Applied Mathematics E-Notes, 15(2015), 162-172 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Absence Of Nontrivial Solutions For A Class Of
Elliptic Equations And Systems In Unbounded
Domains
Kamel Akrouty
Received 15 December 2014
Abstract
In this paper, we are interested in the study of the nonexistence of nontrivial
solutions for a class of ellptic equations, in unbounded domains. This leads us to
extend these results to systems of equations. The method used is based on energy
type identities.
1
Introduction
In this work we study the absence of nontrivial solutions for the following problem
8
n
P
<
@
p (y) @u + f (y; u) = 0 in R
;
:
i=1
u+
@yi
" @u
@
@yi
= 0 on R
considered in H 2 (R
) \ L1 (R
1
C (R
).
We use the notations
), for a bounded and positive function p 2
t = y1 2 R; x = (x1 ; :::; xn
1)
= (y2 ; :::; yn ) 2 ;
H = L2 ( ) ;
ku (t; x)k =
Z
(1)
@ :
2
ju(t; x)j dx
1
2
; the norm of u in H;
Z nX1
@u
kru (t; x)k =
@xi
i=1
2
F (y; u) = F (t; x; u) =
Zu
2
dx;
f (t; x; )d ; 8 x 2
0
Mathematics Subject Classi…cations: 35J25; 35J60.
Laboratory, Larbie Tebessi University, Tebessa, Algeria,
y LAMIS
162
; u 2 R:
K. AKROUT
163
Let L be the operator de…ned by
Lu (t; x) =
n
X1
i=1
and f : R
@
@xi
p (t; x)
@u
@xi
; (t; x) 2 R
;
R ! R a real continuous function, locally Lipschitz in u, such that
f (t; x; 0) = 0; 8 x 2 :
We assume that
u 2 H 2 (R; H) \ L1 (R; L1 ( ))
satis…es the equation
@
@t
p (t; x)
@u
@t
+ Lu (t; x) + f (t; x; u) = 0 a.e. (t; x) 2 R
(2)
under the boundary conditions
(u + "
@u
) (t; ) = 0; (t; ) 2 R
@n
u(t; ) = 0; (t; ) 2 R
@ ; Robin condition,
(3)
@ ; Dirichlet condition,
(4)
@u(t; )
= 0; (t; ) 2 R @ ; Neumann condition.
@
We will extend our result for (1) to the system of m equations of the form
8
n
P
<
@uk
@
+ fk (y; u1 ; :::; um ) = 0 in R
;
@yi pk (y) @yi
i=1
:
k
uk + " @u
= 0 on R @ ;
@
(5)
Rm ! R; are real continuous functions, locally Lipschitz
where 1 k m, fk : R
in ui , verifying
fk (t; x; u1 ; :::; 0; :::; um ) = 0; 8 x 2 ;
Rm ! R such that
9Fm : R
@Fm
= fj (t; x; s1 ; :::; sm ) ; 1
@sj
j
m:
Let Lk be the operators de…ned by
Lk u (t; x) =
n
X1
i=1
@
@xi
pk (t; x)
@u
@xi
; (t; x) 2 R
;
and assume that
uk 2 H 2 (R; H) \ L1 (R; L1 ( ))
are solutions of the system
@
@t
pk (t; x)
@uk
@t
+ Lk uk (t; x) + f (t; x; u1 ; :::; um ) = 0 ; (t; x) 2 R
;
(6)
164
Absence of Nontrivial Solutions
where 1
k
m with boundary conditions
(uk + "
@uk
) (t; ) = 0; (t; ) 2 R
@n
uk (t; ) = 0; (t; ) 2 R
@ ; Robin condition,
(7)
@ ; Dirichlet condition,
(8)
@uk (t; )
= 0; (t; ) 2 R @ ; Neumann condition.
