Nonexistence of nontrivial solutions for the px-Laplacian equations and systems in unbounded domains of Rn Boundary Value Problems 2011, 2011:50 doi:10.1186/1687-2770-2011-50 Akrout Kame
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Nonexistence of nontrivial solutions for the p(x)-Laplacian equations and
systems in unbounded domains of Rn
Boundary Value Problems 2011, 2011:50 doi:10.1186/1687-2770-2011-50
Akrout Kamel (akroutkamel@gmail.com)
ISSN 1687-2770
Article type Research
Article URL http://www.boundaryvalueproblems.com/content/2011/1/50
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Boundary Value Problems
© 2011 Kamel ; licensee Springer.
Trang 2Nonexistence of nontrivial solutions for the p (x) −Laplacian
AKROUT Kamel∗1
1 Department of mathematics and informatics.Tebessa university Algeria.
Email: AKROUT Kamel∗- akroutkamel@gmail.com;
∗Corresponding author
Abstract
In this paper, we are interested on the study of the nonexistence of nontrivial solutions for the p (x) −Laplacian
equations, in unbounded domains of Rn This leads us to extend these results to m-equations systems The method used is based on pohozaev type identities
1 Introduction
Several works have been reported by many authors, comprise results of nonexistence of nontrivial solutions
of the semilinear elliptic equations and systems, under various situations, see [1-8] The Pohozaˇev identity [1] published in 1965 for solutions of the Dirichlet problem proved absence of nontrivial solutions for some elliptic equations of the form ½
−∆u + f (u) = 0 in Ω,
u = 0 on ∂Ω,
when Ω is a star shaped bounded open domain in Rn and f is a continuous function on R satisfying
(n − 2)F (u) − 2nuf (u) > 0,
Trang 3A Hareux and B Khodja [2] established under the assumption
f (0) = 0,
2F (u) − uf (u) ≤ 0.
that the problems ½
−∆u + f (u) = 0 in J × ω,
¡
u or ∂u
∂n
¢
= 0 on ∂ (J × ω) admit only the null solution in H2(J × ω) ∩ L ∞ (J × ω) where J is an interval of R and ω is a connected
unbounded domain of RN such as
∃Λ ∈ R N , kΛk = 1, hn(x), Λi ≥ 0 on ∂ω, hn(x), Λi 6= 0,
(n(x) is the outward normal to ∂ω at the point x)
In this work we are interested in the study of the nonexistence of nontrivial solutions for the p (x)-laplacian
problem
½
−∆ p(x) u = H (x) f (u) in Ω
with
Bu =
½
u Dirichlet condition (1.2)
∂u
∂ν Neumann condition (1.3)
where
∆p(x) u = div
³
|∇u| p(x)−2 ∇u
´
Ω is bounded or unbounded domains of Rn , f is a locally lipshitzian function, H and p are given continuous
real functions of C¡Ω¢verifying
F (t) =R0t f (σ) dσ, f (0) = 0,
H (x) > 0, (x, ∇H (x)) 6= 0 and lim
|x|→+∞ H (x) = 0,
p (x) > 1, (x, ∇p (x)) ≥ 0, ∀x ∈ Ω,
a = sup
x∈Ω
³
1 − n p(x) +(x,∇p(x)) p2(x)
´
.
(1.4)
(., ) is the inner product in R n
We extend this technique to the system of m−equations
½
−∆ p k (x) u = H (x) f k (u1, , u m ) in Ω, 1 ≤ k ≤ m,
Bu k = 0 on ∂Ω, 1 ≤ k ≤ m, (1.6)
with
Bu k =
½
u k Dirichlet condition (1.7)
∂u k
∂ν Neumann condition (1.8)
Trang 4Where {f k } are locally lipshitzian functions verify
f k (s1, , s k−1 , 0, u k+1 , , s m ) = 0, (0 ≤ k ≤ m) ,
∃F m: Rm → R : ∂F m
∂s k (s1, , s m ) = f k (s1, , s m )
H is previously defined and p k functions of C1¡
Ω¢class, verify
p k (x) > 1, (x, ∇p k (x)) ≥ 0, ∀x ∈ Ω.
