Complex Anal. Oper. Theory https://doi.org/10.1007/s11785-018-0809-2
Complex Analysis and Operator Theory
A singular System Involving the Fractional p-Laplacian Operator via the Nehari Manifold Approach Kamel Saoudi1
Received: 24 December 2017 / Accepted: 28 May 2018 © Springer International Publishing AG, part of Springer Nature 2018
Abstract In this work we study the fractional p-Laplacian equation with singular nonlinearity ⎧ ⎪ (−)sp u = λa(x)|u|q−2 u + ⎪ ⎪ ⎪ ⎪ ⎨ (−)sp v = μb(x)|v|q−2 v + ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0, in R N \,
1−α −α 1−β , 2−α−β c(x)|u| |v|
in ,
1−β 1−α |v|−β , 2−α−β c(x)|u|
in ,
where 0 < α < 1, 0 < β < 1, 2−α −β < p < q < ps∗ , ps∗ = N −N ps is the fractional Sobolev exponent, λ, μ are two parameters, a, b, c ∈ C() are non-negative weight functions with compact support in , and (−)s p is the fractional p-Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ and μ. Keywords Fractional p-Laplace operator · Nehari manifold · Singular elliptic system · Multiple positive solutions Mathematics Subject Classification 34B15 · 37C25 · 35R20
Communicated by Daniel Aron Alpay.
B 1
Kamel Saoudi
[email protected] College of sciences at Dammam, Imam Abdulrahman Bin Faisal University, Dammam 31441, Kingdom of Saudi Arabia
K. Saoudi
1 Introduction Let be a bounded domain in Rn with smooth boundary ∂, N > ps, s ∈ (0, 1),and ps∗ = NN−pps . The purpose of this work is to study the existence of multiple solutions of the singular elliptic problem involving the fractional Lapalcien: ⎧ ⎪ (−)sp u = λa(x)|u|q−2 u + ⎪ ⎪ ⎪ ⎪ ⎨ (−)sp v = μb(x)|v|q−2 v + ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0, in R N \,
1−α −α 1−β , 2−α−β c(x)|u| |v|
in ,
1−β 1−α |v|−β , 2−α−β c(x)|u|
in ,
(1.1)
where 0 < α < 1, 0 < β < 1, 2−α −β < p < q < ps∗ , ps∗ = N −N ps is the fractional Sobolev exponent, λ, μ are two parameters, a, b, c ∈ C() are non-negative weight functions with compact support in , and (−)sp is the fractional p-Laplace operator defined as (−)sp u(x) = 2lim
0 R N \B
|u(x) − u(y)| p−2 (u(x) − u(y)) dy, |x − y| N +sp
x ∈ RN
The problems of this type are important in many fields of sciences, notably the fields of physics, probability, finance, electromagnetism, astronomy, and fluid dynamics, it also they can be used to accurately describe the jump Lévy processes in probability theory and fluid potentials for more details see [1–3] and references therein. Before giving our main results, let us briefly recall the literature concerning related fractional problems. Recently, a great deal of attention has been focused on studying of problems involving fractional Laplacian. There are many works on existence of a solution for fractional elliptic equations with regular non-linearities like
(−)s u = λu p + u q in u|∂ = 0, u > 0 in .
(1.2)
where N > 2s, 0 < s < 1, p, q > 0, λ > 0. In [4,5] the authors studied the existence and the multiplicity of non-negative solutions to the sub critical growth problems (1.2). The critical exponent problems (1.2) are studied in [6–12]. Some other results dealing with the existence of solutions concerning Dirichlet problem involving the spectral fractional laplacian has been treated in [13,14] and references therein. Note that, these two fractional operators (i.e. the ’integral’ one and the ’spectral’ one) are different. We refer the interested reader to [15] for a careful comparison of theses two operators. In the Case of the problem involving the fractional p-Laplace existence results via Morse theory has been treated in Iannizzotto et al. [16]. The critical case is treated in Perera et al. [17] with additional new abstract based on a pseudo-index related to the Z2 −cohomological index. These restrictions are used to prove the existence of a range of the validity of the Palais–Smale condition. Note that, in this work, the bifurcation and multiplicity results is obtained for some restrictions on the parameter
A singular System Involving the Fractional...
