Applied Mathematics E-Notes, 16(2016), 21-32 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Existence Of Nontrivial Solutions For A p-Laplacian
Problems With Impulsive E¤ects
El Miloud Hssiniy, Mohammed Massarz, Lakhdar Elbouyahyaouix
Received 11 January 2015
Abstract
In this paper, we study a class of p-Laplacian problems with impulsive conditions depending on a real parameter . Using variational methods and Bonnano’s
critical points theorem [4], we give some appropriate conditions on the nonlinear
term and the impulsive functions to …nd a range of the control parameter for
which the impulsive problem admits at least one nontrivial solution.
1
Introduction
In this work, we study the existence of nontrivial solutions for the following p-Laplacian
problem with the impulsive conditions
8
( (x) p (u0 ))0 + s(x) p (u) = f (x; u(x)); a.e. x 2 (a; b);
>
>
<
0 +
0
(1)
1 u (a )
2 u(a) = 0;
1 u (b ) + 2 u(b) = 0;
>
>
:
( (xj ) p (u0 (xj ))) = Ij (u(xj )); j = 1; 2; : : : ; n;
where p (t) = jtjp 2 t and a; b 2 R with a < b, p > 1, 1 , 2 , 1 , 2 and are positive
constants, ; s 2 L1 ([a; b]) with 0 := ess inf x2[a;b] (x) > 0, s0 := ess inf x2[a;b] s(x) >
0, (a+ ) = (a) > 0, (b ) = (b) > 0, f : [a; b] R ! R, Ij : R ! R are continuous
for j = 1; : : : ; n, and x0 = a < x1 < x2 <
< xn < xn+1 = b: We note that
( (xj )
0
p (u (xj )))
= (x+
j )
0 +
p (u (xj ))
(xj )
0
p (u (xj
));
where z(y + ) and z(y ) denote the right and left limits of z(y) at y respectively.
The theory of impulsive di¤erential equations has become an important area of
investigation in the past two decades because of their applications to various problems arising in communications, control technology, electrical engineering, population
dynamics, biotechnology processes, chemistry and biology (see [2, 3, 8, 12, 14]).
There have been many papers to study impulsive problems by variational method
and critical point theory, we refer the reader to [13, 15, 20, 23] and references cited
Mathematics Subject Classi…cations: 34B15, 34B18, 58E30.
Mohamed I, Faculty of Sciences, Oujda, Morocco
z University Mohamed I, Faculty of Sciences and Technics, Al Hoceima, Morocco
x Regional Centre of Trades Education and Training, Taza, Morocco
y University
21
22
Existence of Nontrivial Solution for a p-Laplacian Problems
therein. In [19] Tian and Ge obtained su¢ cient conditions that guarantee the existence of at least two positive solutions of a p-Laplacian boundary value problem with
impulsive e¤ects
8
0
0
>
> ( (t) p (u (t))) + s(t) p (u(t)) = f (t; u(t)); a.e. t 2 (a; b);
<
u0 (a+ )
u(a) = A;
u0 (b ) + u(b) = B;
>
>
:
( (ti ) p (u0 (tj ))) = Ii (u(ti )); i = 1; 2; : : : ; l;
where a; b 2 R with a < b, p > 1, p (t) = jtjp 2 t, ; s 2 L1 ([a; b]) with ess inf t2[a;b] (t) >
0, ess inf t2[a;b] s(t) > 0, 0 < (a); (b) < +1, A
0; B
0, , , , are positive
constants, Ii 2 C([0; +1); [0; +1)) for i = 1; : : : ; l, f 2 C([a; b] [0; +1); [0; +1)),
f (t; 0) 6= 0 for t 2 [a; b], t0 = a < t1 < t2
< tl < tl+1 = b.