a k = sup
x∈Ω
³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
´
(1.9)
2 Integral identities
Let
L p(x)(Ω) =
½
u measurable real function : R
Ω
|u (x)| p(x) dx < +∞
¾
,
with the norm
|u| L p(x)(Ω)= |u| p(x) = inf
½
λ > 0 :R
Ω
¯
¯u(x) λ
¯
¯p(x) dx ≤ 1
¾
,
and
W 1,p(x)(Ω) =©u ∈ L p(x) (Ω) : |∇u| ∈ L p(x)(Ω)ª,
with the norm
kuk W 1,p(x)(Ω)= |u| L p(x)(Ω)+ |∇u| L p(x)(Ω).
Denote W01,p(x) (Ω) the closure of C ∞
0 (Ω) in W 1,p(x) (Ω) ,
Lemma 1 Let u ∈ W01,p(x) (Ω) ∩ L ∞¡
Ω¢solution of the equation (1.1) − (1.2), we have
R
Ω
h³
1 − n p(x)+(x,∇p(x)) p2(x)
³
1 − ln
³
|∇u| p(x)
´´
− a
´
|∇u| p(x) dx
+H (x) (nF (u) − auf (u)) + (x, ∇H (x)) F (u)] dx
= R
∂Ω
³
1 − 1
p(x)
´
|∇u| p(x) (x, ν) ds
(2.1)
Lemma 2 Let u ∈ W01,p(x) (Ω) ∩ L ∞¡
Ω¢solution of the equation (1.1) − (1.3), we have
R
Ω
h³
1 − n p(x)+(x,∇p(x)) p2(x)
³
1 − ln³|∇u| p(x)´´− a´|∇u| p(x) dx
+H (x) (nF (u) − auf (u)) + (x, ∇H (x)) F (u)] dx
= R
∂Ω
³³
1 − 1
p(x)
´
|∇u| p(x) + H (x) F (u)´(x, ν) ds
(2.2)
Trang 5Proof Multiplying the equation (1.1) by Pn
j=1
x i ∂x ∂u i and integrating the new equation by parts in Ω ∩ B R ,
B R = B (0, R)
Ω∩B R
div³|∇u| p(x)−2 ∇u´
Ã
n
P
j=1
x j ∂x ∂u j
!
dx
= − Pn
i,j=1
R
Ω∩B R
∂
∂x i
³
|∇u| p(x)−2 ∂u ∂x
i
´
x j ∂x ∂u j dx
= R
Ω∩B R
"
|∇u| p(x) + |∇u| p(x)−2 Pn
i,j=1
x j ∂x ∂u i ∂
2u
∂x i ∂x j
#
dx
− Pn
i,j=1
R
∂(Ω∩B R)
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i ds
Introducing the following result
|∇u| p(x)−2Pn
i=1
∂u
∂x i
∂2u
∂x i ∂x j = 1
p(x) ∂x ∂ j
³
|∇u| p(x)´−
∂p
∂xj
p2(x) |∇u| p(x)ln³|∇u| p(x)´
we have
R
Ω∩B R
"
|∇u| p(x)+Pn
j=1
x j
p(x)
∂
∂x j
³
|∇u| p(x)
´
−Pn
j=1
(x,∇p(x))
p2(x) |∇u| p(x)ln
³
|∇u| p(x)
´
dx
∂(Ω∩B R)
n
P
i,j=1
R
∂(Ω∩B R)
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i ds
= R
Ω∩B R
h
1 − n p(x)+(x,∇p(x)) p2(x)
³
1 − ln
³
|∇u| p(x)
´´i
|∇u| p(x) dx
∂(Ω∩B R)
Ã
n
P
i,j=1
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i −Pn
j=1
1
p(x) |∇u| p(x) x j ν j
!