(9)
@
The study of the nonexistence of nontrivial solutions of elliptic equations and systems is the subject of several works of many authors, and various methods are used
to obtain necessary and su¢ cient conditions that guarantee that the systems studied
admit only trivial solutions. The works of Esteban & Lions [4], Pohozaev [9] and Van
Der Vorst [10], contain results concerning semilinear elliptic equations and systems. A
similar result can be found in [7] and [1], where equations of the form
8
n
P
< @
@
@u
(t) @u
;
@t
@t
@xi p (t; x) @xi + f (x; u) = 0 in R
i=1
:
@u
= 0 on R @!;
u + " @n
are considered in H 2 (R !) \ L1 (R !); where
is a bounded domain of Rn ,
0 < " < +1; 2 R , p :
R ! R is continuous and positive function in , and f
:
R ! R is locally Lipschitz continuous function such that
(t) > 0 (resp < 0) ; 8t 2 R;
2F (x; u)
uf (x; u)
0 (resp
0) ;
where F is the primitive of the function f:
Our proof is based on energy type identities established in section 2, which make it
possible to obtain the main nonexistence result in section 3. In section 4 we apply the
results to some examples.
2
Identities of Energy Type
In this section, we give essential lemmas for showing the main result of this paper.
LEMMA 1. Let F and p satisfy
@F
@t (t; x; u)
@p
0
@t (t; x)
0 (resp
0) ; 8 (t; x) 2 R
(resp
0) ; 8 (t; x) 2 R
:
;
(10)
Then the following energy identity,
Z
Z
Z
2
1
@u
1
2
p (t; x)
p(t; x) jruj dx +
F (t; x; u)dx
dx +
2
@t
2
Z
1
+
p(t; s)u2 (t; s) ds = 0
2" @
(11)
K. AKROUT
165
holds for any solution of the Robin problem of (2) and (3).
1;1
PROOF. The assumptions f 2 Wloc
(R
R), p 2 L1 (R
) and f (t; x; 0) =
0; for x 2 ; allow us to deduce the existence of two positive constants C1 and C2 , such
that
2
jp(t; x)j C1 ; jF (t; x; u)j C2 ju (t; x)j :
In addition, consider the functions
Z
Z
1
2
p(t; x) jruj dx and B (t) =
A (t) =
F (t; x; u)dx for t 2 R;
2
where A and B are of class C 1 , and
2
jA (t)j
C1 kru(t; x)k and jB (t)j
Then,
B 0 (t) =
Z
f (t; x; u)
2
C2 ku(t; x)k for t 2 R:
@u @F
+
@t
@t
dx for t 2 R;
and
A0 (t)
Z
=
n 1
1 X @p
@u
@u @ 2 u
+
(t; x)
p(t; x)
@x
@x
@t
2
@t
@t
i
i
i=1
i=1
n
X1
2
!
dx
Z nX1
n 1
@
1 X @p
@u
@u @u
dx +
(t; x)
p(t; x)
@x
@x
@t
2
@t
@x
i
i
i
i=1
i=1
Z nX1
@u @u
p(t; s)
+
(t; s) ds:
i
@xi @t
@ i=1
=
2
De…ne the function M : R ! R by
M (t) =
1
2
Z
p (t; x)
@u
@t
2
dx + A(t) + B(t):
Obviously M is absolutely continuous and di¤erentiable in R, and
M 0 (t)
=
1
2
Z
Z
@u
@P
(t; x)
@t
@t
2
dx
Z
p (t; x)
@u @ 2 u
dx + A0 (t) + B 0 (t)
@t @t2
Z
1
@p
2
=
(t; x) jruj dx
dx +
2
@t
Z
Z nX1
@u @u
@F
(t; x; u) dx +
p(t; s)
(t; s) ds
+
i
@t
@x
@t
i
@ i=1
!
Z
n
X
@u
@
@u
@u
@
p (t; x)
p(t; x)
+ f (t; x; u)
dx:
+
@t
@t
@xi
@xi
@t
i=2
1
2
@P
@u
(t; x)
@t
@t
2
166
Absence of Nontrivial Solutions
Because u is solution of (2) and (3), we deduce that
!
