On the existence of a weak solution for some singular p(x)-biharmonic equation with Navier boundary conditions

In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the p(x)-biharmonic operator: { ∆(|∆u|p(x)−2∆u) = g(x)u−γ(x) ∓ λf(x, u) in Ω, ∆u = u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 3) is a bounded domain with C2 boundary, λ is a positive parameter, γ : Ω → (0, 1) is a continuous function, p ∈ C(Ω) with 1< p− := infx∈Ω p(x)≤ p+ := supx∈Ω p(x)< 2 , as usual, p∗(x)= Np(x) N−2p(x) , g ∈ L p∗(x) p∗(x)+γ(x)−1 (Ω), and f(x, u) is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.

The operator ∆ 2 p(x) u := ∆(|∆u| p(x)−2 ∆u) is called the p(x)-biharmonic operator of fourth order, where p is a continuous non-constant function. This differential operator is a natural generalization of the p-biharmonic operator ∆ 2 p u := ∆(|∆u| p−2 ∆u), where p > 1 is a real constant. However, the p(x)-biharmonic operator possesses more complicated non-linearity than the p-biharmonic operator, due to the fact that ∆ 2 p(x) is not homogeneous. This fact implies some difficulties; for example, we can not use the Lagrange multiplier theorem in many problems involving this operator.
The study of this kind of operators occurs in interesting areas such as electrorheological fluids (see [19]), elastic mechanics (see [25]), stationary thermo-rheological viscous flows of non-Newtonian fluids, image processing (see [6]) and the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium (see [1]).
Problem (P ∓λ ) is a new variant of p(x)-biharmonic equations due to the singular term and the indefinite one. Note that results for p(x)-Laplace equations with singular non-linearity are rare. Meanwhile, elliptic and singular elliptic equations involving the p(x)-Laplace and the p(x)-biharmonic operators can be found in [1, 2, 5, 8-10, 13, 14, 17, 18, 20-23].
Recently, Ayoujil and Amrouss [4] studied the following problem: In the case when max x∈Ω q(x) < min x∈Ω p(x), they proved that the energy functional associated to problem (P) has a nontrivial minimum for any positive λ; see [4,Theorem 3.1]. In the case when min x∈Ω q(x) < min x∈Ω p(x) and q(x) has a subcritical growth, they used Ekeland's variational principle in order to prove the existence of a continuous family of eigenvalues which lies in a neighborhood of the origin. Finally, when they showed (see [4,Theorem 3.8]) that for every Λ > 0 the energy functional Φ λ corresponding to (P) has a Mountain Pass-type critical point which is nontrivial and nonnegative, and hence Λ = (0, +∞), where Λ is the set of the eigenvalues. The same problem, for p(x) = q(x), is studied by Ayoujil and Amrouss in [3]. They established the existence of infinitely many eigenvalues for problem (P) by using an argument based on the Ljusternik-Schnirelmann critical point theory. They showed that sup Λ = +∞, and they pointed out that only under special conditions, which are somehow connected with a kind of monotony of the function p(x), one has inf Λ > 0 (this is in contrast with the case when p(x) is a constant, where one always has inf Λ > 0).
Later, Ge, Zhou and Wu [10] considered the following problem: where V is an indefinite weight and λ is a positive real number. They considered different situations concerning the growth rates, and they proved, using the mountain pass lemma and Ekeland's principle, the existence of a continuous family of eigenvalues. A recent paper concerning this type of problems is [12]. Inspired by the above-mentioned papers, we study problem (P ∓λ ), which contains a singular term and indefinite many more general terms than the one studied in [10]. In this new situation, we will show the existence of a weak solution for problem (P ∓λ ). The paper is organized as follows: In Section 2, we recall some definitions concerning variable exponent Lebesgue spaces, L p(x) (Ω), as well as Sobolev spaces, W k,p(x) (Ω). Moreover, some properties of these spaces will also be exhibited to be used later. Our main results are stated in Section 3. The proofs of our results will be presented in Section 4 and Section 5.