ds
On the other hand
R
Ω∩B R
H (x) f (u)
Ã
n
P
j=1
x j ∂x ∂u j
!
dx = Pn
j=1
R
Ω∩B R
x j H (x) ∂
∂x j (F (u)) dx
= − R
Ω∩B R
(nH (x) + (x, ∇H (x))) F (u) dx + Pn
j=1
R
∂(Ω∩B R)
H (x) F (u) x j ν j ds
these results conduct to the following formula
R
Ω∩B R
h³
1 − n p(x) +(x,∇p(x)) p2(x)
³
1 − ln
³
|∇u| p(x)
´´´
|∇u| p(x) dx
+ (nH (x) + (x, ∇H (x))) F (u)] dx
= R
∂(Ω∩B R)
"
n
P
i,j=1
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i
−Pn
j=1
³
1
p(x) |∇u| p(x) − H (x) F (u)
´
x j ν j
#
ds
(2.3)
Multiplying the present equation (1.1) by au and integrating the obtained equation by parts in Ω,we obtain
R
(Ω∩B R)
h
a |∇u| p(x) − auH (x) f (u)
i
dx = R
∂(Ω∩B R)
a |∇u| p(x) ∂u ∂ν uds = 0, (2.4)
Trang 6Combining (2.3) and (2.4) we obtain
R
Ω∩B R
h³
1 − n p(x)+(x,∇p(x)) p2(x)
³
1 − ln
³
|∇u| p(x)
´´
− a
´
|∇u| p(x) dx
+H (x) (nF (u) − auf (u)) + (x, ∇H (x)) F (u)] dx
= R
∂(Ω∩B R)
"
n
P
i,j=1
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i
−Pn
j=1
³
1
p(x) |∇u| p(x) − H (x) F (u)
´
x j ν j
#
ds
= R
∂Ω∩B R
"
n
P
i,j=1
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i
−Pn
j=1
³
1
p(x) |∇u| p(x) − H (x) F (u)´x j ν j
#
ds
+ R
Ω∩∂B R
"
n
P
i,j=1
|∇u| p(x)−2 ∂u ∂x
i
∂u
∂x j x j ν i
−Pn
j=1
³
1
p(x) |∇u| p(x) − H (x) F (u)
´
x j ν j
#
ds
On (Ω ∩ ∂B R ) we have n i= x i
|x|
so the last integral is major by
M (R) = R R
Ω∩∂B R
³³
1 + 1
p(x)
´
|∇u| p(x) + |H (x)| |F (u)|
´
ds
We remark that if Ω in bounded, so for R is little greater, we get Ω ∩ ∂B R = φ, then M (R) = 0.
If Ω is not bounded, such as |∇u| ∈ W 1,p(x) (Ω), F (u) ∈ L1(Ω) and lim
|x|→+∞ H (x) → 0, we should see
+∞R
0
dr R
Ω∩∂B R
³³
1 + 1
p(x)
´
|∇u| p(x) + |H (x)| |F (u)|´ds < +∞
consequently we can always find a sequence (R n)n, such as
lim
n→+∞ M (R n ) → 0.