Z
Z
2
@F
@u
@P
2
0
+ jruj dx +
M (t) =
(t; x)
(t; x; u) dx
@t
@t
@t
Z nX1
@u @u
(t; s) ds;
+
p(t; s)
i
@xi @t
@ i=1
while on the boundary,
Z
@
=
=
Z
n
X1
p(t; s)
i=1
@u
@xi
i
@u
(t; s) ds
@t
Z
@u @u
1
@u
p(t; s)
(t; s) ds =
p(t; s) u (t; s) ds
@t @
" @
@t
@
Z
Z
@p
1
1 d
(t; s)u2 (t; s) ds:
p(t; s)u2 (t; s) ds +
2" dt
2" @ @t
@
Also
d
dt
1
M (t) +
2"
Z
2
p(t; s)u (t; s) ds
=
@
1
2
Z
Z
@P
(t; x)
@t
@u
@t
2
2
+ jruj
!
dx
@F
(t; x; u) dx
@t
Z
1
@p
+
(t; s)u2 (t; s) ds:
2" @ @t
+
We set
1
N (t) = M (t) +
2"
Z
p(t; s)u2 (t; s) ds:
@
Conditions (2:1) imply that
N 0 (t)
0 (resp
0) ; 8t 2 R,
i.e., M is monotone. However, this function also satis…es
lim M (t) = 0
jtj!+1
because M 2 L2 (R) : Hence, M (t) = 0 for t 2 R, and this gives the desired result.
LEMMA 2. Let F and p verify (2:1). The solution of the Dirichlet problem (2) and
(4) or the Neumann problem (2) and (5), satis…es the following energy identity
1
2
Z
p (t; x)
@u
@t
2
dx +
1
2
Z
2
p(t; x) jruj dx +
Z
F (t; x; u)dx = 0:
K. AKROUT
167
PROOF. For the problem (2) and (4), the fact that u = 0 on the boundary implies
that
Z
Z
d
@u
@u @u
(t; s) ds =
p(t; s) u (t; s) ds
p(t; s)
@ @t
dt
@
@
@
Z
Z
@p @u
@2u
u (t; s) ds
u (t; s) ds
p(t; s)
@t@
@ @t @
@
= 0:
For the problem (2) and (5), the fact that
Z
p(t; s)
@
@u
@
= 0 on the boundary implies that
@u @u
(t; s) ds = 0:
@ @t
The remainder of the proof is similar to that of Lemma 1.
LEMMA 3. Let
and pk satisfy
@Fm
(t; x; u1 ; :::; um )
@t
@pk
(t; x)
@t
0 (resp
0 (resp
0) ; 1
k
0) ; 8 (t; x) 2 R
;
m; 8 (t; x) 2 R
:
Then any solution of the system (6) and (7) satis…es the following energy identity
m Z
m Z
2
1X
1X
@uk
2
pk (t; x)
(t; x) dx +
pk (t; x) jruk j dx
2
@t
2
k=1
k=1
Z
m Z
X
1
+
Fm (t; x; u1 ; :::; um )dx +
pk (t; s)u2k (t; s) ds = 0:
2"
@
(12)
k=1
LEMMA 4. Let and pk verify (12). Then the solutions of the systems (6) and (8),
or (6) and (9) satisfy the following equality
m Z
m Z
2
1X
1X
@uk
2
(t; x) dx +
pk (t; x)
pk (t; x) jruk j dx
2
@t
2
k=1
Z k=1
+
Fm (t; x; u1 ; :::; um )dx = 0:
PROOF. Let us de…ne the function Mm : R ! R by
m
Mm (t) =
1X
2
k=1
Z
2
pk (t; x)
@uk
(t; x) dx + Am (t) + Bm (t);
@t
(13)
168
Absence of Nontrivial Solutions
where the functions Am and Bm are de…ned as follows
Am (t) =
1
2
Bm (t) =
Z X
m
Z
k=1
2
pk (t; x) jruk j dx for t 2 R;
Fm (t; x; u1 ; :::; um )dx for t 2 R:
The rest of the proof is similar to the proofs of the preceding lemmas.
3
The Main Results
In this section, we present three main theorems.
THEOREM 1. Let us suppose that F and f verify
2F (t; x; u)
uf (t; x; u)
0;
(14)
and (10) holds. Then the problem (2) and (3) admit only the null solution.
PROOF. Let us de…ne the function E by
Z
E (t) =
p (t; x) u2 (x; t) dx:
Multiplying equation (2) by u and integrating the new equation on
Z "
=
Z "
n
X1
@
@t
@u
p (t; x)
@t
1
2
@ 2 u2
@p @ u2
+ p (t; x)
@t @t
@t2
i=1
@
@xi
i
+p (t; x) jruj + uf (x; u) dx
2
=
=
@u
p (t; x)
@xi
!
i=1
#
+ f (t; x; u) udx
+ p (t; x)
n
X1 Z
@
p (t; s)
, we obtain
@u
@t
2
@u
u (t; s)
@xi
i ds
! Z
!
