See discussions, stats, and author profiles for this publication at: https://www.researchgate.

net/publication/280114644

Multiplicity of Nontrivial Solutions of a Class of Fractional p-Laplacian

Problem

Article in Zeitschrift für Analysis und ihre Anwendungen · July 2015

DOI: 10.4171/ZAA/1541

CITATIONS READS

14 3,125

1 author:

Abdeljabbar Ghanmi
Higher Institute of Medical Technologies of Tunis
80 PUBLICATIONS 635 CITATIONS

SEE PROFILE

All content following this page was uploaded by Abdeljabbar Ghanmi on 18 July 2015.

The user has requested enhancement of the downloaded file.

Zeitschrift für Analysis und ihre Anwendungen c European Mathematical Society
Journal of Analysis and its Applications
Volume 34 (2015), 309–319
DOI: 10.4171/ZAA/1541

Multiplicity of Nontrivial Solutions of a

Class of Fractional p-Laplacian Problem
Ghanmi Abdeljabbar

Abstract. In this paper, we deal with existence of nontrivial solutions to the frac-
tional p-Laplacian problem of the type
(
(−4)αp u = 1r ∂F∂u
(x,u)
+ λa(x)|u|q−2 u in Ω,
u = 0 in Rn \ Ω,

where Ω is a bounded domain in Rn with smooth boundary ∂Ω, a ∈ C(Ω), p ≥ 2,

np
α ∈ (0, 1) such that pα < n, 1 < q < p < r < n−αp , and F ∈ C 1 (Ω × R, R). Using the
decomposition of the Nehari manifold, we prove that the non-local elliptic problem
has at least two nontrivial solutions.
Keywords. Nontrivial solutions, sign-changing weight function, Nehari manifold
Mathematics Subject Classification (2010). Primary 35J35, secondary 35J50,
35J60

1. Introduction

In this paper, we are concerned with the multiplicity of nontrivial solutions for
the following problem
(
(−4)αp u = 1r ∂F∂u
(x,u)
+ λa(x)|u|q−2 u in Ω,
(P)
u = 0 in Rn \ Ω,

where Ω is a bounded domain in Rn with smooth boundary, a ∈ C(Ω), λ > 0,

np
p ≥ 2, such that n > pα and 1 < q < p < r < p∗α , p∗α = n−αp . The function
1
F ∈ C (Ω × R, R) is positively homogeneous of degree r , that is, F (x, tu) =
tr F (x, u, v)(t > 0) holds for all (x, u) ∈ Ω × R.

G. Abdeljabbar: Département de Mathématiques, Faculté des Sciences de Tunis Cam-

pus Universitaire, 2092 Tunis, Tunisia.; Abdeljabbar.ghanmi@lamsin.rnu.tn
310 G. Abdeljabbar

Throughout this paper the sign changing weight function a satisfies the
following condition

(A) a ∈ C(Ω) with kak∞ = 1 and a± := max(±a, 0) 6≡ 0,

and the fractional p-Laplacian operator may be defined for p ∈ (1, ∞) as

|u(x) − u(y)|p−2 (u(x) − u(y))

Z
(−4)αp u(x) = 2 lim dy, x ∈ Rn .
ε&0 Rn \Bε (x) |x − y|n+pα

Recently, a lot of attention is given to the study of fractional and non-local

operators of elliptic type due to concrete real world applications in finance, thin
obstacle problem, optimization, quasi-geostrophic flow etc. Dirichlet bound-
ary value problem in case of fractional Laplacian using variational methods is
recently studied in [3,5,7,9,10]. Also existence and multiplicity results for nonlo-
cal operators with convex-concave type nonlinearity is shown in [12]. Moreover
multiplicity results with sign-changing weight functions using Nehari manifold
and fibering map analysis is also studied in many papers (see [1, 2, 4, 10]).
In this paper, we propose a very simple variational method to prove the
existence of at least two nontrivial solutions of problem (P). In fact, we use
the decomposition of the Nehari manifold as λ vary to prove our main result.
Before stating our main result, we need the following assumptions:
(H1 ) F : Ω × R −→ R is a C 1 function such that

F (x, tu) = tr F (x, u)(t > 0) for all x ∈ Ω, u ∈ R.