λ. Moreover, by Nehari manifold and fibering maps the multiplicity of solutions has been investigated in [18,19]. In particular, in [18], the authors considered the problem ⎧ 2α ⎪ (−)sp u = λ|u|q−2 u + α+β |u|α−2 u|v|β , in , ⎪ ⎪ ⎪ ⎪ ⎨ 2β (−)sp v = μ|v|q−2 v + α+β |u|α |v|β−2 v, in , ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0, in R N \, where is a bounded domain in Rn with smooth boundary ∂, N > sp, s ∈ (0, 1), p is the fractional sobolev exponent, λ, μ are two p < α + β < p ∗ , ps∗ = NN−sp parameters. They studied the associated Nehari manifold using the fibering maps and showed the existence of non-negative solutions when the pair of parameters (λ, μ) satisfies some condition. On the other hand, for the fractional elliptic problems when p = 2, Saoudi et al. [20] studied the existence of solutions of the following system: ⎧ ⎪ (−)s u = λa(x)|u|q−2 u + ⎪ ⎪ ⎪ ⎪ ⎨ (−)s v = μb(x)|v|q−2 v + ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u = v = 0, in R N \,
1−α −α 1−β , 2−α−β c(x)|u| |v|
in ,
1−β 1−α |v|−β , 2−α−β c(x)|u|
in ,
(1.3)
where is a bounded domain in Rn with smooth boundary ∂, N > 2s, s ∈ (0, 1), 0 < α < 1, 0 < β < 1, 1 < q < 2 < 2s∗ , 2s∗ = N2N −2s is the fractional Sobolev exponent, λ, μ are two parameters, a, b, c ∈ C() are non-negative weight functions, and (−)s is the fractional Laplace operator. With the help of the Nehari manifold and the fibering maps (appropriately modified), the authors proved the existence of at least two non-negatives solutions of (1.3). In the local setting (s = 1), problems (1.1) has quite an extensive literature. We refer the reader to the monographs of Ghergu–Radulescu [21] for a more general presentation of these results and the survey article of Crandall et al. [22]. After this, many authors have been considered the problem for Laplacian, p-Laplacian, N -Laplacian operators, using the technique used in [22] or a combination of this approach with the Nehari’s and Perron’s methods, among others, we would like to mention [23–29]. Motivated by above results, the aim of the present paper is to study system (1.1) in the singular case. Using variational methods and a Nehari manifold decomposition, we prove that system (1.1) admits at least two positive solutions when the pair of parameters (λ, μ) satisfies some condition. In order to state our result, let us introduce some notations. We define W s, p (R N ) := u ∈ L p (R N ) : u measurable , |u|s, p < ∞ ,
K. Saoudi
the usual fractional Sobolev space with the Gagliardo norm
p p 1 ||u||s, p := ||u|| p + |u|s, p p . For a detailed account on the properties of W s, p (R N ) we refer the reader to [30]. Denote
Q = R2N \ (R N \) × (R N \) and we define the space def X = u : R N → R Lebesgue measurable : u\ ∈ L p () and |u(x) − u(y)| p p ∈ L (Q) |x − y| N +sp with the norm u X = u L p () +
Q
|u(x) − u(y)| p d xd y |x − y| N +sp
1/ p .
Through this paper we consider the space X 0 = u ∈ X : u = 0 a.e. in Rn \ , with the norm u X 0 =
Q
|u(x) − u(y)| p d xd y |x − y| N +sp
1/ p .
It is readily seen that (X 0 , ||.||) is a uniformly convex Banach space and that the embedding X 0 → L q () is continuous for all 1 ≤ q ≤ ps∗ , and compact for all 1 ≤ q < ps∗ . The dual space of (X 0 , ||.||) is denoted by (X ∗ , ||.||∗ ), and ., . denotes the usual duality between X 0 and X ∗ . Let E = X 0 × X 0 be the Cartesian product of two Hilbert spaces, which is (In [3] it is claimed that X 0 is a Hilbert space) a reflexive Banach space endowed with the norm 1/ p |u(x) − u(y)| p |v(x) − v(y)| p (u, v) = d xd y + d xd y . (1.4) N + ps N + ps Q |x − y| Q |x − y| Definition 1.1 We say that (u, v) ∈ E is a weak solution of problem (1.1) if u, v > 0 in , one has Q
|u(x) − u(y)| p−2 (u(x) − u(y))(φ(x) − φ(y)) dx dy |x − y| N +sp
A singular System Involving the Fractional...
|v(x) − v(y)| p−2 (v(x) − v(y))(ψ(x) − ψ(y)) + dx dy |x − y| N +sp Q
1−α = λa(x)|u|q−2 uφ + μb(x)|v|q−2 vψ d x + c(x)u −α v 1−β φ d x 2 − α − β 1−β 1−α −β + c(x)u v ψ dx 2−α−β for all (φ, ψ) ∈ E. We give below the precise statements of results that we will prove. Theorem 1.1 Let s ∈ (0, 1), N > sp and be a bounded domain in Rn . If 0 < α < 1, 0 < β < 1, 1 < p < q < ps∗ , then, there exists an explicit number p
p
0 ( p, q, ps∗ , α, β, ||, ||c||∞ , S) such that for 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 ( p, q, ps∗ , α, β, ||, ||c||∞ , S), problem (1.1) has at least two nontrivial positive solutions. This paper is organized as follows: The Sect. 2 is devoted to proof some lemmas in preparation for the proof of our main result. While, existence of two solutions (Theorem 1.1) will be presented in Sects. 3 and 4.