Z
2
@ u2
1 d
@u
2
p (t; x)
dx +
p (t; x)
+ jruj dx
2 dt
@t
@t
Z
Z
@u
u (t; s) ds = 0
+
uf (x; u) dx
p (t; s)
@
@
!
!
Z
Z
2
@ u2
1 d
@u
2
p (t; x)
dx +
p (t; x)
+ jruj dx
2 dt
@t
@t
Z
Z
1
p (t; s) u2 (t; s) ds = 0 :
+
uf (x; u) dx +
" @
K. AKROUT
169
Using identity (11), we have
!
Z
Z
@ u2
@u
d
p (t; x)
dx = 4
p (t; x)
dt
@t
@t
2
dx
2
Z
(2F (x; u)
uf (x; u)) dx:
The assumption (3:1) implies that
Z
d
dt
!
@ u2
p (t; x)
dx
@t
0; 8t 2 R:
We conclude that the function
K (t) =
Z
p (t; x)
@ u2
dx
@t
is monotone. But, this function veri…es
lim K (t) = 0;
jtj!+1
witch implies that
K (t) = 0; 8t 2 R;
In addition
E 0 (t)
Z
@ u2
@p
(t; x) u2 (t; x) dx +
p (t; x)
dx
@t
@t
Z
@p
=
(t; x) u2 (t; x) dx + K (t)
@t
Z
@p
=
(t; x) u2 (t; x) dx:
@t
=
Z
The condition (10) implies that
E 0 (t)
0 (resp
0) ;
i.e. E is monotone. But, this function veri…es
lim E (t) = 0;
jtj!+1
witch implies that
E (t) = 0; 8t 2 R;
We deduce that u
0 in R
.
THEOREM 2. Let F and f verify (14) and assume (10) holds. Then the only
solution of the problems (2) and (4), or (2) and (5) is the null solution.
PROOF. The proof is identical to that of Theorem 1 using appropriate lemmas.
170
Absence of Nontrivial Solutions
THEOREM 3. Let us suppose that Fm and fk ; 1
2Fm (x; u1 ; :::; um )
m
X
k
m, satisfy
uk fk (x; u1 ; :::; um )
0;
k=1
and (12) holds. Then the system (6) and (7) admits only the null solutions.
PROOF. Let us de…ne the functions Em and Km by
Z
Z
@
u2 (t; x) dx:
Em =
pk (t; x) u2k (t; x) dx and Km =
pk (t; x)
@t k
Multiplying equation (6) by uk , integrating the new equation on , and summing on
k from 1 to m, one obtain
!
Z
m Z
2
X
@ u2k
@uk
1 d
pk (t; x)
dx +
(t; x) dx
pk (t; x)
2 dt
@t
@t
k=1
Z
Z
m
m
X
X
2
+
pk (t; x) jruk j dx +
uk (t; x) fk (x; u1 ; :::; um ) dx
+
k=1
m Z
X
1
"
k=1
k=1
pk (t; s) u2k (t; s) ds = 0:
@
By using identity (13), we deduce that
0
Km
=
4
m Z
X
pk (t; x)
2
2Fm (x; u1 ; :::; um )
2
k=1
Z
@uk
(t; x) dx
@t
m
X
uk (t; x) fk (x; u1 ; :::; um ) dx:
k=1
Then, the assumption (3:4) implies that
Km (t) = 0; 8t 2 R:
So
Em (t) = 0; 8t 2 R:
and this gives the desired result.
4
Applications
EXAMPLE 1. Let p; q;
1, m 2 R and
'0 ; '1 ; '2 :
!
! R;
K. AKROUT
171
be nonnegative functions of class C 1 (R) such that
m
@'i
@x1
0, 8i = 0; 1; 2,
f (x; u) = mu + '1 (x) juj
Then the problem de…ned by
8
n
P
<
@
i=1
:
@xi
p 1
u + '2 (x) juj
q 1
u:
@u
'0 (x) @x
+ f (x; u) = 0 in R
i
@u
u + " @n
(y; ) = 0 on R
;
@ ;
admits only the null solution.
