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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

Citation and License

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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/50


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

© 2011 Kamel; licensee Springer.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M1">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M2">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M3">View MathML</a>

that the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M4">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M5">View MathML</a>

(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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M6">View MathML</a>

(1.1)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M7">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M8">View MathML</a>

Ω is bounded or unbounded domains of ℝn, f is a locally lipshitzian function, H and p are given continuous real functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M9">View MathML</a> verifying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M10">View MathML</a>

(1.4)

(., .) is the inner product in ℝn.

We extend this technique to the system of m-equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M11">View MathML</a>

(1.6)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M12">View MathML</a>

Where {fk} are locally lipshitzian functions verify

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M13">View MathML</a>

H is previously defined and pk functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M14">View MathML</a> class, verify

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M15">View MathML</a>

(1.9)

2 Integral identities

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M16">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M17">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M18">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M19">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M20">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M21">View MathML</a> in W1, p(x) (Ω),

Lemma 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M22">View MathML</a>solution of the equation (1.1) - (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M23">View MathML</a>

(2.1)

Lemma 2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M24">View MathML</a>solution of the equation (1.1) - (1.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M25">View MathML</a>

(2.2)

Proof Multiplying the equation (1.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M26">View MathML</a> and integrating the new equation by parts in Ω ∩ BR, BR = B (0, R)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M27">View MathML</a>

Introducing the following result

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M28">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M29">View MathML</a>

On the other hand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M30">View MathML</a>

these results conduct to the following formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M31">View MathML</a>

(2.3)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M32">View MathML</a>

(2.4)

Combining (2.3) and (2.4) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M33">View MathML</a>

On (Ω ∩ ∂BR) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M34">View MathML</a>

so the last integral is major by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M35">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M36">View MathML</a> we should see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M37">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M38">View MathML</a>

In the problem (1.1) - (1.2), u|Ω = 0. Then, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M39">View MathML</a>, we obtain the identity (2.1).

In the problem (1.1) - (1.3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M40">View MathML</a>, we obtain the identity (2.2). ■

Lemma 3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M41">View MathML</a>, solution of the system (1.6) - (1.7). Then for the constants ak of ℝ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M42">View MathML</a>

(2.5)

Lemma 4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M43">View MathML</a>, solutions of the system (1.6) - (1.8). Then for the constants ak of ℝ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M44">View MathML</a>

(2.6)

Proof Multiplying the equation (1.6) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M45">View MathML</a> and integrating the new equation by part in Ω ∩ BR, BR = B (0, R), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M46">View MathML</a>

On the other hand

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M47">View MathML</a>

These results conduct to the following formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M48">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M49">View MathML</a>

which leads to the following identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M50">View MathML</a>

(2.7)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M51">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M52">View MathML</a>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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M53">View MathML</a>

(3.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M54">View MathML</a>

(3.2)

Then, the problem admits only the null solution.

Proof Ω is star shaped, imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M55">View MathML</a>

(3.3)

On the other hand, the condition (3.1) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M56">View MathML</a>

(3.4)

(1.4), (3.3) and (3.4), allow to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M57">View MathML</a>

So, the problem (1.1) - (1.2) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M58">View MathML</a>

(3.5)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M59">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M60">View MathML</a>

Hence u = cte = 0, because u|Ω = 0. ■

theorem 3.2 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M61">View MathML</a>solution of the problem (1.1) - (1.3), Ω is a star shaped and that a, H, F and F verify the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M62">View MathML</a>

(3.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M63">View MathML</a>

(3.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M64">View MathML</a>

(3.8)

Therefore, the problem admits only the null solution.

Proof Similar to the proof of theorem 1. ■

theorem 3.3 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M65">View MathML</a>solution of the system (1.6) - (1.7), Ω is a star shaped and that ak, H, fk and Fm verify the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M66">View MathML</a>

(3.9)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M67">View MathML</a>

(3.10)

So, the system admits only the null solutions.

Proof Ω is a star shaped, implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M68">View MathML</a>

(3.11)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M69">View MathML</a>

(3.12)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M70">View MathML</a>

So the system (1.6) - (1.7) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M71">View MathML</a>

(3.13)

Multiplying (3.13) by uk and integrating on Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M72">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M73">View MathML</a>

Therefore uk = cte = 0, ∀1 ≤ k m, because uk|Ω = 0. ■

theorem 3.4 If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M74">View MathML</a>solution of the system (1.6) - (1.8), Ω is a star shaped and that ak, H, fk and Fm verify the following conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M75">View MathML</a>

(3.14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M76">View MathML</a>

(3.15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M77">View MathML</a>

(3.16)

So, the problem admit only the null solution.

Proof Similar to the that of theorem 3. ■

4 Examples

Example 1 Considering in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M78">View MathML</a>the following problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M79">View MathML</a>

(4.1)

where Ω is a bounded domain of n, c, μ > 0, q > 1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M80">View MathML</a>

By choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M81">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M82">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M83">View MathML</a>

Example 2 Considering in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M84">View MathML</a>the following elliptic system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M85">View MathML</a>

(4.2)

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

By choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M86">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M87">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/50/mathml/M88">View MathML</a>

Competing interests

The author declares that they have no competing interests.

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