2 Nehari Manifold and Fibering Map Analysis In this section, we collect some basic results on a Nehari manifold and give the analysis of the fibering maps. Associated to the problem (1.1) we define the functional E λ,μ : E → R given by |u(x) − u(y)| p |v(x) − v(y)| p 1 1 d x d y + dx dy E λ,μ (u, v) = N +sp p Q |x − y| p Q |x − y| N +sp
1 1 − λa(x)|u|q + μb(x)|v|q d x − c(x)(u + )1−α (v + )1−β d x. q 2−α−β As usual, r + = max{r, 0} and r − = max{−r, 0} for r ∈ R. Notice that E λ,μ is not a C 1 functional in E, and hence classical variational methods are not applicable. Also, notice that (u, v) is a weak solution of problem (1.1), then u, v > 0 in and satisfies the equation ||(u, v)|| p dx − λ
a(x)|u|q dx − μ
b(x)|v|q(x) dx −
c(x)u 1−α v 1−β dx = 0. (2.1)
One can easily verify that the energy functional E λ,μ (u, v) is not bounded below on the space E. But we will show that E λ,μ (u, v) is bounded below on this Nehari manifold and we will extract solutions by minimizing the functional on suitable subsets.
K. Saoudi
The Nehari manifold is defined as Nλ,μ = {(u, v) ∈ E\ ||(u, v)|| − λ − c(x)(u + )1−α (v + )1−β dx = 0}.
p
a(x)|u| dx − μ q
b(x)|v|q(x) dx
We note that Nλ,μ contains every solution of problem (1.1). Now as we know that the Nehari manifold is closely related to the behaviour of the functions u,v : t → E λ,μ (tu, tv) for t > 0 defined by
tp tq ||(u, v)|| p − λa(x)|u|q + μb(x)|v|q dx p q 2−α−β t c(x)|u|1−α |v|1−β dx, − 2−α−β
u,v (t) =
which gives
u,v (t)
λa(x)|u|q + μb(x)|v|q dx ||(u, v)|| − t 1−α−β 1−α −t c(x)|u| |v|1−β dx,
=t
p−1
p
q−1
(2.2)
and
λa(x)|u|q + μb(x)|v|q dx
u,v (t) = ( p − 1)t p−2 ||(u, v)|| p − (q − 1)t q−2 −α−β 1−α 1−β − (1 − α − β)t c(x)|u| |v| dx.
Such maps are called fiber maps and were introduced by Drabek and Pohozaev in [31]. We define the best constant of the embedding S as p
S = inf{||u|| p : u ∈ X 0 , |u| p∗ = 1}.
(2.3)
s
From (2.3), we have p
||(u, v)|| p∗ ≤ S
− 1p
s
(u, v) p .
(2.4)
By Hölder’s inequality and Sobolev inequalities, one has
ps∗ −q
q q ∗ λa(x)|u|q + μb(x)|v|q d x ≤ || ps λ||a||∞ ||u|| p∗ + μ||b||∞ ||v|| p∗ s
≤ || ≤ ||
ps∗ −q ps∗
S
− qp
ps∗ −q ps∗
S
− qp
s
λ||a||∞ ||u||q + μ||b||∞ ||v||q p−q p p p (λ||a||∞ ) p−q + μ||b||∞ ) p−q
A singular System Involving the Fractional...
||u||q + ||v||q p−q ps∗ −q p p p ∗ −q ≤ || ps S p (λ||a||∞ ) p−q + (μ||b||∞ ) p−q ||(u, v)||q (2.5) and using Young’s inequality and Sobolev inequalities, we obtain that 1−α c(x)|u|1−α |v|1−β dx ≤ ||c||∞ |u|2−α−β dx 2−α−β 1−β |v|2−α−β dx + 2−α−β ≤ ||c||∞ S
− 2−α−β p
||(u, v)||2−α−β
(2.6)
Lemma 2.1 Let (u, v) ∈ E\{(0, 0)}, then (tu, tv) ∈ Nλ,μ if and only if u,v (t) = 0. Proof Let (tu, tv) ∈ Nλ,μ . This means that
a(x)|tu| dx − μ b(x)|tv|q(x) dx 0 = ||(tu, tv)|| − λ − c(x)|tu|1−α |tv|1−β dx
p−1 =t λa(x)|u|q + μb(x)|v|q dx ||(u, v)|| p − t q−1 −t 1−α−β c(x)|u|1−α |v|1−β dx p
= u,v (t).
q
This give the proof of the Lemma 2.1.
