On the first eigenvalue for a (p(x),q(x))-Laplacian elliptic system

In this article, we deal about the first eigenvalue for a nonlinear gradient type elliptic system involving variable exponents growth conditions. Positivity, boundedness and regularity of associated eigenfunctions for auxiliaries systems are established.


Introduction and setting of the problem
In the present paper, we focus on finding a non zero first eigenvalue for the system of quasilinear elliptic equations on ∂Ω on a bounded domain Ω ⊂ R N . Here ∆ p(x) u = div(|∇u| p(x)−2 ∇u) and ∆ q(x) v = div(|∇v| q(x)−2 ∇u) are usually named the p(x)-Laplacian and the q(x)-Laplacian operator.
During the last decade, the interest for partial differential equations involving the p(x)-Laplacian operator is increasing. When the exponent variable function p(·) is reduced to be a constant, ∆ p(x) u becomes the well-known p-Laplacian operator ∆ p u. The p(x)-Laplacian operator possesses more complicated nonlinearity than the p-Laplacian. So, one cannot always to transpose to the problems arising the p(x)-Laplacian operator the results obtained with the p-Laplacian. The treatments of solving these problem are often very complicated and needs a mathematical tools (Lebesgue and Sobolev spaces with variable exponents, see for instance [4] and its abundant reference). Among them, finding first eigenvalue of p(x)-Laplacian Dirichlet presents more singular phenomena which do not appear in the constant case. More precisely, it is well known that the first eigenvalue for the p(x)-Laplacian Dirichlet problem may be equal to zero (for details, the reader interested can consult [9]). In [9], the authors consider that Ω is a bounded domain and p is a continuous function from Ω to ]1, +∞[. They given some geometrical conditions insuring that the first eigenvalue is 0. Otherwise, in one dimensional space, monotonicity assumptions on the function p is a necessary and sufficient condition such that the first eigenvalue is strictly positive. In higher dimensional case, assuming monotonicity of an associated function defined by p, the first eigenvalue is strictly positive. The fact of the first eigenvalue is zero, has been observed earliest by [7]. Indeed, the authors illustrate this phenomena by taking Ω = (−2, 2) and p(x) = 3χ [0,1] (x)+ (4 − |x|)χ [1,2] (x). In this condition, the Rayleigh quotient is equal to zero. The main reason derives that the well-known Poincaré inequality is not always fulfilled. However, Fu in [11] shown that when Ω is a bounded Lipschitz domain, p is L ∞ (Ω) the Poincaré inequality holds (i.e. there is a constant C depending on Ω such that for any u ∈ W 1,p(x) 0 (Ω), Ω |u| p(x) ≤ C Ω |∇u| p(x) ). For a use of this result see for instance [2], [12].
Compared the investigation for one equation, elliptic systems haven't a similar growth concerning in the first eigenvalue. First of all, when p(x) and q(x) are constant on Ω, in [3], the following elliptic Dirichlet system is considered Assuming, Ω is a bounded open in R N with smooth boundary ∂Ω and the constant exponents −1 < α, β and 1 < p, q < N satisfying the condition , the author shown the existence of the first eigenvalue λ(p, q) > 0 associated to a positive and unique eigenfunction (u * , v * ). Further more, this result have been extended by Kandilakis and al. [13] for the system with Ω is an unbounded domain in R N with non compact and smooth boundary ∂Ω, the constant exponents 0 < α, β and 1 < p, q < N satisfying (1.5) α+1 p + β+1 q = 1 and (α + 1) N −p N p < q, (β + 1) N −q N q < p. Inspired by [3], Khalil and al. in [14] shown that the first eigenvalue λ p,q of (1.2) is simple and moreover they established stability (continuity) for the function (p, q) −→ λ(p, q).
Motivated by the aforementioned papers, in this work we establish the existence of one-parameter family of nontrivial solutions ((û R ,v R ), λ * R ) for all R > 0 for problem (1.1). In addition, we show that the corresponding eigenfunction (û R ,v R ) is positive in Ω, bounded in L ∞ (Ω) × L ∞ (Ω) and belongs to C 1,γ (Ω) × C 1,γ (Ω) for certain γ ∈ (0, 1) if p, q ∈ C 1 (Ω) ∩ C 0,θ (Ω). Furthermore, by means of geometrical conditions on the domain Ω, we prove that the infimum of the eigenvalues of (1.1) is positive. To the best of our knowledge, it is for the first time when the positive infimum eigenvalue for systems involving p(x)-Laplacian operator is studied. However, we point out that in this paper, the existence of an eigenfunction corresponding to the infimum of the eigenvalues of (1.1) is not established and therefore, this issue still remains an open problem.
The rest of the paper is organized as follows. Section 2 contains hypotheses, some auxiliary and useful results involving variable exponent Lebesgue-Sobolev spaces and our main results. Section 3 and section 4 present the proof of our main results.