In the problem (1.1) − (1.2) , u| ∂Ω = 0.Then, ∇u = ∂u
∂ν n, we obtain the identity (2.1)
In the problem (1.1) − (1.3) , ∂u
∂ν
¯
∂Ω = 0,we obtain the identity (2.2) Lemma 3 Let u k ∈ W 1,p k (x)
0 (Ω) ∩ L ∞¡
Ω¢(1 ≤ k ≤ m) , solutions of the system (1.6) − (1.7) Then for the
constants a k of R, we have
R
Ω
· m P
k=1
³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln
³
|∇u k | p k (x)´´
− a k
´
|∇u k | p k (x)
+H (x)
µ
nF m (u1, , u m ) − Pm
k=1
a k u k f k (u1, , u m)
¶ +
+ (x, ∇H (x)) F m (u1, , u m )] dx
= R
∂Ω
m
P
k=1
³
1 − 1
p k (x)
´
|∇u k | p k (x) (x, ν) ds
(2.5)
Trang 7Lemma 4 Let u k ∈ W01,p (Ω) ∩ L ∞¡
Ω¢ (1 ≤ k ≤ m) , solutions of the system (1.6) − (1.8) Then for the
constants a k of R, we have
R
Ω
· m P
k=1
³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln
³
|∇u k | p k (x)´´
− a k
´
|∇u k | p k (x)
+H (x)
µ
nF m (u1, , u m ) − Pm
k=1
a k u k f k (u1, , u m)
¶ +
+ (x, ∇H (x)) F m (u1, , u m )] dx
= R
∂Ω
·m P
k=1
³
1 − 1
p k (x)
´
|∇u k | p k (x)
+ H (x) F m (u1, , u m)
¸
(x, ν) ds
(2.6)
Proof Multiplying the equation (1.6) by Pn
j=1
x i ∂u ∂x k i and integrating the new equation by part in Ω ∩ B R ,
B R = B (0, R), we get
R
Ω∩B R
h
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln³|∇u k | p k (x)´´i
|∇u k | p k (x) dx
= R
∂(Ω∩B R)
Ã
n
P
i,j=1
|∇u k | p k (x)−2 ∂u k
∂x i
∂u k
∂x j x j ν i − Pn
j=1
1
p k (x) |∇u k | p k (x)
x j ν j
!
ds
On the other hand
R
Ω∩B R
H (x) f k (u1, , u m)
Ã
n
P
j=1
x j ∂u ∂x k j
!
dx
= Pn
j=1
R
Ω∩B R
x j H (x) ∂u k
∂x j
∂
∂u k (F m (u1, , u m )) dx
These results conduct to the following formula
R
Ω∩B R
h³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln
³
|∇u k | p k (x)´´´
|∇u k | p k (x)
+Pn
j=1
x j H (x) ∂u k
∂x j
∂
∂u k (F m (u1, , u m))
#
dx
= R
∂(Ω∩B R)
"
n
P
i,j=1
|∇u k | p k (x)−2 ∂u k
∂x i
∂u k
∂x j x j ν i
−Pn
j=1
1
p k (x) |∇u k | p k (x)
x j ν j
#
ds
Doing the sum on k of 1 to m, we obtain
R
Ω∩B R
m
P
k=1
h³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln
³
|∇u k | p k (x)´´´
|∇u k | p k (x)
+Pn
j=1
x j H (x) ∂
∂x j F m (u1, , u m)
#
dx
= R
∂(Ω∩B R)
"
m
P
k=1
n
P
i,j=1
|∇u k | p k (x)−2 ∂u k
∂x i
∂u k
∂x j x j ν i
+Pm
k=1
n
P
j=1
1
p k (x) |∇u k | p k (x) x j ν j
#
ds
Trang 8which leads to the following identity
R
Ω∩B R
m
P
k=1
h³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln³|∇u k | p k (x)´´´
|∇u k | p k (x)
− (nH (x) + (x, ∇H (x))) F m (u1, , u m )] dx
= R
∂(Ω∩B R)
"
m
P
k=1
n
P
i,j=1
|∇u k | p k (x)−2 ∂u k
∂x i
∂u k
∂x j x j ν i
+
µm P
k=1
1
p k (x) |∇u k | p k (x) + H (x) F m (u1, , u m)
¶
(x, ν)
¸
ds
(2.7)
Now, multiply the equation (1.1) by au and integrating the obtained equation by parts in Ω ∩ B R
R
(Ω∩B R)
h
a k |∇u| p k (x)
− a k u k H (x) f k (u1, , u m)idx = 0 (2.8)
Combining (2.7) and (2.8), we get the identities (2.5) and (2.6).