(H2 ) F (x, 0) = ∂F
∂u
(x, 0) = 0.
±
(H3 ) F (x, u) = max(±F (x, u), 0) 6= 0 for all u 6= 0.
We remark that assumption (H1 ) leads to the so-called Euler identity

∂F
u (x, u) = rF (x, u)
∂u
and
|F (x, u)| ≤ K|u|r for some constant K > 0. (1)
Our main result is the following

Theorem 1.1. Under the assumptions (A) and (H1 )-(H3 ), there exists λ0 > 0
such that for all 0 < λ < λ0 , problem (P) has at least two nontrivial solutions.

This paper is organized as follows. In Section 2, we give some notations

and preliminaries. Proofs of Theorem 1.1 is given in Section 3.
 Multiplicity of Nontrivial Solutions 311

2. Preliminaries
In this preliminary section, for the reader’s convenience, we collect some basic
results that will be used in the forthcoming sections. in the following, For all
1 ≤ r ≤ ∞ denote by k.kr the norm of Lr (Ω). The Gagliardo seminorm is
defined for all measurable function u : Rn → R by
 p1
|u(x) − u(y)|p
Z
|u|α,p := dxdy .
R2n |x − y|n+pα
We define the fractional Sobolev space

W α,p (Rn ) := {u ∈ Lp (Rn ) : u measurable , |u|α,p < ∞}

endowed with the norm

p1
||u||α,p := ||u||pp + |u|pα,p .

For a detailed account on the properties of W α,p (Rn ) we refer the reader to [6].
We shall work in the closed linear subspace

E := {u ∈ W α,p (Rn ) : u(x) = 0 a.e. in Rn \ Ω} ,

which can be equivalenty renormed by setting || · || = | · |α,p , note that these type
of spaces were introduced in [9]. It is readily seen that (E, || · ||) is a uniformly
convex Banach space and that the embedding E ,→ Lr (Ω) is continuous for all
1 ≤ r ≤ p∗α , and compact for all 1 ≤ r < p∗α . The dual space of (E, || · ||)
is denoted by (E ∗ , || · ||∗ ), and ≺ ·, ·  denotes the usual duality between E
and E ∗ .
Definition 2.1. We say that u ∈ E is a weak solution of (P) if for every v ∈ E
we have
|u(x) − u(y)|p−2 (u(x) − u(y))(v(x) − v(y))
Z
dxdy
R2n |x − y|n+pα
Z Z
1 ∂F
= (x, u(x))v(x)dx + λ a(x)|u(x)|q−2 u(x)v(x)dx.
r Ω ∂u Ω

The Euler functional Jλ : E → R associated to the problem (P) is defined as

Z Z
1 p 1 λ
Jλ (u) = ||u|| − F (x, u)dx − a(x)|u(x)|q dx.
p r Ω q Ω
Then Jλ is Fréchet differentiable and, for all u ∈ E, we have
Z Z
0
≺ Jλ (u), u = ||u|| − F (x, u)dx − λ a(x)|u(x)|q dx,
p
Ω Ω
312 G. Abdeljabbar

which shows that the weak solutions of (P) are critical points of the func-
tional Jλ . It is easy to see that the energy functional Jλ is not bounded below
on the space E, but is bounded below on an appropriate subset of E and a min-
imizer on subsets of this set gives raise to solutions of (P). In order to obtain
the existence result, we introduce the Nehari manifold

Nλ := {u ∈ E :≺ Jλ0 (u), u = 0}.

Then, u ∈ Nλ if and only if

Z Z
||u|| − F (x, u)dx − λ a(x)|u(x)|q dx = 0.
p
(2)
Ω Ω

We note that Nλ contains every non zero solution of (P).

Lemma 2.2. Jλ is coercive and bounded below on Nλ .

Proof. Let u ∈ Nλ , then we have

   Z
1 1 p 1 1
Jλ (u) = − ||u|| − λ − a(x)|u(x)|q dx ≥ c1 ||u||p − c2 ||u||q .
p r q r Ω

Hence, Jλ is bounded below and coercive on Nλ .