From Lemma 2.1, we have that the elements in Nλ,μ correspond to stationary points of the maps u,v (tu, tv) and in particular, (u, v) ∈ Nλ,μ if and only if u,v (1) = 0. Hence, it is natural to split Nλ,μ into three parts corresponding to local minima, local maxima and points of inflection u,v (t) defined as follows: + = {(u, v) ∈ Nλ,μ : u,v (1) > 0} = (tu, tv) ∈ E\{0, 0} : u,v (t) Nλ,μ = 0, u,v (t) > 0 , − = {(u, v) ∈ Nλ,μ : u,v (1) < 0} = (tu, tv) ∈ E\{0, 0} : u,v (t) Nλ,μ = 0, u (t) < 0 , 0 Nλ,μ = {(u, v) ∈ Nλ,μ : u,v (1) = 0} = (tu, tv) ∈ E\{0, 0} : u,v (t) = 0, u,v (t) = 0 . Our first result is the following
K. Saoudi
Lemma 2.2 E λ,μ is coercive and bounded below on Nλ,μ . Proof Since (u, v) ∈ Nλ,μ , using the embedding of X 0 in L 2−α−β (), we get E λ,μ (u, v) =
1 1 1 1 − (u, v) p − − c(x)|u|1−α |v|1−β dx. p q 2−α−β q
Therefore by (2.6), we get E λ,μ (u, v) ≥
1 1 1 1 − 2−α−β p − (u, v) p − − ||c||∞ S ||(u, v)||2−α−β p q 2−α−β q
Since 2 − α − β < p, then E λ,μ is coercive and bounded below on Nλ,μ .
− + Lemma 2.3 Given (u, v) ∈ Nλ,μ (respectively Nλ,μ ) with u, v ≥ 0, for all (φ, ψ) ∈ E with (φ, ψ) ≥ 0, there exist ε > 0 and a continuous function t = t (k) > 0 such that for all k ∈ R with |k| < ε we have − + t (0) = 1 and t (k)(u + kφ, v + kψ) ∈ Nλ,μ (respectively Nλ,μ ).
Proof We introduce the function f : R × R −→ R defined by:
λa(x)(u + kφ)q ||(u + kφ, v + kψ)|| − t + μb(x)(v + kψ)q d x − c(x)(u + kφ)1−α (v + kψ)1−β d x.
f (t, k) = t
p+α+β−2
p
q+α+β−2
Hence, f t (t, k) = ( p + α + β − 2)t p+α+β−3 ||(u + kφ, v + kψ)|| p
q+α+β−3 λa(x)(u + kφ)q + μb(x)(v + kψ)q d x −(q + α + β − 2)t
− is continuous on R × R. Since (u, v) ∈ Nλ,μ ⊂ Nλ,μ , we have f (1, 0) = 0, and
f t (1, 0) = ( p + α + β − 2)||(u, v)|| − (q + α + β − 2) + μb(x)v q d x < 0 p
λa(x)u q
Therefore, applying the implicit function theorem to the function f at the point (1, 0) we obtain a δ > 0 and a positive continuous function t = t (k) > 0, k ∈ R |k| < δ satisfying t (0) = 1, t (k)(u + kφ, v + kψ) ∈ Nλ,μ , ∀ k ∈ R, |k| < δ.
A singular System Involving the Fractional...
Hence, taking ε > 0 possibly small enough (ε < δ), we get − t (k)(u + kφ, v + kψ) ∈ Nλ,μ , ∀ k ∈ R, |k| < ε. + The case (u, v) ∈ Nλ,μ may be obtained in the same way. This completes the proof of the Lemma 2.3.
Now, we prove the following crucial Lemma: Lemma 2.4 There exists 0 , given by 0 =
p−q ||c||∞ (q + α + β − 2) α+β−2
×S p+α+β−2
q + p−q
p 2−α−β− p
∗
ps −q q +α+β −2 ∗ || ps p+α+β −2
p − p−q
,
such that for p p 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 there exist t0 and t1 such that
u,v (t0 ) = u,v (t1 ) and
u,v (t0 ) < 0 < u,v (t1 ); + − and (t1 u, t1 v) ∈ Nλ,μ . that is, (t0 u, t0 v) ∈ Nλ,μ
Proof We introduce the function ψ : R+ −→ R defined by: ψu,v (t) = t
p−q
||(u, v)|| − t p
2−α−β−q
c(x)|u|1−α |v|1−β dx.
That is, the first derivative of the function ψ is given by ψu,v (t) = ( p − q)t p−q−1 ||(u, v)|| p − (2 − α − β − q)t 1−α−β−q × c(x)|u|1−α |v|1−β dx.