The rest of the proof is similar to the that of lemma 1
3 Principal Result
theorem 3.1 If u ∈ W01,p(x) (Ω) ∩ L ∞¡
Ω¢be a solution of the problem (1.1) − (1.2), Ω is star shaped and that a, H, f and F verify the following assumptions
(3.1) nF (u) − auf (u) ≤ 0, ∀x ∈ Ω, (3.2) (x, ∇H (x)) F (u) ≤ 0, ∀x ∈ Ω.
Then, the problem admits only the null solution.
Proof Ω is star shaped, imply that
R
∂Ω
³
1 − 1
p(x)
´
|∇u| p(x) (x, ν) ds ≥ 0 (3.3)
On the other hand, the condition (3.1) give
R
Ω
h³
1 − n p(x)+(x,∇p(x)) p2(x)
³
1 − ln
³
|∇u| p(x)
´´
− a
´
|∇u| p(x) dx
+H (x) (nF (u) − auf (u)) + (x, ∇H (x)) F (u)] dx ≤ 0 (3.4)
(1.4) , (3.3) and (3.4) , allow to get
F (u) = 0 in Ω.
So, the problem (1.1) − (1.2) becomes
(
−div³|∇u| p(x)−2 ∇u´= 0 in Ω,
Trang 9Multiplying the equation (3.5) by u and integrating over Ω,we get
R
Ω
|∇u| p(x) dx = 0.
So
|∇u| = 0,
Hence u = cte = 0, because u| ∂Ω= 0
theorem 3.2 If u ∈ W01,p(x) (Ω) ∩ L ∞¡
Ω¢solution of the problem (1.1) − (1.3), Ω is a star shaped and that
a, H, f and F verify the following conditions
(3.6) nF (u) − auf (u) ≤ 0, ∀x ∈ Ω, (3.7) (x, ∇H (x)) F (u) ≤ 0, ∀x ∈ Ω.
(3.8) H (x) F (u) ≥ 0 , ∀x ∈ ∂Ω.
Therefore, the problem admits only the null solution.
Proof Similar to the proof of theorem 1
theorem 3.3 If u k ∈ W 1,p k (x)
0 (Ω) ∩ L ∞¡
Ω¢solution of the system (1.6) − (1.7), Ω is a star shaped and that
a k , H, f k and F m verify the following conditions
(3.9) nF m (u1, , u m ) − Pm
k=1
a k u k f k (u1, , u m ) ≤ 0 , ∀x ∈ Ω, (3.10) (x, ∇H (x)) F m (u1, , u m ) ≤ 0 , ∀x ∈ Ω.
So, the system admits only the null solutions.
Proof Ω is a star shaped, implies that
R
∂Ω
m
P
k=1
³
1 − 1
p k (x)
´
|∇u k | p k (x)
(x, ν) ds ≥ 0 (3.11)
On the other hand, the conditions (3.9) and (3.10) , give
R
Ω
· m P
k=1
³
1 − n
p k (x)+(x,∇p k (x))
p2
k (x)
³
1 − ln
³
|∇u k | p k (x)´´
− a k
´
|∇u k | p k (x)
+H (x)
µ
nF m (u1, , u m ) − Pm
k=1
a k u k f k (u1, , u m)
¶ +
+ (x, ∇H (x)) F m (u1, , u m )] dx ≤ 0.
(3.12)
(1.4) , (3.11) and (3.12) , allow to have
F m (u1, , u m ) = 0 in Ω.
Trang 10So the system (1.6) − (1.7) becomes
(
−div
³
|∇u k | p k (x)−2
∇u k
´
= 0 in Ω, 1 ≤ k ≤ m,
u k = 0 on ∂Ω, 1 ≤ k ≤ m (3.13) Multiplying (3.13) by u k and integrating on Ω, we have
R
Ω
|∇u k | p k (x)
dx = 0
So
|∇u k | = 0
Therefore u k = cte = 0, ∀1 ≤ k ≤ m, because u k | ∂Ω= 0
theorem 3.4 If u k ∈ W 1,p k (x)
0 (Ω) ∩ L ∞¡
Ω¢solution of the system (1.6) − (1.8), Ω is a star shaped and that
a k , H, f k and F m verify the following conditions
(3.14) nF m (u1, , u m ) − Pm
k=1
a k u k f k (u1, , u m ) ≤ 0 , ∀x ∈ Ω, (3.15) (x, ∇H (x)) F m (u1, , u m ) ≤ 0 , ∀x ∈ Ω,
(3.16) H (x) F m (u1, , u m ) ≥ 0 , ∀x ∈ ∂Ω.
