Existence of solutions for perturbed fourth order elliptic equations with variable exponents

Using variational methods, we study the existence and multiplicity of solutions for a class of fourth order elliptic equations of the form { ∆p(x)u−M (∫ Ω 1 p(x) |∇u| p(x) dx ) ∆p(x)u = f (x, u) in Ω, u = ∆u = 0 on ∂Ω, where Ω ⊂ RN , N ≥ 3, is a smooth bounded domain, ∆p(x)u = ∆(|∆u| p(x)−2∆u) is the operator of fourth order called the p(x)-biharmonic operator, ∆p(x)u = div ( |∇u|p(x)−2∇u ) is the p(x)-Laplacian, p : Ω → R is a log-Hölder continuous function, M : [0,+∞) → R and f : Ω ×R → R are two continuous functions satisfying some certain conditions.

We point out that if p(.) is a constant then problem (1.1) has been studied by many authors in recent years, we refer to some interesting papers [4,11,21,26,27,31,32,[36][37][38]40].In [38], Wang and An considered the following fourth-order elliptic equation where Ω ⊂ R N , N ≥ 1, is a smooth bounded domain, M : [0, +∞) → R and f : Ω × R → R are two continuous functions.This problem is related to the stationary analog of the evolution equation of Kirchhoff type where ∆ 2 is the biharmonic operator, ∇u denotes the spatial gradient of u, see [8] for the meaning of the problem from the point of view of physics and engineering.By assuming that M is bounded on [0, +∞) and the nonlinear term f satisfies the Ambrosetti-Rabinowitz type condition, Wang et al. obtained in [38] at least one nontrivial solution for problem (1.2) using the mountain pass theorem.Moreover, the authors also showed the existence at least two solutions in the case when f is asymptotically linear at infinity.After that, Wang et al. [37] studied problem (1.2) in the case when M is unbounded function, i.e.M(t) = a + bt, where a > 0, b ≥ 0 by using the mountain pass techniques and the truncation method.Some extensions regarding these results can be found in [4,11,21,31,36,40] in which the authors considered problem (1.2) in R N or the nonlinearities involved critical exponents.In [26,27,32], problem (1.1) was studied in the general case when p(.) = p ∈ (1, +∞) is a constant.
In recent years, the study of differential equations and variational problems with nonstandard p(x)-growth conditions has received more and more interest.The reason of such interest starts from the study of the role played by their applications in mathematical modelling of non-Newtonian fluids, in particular, the electrorheological fluids and of other phenomena related to image processing, elasticity and the flow in porous media, we refer the readers to [5,35,43] for more details.Some results on problems involving p(x)-Laplace operator or p(x)biharmonic operator can be found in [6,7,9,10,12,16,17,28,30,33].These types of operators where p(.) is a continuous function possess more complicated properties than the constant cases, mainly due to the fact that they are not homogeneous.We also find that Kirchhoff type problems with variable exponents has received a lot of attention in recent years, see for example [1, 2, 13-15, 18-20, 41].
Motivated by the contributions cited above, in this paper we study the existence and multiplicity of solutions for perturbed fourth order elliptic equations with variable exponents of the form (1.1).More precisely, we consider problem (1.1) in two case when f is sublinear or superlinear at infinity.In the sublinear case, we obtain an existence result using the minimum principle while in the superlinear case we prove some existence and multiplicity results with the help of the Mountain Pass Theorem, Fountain Theorem and Dual Fountain Theorem.To the best of our knowlegde, the present paper is the first contribution to the study of this type of problems in Sobolev spaces with variable exponents.