Now as we know that the Nehari manifold is closely related to the behavior
of the functions Φu : [0, ∞) → R defined as

Φu (t) = Jλ (tu).

Such maps are called fiber maps and were introduced by Drabek and Pohozaev
in [7].
For u ∈ E, we have

tp tr tq
Z Z
p
Φu (t) = ||u|| − F (x, u)dx−λ a(x)|u(x)|q dx,
p r Ω q Ω
Z Z
0 p−1 p r−1 q−1
Φu (t) = t ||u|| −t F (x, u)dx−λt a(x)|u(x)|q dx,
Ω Z Ω Z
00 p−2 p r−2 q−2
Φu (t) = (p−1)t ||u|| −(r−1)t F (x, u)dx−λ(q−1)t a(x)|u(x)|q dx.
Ω Ω

Then, it is easy to see that tu ∈ Nλ if and only if Φ0u (t) = 0 and in particular,
u ∈ Nλ if and only if Φ0u (1) = 0. Thus it is natural to split Nλ into three parts
corresponding to local minima, local maxima and points of inflection. For this
 Multiplicity of Nontrivial Solutions 313

we set

Nλ+ = {u ∈ Nλ : Φ00u (1) > 0} = {tu ∈ E : Φ0u (t) = 0, Φ00u (t) > 0} ,
Nλ− = {u ∈ Nλ : Φ00u (1) < 0} = {tu ∈ E : Φ0u (t) = 0, Φ00u (t) < 0} ,
Nλ0 = {u ∈ Nλ : Φ00u (1) = 0} = {tu ∈ E : Φ0u (t) = 0, Φ00u (t) = 0} .

Before studying the behavior of Nehari manifold using fibering maps, we intro-
duce some notations
 Z   Z 
+ −
F = u ∈ E : F (x, u)dx > 0 , F = u∈E : F (x, u)dx < 0 ,
Ω Ω
 Z   Z 
+ q − q
A = u ∈ E : a(x)|u(x)| dx > 0 , A = u ∈ E : a(x)|u(x)| dx < 0 .
Ω Ω
R
Now
R we study the fiber map Φu according to the sign of Ω
a(x)|u(x)|q dx and
Ω
F (x, u)dx.

Case 1. u ∈ F − ∩ A− .
In this case Φu (0) = 0 and Φ0u (t) > 0, ∀t > 0 which implies that Φu is strictly
increasing and hence no critical point.

Case 2. u ∈ F + ∩ A− .
In this case, firstly we define mu : [0, ∞) → R by
Z
p−q p r−q
mu (t) = t ||u|| − t F (x, u)dx.
Ω

Clearly, for t > 0, tu ∈ Nλ if and only if t is a solution of

Z
mu (t) = λ a(x)|u(x)|q dx.
Ω

As we have mu (t) → −∞ as t → ∞ and

Z
m0u (t) p−q−1
= (p − q)t p
||u|| − (r − q)t r−q−1
F (x, u)dx.
Ω

0 −
Therefore, mR u (t) > 0 asq t → 0. Since u ∈ A , there0 exists Tq−1such that
mRu (t) = λ Ω a(x)|u(x)| dx. Thus, for 0 < t < T, Φu (t) = t mu (t)−
λ Ω a(x)|u(x)|q dx > 0 and for t > T, Φ0u (t) < 0. Hence, Φu (t) is increas-

ing on (0, T ), decreasing on (T, ∞). Since Φu (t) > 0 for t close to 0 and
Φu (t) → −∞ as t → ∞, we get Φu has exactly one critical point t1 , which is a
global maximum point. Hence t1 u ∈ Nλ− .
314 G. Abdeljabbar

Case 3. u ∈ F − ∩ A+ .
In this case, mu (0) = 0, m0u (t) > 0, ∀t > 0, which implies that mu is strictly
increasing
R and since u ∈ A+ , there exists a unique t1 > 0 such that mu (t1 ) =
λ Ω a(x)|u(x)|q dx. This implies that Φu is decreasing on (0, t1 ), increasing on
(t1 , ∞) and Φ0u (t1 ) = 0. Thus, Φu has exactly one critical point t1 , corresponding
to global minimum point. Hence t1 u ∈ Nλ+ .