Notations and preliminaries
To study p(x)-biharmonic problems, we need some results on the spaces L p(x) (Ω), W 1,p(x) (Ω) and W k,p(x) (Ω) (for details, see [9,18]) and some properties of the p(x)-biharmonic operator, which will be needed later. Set Let p be a Lipschitz continuous function on Ω. We set 1 < p − := min x∈Ω p(x) ≤ p + = max x∈Ω p(x) < ∞ and We recall the following so-called Luxemburg norm on this space defined by the formula Clearly, when p(x) = p, a positive constant, the space L p(x) (Ω) reduces to the classical Lebesgue space L p (Ω), and the norm |u| p(x) reduces to the standard norm For any positive integer k, as in the constant exponent case, let Then W k,p(x) (Ω) is a separable and reflexive Banach space equipped with the norm Then the Hölder-type inequality holds.
The modular on the space and it satisfies the following propositions.
Proposition 2.1 (see [16]). For all u ∈ L p(x) (Ω), we have the following assertions: Proposition 2.2 (see [7]). Let p and q be two measurable functions such that p ∈ L ∞ (Ω) and Then For more details concerning the modular, see [9,16].

Definition 2.3.
Assuming that E and F are Banach spaces, we define the norm on the space In order to discuss problems (P ∓λ ), we need some theories on the space . From Definition 2.3 we know that for any u ∈ X, ‖u‖ = ‖u‖ 1,p(x) + ‖u‖ 2,p(x) , and thus Zang and Fu [24], proved the equivalence of the norms, and they even proved that the norm |∆u| p(x) is equivalent to the norm ‖u‖ (see [24,Theorem 4.4]). Let us choose on X the norm defined by ‖u‖ = |∆u| p(x) . Note that (X, ‖ ⋅ ‖) is also a separable and reflexive Banach space and that the modular is defined as ρ p(x) : X → ℝ by ρ p(x) (∆u) = ∫ Ω |∆u| dx and satisfies the same properties as in Proposition 2.1. Hereafter, let
Throughout this paper, the letters k, c, C, C i , i = 1, 2, . . . , denote positive constants which may change from line to line.

Hypotheses and main results
Let us impose the following hypotheses on the non-linearity f : Ω × ℝ → ℝ: (f1) f is a C 1 function such that f(x, 0) = 0. (f2) There exists Ω 1 ⋐ Ω with |Ω 1 | > 0, and a nonnegative function h 1 on Ω 1 such that (f3) There exists a positive function h on Ω such that h ∈ L s(x) (Ω) and where s, s 1 , r and r 1 ∈ C(Ω) are such that 1 < max{r(x), r 1 Some remarks regarding the hypotheses are in order.
Here we state our main results asserted in the following two theorems.