Preliminaries
We recall in what follows some definitions and basic properties of the generalized Lebesgue-Sobolev spaces L p(x) (Ω) and W k,p(x) (Ω) where Ω is an open subset of R N .In that context, we refer to the books of Diening et al. [22] and Musielak [34], the papers of Fan et al. [24,25], Zang et al. [42], Ayoujil et al. [6,7] and Boureanu et al. [10].Set For any h ∈ C + (Ω) we define For any p(x) ∈ C + (Ω), we define the variable exponent Lebesgue space We recall the following so-called Luxemburg norm on this space defined by the formula Variable exponent Lebesgue spaces resemble classical Lebesgue spaces in many respects: they are Banach spaces, the Hölder inequality holds, they are reflexive if and only if 1 < p − ≤ p + < ∞ and continuous functions are dense if p + < ∞.The inclusion between Lebesgue spaces also generalizes naturally: if 0 < |Ω| < ∞ and p 1 , p 2 are variable exponents so that p 1 (x) ≤ p 2 (x) a.e.x ∈ Ω then there exists the continuous embedding L p 2 (x) (Ω) → L p 1 (x) (Ω).We denote by L p (x) (Ω) the conjugate space of L p(x) (Ω), where 1 p(x) + 1 p (x) = 1.For any u ∈ L p(x) (Ω) and v ∈ L p (x) (Ω) the Hölder inequality holds true.
An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the L p(x) (Ω) space, which is the mapping ρ p(x) : L p(x) (Ω) → R defined by If u ∈ L p(x) (Ω) and p + < ∞ then the following relations hold As in the constant case, for any positive integer k, the Sobolev space with variable exponent W k,p(x) (Ω) is defined by , also becomes a separable and reflexive Banach space.Due to the log-Hölder continuity of the exponent p, the space C ∞ (Ω) is dense in W k,p(x) (Ω).Moreover, we have the following embedding results.Proposition 2.1 (see [24,25]).For p, r ∈ C + (Ω) such that r(x) ≤ p * k (x) for all x ∈ Ω, there is a continuous embedding W k,p(x) (Ω) → L r(x) (Ω), If we replace ≤ with <, the embedding is compact.
We denote by W k,p(x) 0 (Ω) the closure of C ∞ 0 (Ω) in W k,p(x) (Ω).Note that the weak solutions of problem (1.1) are considered in the generalized Sobolev space According to [42], the norm .X is equivalent to the norm |∆.| p(x) in the space X.Consequently, the norms .2,p(x) , .X and |∆.| p(x) are equivalent.For this reason, we can consider in the space X the following equivalent norms Let us define the functional Λ : X → R by then using similar arguments as in [10, Proposition 1] we obtain the following modular-type inequalities.

Main results
In this section, we will discuss the existence and multiplicity of weak solutions of problem (1.1).Let us denote by c i , i = 1, 2, . . .general positive constants whose value may change from line to line.We will look for weak solutions of problem (1.1) in the space X := W 1,p(x) 0 (Ω) ∩ W 2,p(x) (Ω) with the norm mentioned as in Section 2. First, let us make the definition of a weak solution of problem (1.1) as follows.
Let us define the functional J : X → R by where and M(t) = t 0 M(s) ds.Using some simple computations, we can show that J ∈ C 1 (X, R) and its derivative is given by the formula for all u, v ∈ X.Thus, we will seek weak solutions of problem (1.1) as the critical points of the functional J.We first obtain an existence result for problem (1.1) in the case when f is sublinear at infinity.We also consider case when the Kirchhoff function M are allowed to be degenerate at zero.
Then problem (1.1) has a nontrivial weak solution.
Proof.From (F 0 ), there exists We also obtain from (M 1 ) and (M 2 ) that where M(t) = t 0 M(s) ds and t is a positive constant depending on > 0. For t 0 given as above, let us define the set Then it follows that X is a closed subspace of the reflexive Banach space X, so X is a reflexive Banach space too.Moreover, for any u ∈ X, we have By relations (3.3) and (3.4), by the Sobolev embedding, we deduce that for any u ∈ X with u > 1 large enough, Since 1 < q + < p − it follows that the functional J is coercive in X.Moreover, we find that J is weakly lower semicontinuous in X and thus, J attains its infimum in X and there exists Next, we show that u 0 = 0 i.e. u 0 is a nontrivial weak solution of problem (1.1).Let Then, for any 0 < t < δ we deduce from (F 0 ) and (3.4) that |φ| r(x) dx.
In the next part of this paper, we will study the existence and multiplicity of weak solutions for problem (1.1) in the case when f is superlinear at infinity.In the sequel, we always assume that the following conditions hold: (M 1 ) There exists m 0 > 0 such that M(t) ≥ m 0 , ∀t ≥ 0; (M 2 ) There exists µ ∈ (0, 1) such that where M(t) = t 0 M(s) ds.Definition 3.3.A functional J is said to satisfy the Palais-Smale condition (or (PS) condition) in a space X, if any sequence {u n } ⊂ X such that {J(u n )} is bounded and J (u n ) → 0 as n → ∞, has a convergent subsequence.Lemma 3.4.If M satisfies (M 1 )-(M 2 ), f satisfies (F 0 ) and the Ambrosetti-Rabinowitz type condition, namely, (F 1 ) there exist T 0 > 0 and θ > p + 1−µ such that then the functional J satisfies the (PS) condition.
Proof.Suppose that {u n } ⊂ X, |J(u n )| ≤ c and J (u n ) → 0 in X * as n → ∞.We will show that {u n } is bounded in X.By contradiction, we assume that u n → +∞.For n large enough, by the conditions (F 1 ), (M 1 ), (M 2 ) and Proposition 2.2 we have where Dividing by u n p − in the last inequality and letting n → ∞ we obtain a contradiction.It follows that the sequence {u n } is bounded in X.Without loss of generality, we assume that {u n } converges weakly to u in X.Then {u n } converges strongly to u in L r(x) (Ω) for all r(x) < p * 2 (x).Since J (u n ) → 0 in X * we deduce that J (u n )(u n − u) → 0 as n → ∞.We also have J (u)(u n − u) → 0 as n → ∞ because {u n } converges weakly to u in X.Thus, Using (F 0 ) and the Hölder inequality, we have Since the sequence {u n } converges weakly to u ∈ X = W 1,p(x) 0 (Ω) ∩ W 2,p(x) (Ω), it is bounded in X and converges weakly to u in W 1,p(x) 0 (Ω), so we deduce that (3.7) Let us recall the following elementary inequalities (see [6]) then, it follows from (3.8) and (3.9) that U p(x) V p(x) V p(x) where A (i) , C (i) : R i × R i → R, i = 1, N are defined by the following formulas
Proof.Our idea is to apply the mountain pass theorem [3].By Lemma 3.4, J satisfies the Palais-Smale condition in X.Since p + < q − ≤ q(x) < p * 2 (x), the embedding X → L p + (Ω) is continuous and compact and then there exists c 7 > 0 such that Let > 0 be small enough such that c p + 7 < 1 2p + min {1, m 0 }.By the assumptions (F 0 ) and (F 2 ), there exists c > 0 depending on such that Hence, for all u ∈ X with u < 1, we have where c is a positive constant.Since q − > p + , we conclude that there exist α > 0 and ρ > 0 such that J(u) ≥ α > 0 for all u ∈ X with u = ρ.
On the other hand, from (F 1 ) it follows that From (M 2 ) we can easily obtain that where t 0 is an arbitrary positive constant.Hence, for w ∈ X\{0} and t > 1, we have 1−µ > p + .Since J(0) = 0, we conclude that J satisfies all assumptions of the mountain pass theorem [3].So, J admits at least one nontrivial critical point and problem (1.1) has a nontrivial weak solution.