In [1], by virtue of Ricceri’s three critical points theorem [18], Bai and Dai studied
the existence of at least three solutions for the following p-Laplacian impulsive problem
8
( (t) p (u0 (t)))0 + s(t) p (u(t)) = f (t; u(t)); a.e. t 2 (a; b);
>
>
<
0 +
0
1 u (a )
2 u(a) = 0;
1 u (b ) + 2 u(b) = 0;
>
>
:
( (ti ) p (u0 (tj ))) = Ii (u(ti )); i = 1; 2; : : : ; l:
In [5], Bonnano et al. considered the second-order impulsive di¤erential equations with
Dirichlet boundary conditions, depending on two real parameters
8
u00 (t) + a(t)u0 (t) + b(t)u(t) = g(t; u(t)); t 2 [0; T ] ; t 6= tj ;
>
>
<
u(0) = u(T ) = 0;
>
>
:
u0 (tj ) = u0 (t+
u0 (tj ) = Ij (u(tj )), j = 1; 2; :::; n;
j )
where ; > 0, g : [0; T ] R ! R, a; b 2 L1 ([0; T ]) satisfy the conditions ess inf t2[0;T ] a(t)
0, ess inf t2[0;T ] b(t)
0, 0 = t0 < t1 < t2 <
< tn < tn+1 = T , u0 (tj ) =
0
0
0 +
0
u (tj ) u (tj ) = limt!t+ u (t) limt!t u (t), and Ij : R ! R are continuous for every
j
j
j = 1; 2; : : : ; n. Under an appropriate growth condition of the nonlinear function, and a
small perturbations of impulsive terms, they established the existence of at least three
solutions by choosing in a suitable way and for every lying in a precise interval.
Recently, the authors in [10] studied the following nonlinear perturbed problem
8
0
0
>
> ( (t) p (u (t))) + s(t) p (u(t)) = f (t; u(t)) + g(t; u(t)); a.e. t 2 (a; b);
<
0 +
0
1 u (a )
2 u(a) = 0;
1 u (b ) + 2 u(b) = 0;
>
>
:
( (ti ) p (u0 (tj ))) = Ii (u(ti )); i = 1; 2; : : : ; l:
They utilized Bonnano’s theorem [6], to establish precise values of and for which
the above problem admits at least three weak solutions.
Motivated by the above mentioned works, our goal in this paper is to obtain some
su¢ cient conditions to guarantee that problem (1) admits at least one nontrivial solution when the parameter lies in di¤erent intervals. Our analysis is mainly based on
the critical point theorems obtained by Bonanno [4]. This theorem has been used in
several works to obtain existence results for di¤erent kinds of problems. For review on
the subject, we refer the reader to [7, 9, 11].
Hssini et al.
2
23
Preliminaries
Our main tools are two consequences of a local minimum theorem ([4], Theorem 3.1)
which is a more general version of the Ricceri variational principle (see [17]). Given a
set X and two functionals ; : X ! R; we put
(r1 ; r2 ) :=
u2
supv2
inf
1
1 (]r
r2
(]r1 ;r2 [)
(u)
1 (r1 ; r2 )
:=
u2
1 (]r
1 (]
(u)
1 ;r2 [)
(u)
;
(u)
supv2
sup
(v)
1 ;r2 [)
1;r1 [)
(2)
(v)
;
r1
(3)
and
(u)
(r) :=
sup
supv2
(u)
1 (]r;+1[)
u2
1 (]
(v)
1;r[)
(4)
r
for all r; r1 ; r2 2 R; with r1 < r2 .
THEOREM 1 ([4], Theorem 5.1). Let X be a re‡exive real Banach space, : X !
R be a sequentially weakly lower semicontinuous, coercive and continuously Gateaux
di¤erentiable function whose Gateaux derivative admits a continuous inverse on X ,
and
: X ! R be a continuously Gateaux di¤erentiable function whose Gateaux
derivative is compact. Put I =
and assume that there are r1 ; r2 2 R; with
r1 < r2 ; such that
(r1 ; r2 ) < 1 (r1 ; r2 );
(5)
i
h
1
1
where ; 1 are given by (2) and (3). Then, for each 2
(r1 ;r2 ) ; (r1 ;r2 ) ; there
1
is a u0; 2
I 0 (u0; ) = 0:
1
(]r1 ; r2 [) such that I (u0; )
1
I (u) for all u 2
(]r1 ; r2 [) and
THEOREM 2 ([4], Theorem 5.3). Let X be a real Banach space;
: X ! R
be a continuously Gateaux di¤erentiable function whose Gateaux derivative admits a
continuous inverse on X . : X ! R be a continuously Gateaux di¤erentiable function
whose Gateaux derivative is compact. Fix inf X < r < supX and assume that
(r) > 0;
where is given by (4), and for each
1
; there is a u0; 2
Then, for > (r)
1
u2
(]r; +1[) and I 0 (u0; ) = 0:
(6)
1
> (r)
the function I =
1
(]r; +1[) such that I (u0; )
Let X be the Sobolev space W 1;p ([a; b]) equipped with the norm
kuk :=
Z
a
b
0
p
(x) ju (x)j dx +
Z
a
b
p
!1=p
s(x)ju(x)j dx
;
is coercive.
