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Nonexistence of nontrivial solutions for the p ( x )- Laplacian equations and systems in unbounded domains of n

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 n . 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 [18]. The Pohozaĕv identity [1] published in 1965 for solutions of the Dirichlet problem proved absence of nontrivial solutions for some elliptic equations of the form

- Δ u + f ( u ) = 0  in  Ω , u = 0  on  Ω ,

when Ω is a star shaped bounded open domain in n and f is a continuous function on satisfying

( n - 2 ) F ( u ) - 2 n u f ( u ) > 0 ,
  1. A.

    Hareux and B. Khodja [2] established under the assumption

    f ( 0 ) = 0 , 2 F ( u ) - u f ( u ) 0 .

that the problems

- Δ 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 and ω is a connected unbounded domain of N such as

Λ N , Λ  = 1 , n ( x ) , Λ 0 on  ω , n ( x ) , Λ 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

- Δ p ( x ) u = H ( x ) f ( u ) in  Ω B u = 0  on  Ω
(1.1)

with

B u = u Dirichlet condition 1 . 2 u ν Neumann condition 1 . 3

where

Δ p x u = d i v u p ( x ) - 2 u

Ω is bounded or unbounded domains of n , f is a locally lipshitzian function, H and p are given continuous real functions of C ( Ω ̄ ) verifying

F ( t ) = 0 t f ( σ ) d σ , f ( 0 ) = 0 , H ( x ) > 0 , ( x , H ( x ) ) 0  and  lim x  +  H ( x )  = 0 , p ( x ) > 1 , ( x , p ( x ) ) 0 , x Ω ̄ , a = sup x Ω ̄ 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) .
(1.4)

(., .) is the inner product in n .

We extend this technique to the system of m-equations

- Δ p k x u = H ( x ) f k ( u 1 , . . . , u m )  in  Ω , 1 k m , B u k = 0  on  Ω , 1 k m ,
(1.6)

with

B u k = u k Dirichlet condition  1 . 7 u k ν Neumann condition  1 . 8

Where {f k } are locally lipshitzian functions verify

f k ( s 1 , . . . , s k - 1 , 0 , u k + 1 , . . . , s m ) = 0 , ( 0 k m ) , F m : m : F m s k ( s 1 , . . . , s m ) = f k ( s 1 , . . . , s m ) .

H is previously defined and p k functions of C 1 ( Ω ¯ ) class, verify

p k ( x ) > 1 , ( x , p k ( x ) ) 0 , x Ω ̄ . a k = sup x Ω ̄ 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x )
(1.9)

2 Integral identities

Let

L p ( x ) ( Ω ) = u  measurable real function : Ω u ( x ) p ( x ) d x <  +  ,

with the norm

u L p ( x ) ( Ω ) = u p ( x ) = inf λ > 0 : Ω u ( x ) λ p ( x ) d x 1 ,

and

W 1 , p ( x ) ( Ω ) = { u L p x ( Ω ) : u L p x ( Ω ) } ,

with the norm

u W 1 , p ( x ) ( Ω ) = u L p ( x ) ( Ω ) + u L p ( x ) ( Ω ) .

Denote W 0 1 , p ( x ) ( Ω ) the closure of C 0 ( Ω ) in W1, p(x)(Ω),

Lemma 1 Letu W 0 1 , p x Ω L Ω ̄ solution of the equation (1.1) - (1.2), we have

Ω 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) - a u p ( x ) d x + H ( x ) ( n F ( u ) - a u f ( u ) ) + ( x , H ( x ) ) F ( u ) d x = Ω 1 - 1 p x u p ( x ) ( x , ν ) d s
(2.1)

Lemma 2 Letu W 0 1 , p ( x ) ( Ω ) L ( Ω ̄ ) solution of the equation (1.1) - (1.3), we have

Ω 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) - a u p ( x ) d x + H ( x ) ( n F ( u ) - a u f ( u ) ) + ( x , H ( x ) ) F ( u ) d x = Ω 1 - 1 p x u p ( x ) + H ( x ) F ( u ) ( x , ν ) d s
(2.2)

Proof Multiplying the equation (1.1) by j = 1 n x i u x i and integrating the new equation by parts in Ω ∩ B R , B R = B (0, R)