(t) = 0 it can be easily verified that ψ , attains it is maximum at Set ψu,v u,v
tmax (u, v) =
1 (q + α + β − 2) c(x)|u|1−α |v|1−β dx β+α−2− p . (q − p)||(u, v)|| p
K. Saoudi (t) > 0 for all 0 < t < t Moreover, ψu,v max (u, v) and u,v (t) < 0 for all t > tmax (u, v). Thus u,v (t) attains its maximum at tmax (u, v), that is,
⎡
q +α+β −2 ψu,v (tmax ) = ⎣ q−p ×
||(u, v)|| c(x)|u|
p−q p+β+α−2
−
q +α+β −2 q−p
p q+α+β−2 p+β+α−2
1−α |v|1−β
p+α+β −2 = q +α+β −2 ×
dx
||(u, v)||
⎤ ⎦
p−q p+β+α−2
p q+α+β−2 p+β+α−2
1−α |v|1−β dx c(x)|u|
p+β+α−2
.
q− p p+β+α−2
q +α+β −2 q−p
2−α−β−q
.
q− p p+β+α−2
Now, (u, v) ∈ Nλ,μ if and only if ψu,v (tmax ) −
λa(x)|u|q + μb(x)|v|q dx = 0
and using (2.5) we see that
λa(x)|u|q + μb(x)|v|q dx ψu,v (tmax ) −
≥ ψu,v (tmax ) − || p+α+β −2 ≥ q +α+β −2 − ||
ps∗ −q ps∗
S
− qp
ps∗ −q ps∗
S
− qp
p−q p p p (λ||a||∞ ) p−q + (μ||b||∞ ) p−q ||(u, v)||q
q +α+β −2 q−p p
p−q p+β+α−2
p
(λ||a||∞ ) p−q + (μ||b||∞ ) p−q
||(u, v)|| c(x)|u|
p−q p
p q+α+β−2 p+β+α−2
1−α |v|1−β
dx
q− p p+β+α−2
||(u, v)||q > 0
if and only if p
p
(λ||a||∞ ) p−q + (μ||b||∞ ) p−q <
p−q ||c||∞ (q + α + β − 2) ∗
ps −q q +α+β −2 ∗ || ps p+α+β −2
We can also see that ψu,v (tmax ) −
p
p 2−α−β− p
−
p p−q
α+β−2
S p+α+β−2
q + p−q
= 0 .
q q λa(x)|u| + μb(x)|v| dx = 0 if and only p
(t) = 0. So for (λ||a|| ) p−q + (μ||b|| ) p−q < , there exist exactly two if φu,v ∞ ∞ 0 (t ) > 0 and ψ (t ) < 0 that is, (t u, t v) ∈ N + and points 0 < t0 < t1 with ψu,v 0 0 0 u,v 1 λ,μ
A singular System Involving the Fractional... − (t1 u, t1 v) ∈ Nλ,μ . Thus, φu,v has a local minimum at t = t0 and a local maximum at t = t1 , that is φu,v is decreasing in (0, t0 ) and increasing in (t0 , t1 ). The proof of Lemma 2.4 is now completed.
As a consequence of Lemma 2.4, we have the following result: Lemma 2.5 There exists 0 , given by 0 =
p−q ||c||∞ (q + α + β − 2) α+β−2
S p+α+β−2
q + p−q
p 2−α−β− p
∗
ps −q q +α+β −2 ∗ || ps p+α+β −2
p − p−q
, p
p
± such that for 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 we have Nλ,μ = ∅ and 0 Nλ,μ = {0}. ± Proof Firstly, using Lemma 2.3, we conclude that Nλ,μ are non-empty for all (λ, μ) p
p
with 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 . Now, we proceed by contradiction to p p 0 = {0} for all (λ, μ) with 0 < (λ||a|| ) p−q + (μ||b|| ) p−q < . prove that Nλ,μ ∞ ∞ 0 0 . Then, from the definition of N 0 , it follows that Let (u, v) ∈ Nλ,μ λ,μ ||(u, v)|| p − λ
a(x)|u|q dx − μ
b(x)|v|q dx −
c(x)|u|1−α |v|1−β dx = 0. (2.7)
So, using (2.7) combined with (2.1), we obtain 0 = ||(u, v)|| p − λ a(x)|u|q dx − μ b(x)|v|q dx − c(x)|u|1−α |v|1−β dx p = ( p − q)||(u, v)|| − (q + α + β − 2) c(x)|u|1−α |v|1−β dx.
Using (2.6), we obtain that ||(u, v)|| p ≤
(q + α + β − 2) − 2−α−β p ||c||∞ S ||(u, v)||2−α−β p−q
which implies that ||(u, v)|| ≥
2−α−β p−q S p ||c||∞ (q + α + β − 2)
−
1 p+α+β−2
.