In this case we have
F (x; u) =
1
1
p
mu2 +
' (x) juj
2
p+1 1
1
1 @'1
@F
1
p
(x; u) = mu2 +
(x) juj
@x1
2
p + 1 @x1
It su¢ ces to check that
2F (x; u)
@F
(x; u)
@x1
u+
1
q
' (x) juj
q+1 2
1
u+
0 if m
0;
1
u;
1 @'2
q
(x) juj
q + 1 @x1
@F
(x; u) 0 if m 0;
@x1
2
2
p+1
1) juj
+ '2 (x) (
uf (x; u) = '1 (x) (
p+1
q+1
1
u:
q+1
1) juj
0;
and apply Theorem 1.
EXAMPLE 2. Let be a bounded open of set Rn ; p; q 1;Then, the system
8
p 1
q+1
u + (p + 1) (x) u juj
jvj
= 0 in R
;
<
q 1
p+1
v + (q + 1) (x) v jvj
juj
= 0 in R
;
:
@u
@v
u + " @n
(t; ) = v + " @n
(t; ) = 0 on R @ ;
where
:
! R; is nonnegative,
@
@x1
0 (resp
0)
admits only the trivial solutions, u v 0:
Indeed, there exist a function F de…ned as follows
F (x; u; v) = (x) juj
which satis…es
p+1
jvj
q+1
@F
p
= f1 (x; u; v) = (p + 1) (x) u juj
@u
;
1
jvj
q+1
;
172
Absence of Nontrivial Solutions
@F
q 1
p+1
= f2 (x; u; v) = (q + 1) (x) v jvj
juj
;
@v
@F
@
p+1
q+1
=
(x) juj
jvj
0 (resp
0) ;
@x1
@x1
2F (x; u; v)
uf1 (x; u; v)
vf2 (x; u; v) =
p+1
(x) (p + q) juj
q+1
jvj
0:
Theorem 3 gives the result.
References
[1] K. Akrout and B. Khodja, Absence of nontrivial solutions for a class of partial differential equations and systems in unbounded domains, Electron. J. Qual. Theory
Di¤er. Equ., 2009, 10 pp.
[2] S. Benmehidi and B. Khodja, Some results of nontrivial solutions for a nonlinear
PDE in Sobolev space, Electron. J. Qual. Theory Di¤er. Equ. 2009, No. 44, 14 pp.
[3] H. Brezis, Analyse Fonctionnelle. Théorie et Applications, Collection Mathématiques Appliquées Pour La Maîtrise, Masson, Paris, 1983.
[4] M. J. Esteban and P. Lions, Existence and non-existence results for semi linear
elliptic problems in unbounded domains, Proc. Roy. Soc. Edimburgh. 93-A(1982),
1–14.
[5] N. Kawarno, W. M. Ni, and S. Yotsutani, Syotsutani, A generalized Pohozaev
identity and its applications, J. Math. Soc. Japan, 42(1990), 541–564.
[6] B. Khodja and A. Moussaoui, Nonexistence result for semilinear systems in unbounded domains, Electron. J. Di¤erential Equations 2009, 11 pp.
[7] B. Khodja, Nonexistence of solutions for semilinear equations and systems in cylindrical domains, Comm. Appl. Nonlinear Anal., 7(2000), 19–30.
[8] W. M. Ni. and J. Serrin, Nonexistence theorems for quasilinear partial di¤erential
equations. Proceedings of the conference commemorating the 1st centennial of
the Circolo Matematico di Palermo (Italian) (Palermo, 1984). Rend. Circ. Mat.
Palermo (2) Suppl. No. 8(1985), 171–185.
[9] S. I. Pohozaev, Eeigenfunctions of the equation
viet.Math.Dokl., (1965), 1408–1411.
u +
f (u) = 0; So-
[10] R. C. A. M. Van Der Vorst, Variational identities and applications to di¤erential
systems, Arch. Rational Mech. Anal., 116(1992), 375–398.
[11] C. S. Yarur, Nonexistence of positive singular solutions for a class of semilinear
elliptic systems, Electron. J. Di¤erential Equations 1996, No. 08, approx. 22 pp.