Hypotheses -Main results and some auxiliary results
Let L p(x) (Ω) be the generalized Lebesgue space that consists of all measurable real-valued functions u satisfying The variable exponent Sobolev space W 1,p(·) 0 (Ω) is defined by (Ω) a Banach space and the following embedding q(x) = 1, (H.4): p and q are two variable exponents of class C 1 (Ω) satisfying

2.2.
Main results. Throughout this paper, we set X (Ω) the functionals A and B as follows: and denote by (z, w) = z 1,p(x) + w 1,q(x) . The same reasoning exploited in [8] implies that A and B are of class C 1 (X p(x),q(x) 0 (Ω), R). The Fréchet derivatives of A and B at (z, w) in X p(x),q(x) 0 (Ω) are given by (Ω). Let R > 0 be fixed, we set It is obvious to notice that the set X R is not empty. Indeed, let (z 0 , w 0 ) ∈ X Now, define the Rayleigh quotients The constant λ * R in (2.6) can be written as follows Our first main result provides the existence of a one -parameter family of solutions for the system (1.1). Theorem 1. Assume that (H.1) -(H.4) hold. Then, the system (1.1) has a oneparameter family of nontrivial solutions ((û R ,v R ), λ * R ) for all R ∈ (0, +∞). Moreover, if one of the following conditions holds: (a.1): There is vectors l 1 , l 2 ∈ R N \ {0} such that for all x ∈ Ω, f (t 1 ) = p(x + t 1 l 1 ) and g(t 2 ) = q(x + t 2 l 2 ) are monotone for ∈ Ω such that for all w 1 , w 2 ∈ R\{0} with w 1 , w 2 = 1, the functions f (t 1 ) = p(x 0 +t 1 w 1 ) and g(t 2 ) = p(x 2 +t 2 w 2 ) are monotone The proof of Theorem 1 will be done in section 3 while in section 4 we will present the proof of Theorem 2.
(Ω) the inequality Recall that if there exist a constant L > 0 and an exponent θ ∈ (0, 1) such that then the function p is said to be Hölder continuous on Ω and we denote p ∈ C 0,θ (Ω). For a later use, we have the next result.
Proof. Recall that for s > 0 we have Multiplying by s one get for all s = 1. Thus, it follows that Hence, for 0 < s < 1 one has

Proof of Theorem 1
Taking account of the assumption (H.3), we note that the system (1.1) is arising from a nonlinear eigenvalue type problem. Solvability of general class of nonlinear eigenvalues problems of type A ′ (x) = λB ′ (x) have been treated by M.S Berger in [1]. We recall this main tool.
Suppose that the C 1 functionals A and B defined on the reflexive Banach space X have the following properties: (1) A is weakly lower semicontinuous and coercive on X ∩ {B(x) ≤ const.}; (2) B is continuous with respect to weak sequential convergence and B ′ (x) = 0 only at x = 0.
Remark 2. In the statement (ii) of the theorem 3, the condition "B ′ (x) = 0 only at x = 0" may be replaced by "B(x) = 0 only at x = 0 ". Indeed, in the proof of Theorem 3, assume that the minimizing problem inf {B(x)=R} A(x) is attained at x R ∈ X then because A and B are differentiable there exists (λ 1 , λ 2 ) a pair of Lagrange multipliers such that Consequently, λ 1 and λ 2 are not both zero. In fact, if λ 2 = 0 and λ 1 = 0 then we get λ 2 (B ′ (x R ), x R ) = 0. So, for instance, assume that the following condition obeys "there exists γ > 0 such that 3.1. Properties on A and B.

3) and (H.4)) the above inequality implies that
(Ω). By the first part in (H.4) and (2.1) the embeddings W Using (H.3) and the definition of B, we have By Hölder inequality one has Therefore, the strong convergence in (3.1) ensures that → 0 as n → +∞.
A quite similar argument provides → 0 as n → +∞.
(iii) From (2.5), it is clear to notice that for any (z, w) ∈ X p(x),q(x) 0 (Ω), by taking ϕ = 1/p(x)z and ψ = 1/q(x)w, the following identity holds Then the statement (iii) follows. This conclude the proof of the Lemma.