Case 4. u ∈ F + ∩ A+ .
In this case, we claim that there exists µ0 > 0 such that for λ ∈ (0, µ0 ), Φu
has exactly two critical points t1 and t2 . Moreover, t1 is a local minimum point
and t2 is a local maximum point. Thus t1 u ∈ Nλ+ and t2 u ∈ Nλ− . We prove this
claim in the following Lemma:
Lemma 2.3. There exists µ0 > 0 such that for λ ∈ (0, µ0 ), Φu take positive
value for all non-zero u ∈ E. Moreover, if u ∈ F + ∩ A+ , then Φu has exactly
two critical points.
Proof. Let u ∈ E, define
tp tr
Z
Mu (t) = ||u||p − F (x, u)dx.
p r Ω

Then Z
Mu0 (t) =t p−1 p
||u|| − t r−1
F (x, u)dx
Ω
 1
 r−p
||u||p
and Mu attains its maximum value at T = R . Moreover,
Ω (x,u)dx
F

p
||u||r
   r−p
1 1
Mu (T ) = − R
p r Ω
F (x, u)dx
and p(r−2)
||u|| r−p
Mu00 (T ) = (p − r) R p−2 < 0.

r−p
Ω
F (x, u)dx
For 1 ≤ ν < p∗α
we denoted by Sν be the Sobolev constant of embedding
ν
E ,→ L (Ω), then, by (1) we have
r−p
Mu (T ) ≥ p = δ, (3)
rp(KSrr ) r−p
which is independent of u. We now show that there exists µ0 > 0 such that
Φu (T ) > 0. Using condition (A) and the Soblev imbedding, we get
q
Tq Sqq Sqq ||u||p Sqq
Z  r−p
q
q q q q R
a(x)|u(x)| dx ≤ ||u|| T = ||u|| = Mu (T ) p .
q Ω q q Ω
F (x, u)dx q
 Multiplicity of Nontrivial Solutions 315

Thus
Tq
Z  p−q 
q q
Φu (T ) = Mu (T )−λ a(x)|u(x)|q dx ≤ Mu (T )−λcMu (T ) p = δ p δ p −λc ,
q Ω
where δ is the constant given in (3). Let
p−q
qδ p
µ0 = .
Sqq
Then, the choice of such µ0 completes the proof.
Corollary 2.4. If λ < µ0 , then there exists δ1 > 0 such that Jλ (u) > δ1 for all
u ∈ Nλ− .
Proof.
R Let u ∈ Nλ− , then Φu has a positive global maximum at T = 1 and
Ω
a(x)|u(x)|q dx > 0. Thus, if λ < µ0 , then we have
q
 p−q 
Jλ (u) = Φu (1) = Φu (T ) ≥ δ δp p − λc > 0,

where δ is the same as in Lemma 2.3, and so the result follows immediately.
Lemma 2.5. There exists µ1 such that if 0 < λ < µ1 , then Nλ0 = ∅.
Proof. Let
 p−q
 r−p
r−p p−q
µ1 = q ,
Sq (r − q) KSrr (r − q)
where K is given by (1).
Suppose otherwise, that 0 < λ < µ1 such that Nλ0 6= ∅. Then, for u ∈ Nλ0 ,
we have
Z Z
00
0 = Φu (1) = (p − 1)||u|| − (r − 1) F (x, u)dx − λ(q − 1) a(x)|u(x)|q dx.
p
Ω Ω
R
So, it follows from (2) that (r−p)||u||p = λ(r−q) Ω a(x)|u|q dx ≤ λ(r−q)Sqq ||u||q ,
and so  1
 p−q
qr −q
||u|| ≤ λSq . (4)
r−p
R
On the other hand, by (1) we get (p − q)||u||p = λ(r − q) Ω F (x, u)dx ≤
K(r − q)Srr ||u||r , then
 1
 r−p
p−q
||u|| ≥ . (5)
KSrr (r − q)

Combining (4) and (5) we obtain λ ≥ µ1 , which is a contradiction.

