Abstract:In this article, we study the following p-fractional Laplacian equationwhere Ω is a bounded domain in R n with smooth boundary, n > pα, p ≥ 2, α ∈ (0, 1), λ > 0 and b : Ω ⊂ R n → R is a sign-changing continuous function. We show the existence and multiplicity of non-negative solutions of (P λ ) with respect to the parameter λ, which changes according to whether 1 < β < p or p < β < p * = np n−pα respectively. We discuss both the cases separately. Non-existence results are also obtained.
“…As for the fractional case s ∈ (0, 1), we refer to [30] and [27], where variational techniques were used. The use of variational techniques allows for somewhat relaxed hypotheses, however they only give existence of positive solutions of (1.4) for positive values of λ.…”
In this work we obtain a Liouville theorem for positive, bounded solutions of the equationwhere (−∆) s stands for the fractional Laplacian with s ∈ (0, 1), and the functions h and f are nondecreasing. The main feature is that the function h changes sign in R, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
“…As for the fractional case s ∈ (0, 1), we refer to [30] and [27], where variational techniques were used. The use of variational techniques allows for somewhat relaxed hypotheses, however they only give existence of positive solutions of (1.4) for positive values of λ.…”
In this work we obtain a Liouville theorem for positive, bounded solutions of the equationwhere (−∆) s stands for the fractional Laplacian with s ∈ (0, 1), and the functions h and f are nondecreasing. The main feature is that the function h changes sign in R, therefore the problem is sometimes termed as indefinite. As an application we obtain a priori bounds for positive solutions of some boundary value problems, which give existence of such solutions by means of bifurcation methods.
“…The Brezis Nirenberg type problem involving p-fractional Laplacian has been studied in [29] whereas existence has been investigated via Morse theory in [30]. Problems involving p-fractional Laplacian has been studied in [24,25] using Nehari manifold. A vast amount of literature can be found for the case p = 2, i.e., fractional Laplacian (−∆) s , which are contributed in recent years.…”
We study the following nonlinear system with perturbations involving p-fractional Laplacian R), i = 1, 2 and f 1 , f 2 : R n → R are perturbations. We show existence of atleast two nontrivial solutions for (P ) using Nehari manifold and minimax methods.
“…Here, we first recall the variational framework for problem (1.1), in which most of results can be referred to [42,45]. It is worth mentioning that the functional setting was first introduced by Autuori & Pucci in [46] as p = 2 in R N and Servadei & Valdinoci in [47][48][49] …”
In this paper, we are interested in the multiplicity of solutions for a non-homogeneous
p
-Kirchhoff-type problem driven by a non-local integro-differential operator. As a particular case, we deal with the following elliptic problem of Kirchhoff type with convex–concave nonlinearities:
a
+
b
∬
R
2
N
|
u
(
x
)
−
u
(
y
)
|
p
|
x
−
y
|
N
+
s
p
d
x
d
y
θ
−
1
(
−
Δ
)
p
s
u
=
λ
ω
1
(
x
)
|
u
|
q
−
2
u
+
ω
2
(
x
)
|
u
|
r
−
2
u
+
h
(
x
)
in
R
N
,
where
(
−
Δ
)
p
s
is the fractional
p
-Laplace operator,
a
+
b
>0 with
a
,
b
∈
R
0
+
, λ>0 is a real parameter,
0
<
s
<
1
<
p
<
∞
with
sp
<
N
, 1<
q
<
p
≤
θp
<
r
<
Np
/(
N
−
sp
),
ω
1
,
ω
2
,
h
are functions which may change sign in
R
N
. Under some suitable conditions, we obtain the existence of two non-trivial entire solutions by applying the mountain pass theorem and Ekeland's variational principle. A distinguished feature of this paper is that
a
may be zero, which means that the above-mentioned problem is degenerate. To the best of our knowledge, our results are new even in the Laplacian case.
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