A priori bound for A.
Lemma 5. Let R a fixed and strictly positive real. There exists a constant K(R) > 0 depending on R such that Proof. First, observe from Lemma 2 that if ∇z L p(x) (Ω) < 1, we have Then if follows that Hence it holds A quite similar argument shows that (Ω), Young inequality and (H.3) imply (3.6) Assume that (z, w) ∈ X R is such that From the hypothesis (H.4) on p − , p + , q − and q + , it follows that (3.10) Or again Thus, from (3.11), we conclude that Now, we deal with the case when (z, w) ∈ X R is such that This implies that If Ω |∇z| p(x) dx ≥ 1 we have A(z, w) > 1 q + . We notice that if max ∇z L p(·) (Ω) , ∇w L q(·) (Ω) ≥ 1, from (3.13) and (3.14), it is clearly that (3.15) A(z, w) > max( 1 p + , 1 q + ). Thus, according to (3.12) and (3.15), for all (z, w) ∈ X R , one has Consequently, there exists a constant K(R) > 0 depending on R such that (3.2) holds.
(ii). Next we show that λ cannot be an eigenvalue for λ < λ * . Indeed, suppose by contradiction that λ is an eigenvalue of problem (1.1). Then there exists (u, v) ∈ X

On the basis of (H.3), (H.4), (2.7) and (3.17), we get
which is not possible and the conclusion follows.
(iii). Now, we claim that the infimum in (2.8) is achieved at an element of X R . Indeed, thanks to the lemma 4, B is weakly continuous on X p(x),q(x) 0 (Ω), then the nonempty set X R is weakly closed. So, since A is weakly lower semicontinuous, we conclude that there exists an element of X R which we denote (û,v R ) such that where A ′ and B ′ are defined as in (2.4) and (2.5) respectively.

Indeed, on account of (H.4), (3.19) and (3.20) we have
showing that (3.21) holds. In the same manner we can prove that (3.22) Adding together (3.21) and (3.22), on account of (H.3) and (3.14), we achieve that Then, bearing in mind (3.15) it turns out that λ R = λ * R , showing that λ * R is at least one eigenvalue of (1.1).
Then, combining this last point with the characterization (3.18), we get and (Ω).
On other words, it means that ((û R ,v R ), λ * R ) is a solution of the system (1.1).
(Ω) \ {0} such that B(z, w) > 0 and assume that R (z,w) = B(z, w). According to (iii) in Propostion 1, the constant exists and then At this point, combining with (3.27) yields This ends the proof of Theorem 1.
For a better reading, we divide the proof of Theorem 2 in several lemmas. Lemma 6. Assume hypotheses (H.1)-(H.4) hold. Then, for any fixed k in N, there exist x k , y k ∈ Ω such that the following estimates hold: where |Ω| denotes the Lebesgue measure of a set Ω in R N .
Proof. Before starting the proof, let us note that Let us prove (4.5).
. Therefore, by Hölder's inequality and Mean value Theorem, there exist x k and t k ∈ Ω such that This shows that the inequality (4.5) holds true. Here p ′ and p are conjugate variable exponents functions.
Next, we show (4.6). By (4) and Young's inequality, we get (4.7) | Observe from (3.13) that Using the mean value theorem, there exists x k ∈ Ω such that Similarly, we can find y k ∈ Ω such that Then, combining (4.7), (4.8) and (4.9), the inequality (4.6) holds true, ending the proof of the lemma 6.
By using the Lemma we can prove the next result.
Before continuing, we distinguish the cases where û R p(x)d k+1 , v R q(x)d k+1 , û R p(x)d k and v R p(x)d k are each either less than one or either greater than one. Using (H.4) and (4.1) we obtain (4.24) ln max{ û R d k+1 with (4.26) a = ln dd, b = ln C 3 .
Next, we show thatû R andv R are strictly positive in Ω.
Hence, from the above notation, we get Since µ 1 U =û R = 0 and µ 2 V =v R = 0 on ∂Ω, we are allowed to apply [19, Lemma 2.3] and we deduce that Thereby the positivity of (û R ,v R ) in Ω is proven.
To end the proof of Lemma 8, we claim a regularity property forû R andv R .