316 G. Abdeljabbar

Here and always, we define λ0 as

λ0 = min(µ0 , µ1 ). (6)
We remark that if 0 < λ < λ0 , then all the above Lemmas hold true.
Lemma 2.6. Let u be a local minimizer for Jλ on subsets Nλ+ or Nλ− of Nλ
such that u 6∈ Nλ0 , then u is a critical point of Jλ .
Proof. Since u is a minimizer for Jλ under the constraint Iλ (u) :=≺ Jλ0 (u), u 
= 0, by the theory of Lagrange multipliers, there exists µ ∈ R such that
Jλ0 (u) = µIλ0 (u). Thus
≺ Jλ0 (u), u = µ ≺ Iλ0 (u), u = µΦ00u (1) = 0,
but u 6∈ Nλ0 and so Φ00u (1) 6= 0. Hence µ = 0. This completes the proof.

3. Proof of our result

Throughout this section, we assume that the parameter λ satisfies 0 < λ <
λ0 , where λ0 is the constant given by (6). That leads us consequently to the
following results on the existence of minimizers in Nλ+ and Nλ− .
Lemma 3.1. If 0 < λ < λ0 , then Jλ achieves its minimum on Nλ+ .
Proof. Since Jλ is bounded below on Nλ and so on Nλ+ , there exists a minimizing
sequence {uk } ⊂ Nλ+ such that
lim Jλ (uk ) = inf+ Jλ (u).
k→∞ u∈Nλ

As Jλ is coercive on Nλ , {uk } is a bounded sequence in E. Therefore, for all

1 ≤ ν < p∗s we have

uk * uλ weakly in E
uk → uλ strongly in Lν (Rn ).
R
If we choose u ∈ E such that Ω a(x)|u(x)|q dx > 0, then there exists t1 > 0
such that t1 u ∈ Nλ+ and Jλ (t1 u) < 0, Hence, inf u∈N + Jλ (u) < 0.
λ
On the other hand, since {uk } ⊂ Nλ we have
   Z
1 1 p 1 1
Jλ (uk ) = − ||uk || − λ − a(x)|uk (x)|q dx,
p r q r Ω

and so λ( 1q − 1r ) Ω a(x)|uk (x)|q dx = ( p1 − 1r )||uk ||p − Jλ (uk ). Letting k tends to

R

infinity, we get Z
a(x)|uλ (x)|q dx > 0. (7)
Ω
 Multiplicity of Nontrivial Solutions 317

Next we claim that uk → uλ . Suppose this is not true, then

||uλ ||p < lim inf ||uk ||p .

k→∞

Since Φ0uλ (t1 ) = 0, it follows that Φ0uk (t1 ) > 0 for sufficiently large k. So, we
must have t1 > 1 but t1 uλ ∈ Nλ+ and so

Jλ (t1 uλ ) < Jλ (uλ ) ≤ lim Jλ (uk ) = inf+ Jλ (u),

k→∞ u∈Nλ

which is a contradiction. Since Nλ0 = ∅, then uλ ∈ Nλ+ . Finally, uλ is a

minimizer for Jλ on Nλ+ .

Lemma 3.2. If 0 < λ < λ0 , then Jλ achieves its minimum on Nλ− .

Proof. Let u ∈ Nλ− , then from Corollary 2.4, there exists δ1 > 0 such that
Jλ (u) ≥ δ1 . So, there exists a minimizing sequence {uk } ⊂ Nλ− such that

lim Jλ (uk ) = inf− Jλ (u) > 0. (8)

k→∞ u∈Nλ

On the other hand, since Jλ is coercive, {uk } is a bounded sequence in E.

Therefore, for all 1 ≤ ν < p∗s we have

uk * vλ weakly in E
uk → vλ strongly in Lν (Rn ).

Since u ∈ Nλ , then we have

   Z
1 1 p 1 1
Jλ (uk ) = − ||uk || + − F (x, uk )dx. (9)
p q q r Ω

Letting k goes to infinity, it follows from (8) and (9) that

Z
F (x, vλ )dx > 0. (10)
Ω

Hence, vλ ∈ F + and so Φvλ has a global maximum at some point T and con-
sequently, T vλ ∈ Nλ− . on the other hand, uk ∈ Nλ− implies that 1 is a global
maximum point for Φuk , i.e.