I (u) for all
24
Existence of Nontrivial Solution for a p-Laplacian Problems
which is equivalent to the usual one. We de…ne the norm in C 0 ([a; b]) as
kuk1 = max ju(x)j:
x2[a;b]
Since p > 1, X is compactly embedded in C 0 ([a; b]).
LEMMA 1 ([19], lemma 2.6). For u 2 X, we have kuk1 M kuk, where
(
)
(b a) 1=p (b a)1=q
1 1
1=q
;
M = 2 max
;
+ = 1:
1=p
1=p
p q
s0
0
Throughout the sequel, we assume that the functions f and Ij satisfy the following
assumptions:
(F) f : [a; b] R ! R is an L1 -Carathéodory function namely: t ! f (t; x) is measurable for every x 2 R, x ! f (t; x) is continuous for almost every t 2 [a; b], and for
every % > 0 there exists a function l% 2 L1 ([a; b]) such that
sup jf (x; )
l% (x) for a.e. x 2 [a; b]:
j j %
(H) The impulsive functions Ij have sublinear growth, i.e., there exist constants
aj ; bj > 0 and j 2 [0; p 1) such that
jIj (x)j
aj + bj jxj
j
for all x 2 R; j = 1; 2; : : : ; n:
DEFINITION 1. We say that u 2 X is a weak solution of problem (1) if, for v 2 X,
Z b
Z b
2 u(a)
0
0
(x) p (u (x))v (x)dx +
s(x) p (u(x))v(x)dx + (a) p
v(a)
1
a
a
0
1
Z b
n
X
2 u(b)
@
+ (b) p
v(b)
Ij (u(xj ))v(xj )A = 0:
f (x; u(x))v(x)dx
a
1
Now, Put
F (x; ) =
Z
0
j=1
f (x; t)dt for all (x; ) 2 [a; b]
R:
We introduce the functional I : X ! R de…ned, for each u 2 X; by
I (u) =
where
8
>
<
>
:
(u) = p1 kukp +
(u) =
Rb
a
(u)
(u);
1
(a) p
p
2
1 ju(a)j
p p
1
F (x; u)dx
Pn
+
R u(xj )
j=1 0
1
(b) p
p
2
1 ju(b)j ;
p p
1
Ij (t)dt:
(7)
Hssini et al.
25
By the property of f and the continuity of Ij (j = 1; 2; :::; n); we have that and
are well de…ned and Gâteaux di¤erentiable functionals, whose Gâteaux derivatives at
u 2 X are given by
Z b
Z b
0
(u)v =
(x) p (u0 (x))v 0 (x)dx +
s(x) p (u(x))v(x)dx
a
a
2 u(a)
+ (a)
p
and
0
v(a) + (b)
1
(u)v =
Z
2 u(b)
v(b)
p
1
b
f (x; u)vdx
a
n
X
Ij (u(xj ))v(xj )
j=1
for all v 2 X:
We need the following Proposition in the proofs of our main results.
PROPOSITION 1 ([10], Proposition 2.4). Let T : X ! X be de…ned by
Z b
Z b
0
0
T (u)h =
(x) p (u (x))h (x)dx +
s(x) p (u(x))h(x)dx
a
a
+ (a)
2 u(a)
p
h(a) + (b)
1
2 u(b)
h(b);
p
1
for every u; h 2 X. Then the operator T admits a continuous inverse on X .
3
Main Results
For the sake of convenience, we put
k :=
and
c
:=
2p (p
n
X
aj
+
c
j=1
2(p + 1) 0
+ 1)jj jj1 + (p + 2)(b
bj
j +1
c
j
1
and
(8)
a)p jjsjj1
(d) =
0d
k(b
Mp
a)p
2=p
p
1
;
where aj , bj , j are given by (H), M is given in Lemma 1 and c, d are two positive
constants. Moreover, given a nonnegative constant and a positive constant such
that
!