- Ω B R d i v u p ( x ) - 2 u j = 1 n x j u x j d x = - i , j = 1 n Ω B R x i u p ( x ) - 2 u x i x j u x j d x = Ω B R u p ( x ) + u p ( x ) - 2 i , j = 1 n x j u x i 2 u x i x j d x - i , j = 1 n ( Ω B R ) u p ( x ) - 2 u x i u x j x j ν i d s

Introducing the following result

u p ( x ) - 2 i = 1 n u x i 2 u x i x j = 1 p ( x ) x j u p ( x ) - p x j p 2 ( x ) u p ( x ) ln u p ( x )

we have

Ω B R u p ( x ) + j = 1 n x j p ( x ) x j u p ( x ) - j = 1 n ( x , p ( x ) ) p 2 ( x ) u p ( x ) ln u p ( x ) d x - ( Ω B R ) i , j = 1 n ( Ω B R ) u p ( x ) - 2 u x i u x j x j ν i d s = Ω B R 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) u p ( x ) d x - ( Ω B R ) i , j = 1 n u p ( x ) - 2 u x i u x j x j ν i - j = 1 n 1 p ( x ) u p ( x ) x j ν j d s

On the other hand

Ω B R H ( x ) f ( u ) j = 1 n x j u x j d x = j = 1 n Ω B R x j H ( x ) x j ( F ( u ) ) d x = - Ω B R ( n H ( x ) + ( x , H ( x ) ) ) F ( u ) d x + j = 1 n ( Ω B R ) H ( x ) F ( u ) x j ν j d s

these results conduct to the following formula

Ω B R 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) u p x d x + ( n H ( x ) + ( x , H ( x ) ) ) F ( u ) d x = ( Ω B R ) i , j = 1 n u p ( x ) - 2 u x i u x j x j ν i - j = 1 n 1 p ( x ) u p ( x ) - H ( x ) F ( u ) x j ν j d s
(2.3)

Multiplying the present equation (1.1) by au and integrating the obtained equation by parts in Ω, we obtain

( Ω B R ) a u p ( x ) - a u H ( x ) f ( u ) d x = ( Ω B R ) a u p ( x ) u ν u d s = 0 ,
(2.4)

Combining (2.3) and (2.4) we obtain

Ω B R 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) - a u p ( x ) d x + H ( x ) ( n F ( u ) - a u f ( u ) ) + ( x , H ( x ) ) F ( u ) d x = ( Ω B R ) i , j = 1 n u p ( x ) - 2 u x i u x j x j ν i - j = 1 n 1 p ( x ) u p ( x ) - H ( x ) F ( u ) x j ν j d s = Ω B R i , j = 1 n u p ( x ) - 2 u x i u x j x j ν i - j = 1 n 1 p ( x ) u p ( x ) - H ( x ) F ( u ) x j ν j d s + Ω B R i , j = 1 n u p ( x ) - 2 u x i u x j x j ν i - j = 1 n 1 p ( x ) u p ( x ) - H ( x ) F ( u ) x j ν j d s

On (Ω ∩ ∂B R ) we have n i = x i x

so the last integral is major by

M ( R ) = R Ω B R 1 + 1 p ( x ) u p ( x ) + H ( x ) F ( u ) d s

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| W1, p(x)(Ω), F(u) L1 (Ω) and lim x + H ( x ) 0, we should see

0 + d r Ω B R 1 + 1 p ( x ) u p ( x ) + H ( x ) F ( u ) d s < +

consequently we can always find a sequence (R n ) n , such as

lim n + R n + and 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 0 1 , p k ( x ) ( Ω ) L ( Ω ̄ ) ( 1 k m ) , solution of the system (1.6) - (1.7). Then for the constants a k of , we have

Ω k = 1 m 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) - a k u k p k ( x ) + H ( x ) n F m ( u 1 , . . . , u m ) - k = 1 m a k u k f k ( u 1 , . . . , u m ) + + ( x , H ( x ) ) F m ( u 1 , . . . , u m ) d x = Ω k = 1 m 1 - 1 p k ( x ) u k p k ( x ) ( x , ν ) d s
(2.5)

