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 ℝ^{n}. This leads us to extend these results to mequations 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
when Ω is a star shaped bounded open domain in ℝ^{n }and f is a continuous function on ℝ satisfying
A. Hareux and B. Khodja [2] established under the assumption
that the problems
admit only the null solution in H^{2}(J × ω) ∩ L^{∞}(J × ω). where J is an interval of ℝ and ω is a connected unbounded domain of ℝ^{N }such as
(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
with
where
Ω is bounded or unbounded domains of ℝ^{n}, f is a locally lipshitzian function, H and p are given continuous real functions of verifying
(., .) is the inner product in ℝ^{n}.
We extend this technique to the system of mequations
with
Where {f_{k}} are locally lipshitzian functions verify
H is previously defined and p_{k }functions of class, verify
2 Integral identities
Let
with the norm
and
with the norm
Denote the closure of in W^{1, p(x) }(Ω),
Lemma 1 Let solution of the equation (1.1)  (1.2), we have
Lemma 2 Let solution of the equation (1.1)  (1.3), we have
Proof Multiplying the equation (1.1) by and integrating the new equation by parts in Ω ∩ B_{R}, B_{R }= B (0, R)
Introducing the following result
we have
On the other hand
these results conduct to the following formula
Multiplying the present equation (1.1) by au and integrating the obtained equation by parts in Ω, we obtain
Combining (2.3) and (2.4) we obtain
so the last integral is major by
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) ∈ L^{1 }(Ω) and we should see
consequently we can always find a sequence (R_{n})_{n}, such as
In the problem (1.1)  (1.2), u_{∂Ω }= 0. Then, , we obtain the identity (2.1).
In the problem (1.1)  (1.3), , we obtain the identity (2.2). ■
Lemma 3 Let , solution of the system (1.6)  (1.7). Then for the constants a_{k }of ℝ, we have
Lemma 4 Let , solutions of the system (1.6)  (1.8). Then for the constants a_{k }of ℝ, we have
Proof Multiplying the equation (1.6) by and integrating the new equation by part in Ω ∩ B_{R}, B_{R }= B (0, R), we get
On the other hand
These results conduct to the following formula
Doing the sum on k of 1 to m, we obtain
which leads to the following identity
Now, multiply the equation (1.1) by au and integrating the obtained equation by parts in Ω ∩ B_{R}
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 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
Then, the problem admits only the null solution.
Proof Ω is star shaped, imply that
On the other hand, the condition (3.1) give
(1.4), (3.3) and (3.4), allow to get
So, the problem (1.1)  (1.2) becomes
Multiplying the equation (3.5) by u and integrating over Ω, we get
So
Hence u = cte = 0, because u_{∂Ω }= 0. ■
theorem 3.2 If solution of the problem (1.1)  (1.3), Ω is a star shaped and that a, H, F and F verify the following conditions
Therefore, the problem admits only the null solution.
Proof Similar to the proof of theorem 1. ■
theorem 3.3 If 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
So, the system admits only the null solutions.
Proof Ω is a star shaped, implies that
On the other hand, the conditions (3.9) and (3.10), give
(1.4), (3.11) and (3.12), allow to have
So the system (1.6)  (1.7) becomes
Multiplying (3.13) by u_{k }and integrating on Ω, we have
So
Therefore u_{k }= cte = 0, ∀1 ≤ k ≤ m, because u_{k}_{∂Ω }= 0. ■
theorem 3.4 If 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
So, the problem admit only the null solution.
Proof Similar to the that of theorem 3. ■
4 Examples
Example 1 Considering in the following problem
where Ω is a bounded domain of ℝ^{n}, c, μ > 0, q > 1 and
By choosing
we obtain
So, the problem (4.1) doesn't admit non trivial solutions if
Example 2 Considering in the following elliptic system
where Ω is a bounded domain of ℝ^{n}, c, μ, γ, δ > 0 and p, q > 1.
By choosing
we obtain
So, the system (4.2) doesn't admit non trivial solutions if
Competing interests
The author declares that they have no competing interests.
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