Similarly, since (u, v) ∈ Nλ,μ , we have ||(u, v)|| p =
q +α+β −2 p+α+β −2
λa(x)|u|q + μb(x)|v|q d x.
(2.8)
K. Saoudi
By (2.5) we obtain ||(u, v)|| p ≤
p−q ps∗ −q p p p q +α+β −2 ∗ −q || ps S p (λ||a||∞ ) p−q + (μ||b||∞ ) p−q p+α+β −2 ||(u, v)||q .
Thus, ||(u, v)|| ≤
∗
ps −q q +α+β −2 ∗ −q || ps S p p+α+β −2
1 2−q
p
p
(λ||a||∞ ) p−q + (μ||b||∞ ) p−q
1
p
.
(2.9) 0 , we get From (2.8) and (2.9) and since (u, v) ∈ Nλ,μ
(λ||a||∞ )
p p−q
+ (μ||b||∞ )
p p−q
≥
p−q ||c||∞ (q + α + β − 2) ∗
ps −q q +α+β −2 ∗ || ps α+β p
p 2−α−β−2
−
p p−q
α+β−2
S p+α+β−2
q + p−q
= 0
p
which contradicts 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 . Therefore the proof of the Lemma 2.5 is now completed.
p
p
By Lemmas 2.2 and 2.3, for 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 , we can write + − ∪ Nλ,μ and define Nλ,μ = Nλ,μ + cλ,μ =
inf
+ (u,v)∈Nλ,μ
− E λ,μ (u, v) and cλ,μ =
inf
− (u,v)∈Nλ,μ
E λ,μ (u, v)
+ 3 Existence of Minimizer on Nλ,μ + In this section, we will show that the minimum of E λ,μ is achieved in Nλ,μ . Also, we show that this minimizer is also the first solution of (1.1). p
p
+ , Lemma 3.1 If 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 , then for all (u, v) ∈ Nλ,μ + cλ,μ < 0. + + Proof Let (u + 0 , v0 ) ∈ Nλ,μ , then we have φu 0 (1) > 0 which gives from (2.1),
c(x)|u|1−α |v|1−β dx <
p−q ||(u, v)|| p . 2−α−β −q
(3.1)
A singular System Involving the Fractional...
Hence, using (2.1) with (3.1), we have
1 1 1 1 p E λ,μ (u, v) ≤ c(x)|u|1−α(x) |v|1−β(x) dx − ||(u, v)|| − − p q 2−α−β q 1 p−q 1 1 1 − ||(u, v)|| p . ≤ (3.2) − − p q 2−α−β q 2−α−β −q
Thus, by (3.2), we get E λ,μ (u, v) < −
(q − p)( p + α + β − 2) ||(u, v)|| p < 0. pq(2 − α − β)
+ + < 0 follows from the definition of cλ,μ . This completes the proof of Therefore cλ,μ the Lemma 3.1.
p
p
Theorem 3.1 If 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 , then there exists + + + + (u + 0 , v0 ) ∈ Nλ,μ satisfying E λ,μ (u 0 , v0 ) = inf (u,v)∈N + E λ,μ (u, v). λ,μ
+ Proof Since E λ,μ is bounded below on Nλ,μ and so on Nλ,μ . Then, there exists + + + {(u n , vn )} ⊂ Nλ,μ be a sequence such that + E λ,μ (u + n , vn ) →
inf
+ (u,v)∈Nλ,μ
E λ,μ (u, v) as n → ∞.
Since E λ,μ is coercive, {(u n , vn )} is bounded in E. Then there exists a subsequence, + + + still denoted by (u + n , vn ) and (u 0 , v0 ) ∈ E such that, as n → ∞, + + + u+ n u 0 , vn v0 weakly in X 0 + + + r ∗ u+ n → u 0 , vn → v0 strongly in L () for all 1 ≤ r < ps , + + + u+ n → u 0 , vn → v0 a.e. in .
By Vitali’s theorem (see [32, pp. 133]), we claim that 1−α 1−α a(x)|u + | d x = a(x)|u + d x. lim n 0| n→∞
(3.3)
1−α d x, n ∈ N } is equi-absolutelyIndeed, we only need to prove that { a(x)|u + n| continuous. Note that {u n } is bounded, by the Sobolev embedding theorem, so exists a constant C > 0 such that |u n | ps∗ ≤ C < ∞. Moreover, by Hölder inequalities we have ∗
a(x)u 1−α d x ≤ a∞
∗
s
From (3.4), for every ε > 0, setting δ=
ps
|u|1−α d x ≤ a∞ || ps +α−1 |u|1−α p∗ .
