| dc.description.abstract | In this doctoral thesis we consider a problem involving the fractional p-Laplacian
operator
(−∆p)
su −µ
|u|
p−2u
|x|
p s
=
|u|
p
∗
s
(β)−2u
|x|
β
+
|u|
p
∗
s
(α)−2u
|x|
α
(x ∈ R
N
) (0.7)
where 0 < s < 1, 1 < p < +∞, N > sp, 0 < α < sp, 0 < β < sp, β 6= α, µ is a real parameter,
and p
∗
s
(α) = (p(N −α)/(N − p s) is the critical Hardy-Sobolev exponent; in particular, if α = 0
then p
∗
s
(0) = p
∗
s = N p/(N − sp) is the critical Sobolev exponent. The fractional p-Laplacian
operator is a nonlinear and nonlocal operator defined for differentiable functions by
(−∆p)
su(x) := 2 lim
²→0
+
Z
RN \B²(x)
|u(x)−u(y)|
p−2
(u(x)−u(y))
|x − y|
N+sp
d y (x ∈ R
N
). (0.8)
We prove that for the parameters in the above specified intervals and with
0 6 µ < µH := inf
u∈D
s,p
(R
N )
u6=0
[u]
p
s,p
Z
RN
u
p
|x|
p s
d x
,
there exists a weak solution u ∈ D
s,p
(R
N ) to problem (0.7). The function space where we look
for solution is the fractional homogeneous Sobolev space
D
s,p
(R
N
) :=
n
u ∈ L
p
∗
s (R
N
): [u]s,p < ∞o
,
where [u]s,p denotes the Gagliardo seminorm,
u ∈C
∞
0
(R
N
) 7−→ [u]s,p :=
µÏ
R2N
|u(x)−u(y)|
p
|x − y|
N+sp
d x d y¶ 1
p
.
A fundamental step to prove the existence result to problem (0.7) is the proof of the
independent result relative to the best Hardy constant, given by
1
K(µ,α)
:= inf
u∈D
s,p
(R
N )
u6=0
[u]
p
s,p −µ
Z
RN
|u|
p
|x|
p s
d x
ÃZ
RN
|u|
p
∗
s
(α)
|x|
α
d x! p
p
∗
s
(α)
, (0.9)
which is achieved by a nontrivial function u ∈ D
s,p
(R
N ), under the condition µ ∈ (0,µH ).
In the case p = 2 the fractional 2-Laplacian operator defined in (0.8) is denoted by
(−∆)
su(x) := (−∆2)
su(x). In this case, we consider the function space H
s
(R
N ), defined as
the closure of the space C
∞
0
(R
N ) with respect to the norm
kukHs
(RN )
:=
µZ
RN
|(−∆)
s/2u|
2
d x¶ 1
2
=
µÏ
R2N
|u(x)−u(y)|
2
|x − y|
N+2s
d x d y¶ 1
2
.
We also show that if u ∈ H
s
(R
N ) is a weak solution to problem
(−∆)
su −µ
u
|x|
2s
= |u|
q−2u +
|u|
2
∗
s
(α)−2u
|x|
α
(x ∈ R
N
), (0.10)
where 0 < s < 1, 0 < α < 2s < N, 2∗
s
(α) = 2(N − α)/(N − 2s), µ is a real parameter and q 6= 2
∗
s
,
then u ≡ 0. Therefore, problem (0.10) does not have nontrivial solution when q 6= 2
∗
s
.
The proof of this non-existence result is an immediate consequence of a Pohozaev-type
identity for problem (0.10) that we state in the following way: Suppose that u ∈ H
s
(R
N ) is a
weak solution to problem (0.10). Then the harmonic extension of u on the half-space R
N+1
+ ,
denoted by w = E(u), verifies the identity
(N −2s)
2
Ï
R
N+1
+
y
1−2s
|∇w|
2
d x d y =
1
ks
Z
RN
Ã
NF(x,u)+
X
N
i=1
xi
Z u
0
fxi
(x,t)d t!
d x, (0.11)
where ks =
Γ(s)
2
1−2sΓ(1− s)
, u = w(·, 0), f (x,u) = µ
u
|x|
2s
+ |u|
q−2u +
|u|
2
∗
s
(α)−2u
|x|
α
and F(x,s) =
Z s
0
f (x,t)d t.
Finally, still in the case p = 2 of the fractional Laplacian operator, let
0 6 µ < µ¯ := 2
2s
Γ
2
¡ N+2s
4
¢
Γ
2
¡ N−2s
4
¢,
where µ¯ is the best constant of the continuous embedding H
s
(R
N ) ,→ L
2
(R
N ,|x|
−2s
). We
prove that every positive solution u ∈ H
s
(R
N ) to problem
(−∆)
su −µ
u
|x|
2s
= |u|
2
∗
s −2u +
|u|
2
∗
s
(β)−2u
|x|
β
(x ∈ R
N
) (0.12)
is radially symmetric and decreasing with respect to some point x0 ∈ R
N , that is, for every
positive solution to problem (0.12) there exists an strictly decreasing function v : (0,+∞) →
(0,+∞) such that
u(x) = v(r ), r = |x − x0|. | |
