Research

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

Akrout Kamel

### Author affiliations

Department of mathematics and informatics. Tebessa university. Algeria

Boundary Value Problems 2011, 2011:50  doi:10.1186/1687-2770-2011-50

 Received: 12 January 2011 Accepted: 30 November 2011 Published: 30 November 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 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 ℝn and f is a continuous function on ℝ satisfying

( n - 2 ) F ( u ) - 2 n u f ( u ) > 0 ,

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 ℝ and ω is a connected unbounded domain of ℝN 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 ℝn, 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 ℝn.

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 {fk} 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 pk 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 Let u 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 Let u 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 ∩ BR, BR = 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 ∩ ∂BR) 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 ∩ ∂BR = 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 (Rn)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 ak of ℝ, 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 ak of ℝ, 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 ∩ BR, BR = 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 ∩ BR

( 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 If u 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 If u 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 ak, H, fk and Fm 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 uk and integrating on Ω, we have

Ω u k p k ( x ) d x = 0

So

u k = 0

Therefore uk = cte = 0, ∀1 ≤ k m, because uk|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 ak, H, fk and Fm 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 and p 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

### Competing interests

The author declares that they have no competing interests.

### References

1. Pohozaev, SI: Eeigenfunctions of the equation Δu + λf (u) = 0. Soviet Math Dokl. 1408–1411 (1965)

2. Haraux, A, Khodja, B: Caractère triviale de la solution de certaines équations aux dérivées partielles non linéaires dans des ouverts cylindriques de ℝN. Portugaliae Mathematica. 42(Fasc 2), 1–9 (1982)

3. Esteban, MJ, Lions, P: Existence and non-existence results for semi linear elliptic problems in unbounded domains. Proc Roy Soc Edimburgh. 93-A, 1–14 (1982)

4. Kawarno, N, NI, W, Syotsutani: Generalised Pohozaev identity and its applications. J Math Soc Japan. 42(3), 541–563 (1990). Publisher Full Text

5. Khodja, B: Nonexistence of solutions for semilinear equations and systems in cylindrical domains. Comm Appl Nonlinear Anal. 19–30 (2000)

6. NI, W, Serrin, J: Nonexistence thms for quasilinear partial differential equations. Red Circ Mat Palermo, suppl Math. 8, 171–185 (1985)

7. Van Der Vorst, RCAM: Variational identities and applications to differential systems. Arch Rational; Mech Anal. 116, 375–398 (1991). PubMed Abstract

8. Yarur, C: Nonexistence of positive singular solutions for a class of semilinear elliptic systems. Electronic Journal of Diff Equations. 8, 1–22 (1996)