So, the problem admit only the null solution.
Proof Similar to the that of theorem 3
4 Examples
Example 1 Considering in W01,p(x) (Ω) ∩ W01,q¡Ω¢ the following problem
(
− div³|∇u| p(x)−2 ∇u´= c
(1+|x|) µ u |u| q−1 in Ω,
where Ω is a bounded domain of R n , c, µ > 0, q > 1 and p (x) =
q
1 + |x|2> 1.
By choosing
a = sup
Ω
µ
1 − n+(n−1)|x|2
(1+|x|2)√ 1+|x|2
¶
,
we obtain
(x, ∇H (x)) F (u) = q(1+|x|) −cµ|x| µ+1 |u| q+1 < 0,
(x, ∇p (x)) = √ |x|2
1+|x|2 ≥ 0,
nF (u) − auf (u) =³ n
q+1 − a´|u| q+1 ≤ 0 if q ≥ n−a
a
So, the problem (4.1) doesn’t admit non trivial solutions if
q ≥ n−a
a
Trang 11Example 2 Considering in W01,p(x) (Ω) ∩ W01,γ¡Ω¢, the following elliptic system
−∆ p(x) u = (1+|x|) cγ µ u |u| γ−1 |v| δ in Ω,
−∆ q(x) v = cδ
(1+|x|) µ v |v| δ−1 |u| γ in Ω,
u = 0 on ∂Ω
(4.2)
where Ω is a bounded domain of R n , c, µ, γ, δ > 0 and p, q > 1.
By choosing
a1= sup
x∈Ω
³
1 − n p(x)+(x,∇p(x)) p2(x)
´
and
a2= sup
x∈Ω
³
1 − n p(x)+(x,∇q(x)) q2(x)
´
we obtain
(x, ∇H (x)) F (u, v) = (1+|x|) −cµ µ+1 |u| γ |v| δ < 0,
nF (u, v) − a1uf1(u, v) − a2vf2(u, v) = (n − γa1− δa2) |u| γ |v| δ
So, the system (4.2) doesn’t admit non trivial solutions if
γa1+ δa2≥ n
Competing interests
The author declares that they have no competing interests
References
1 S I Pohozaev, Eeigenfunctions of the equation ∆u + λf (u) = 0, Soviet.Math.Dokl.(1965), 1408-1411.
2 A Haraux & B Khodja, Caract`ere triviale de la solution de certaines ´equations aux d´eriv´ees partielles
non lin´eaires dans des ouverts cylindriques de R N , Portugaliae Mathematica, Vol.42, Fasc.2,1982,1–9.
3 M J Esteban & P Lions, Existence and non-existence results for semi linear elliptic problems in
unbounded domains, Proc.Roy.Soc.Edimburgh 93-A(1982),1-14.
4 N Kawarno, W NI & Syotsutani, Generalised Pohozaev identity and its applications J Math Soc Japan Vol 42 N◦3 (1990), 541-563
5 B Khodja, Nonexistence of solutions for semilinear equations and systems in cylindrical domains Comm Appl Nonlinear Anal (2000), 19-30
Trang 126 W NI & J Serrin, Nonexistence thms for quasilinear partial differential equations Red Circ Mat Palermo, suppl Math 8 (1985), 171-185
7 R C A M Van Der Vorst, Variational identities and applications to differential systems,
Arch.Rational; Mech.Anal.116 (1991) 375-398
8 C Yarur, Nonexistence of positive singular solutions for a class of semilinear elliptic systems Electronic Journal of Diff Equations, 8 (1996), 1-22.