Existence and Nonexistence Results for Weighted Fourth Order Eigenvalue Problems With Variable Exponent

The study of problems of elliptic equations and variational problems with p(x)growth condition has attracted more and more attention in recent years. It possesses a solid background in physics and originates from the study on electrorheological fluids (see [16]) and elastic mechanics (see [18]). It also has wide applications in different research fields, such as image processing model (see e.g. [11,6], stationary thermorheological viscous flows (see [2]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [1]). In the present study, we deal with the following nonlinear eigenvalue problem with indefinite weight


Introduction and statement of main results
The study of problems of elliptic equations and variational problems with p(x)growth condition has attracted more and more attention in recent years.It possesses a solid background in physics and originates from the study on electrorheological fluids (see [16]) and elastic mechanics (see [18]).It also has wide applications in different research fields, such as image processing model (see e.g.[11,6], stationary thermorheological viscous flows (see [2]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [1]).
In the present study, we deal with the following nonlinear eigenvalue problem with indefinite weight △ |△u| p(x)−2 △u = λV (x)|u| q(x)−2 u in Ω, where Ω ⊂ R N is a smooth bounded domain, V ∈ L r(x) (Ω) is an indefinite weight which can change sign in Ω and p, q, r ∈ C + ( Ω) := h; h ∈ C(Ω) and h(x) > 1 for all x ∈ Ω .

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A. Ayoujil The interest in analyzing this kind of problems is motivated by some recent advances in the study of fourth order nonlinear eigenvalue problems involving variable exponents in the last few years.We refer especially to the results in [3,4,10,13,14].
For instance, the case V ≡ 1 and p(x) = q(x) has been studied in [3] and in [4] when p(x) = q(x).In particular, in [3], by the Ljusternik-Schnirelmann principle on C 1 -manifolds, the authors proved among others things the existence of a sequence of eigenvalues and that sup Λ = +∞, where Λ is the set of all nonnegative eigenvalues.In [4], using the mountain pass lemma and Ekeland's variational principle, they established several existence criteria for eigenvalues.
Motived by the above-mentioned papers, our purpose in this paper is to extend the results of [5] to a fourth order nonlinear problem with sign-changing potential.Our approach follows closely the one in the mentioned paper.
Hereafter, we analyze the problem (1.1) under the following assumptions:: , for all x ∈ Ω, ( where Here, we seek solutions for problem (1.1) belonging to the space X := W 2,p(x) (Ω) ∩W 1,p(x) 0 (Ω) in the sense below.Definition 1.1.By a weak solution for (1.1) we understand a function u ∈ X such that Moreover, we say that λ ∈ R is an eigenvalue of problem 1.1 if the weak solution u defined above is not trivial.
We point out that, in the case of positive weight V , any possible eigenvalue of problem 1.1 is necessarily positive.
Define the functionals Φ, Ψ, The energy functional corresponding to problem 1.1 is defined as I λ : X → R, Existence and Nonexistence Results for Weighted Fourth Order...

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Standard arguments imply that I λ ∈ C 1 (X, R) and for all u, v ∈ X, we have Thus, the weak solutions of (1.1) are exactly the critical points of I λ .
The main results of this work are the following.
In the case when V is a sign-changing function, we define ) So, we have Theorem 1.2.Suppose that H(p,q,r) and hold.Then, we have This article is composed of three sections.Section 2 contains some useful results on Sobolev spaces with variable exponents.The proofs are given in Section 3.

Preliminary results
In order to guarantee the integrity of the paper, we first recall some facts on variable exponent spaces L p(x) (Ω) and W k,p(x) (Ω).For details, see [8,9,15].
For p ∈ C + (Ω), define the space Equipped with the so-called Luxemburg norm (Ω) becomes a separable, reflexive and Banach space.An important role in manipulating the generalized Lebesgue spaces is played by the mapping ρ : L p(x) (Ω) → R, called the modular of the L p(x) (Ω) space, defined by We recall the following, (see [8,15], Proposition 2.1.For all u ∈ L p(x) (Ω), we have