ε ||a||∞ C 1−α
ps∗ ps∗ +γ −1
,
(3.4)
K. Saoudi
when A ⊂ with mes A < δ, we have A
ps∗ +α−1 1−α
1−α ps∗ meas E a(x)|u + | d x ≤ ||a|| ||u|| ∗ ∞ n p s
≤ ||a||∞ C 1−α δ
ps∗ +α−1 ps∗
< ε.
Thus, our claim is true. Similarly, lim
n→∞
b(x)|vn+ |1−β d x =
b(x)|v0+ |1−β d x.
(3.5)
On the other hand, by [33] there exists l ∈ L r (R N ) such that + |u + n (x)| ≤ l(x), |vn (x)| ≤ l(x), as k → ∞
for any 1 ≤ r < ps∗ . Therefore by Dominated convergence Theorem we have that
+q λ|u n | + μ|vn+ |q d x →
+q λ|u 0 | + μ|v0+ |q d x.
+ + Moreover, by Lemma (2.4), there exists t0 such that (t0 u + 0 , t0 v0 ) ∈ Nλ,μ . Now, we + + + shall prove u + n → u 0 strongly in X 0 , vn → v0 strongly in X 0 . Suppose otherwise, then either + + + ||(u + 0 , v0 )|| E ≤ lim inf ||(u n , vn )|| E . n→∞
+ + Thus, since (u + n , vn ) ∈ Nλ,μ , one has
lim +
n→∞
(t ) u n ,vn+ 0
p + )|| p − t q−1 q + q t0 ||(u + , v λa(x)|u + n n n | + μb(x)|vn )| dx 0 n→∞ 2−α−β 1−α |v + |1−β dx −t0 c(x)|u + | n n
p q−1 + p + q q λa(x)|u + > t0 ||(u + 0 , v0 )|| − t0 0 | + μb(x)|v0 )| dx 2−α−β + 1−α + 1−β −t0 c(x)|u 0 | |v0 | dx = + + (t0 ) = 0 = lim
u 0 ,v0
+ + Therefore, u + ,v + (t0 ) > 0 for n large enough. Since (u + n , vn ) ∈ Nλ,μ , we have n
n
+ + (1) is increasing for t ∈ tmax (u + n , vn ) > 1. Moreover u + ,v + (1) = 0 and u + n ,vn n
n
+ + (t) < 0 for all t ∈ (0, 1] and n sufficiently (0, tmax (u + n , vn )). This implies that u + n ,vn + + + + large. We obtain 1 < t0 < tmax (u 0 , v0 ). But (t0 u + 0 , t0 v0 ) ∈ Nλ,μ and + E λ,μ (t0 u + 0 , t0 v0 ) =
inf
+ 1
+ E λ,μ (tu + 0 , tv0 )
A singular System Involving the Fractional...
which implies that + + + + + + E λ,μ (t0 u + 0 , t0 v0 ) < E λ,μ (u 0 , v0 ) = lim E λ,μ (u n , vn ) = cλ,μ , n→∞
+ + + which gives a contradiction. Thus, u + n → u 0 strongly in X 0 , vn → v0 strongly in + + X 0 and E λ,μ (u 0 , v0 ) = inf (u,v)∈N + E λ,μ (u, v). The proof of the Theorem 3.1 is λ,μ now completed.
− 4 Existence of Minimizer on Nλ,μ
In this section, we shall show the existence of second solution by proving the existence − . of minimizer of E λ,μ on Nλ,μ 2
2
+ Lemma 4.1 If 0 < (λ||a||∞ ) 2−q + (μ||b||∞ ) 2−q < 0 , then for all (u, v) ∈ Nλ,μ , − cλ,μ > k0 for some k0 = k0 (α, β, p, q, a, b, λ, μ, ||) > 0. − − Proof Let (u − 0 , v0 ) ∈ Nλ,μ , then we have φu − ,v − (1) < 0 which gives from (2.1), 0
c(x)|u|1−α |v|1−β dx >
0
p−q ||(u, v)|| p . 2−α−β −q
(4.1)
Therefore using (2.6), we obtain ||(u, v)|| > S
2−α−β − p( p+α+β−2)
p−q 2−α−β −q
−
1 p+α+β−2
.
(4.2)
On the other hand using (2.5), one has ps∗ −q 1 1 1 1 ∗ − (u, v) p − − || ps p 2−α−β q 2−α−β p−q p p p − qp p−q p−q (λ||a||∞ ) S + (μ||b||∞ ) ||(u, v)||q . ps∗ −q 1 1 1 1 ∗ q = ||(u, v)|| − (u, v) p−q − − || ps p 2−α−β q 2−α−β ⎤ p−q p p p − qp ⎦. (λ||a||∞ ) p−q + (μ||b||∞ ) p−q S
E λ,μ (u, v) ≥
⎡
> ||(u, v)||q ⎣ −
1 1 − p 2−α−β
1 1 − q 2−α−β
||
ps∗ −q ps∗
S
S − qp
( p−q) p
p−q 2−α−β −q
(λ||a||∞ )
p p−q
q− p p+α+β−2
+ (μ||b||∞ )
p p−q
p−q p
⎤ ⎦.