Proof of Theorem 3.3
The study of the existence of solutions to problem (P −λ ) is done by looking for critical points to the functional I −λ : X → ℝ defined by in the Sobolev space X. The proof of the Theorem 3.3 is organized in several lemmas. Firstly, under Remark 3.1 one has |u λ | (1−γ(x))p(x) ≤ C 2 ‖u λ ‖ for all u λ ∈ X (4.1) and Now, we are in a position to show that I −λ possesses a nontrivial global minimum point in X. Proof. First, we recall that in view of assumptions (f3), (f4), inequality (4.1), Remark 3.1 and Proposition 2.1 one has for any u λ ∈ X with ‖u λ ‖ > max (1, A),  Proof. Let φ ∈ C ∞ 0 (Ω) such that supp(φ) ⊂ Ω 1 ⋐ Ω, φ = 1 in a subset Ω ⊂ supp(φ) and 0 ≤ φ ≤ 1 in Ω 1 . Using assertions on the functions g and F and assumption (f2), we have Finally, we point out, using the hypothesis on φ and the definition of the modular on X, that then ρ p(x) (∆φ) = 0, and consequently ‖φ‖ = 0, which contradicts the choice of φ and gives the proof of Lemma 4.2.
Proof. Let {u n } be a minimizing sequence, that is to say I −λ (u n ) → m λ . Suppose {u n } is not bounded, so ‖u n ‖ → +∞ as n → +∞. Since I λ is coercive, we have This contradicts the fact that {u n } is a minimizing sequence, so {u n } is bounded in X, and therefore, up to a subsequence, there exists u λ ∈ X such that u n ⇀ u λ weakly in X, in Ω.
Since J : X → ℝ is sequentially weakly lower semi-continuous (see [12]), we have On the other hand, by Vital's theorem (see [16, p. 113]), we can claim that Indeed, we only need to prove that is equi-absolutely-continuous. Note that {u n } is bounded in X, so Remark 3.1 implies that {u n } is bounded in L p * (x) (Ω). For every ε > 0, using Proposition 2.1 and the absolutely-continuity of Consequently, by the Hölder inequality and Proposition 2.1 one has In what follows, we remark, using assumptions (f2) and (f3), that for all ε > 0 there exists C ε such that Then by the Hölder inequality one has Besides, if u n ⇀ u λ in X, then we have strong convergence in L s (x)r(x) (Ω) and L s 1 (x)r 1 (x) (Ω). So the Lebesgue dominated convergence theorem and Proposition 2.2 enable us to state the following assertion: If Using (4.2), (4.3) and (4.4), we deduce that I −λ is weakly lower semi-continuous, and consequently The proof of Lemma 4.3 is now completed.
Proof of Theorem 3.3. Now, let us show that the weak limit u λ is a weak solution of problem (P −λ ) if λ > 0 is sufficiently large. Firstly, observe that I −λ (0) = 0. So, in order to prove that the solution is nontrivial, it suffices to prove that there exists λ * > 0 such that For this purpose, we consider the variational problem with constraints and define Λ * := inf{λ > 0 : (P −λ ) admits a nontrivial weak solution}.
Therefore, the above remarks show that λ * ≥ Λ * and that problem (P −λ ) has a solution for all λ > λ * . We now argue that problem (P −λ ) has a solution for all λ > Λ * . Fixing λ > Λ * , by the definition of Λ * we can take μ ∈ (Λ * , λ) such that I −μ has a nontrivial critical point u μ ∈ X. Since μ < λ, we obtain that u μ is a subsolution of problem (P −λ ). We now want to construct a super-solution of problem (P −λ ) which dominates u μ . For this purpose, we introduce the constrained minimization problem inf{I −λ (w) : w ∈ X and w ≥ u μ }.
By using the previous arguments to treat (4.5), follows that the above minimization problem has a solution u λ > u μ . Moreover, u λ is also a weak solution of problem (P −λ ) for all λ > Λ * . With the arguments developed in [15], we deduce that problem (P −λ ) has a solution if λ = Λ * . Now, it remains to show that ∆u λ = 0 on ∂Ω. Due to the above arguments, one has where Relation (4.6) implies that Let ζ be the unique solution of the problem Relation (4.7) yields Using the Green formula, we have Therefore, On the other hand, for allũ λ ∈ C ∞ 0 (Ω) there exists a unique v ∈ C ∞ 0 (Ω) such that ∆v =ũ λ in Ω. Thus, relation (4.8) can be rewritten as Applying the fundamental lemma of the calculus of variations, we deduce that Since ζ = 0 on ∂Ω, we conclude that ∆u λ = 0 on ∂Ω. Thus, u λ is a nontrivial weak solution of problem (P −λ ) such that ∆u λ = 0. This completes the proof of Theorem 3.3.

Proof of Theorem 3.4
The proof of Theorem 3.4 is organized in several lemmas. Firstly, we show the existence of a local minimum for I +λ in a small neighborhood of the origin in X.
In the sequel, put m 1 λ = inf v λ ∈X I +λ (v λ ). As a last proposition, we have the following. with g(x) > 0 for almost every x ∈ Ω, and assume that assertions (f5) and (f6) hold. Then I +λ reaches its global minimizer in X, that is, there exists v λ ∈ X such that I +λ (v λ ) = m 1 λ < 0.
Proof. The proof of Lemma 5.3 is word for word as the one of Lemma 4.3.