Lemma 4 Let u k W 0 1 , p ( Ω ) L ( Ω ̄ ) ( 1 k m ) , solutions of the system (1.6) - (1.8). Then for the constants a k of , we have

Ω k = 1 m 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) - a k u k p k ( x ) + H ( x ) n F m ( u 1 , . . . , u m ) - k = 1 m a k u k f k ( u 1 , . . . , u m ) + + ( x , H ( x ) ) F m ( u 1 , . . . , u m ) ] d x = Ω k = 1 m 1 - 1 p k ( x ) u k p k ( x ) + H ( x ) F m ( u 1 , . . . , u m ) ( x , ν ) d s
(2.6)

Proof Multiplying the equation (1.6) by j = 1 n x i u k x i and integrating the new equation by part in Ω ∩ B R , B R = B (0, R), we get

Ω B R 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) u k p k ( x ) d x = ( Ω B R ) i , j = 1 n u k p k ( x ) - 2 u k x i u k x j x j ν i - j = 1 n 1 p k ( x ) u k p k ( x ) x j ν j d s

On the other hand

Ω B R H ( x ) f k ( u 1 , . . . , u m ) j = 1 n x j u k x j d x = j = 1 n Ω B R x j H ( x ) u k x j u k ( F m ( u 1 , . . . , u m ) ) d x

These results conduct to the following formula

Ω B R 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) u k p k ( x ) + j = 1 n x j H ( x ) u k x j u k ( F m ( u 1 , . . . , u m ) ) d x = ( Ω B R ) i , j = 1 n u k p k ( x ) - 2 u k x i u k x j x j ν i - j = 1 n 1 p k ( x ) u k p k ( x ) x j ν j d s

Doing the sum on k of 1 to m, we obtain

Ω B R k = 1 m 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) u k p k ( x ) + j = 1 n x j H ( x ) x j F m ( u 1 , . . . , u m ) d x = ( Ω B R ) k = 1 m i , j = 1 n u k p k ( x ) - 2 u k x i u k x j x j ν i + k = 1 m j = 1 n 1 p k ( x ) u k p k ( x ) x j ν j d s

which leads to the following identity

Ω B R k = 1 m 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) u k p k ( x ) - ( n H ( x ) + ( x , H ( x ) ) ) F m ( u 1 , . . . , u m ) d x = ( Ω B R ) k = 1 m i , j = 1 n u k p k ( x ) - 2 u k x i u k x j x j ν i + k = 1 m 1 p k x u k p k ( x ) + H ( x ) F m ( u 1 , . . . , u m ) ( x , ν ) d s
(2.7)

Now, multiply the equation (1.1) by au and integrating the obtained equation by parts in Ω ∩ B R

( Ω B R ) a k u p k ( x ) - a k u k H ( x ) f k ( u 1 , . . . , u m ) d x = 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 Ifu W 0 1 , 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

n F ( u ) - a u f ( u ) 0 , x Ω ,
(3.1)
( x , H ( x ) ) F ( u ) 0 , x Ω .
(3.2)

Then, the problem admits only the null solution.

Proof Ω is star shaped, imply that

Ω 1 - 1 p ( x ) u p ( x ) ( x , ν ) d s 0 .
(3.3)

On the other hand, the condition (3.1) give

Ω 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) 1 - ln u p ( x ) - a u p ( x ) d x + H ( x ) ( n F ( u ) - a u f ( u ) ) + ( x , H ( x ) ) F ( u ) d x 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

- d i v u p ( x ) - 2 u = 0  in  Ω , u = 0  on  Ω .
(3.5)

Multiplying the equation (3.5) by u and integrating over Ω, we get

Ω u p ( x ) d x = 0 .

So

u = 0 ,

Hence u = cte = 0, because u| Ω= 0. ■

theorem 3.2 Ifu W 0 1 , 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

n F ( u ) - a u f ( u ) 0 , x Ω ,
(3.6)
( x , H ( x ) ) F ( u ) 0 , x Ω .
(3.7)
H ( x ) F ( u ) 0 , x Ω .
(3.8)

Therefore, the problem admits only the null solution.