p
p
(b) 2p 1
(a) 2p 1
0 (1 + C1 )
p
+
6=
; where C1 = M
:
p 1
p 1
Mp
k(b a)p 1
1
1
We set
A ( ) :=
Rb
a
maxjtj
F (x; t)dx +
p
Mp
2
+ ( )
0 (1
( )
p
+ C1 )
k(b a)p 1
Rb
a+b
2
F (t; ) dt
:
26
Existence of Nontrivial Solution for a p-Laplacian Problems
THEOREM 3. Assume that there exists a nonnegative constant
constants 2 and with
p
1
<
0M
p
p
(b
k
(1 + C1 )
<
a)p 1
1
and two positive
p
2;
(9)
such that
(A1) A ( 2 ) < A ( 1 ),
[0; ]:
0 for every (x; t) 2 [a; a+b
2 ]
i
h
Then, for each 2 p1 A (1 ) ; A (1 ) , problem (1) admits at least one nontrivial weak
1
2
solution u 2 X such that
p
p
(A2) F (x; t)
1
pM p
< (u) <
2
pM p
:
PROOF. Let and be the functionals de…ned in (7). It is well known that is
coercive and sequentially weakly lower semicontinuous. From Proposition 1, of course,
0
admits a continuous inverse on X . Moreover, has a compact derivative, it results
sequentially weakly continuous. Hence
and
satisfy all regularity assumptions
requested in Theorem 1. So, our aim is to verify condition (5). To this end, let
r1 =
p
1
pM p
;
r2 =
p
2
pM p
Clearly u0 2 X. Moreover, one has
2p p
ku0 k =
(b a)p
p
Z
a+b
2
2p p
(x)dx +
(b a)p
a
2
b a (x
; and u0 (x) =
a);
;
Z
a+b
2
x 2 [a; a+b
2 [;
x 2 [ a+b
2 ; b]:
p
(x
a) s(x)dx +
a
p
Z
(10)
b
s(x)dx:
a+b
2
Using (8), we observe that
p
p
0
(b
From the de…nition of
a)p
1
ku0 kp
0
k(b
a)p
1
:
(11)
, we have
1
kukp
p
(u)
1
(1 + C1 ) kukp :
p
In particular, we infer
p
p
0
p(b
a)p
1
(u0 )
(1 + C1 )
Hence, it follows from (9) that
r1 < (u0 ) < r2 :
0
pk(b
a)p
1
:
(12)
Hssini et al.
27
1
Now, let u 2 X such that u 2
(]
1; r2 [). By Lemma 1, we obtain
ju(x)j
2
for each x 2 [a; b]:
(13)
Moreover, thanks to (H), we get
n Z
X
j=1
n
X
u(tj )
Ij (x) dx
0
aj kuk1 +
j=1
j
bj
kuk1j +1 ;
+1
(14)
which combined with (13) yields that
sup
u2
1 (]
(u)
=
0
Z
@
sup
1;r2 [)
1 (]
u2
Z
F (x; u)dx
a
b
j=1
max F (x; t)dx +
a jtj
Z
1;r2 [)
2
n
X
j=1
b
2
2
max F (x; t)dx +
a jtj
n Z
X
b
u(xj )
0
Ij (t)dtA
bj
kuk1j +1
+
1
j
aj kuk1 +
2
1
:
(15)
2
Arguing as before, we obtain
sup
1 (]
u2
(u)
1;r1 [)
Z
b
max F (x; t)dx +
a jtj
2
1
1
:
(16)
1
On the other hand, due to Lemma 1, (H), (A2) and (11), we have
Z
(u0 )
Z
Z
b
F (t; ) dt
a+b
2
b
F (t; ) dt
a+b
2
n Z
X
j=1
n
X
j=1
b
F (t; ) dt
( )
a+b
2
u0 (xj )
Ij (t)dt
0
aj ku0 k1 +
j
bj
ku0 k1j +1
+1
( ):
(17)
Therefore, from (12) and (15)–(17), we get
supu2
(r1 ; r2 )
Rb
a
=
1 (]r
1 ;r2 [)
r2
maxjtj
pA ( 2 ):
2
(u)
(u0 )
(u0 )
F (x; t)dx +
p
2
pM p
2
2
2
+ ( )
0 (1
+ C1 )
pk(b a)p
( )
p
1
Rb
a+b
2
F (t; ) dt
28
Existence of Nontrivial Solution for a p-Laplacian Problems
We also obtain
(u0 )
supu2
1 (r1 ; r2 )
(u0 )
Rb
a+b
2
=
1 (]
F (t; ) dt
r1
( )
2
1
( )
0 (1 + C1 )
pk(b a)p
each
that
2
1
A (
Rb
a
1
p
maxjtj
1
F (x; t)dx
p
1
pM p
1
pA ( 1 ):
So, by our assumption
it follows
that (r1 ; r2 ) <
i
h
1
p
(u)
1;r1 [)
; 1
1) A (
2)
1 (r1 ; r2 ):
Hence, from Theorem 1 for
; the functional I admits at least one critical point u such
p
1
pM p
< (u) <
p
2
pM p
;
and the proof of Theorem 3 is achieved.
