a class of critical (
p
,
q
)-Laplacian problems in a bounded domain. Our results extend and complement those in the literature for model cases.
We consider the following fractional Schrödinger equation involving critical exponent:
(
-
Δ
)
s
u
+
V
(
y
)
u
=
Q
(
y
)
u
2
s
∗
-
1
,
u
>
0
,
in
R
N
,
u
∈
D
s
(
R
N
)
,
where
2
s
∗
=
2
N
N
-
2
s
,
(
...y
′
,
y
′
′
)
∈
R
2
×
R
N
-
2
and
V
(
y
)
=
V
(
|
y
′
|
,
y
′
′
)
and
Q
(
y
)
=
Q
(
|
y
′
|
,
y
′
′
)
are bounded nonnegative functions in
R
+
×
R
N
-
2
. By using finite-dimensional reduction method and local Pohozaev-type identities, we show that if
2
+
N
-
N
2
+
4
4
<
s
<
min
{
N
4
,
1
}
and
Q
(
r
,
y
′
′
)
has a stable critical point
(
r
0
,
y
0
′
′
)
with
r
0
>
0
,
Q
(
r
0
,
y
0
′
′
)
>
0
and
V
(
r
0
,
y
0
′
′
)
>
0
, then the above problem has infinitely many solutions, whose energy can be arbitrarily large.
We extend the global compactness result by Struwe (1984) to any fractional Sobolev spaces Ḣs(Ω), for 0<s<N/2 and Ω⊂RN a bounded domain with smooth boundary. The proof is a simple direct ...consequence of the so-called profile decomposition of Gérard (1998).
On finding solutions of a Kirchhoff type problem Huang, Yisheng; Liu, Zeng; Wu, Yuanze
Proceedings of the American Mathematical Society,
July 1, 2016, 20160701, 2016-7-00, Letnik:
144, Številka:
7
Journal Article
Recenzirano
Odprti dostop
Consider the Kirchhoff type problem \displaystyle (\mathcal {P})\qquad \quad \left \{\aligned-\bigg (a+b\int _{\mat... ...ad \text {on }\partial \mathbb {B}_R, \endaligned \right .\qquad \qquad ...\qquad where \mathbb{B}_R\subset \mathbb{R}^N(N\geq 3) is a ball, 2\leq q<p\leq 2^*:=\frac {2N}{N-2} and a, b, \lambda , \mu are positive parameters. By introducing some new ideas and using the well-known results of the problem (\mathcal {P}) in the cases of a=\mu =1 and b=0, we obtain some special kinds of solutions to (\mathcal {P}) for all N\geq 3 with precise expressions on the parameters a, b, \lambda , \mu , which reveals some new phenomenons of the solutions to the problem (\mathcal {P}). It is also worth pointing out that it seems to be the first time that the solutions of (\mathcal {P}) can be expressed precisely on the parameters a, b, \lambda , \mu , and our results in dimension four also give a partial answer to Naimen's open problems J. Differential Equations 257 (2014), 1168-1193. Furthermore, our results in dimension four seem to be almost ``optimal''.
PDF