Proof Similar to the proof of theorem 1. ■

theorem 3.3 If u k W 0 1 , p k ( x ) ( 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

n F m ( u 1 , . . . , u m ) - k = 1 m a k u k f k ( u 1 , . . . , u m ) 0 , x Ω ,
(3.9)
( x , H ( x ) ) F m ( u 1 , . . . , u m ) 0 , x Ω .
(3.10)

So, the system admits only the null solutions.

Proof Ω is a star shaped, implies that

Ω k = 1 m 1 - 1 p k ( x ) u k p k ( x ) ( x , ν ) d s 0 .
(3.11)

On the other hand, the conditions (3.9) and (3.10), give

Ω k = 1 m 1 - n p k ( x ) + ( x , p k ( x ) ) p k 2 ( x ) 1 - ln u k p k ( x ) - a k u k p k ( x ) + H ( x ) n F m ( u 1 , . . . , u m ) - k = 1 m a k u k f k ( u 1 , . . . , u m ) + + ( x , H ( x ) ) F m ( u 1 , . . . , u m ) d x 0 .
(3.12)

(1.4), (3.11) and (3.12), allow to have

F m ( u 1 , . . . , u m ) = 0  in  Ω .

So the system (1.6) - (1.7) becomes

- d i v 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

Ω u k p k ( x ) d x = 0

So

u k = 0

Therefore u k = cte = 0, 1 ≤ km, because u k | Ω= 0. ■

theorem 3.4 If u k W 0 1 , p k ( x ) ( 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

n F m ( u 1 , . . . , u m ) - k = 1 m a k u k f k ( u 1 , . . . , u m ) 0 , x Ω ,
(3.14)
( x , H ( x ) ) F m ( u 1 , . . . , u m ) 0 , x Ω ,
(3.15)
H ( x ) F m ( u 1 , . . . , u m ) 0 , x Ω .
(3.16)

So, the problem admit only the null solution.

Proof Similar to the that of theorem 3. ■

4 Examples

Example 1 Considering in W 0 1 , p ( x ) ( Ω ) W 0 1 , q ( Ω ̄ ) the following problem

- d i v u p ( x ) - 2 u = c ( 1 + x ) μ u u q - 1 i n Ω , u = 0 o n Ω ,
(4.1)

where Ω is a bounded domain of n , c, μ > 0, q > 1 andp x = 1 + x 2 >1.

By choosing

a = sup Ω 1 - n + ( n - 1 ) x 2 1 + x 2 1 + x 2 ,

we obtain

( x , H ( x ) ) F ( u ) = - c μ x q ( 1 + x ) μ + 1 u q + 1 < 0 , ( x , p ( x ) ) = x 2 1 + x 2 0 , n F ( u ) - a u f ( u ) = n q + 1 - a u q + 1 0 i f q n - a a .

So, the problem (4.1) doesn't admit non trivial solutions if

q n - a a .

Example 2 Considering in W 0 1 , p ( x ) ( Ω ) W 0 1 , γ ( Ω ̄ ) , the following elliptic system

- Δ p ( x ) u = c γ ( 1 + x ) μ u u γ - 1 v δ i n Ω , - Δ q ( x ) v = c δ ( 1 + x ) μ v v δ - 1 u γ i n Ω , u = 0 o n Ω
(4.2)

where Ω is a bounded domain of n , c, μ, γ, δ > 0 and p, q > 1.

By choosing

a 1 = sup x Ω ̄ 1 - n p ( x ) + ( x , p ( x ) ) p 2 ( x ) a n d a 2 = sup x Ω ̄ 1 - n p ( x ) + ( x , q ( x ) ) q 2 ( x )

we obtain

( x , H ( x ) ) F ( u , v ) = - c μ 1 + x μ + 1 u γ v δ < 0 , n F ( u , v ) - a 1 u f 1 ( u , v ) - a 2 v f 2 ( u , v ) = ( n - γ a 1 - δ a 2 ) u γ v δ

So, the system (4.2) doesn't admit non trivial solutions if

γ a 1 + δ a 2 n

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Kamel, A. Nonexistence of nontrivial solutions for the p ( x )- Laplacian equations and systems in unbounded domains of n . Bound Value Probl 2011, 50 (2011). https://doi.org/10.1186/1687-2770-2011-50

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