Jλ (tuk ) = Φuk (t) ≤ Φuk (1) = Jλ (uk ). (11)

Next we claim that uk → uλ . Suppose this is not true, then

||uλ ||p < lim inf ||uk ||p ,

k→∞
318 G. Abdeljabbar

and from (11) it follows that

Tp Tr Tq
Z Z
p
Jλ (T vλ ) = ||vλ || − F (x, vλ )dx − λ a(x)|vλ |q dx
p r Ω q Ω
 p r Z
Tq
Z 
T p T q
< inf ||uk || − F (x, uk )dx − λ a(x)|uk | dx
k→∞ p r Ω q Ω
≤ lim Jλ (T uk )
k→∞
≤ lim Jλ (uk )
k→∞
= inf− Jλ (u),
u∈Nλ

which is a contradiction. Hence, uk → vλ . Since Nλ0 = ∅, then vλ ∈ Nλ− .

Proof of Theorem 1.1. By Lemma 3.1 and Lemma 3.2, Problem (P) has two
weak solutions uλ ∈ Nλ+ and vλ ∈ Nλ+ . On the other hand, from (7) and
(10), these solutions are nontrivial. Since Nλ− ∩ Nλ+ = ∅, then uλ and vλ are
distinct.

Acknowledgment. I thank the referees for their careful reading of the paper
and for their helpful comments and suggestions.

References

[1] Abdeljabbar, G., Existence of nontrivial solutions of p-Laplacian equation with

sign-changing weight functions. ISRN Math. Anal. 2014 (2014), Art. ID 461965,
7 pp.
[2] Ambrosetti, A., Garca Azorero, J. and Peral, I., Multiplicity results for some
nonlinear elliptic equations. J. Funct. Anal. 137 (1996)(1), 219 – 242.
[3] Barrios, B., Colorado, E., Servadei, R. and Soria, F., A critical fractional
equation with concave-convex power nonlinearities. Ann. Inst. H. Poincaré
Anal. Non Linéaire (2014), (available at http://dx.doi.org/10.1016/j.anihpc.
2014.04.003).
[4] Chen, C., Kuo, Y. and Wu, T., The Nehari manifold for a Kirchhoff type
problem involving sign-changing weight functions. J. Diff. Equ. 250 (2011),
1876 – 1908.
[5] Di Castro, A., Kuusi, T. and Palatucci, G., Local behavior of fractional p-
minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire (2015), (available at
http://dx.doi.org/10.1016/j.anihpc.2015.04.003).
[6] Di Nezza, E., Palatucci, G. and Valdinoci, E., Hitchhiker’s guide to the frac-
tional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521 – 573.
 Multiplicity of Nontrivial Solutions 319

[7] Drabek, P. and Pohozaev, S. I., Positive solutions for the p-Laplacian: appli-
cation of the fibering method. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997),
703 – 726.
[8] Rabinowitz, P. H., Minimax Methods in Critical Point Theory with Applica-
tions to Differential Equations. Providence: Amer. Math. Soc. 1986.
[9] Servadei, R. and Valdinoci, E., Mountain pass solutions for non-local elliptic
operators. J. Math. Anal. Appl. 389 (2012)(2), 887 – 898.
[10] Servadei, R. and Valdinoci, E., Variational methods for non-local operators of
elliptic type. Discrete Contin. Dyn. Syst. 33 (2013)(5), 2105 – 2137.
[11] Silvestre, L., Hölder estimates for solutions of integro differential equations like
the fractional Laplace. Indiana Univ. Math. J. 55 (2006), 1155 – 1174.
[12] Su, X. and Wei, Y., Multiplicity of solutions for non-local elliptic equations
driven by fractional Laplacian. Calc. Var. 52 (2015), 95 – 124.
[13] Yu, X., The Nehari manifold for elliptic equation involving the square root of
the Laplacian. J. Diff. Equ. 252 (2012), 1283 – 1308.

Received July 22, 2014; revised November 17, 2014

View publication stats