Now, we point out the following consequence of Theorem 3.
THEOREM 4. Assume that there exist two constants
0M
(b
p
a)p 1
p
<
k
(1 + C1 )
p
and
with
;
such that assumption (A2) in Theorem 3 holds. Furthermore, suppose that
Rb
a
maxjtj
F (x; t)dx +
2
k(b a)p 1
<
p
0 M (1 + C1 )
Rb
a+b
2
F (t; ) dt
( )
( )
:
(18)
problem (1) admits at least one nontrivial weak solution u such that ju(x)j <
x 2 [a; b]:
for all
p
p
Then, for each
3
p
p
15
0 M (1 + C1 )
2
Rb
p
( )
a+b F (t; ) dt
2
p
( )
; Rb
a
maxjtj
k(b
a)p
1
F (x; t)dx +
PROOF. Our aim is to apply Theorem 3. To this end we pick
1
2
2
4;
= 0 and
2
= :
Hssini et al.
29
From (18); one has
A ( )
Rb
a
=
h
<
maxjtj
0
M
i hR
p
b
M p (1+C1 )
p k(b
a)p
1
a
p
M
a
=
=
+ ( )
p
1
Rb
2
F (x; t)dx +
maxjtj
F (x; t)dx +
p
M
Rb
( )
p
a+b
2
0 (1 + C1 )
k(b a)p 1
maxjtj
2
F (x; t)dx +
+ C1 ) p
k(b a)p 1
Rb
0 (1
2
a+b
2
<
F (t; ) dt
F (t; ) dt
i
( )
( )
p
0 (1 + C1 )
k(b a)p 1
A (0):
Hence, Theorem 3 ensures the existence of nontrivial weak solution u of problem (1)
such that
p
1
kukp
;
(u) <
p
pM p
and clearly by Lemma 1, ju(x)j <
for all x 2 [a; b]:
Finally, we also give an application of Theorem 2.
THEOREM 5. Assume that there exist two constants
p
such that
Z
<
0M
(b
2
a jtj
and
<
F (x; )
j jp
j!+1
lim sup
j
Then, for each
p
Z
with
;
b
F (x; )dx
( )
a+b
2
( );
(19)
0 uniformly in x:
(20)
> ; where
p
=
p
a)p 1
b
max F (x; t)dx +
p
and
hR
b
a
Mp
maxjtj
F (x; t)dx +
0 (1
p
+ C1 )
k(b a)p 1
2
+ ( )
( )
Rb
a+b
2
i;
F (x; )dx
problem (1) admits at least one nontrivial weak solution u such that jjujj >
M (1+C1 )1=p
:
PROOF. The functionals and
given by (7) satisfy all regularity assumptions
requested in Theorem 2. Moreover, by standard computations, condition (20) implies
30
Existence of Nontrivial Solution for a p-Laplacian Problems
that I ; > 0; is coercive. To apply Theorem 2, it su¢ ces to verify condition (6).
p
Indeed, put u0 (x) as in (10) and r = pM p : Arguing as in the proof of Theorem 3 we
obtain
(u)
(r)
Rb
a+b
2
supv2
(u)
F (t; ) dt
1 (]
1;r[)
(v)
r
( )
0 (1
( )
+ C1 )
pk(b a)p
2
Rb
a
p
maxjtj
F (x; t)dx
:
p
1
pM p
So, from our assumption it follows that (r) > 0: Hence, in view of Theorem 2 for each
> ; I admits at least one local minimum u such that
p
pM p
< (u)
1
(1 + C1 )jjujjp ;
p
and our conclusion is achieved.
Acknowledgments. The authors would like to thank the anonymous referee for
his/her valuable suggestions and comments, which greatly improve the manuscript.
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