K. Saoudi p
p
Thus, if 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 , then E λ,μ (u, v) > k0 for all − − for some k0 = k0 (α, β, p, q, a, b, λ, μ, ||) > 0. Therefore cλ,μ > k0 (u, v) ∈ Nλ,μ − follows from the definition of cλ,μ . This completes the proof of Lemma 4.1.
p
p
Theorem 4.1 If 0 < (λ||a||∞ ) p−q + (μ||b||∞ ) p−q < 0 , then there exists − − − − (u − 0 , v0 ) ∈ Nλ,μ satisfying E λ,μ (u 0 , v0 ) = inf (u,v)∈N − E λ,μ (u, v). λ,μ
− . Then, there exists Proof Since E λ,μ is bounded below on Nλ,μ and so on Nλ,μ − − − {(u n , vn )} ⊂ Nλ,μ be a sequence such that − E λ,μ (u − n , vn ) →
inf
− (u,v)∈Nλ,μ
E λ,μ (u, v) as n → ∞.
Since E λ,μ is coercive, {(u n , vn } is bounded in E. Then there exists a subsequence, − − − still denoted by (u − n , vn ) and (u 0 , v0 ) ∈ E such that, as n → ∞, − − − u+ n u 0 , vn v0 weakly in X 0 − − − r ∗ u− n → u 0 , vn → v0 strongly in L () for all 1 ≤ r < ps , − − − u− n → u 0 , vn → v0 a.e. in .
Moreover, as in Lemma 3.1, we have − 1−α 1−α |u n | d x = |u − d x, lim 0| n→∞
lim
n→∞
|vn− |1−β d x =
|v0− |1−β d x.
and
q + q λa(x)|u + n | + μb(x)|vn | d x →
+ q q λa(x)|u + 0 | + μb(x)|v0 | d x.
− − Moreover, by Lemma (2.4), there exists t1 such that (t1 u − 0 , t1 v0 ) ∈ Nλ,μ . Now, we − − − prove u − n → u 0 strongly in X 0 , vn → v0 strongly in X 0 . Suppose otherwise, then either − − − ||(u − 0 , v0 )|| E ≤ lim inf ||(u n , vn )|| E n→∞
Thus, since have
− (u − n , vn )
∈
− Nλ,μ ,
− E λ,μ (tu − 0 , tv0 )
− ≤ E λ,μ (u − 0 , v0 ), for all t ≥ 0 we
− − − − − − E λ,μ (t1 u − 0 , t1 v0 ) < lim E λ,μ (t1 u n , t1 vn ) ≤ lim E λ,μ (u n , vn ) = cλ,μ . n→∞
n→∞
− − − which gives a contradiction. Thus, u − n → u 0 strongly in X 0 , vn → v0 strongly in − − X 0 and E λ,μ (u 0 , v0 ) = inf (u,v)∈N − E λ,μ (u, v). The proof of the Theorem 4.1 is λ,μ now completed.
A singular System Involving the Fractional...
Proof of Theorem (1.1) Let us start by proving the existence of non-negative solutions. + + − − First, by Theorems 3.1, 4.1, we conclude that there exist (u + 0 , v0 ) ∈ Nλ,μ , (u 0 , v0 ) ∈ − satisfying Nλ,μ + E λ,μ (u + 0 , v0 ) =
inf
E λ,μ (u, v)
inf
E λ,μ (u, v).
+ (u,v)∈Nλ,μ
and − E λ,μ (u − 0 , v0 ) =
− (u,v)∈Nλ,μ
+ + + + + + Moreover, since E λ,μ (u + 0 , v0 ) = E λ,μ (|u 0 |, |v0 |) and (|u 0 |, |v0 |) ∈ Nλ,μ and − − − − − − − similarly E λ (u 0 , v0 ) = E λ,μ (|u 0 |, |v0 |) and (|u 0 |, |v0 |) ∈ Nλ,μ , so we may assume ± ± ± (u ± 0 , v0 ) ≥ 0. By Lemma 2.3, we may assume that (u 0 , v0 ) are non-trivials nonnegatives solutions of problem (1.1). Finally, it remain to show that the solutions found − + ± ∩ Nλ,μ = ∅, then, (u ± in Theorems 3.1, 4.1, are distinct. Since Nλ,μ 0 , v0 ) are distinct. The proof of the Theorem 1.1 is now completed.
Acknowledgements The author would like to thank the anonymous referees for their carefully reading this paper and their useful comments.
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