(b)
As in the constant exponent case, for any positive integer k, set We can define the norm on W k,p(x) (Ω) by and W k,p(x) (Ω) also becomes a separable, reflexive and Banach space.We denote by W Definition 2.1.Assume that spaces E, F are Banach spaces, we define the norm on the space E ∩ F as u = u E + u F .
From Definition 2.1, we can know that for any u ∈ X, u In the Zang and Fu's paper [17], the equivalence of the norms was proved, and it was even proved that the norm |△u| p(x) is equivalent to the norm u X (see [17,Theorem 4.4]).
Let us choose on X the norm .defined by Note that, (X, .) is also a separable and reflexive Banach space.Similar to Proposition 2.2, we have the following.A. Ayoujil The following result (see [3,Theorem 3.2]), which will be used later, is an embedding result between the spaces X and L q(x) (Ω).
Then, there is a continuous and compact embedding X into L q(x) (Ω).
We recall also the following proposition, which will be needed later.
(i) The convexity of Φ ensures this assertion.
(ii) Let (u n ) be a sequence in X such that u n ⇀ u in X. Denote by r ′ (x) the conjugate exponent of the function r(x) r ′ (x) = r(x) r(x)−1 .Then, as q(x)r ′ (x) < p * 2 (x), Theorem 2.1 implies u n ⇀ u in L q(x)r ′ (x) (Ω) .This, together with the continuity of the Nemytski operator N V,q defined by The proof is complete.

Proofs
At first, we start with the following Lemma which plays a crucial role for proving Theorem1.1.Lemma 3.1.Suppose that assumptions H(p, q, r) and (1.3) hold, then and Proof: Applying the Hölder's inequality, we obtain .
By help of proposition 2.4, it follows On the other hand, from (1.2), we have p(x) < q(x)r ′ (x) < p * 2 (x) for all x ∈ Ω.So, in view proposition 2.1, X is continuously embedded in L q(x)r ′ (x) (Ω).Then, there exists c > 0 such that For any u ∈ X with u ≤ 1 small enough, by relations (3.3) and (3.4), we infer (3.5) Since p + < q − ≤ q + , passing to the limit as u → 0 in the above inequality, we deduce that the assertion (3.1) holds true.
Next, we show that relation the assertion (3.2) holds.From (1.3), there exists a positive constant δ such that q + − 1 2 p -< δ < q -, and thus we have Let s(x) be any measurable function s(x) such that holds for almost all x ∈ Ω and A. Ayoujil Clearly, s ∈ L ∞ (Ω) and 1 < s(x) < r(x).Moreover, It is easy to see where t(x) := r(x)s(x) r(x)−s(x) and s ′ (x) = s(x) s(x)−1 .Let u ∈ X with u > 1. Thanks to Hölder's inequality again, we have Without loss of generality we may suppose that V |u| δ s(x) > 1.Then, from Proposition 2.1 and using Hölder's inequality, we |u| δs(x) In view of proposition 2.4, we write Hence, substituting the above inequalities into (3.10) and thanks to Young's inequality, it follows From (3.9), we infer by Theorem 2.1 that X is continuously embedded both in

.13)
Existence and Nonexistence Results for Weighted Fourth Order...
(ii) Let (u n ) ∈ X \ {0} be a minimizing sequence for λ * , that is By 3.2, (u n ) is bounded in X which is reflexive.Then, there exists u ∈ X such that u n ⇀ u in X.This together with proposition 2.5 yields that A. Ayoujil It remains to show that u is nontrivial.Let suppose by contradiction that u = 0.Then, lim Ψ(u n ) = 0 and so, via (3.14), we deduce This fact combined with proposition 2.3 implies that u n → 0. According to (3.2), we get and this is a contradiction.Thus, u = 0.
Precise that if λ > 0 is an eigenvalue of problem (1.1) with weight V then, −λ is an eigenvalue of problem (1.1) with weight ?V .Hence, it is enough to show Theorem 1.1 only for λ > 0. So, the problem 1.1 has only to be considered in X + and in this case, the proof is similar to that of Theorem 1.1 and thus it will be omitted here.

4 .
The following statements are equivalent each other: (a) lim n→∞ |u n − u| p(x) = 0, Existence and Nonexistence Results for Weighted Fourth Order...