Inﬁnitely many solutions for a nonlinear Navier boundary systems

: In this article, we study the following ( p ( x ) ,q ( x )) -biharmonic type system We prove the existence of inﬁnitely many solutions of the problem by applying a general variational principle due to B. Ricceri and the theory of the variable exponent Sobolev spaces.

There are many works devoted to the existence of solutions for variable exponent problems, both on bounded domain and unbounded domain, we refer to [1,4,7,19] as examples. For existence results on elliptic systems, we refer to [8,16,18].
The investigation of existence and multiplicity of solutions for problems involving biharmonic, p-biharmonic and p(x)-biharmonic operators has drawn the attention of many authors, see [2,3,5,6,11] and references therein. Candito and Livrea [5] considered the nonlinear elliptic Navier boundary-value problem There the authors established the existence of infinitely many solutions.
In the present paper, we look for the existence of infinitely many solutions of system (1.1). More precisely, we will prove the existence of well precise intervals of parameters such that problem (1.1) admits either an unbounded sequence of solutions provided that F (x, u, v) has a suitable behaviour at infinity or a sequence of nontrivial solutions converging to zero if a similar behaviour occurs at zero.
In the case when p(x) ≡ p and q(x) ≡ q are two constants, we know that the problem (1.1) has infinitely many solutions from [12]. Here we point out that the p(x)-biharmonic operator possesses more complicated nonlinearities than pbiharmonic, for example, it is inhomogeneous and usually it does not have the socalled first eigenvalue, since the infimum of its principle eigenvalue is zero.
This article is organized as follows. In Section 2, we introduce the generalized Lebesgue-Sobolev spaces and some important related results. In section 3, we give the main results of this paper. In section 4, we use the general variational principle by B. Ricceri to prove the main results.

Preliminaries
To study p(x)-Laplacian problems, we need some results on the spaces L p(x) (Ω), W k,p(x) (Ω) and properties of p(x)-Laplacian used later.
Define the generalized Lebesgue space by where p ∈ C + (Ω) and One introduces in L p(x) (Ω) the norm The space (L p(x) (Ω), |.| p(x) ) is a Banach space.
The Sobolev space with variable exponents W k,p(x) (Ω) is defined as also becomes a Banach, separable and reflexive space. For more details, we refer the reader to [9,10,13]. We denote by W k,p(x) 0 (Ω) the closure of C ∞ 0 (Ω) in W k,p(x) (Ω). In this paper, we shall look for weak solutions of problem (1.1) in the space X defined by which is separable and reflexive Banach spaces with the norm According to [17], the norm |.| 2,p(x) is equivalent to the norm |△.| p(x) in the space (Ω). Consequently, the norms |.| 2,p(x) , |△.| p(x) and . p(x) are equivalent.
The following proposition will plays an important role in our arguments.
Using the similar proof method with [9], we have the following result.
Let us recall for the reader's convenience a smooth version of a previous result of Ricceri [14].
Proposition 2.4. [14] Let X be a reflexive real Banach space, let Φ, Ψ : X → R be two Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf X Φ, let us put and Then, one has (a) for every r > inf X Φ and every λ ∈]0, 1 ϕ(r) [, the restriction of the functional (b) If γ < +∞ then, for each λ ∈]0, 1 γ [, the following alternative holds: either (c) If δ < +∞ then, for each λ ∈]0, 1 δ [, the following alternative holds: either (c1) there is a global minimum of Φ which is a local minimum of I λ ,or (c2) there is a sequence of pairwise distinct critical points (local minima) of I λ which weakly converges to global minimum of Φ.
For each(u, v) ∈ X, we define Then, the operator L : 2) satisfies the assertions of the following proposition.
1. L is continuous, bounded and strictly monotone.

Main results
where Γ denotes the Gamma function and K is given by (2.1).
Define the functional J λ : X → R, by The functionals Φ, Ψ : X → R are well defined, Gâteaux differentiable functionals whose Gâteaux derivatives at (u, v) ∈ X are given by for all (ϕ, ψ) ∈ X.
In view of (2.2) and proposition 2.5, we see that Φ ∈ C 1 (X, R) and (u, v) ∈ X is a weak solution of (1.1) if and only if (u, v) is a critical point of the functional J λ .
Since X is compactly embedded in C(Ω) × C(Ω), we can see that Φ, Ψ : X → R are sequentially weakly lower semi-continuous. Moreover Φ is coercive.
Our main results are the following two theorems. (i2) There exist x 0 ∈ Ω, 0 < R 1 < R 2 as considered in (3.1) such that, if we put
Then, for every problem (1.1) admits a sequence (u n ) of weak solutions such that u n ⇀ 0.
When u p(x) > 1, we have 1 p + u p − p(x) < r n , so u p(x) < (p + r n ) 1 p − . Hence, for n large enough (r n > 1), Using (2.1) and (4.3), we obtain, for all x ∈ Ω |u(x)| < K(p + r n ) Therefore, for n large enough (r n > 1), It follows from (4.2) and (4.4) that (4.5) From (4.5), it is clear that Λ ⊆]0, 1 γ [. For λ ∈ Λ, we claim that the functional J λ is unbounded from below. Indeed, Lβ, we can consider a sequence (η n ) of positive numbers and δ > 0 such that lim n→+∞ η n = +∞ and Now we consider the function u n defined by then (u n , u n ) ∈ X and Then, for n large enough (4.9) By (i1), we have Combining (4.6), (4.9) and (4.10), we obtain J λ (u n , u n ) = Φ(u n , u n ) − λΨ(u n , u n ) Let (b n ) be a sequence of positive numbers such that b n → 0 + and It follows from (4.1) and (4.13) that (4.14) By (4.14), we see that Λ ⊆]0, 1 δ [. Now, for λ ∈ Λ, we claim that J λ has not a local minimum at zero. Indeed, Lβ 0 , we can consider a sequence (η n ) of positive numbers and δ > 0 such that η n → 0 + and for n large enough. Let (u n ) be the sequence defined in (4.7). By combining (4.9), (4.10) and (4.15), and taking into account (i3), we have J λ (u n , u n ) = Φ(u n , u n ) − λΨ(u n , u n ) for n large enough. This together with the fact that (u n , u n ) → 0 show that J λ has not a local minimum at zero, and the claim follows. The alternative of proposition 2.4 case (c) ensures the existence of sequence (u n ) of pairwise distinct critical points (local minima) of J λ which weakly converges to 0. This completes the proof of Theorem 3.3. ✷ It is easy to verify that F is non-negative and F ∈ C 1 (R 2 ). for all n ∈ N * , the restriction of F on B((a n+1 , a n+1 ), 1) attains its maximum in (a n+1 , a n+1 ) and