N. T. Chung
In what follows, we will study the multiplicity of weak solutions for problem (1.1) by using the Fountain Theorem and the Dual Fountain Theorem.For the reader's convenience, we recall these results as follows.
Then, there exists a subsequence of {u k }, still denoted by {u k } such that {u k } converges weakly to u in X and lim k→∞ e * j , u k = e * j , u = 0, j = 1, 2, . . ., which implies that u = 0 and thus, {u k } converges weakly to 0 in X.Since the embedding X → L s(x) (Ω) is compact, {u k } converges strongly to 0 in L s(x) (Ω).Therefore, we have β k → 0 as k → ∞.
Definition 3.8.We say that J satisfies the (PS) * c condition (with respect to (Y n )) if any sequence {u n j } ⊂ X such that u n j ∈ Y n j , J(u n j ) → c and (J| Y n j ) (u n j ) → 0 as n j → +∞, contains a subsequence converging to a critical point of J. Proposition 3.9 (see [39,Dual Fountain Theorem]).Assume that (X, • ) is a separable Banach space, J ∈ C 1 (X, R) is an even functional satisfying the (PS) * c condition.Moreover, for each k = 1, 2, . . ., there exist ρ k > r k > 0 such that Then J has a sequence of negative critical values tending to 0. Theorem 3.10.Assume that the conditions (M 1 ), (M 2 ), (F 0 ), (F 1 ) hold, and f satisfies (F 3 ) f (x, −t) = − f (x, t) for all x ∈ Ω and t ∈ R.
Proof of Theorem 3.10.According to (F 3 ) and Lemma 3.4, J is an even functional and satisfies the (PS) condition.We will prove Theorem 3.10 by using the Fountain Theorem, see Proposition 3.7.Indeed, we will show that if k is large enough, then there exist ρ k > r k > 0 such that (A 1 ) and (A 2 ) hold.Thus, the assertion of conclusion can be obtained.
In order to prove Theorem 3.11, we need to verify the following lemma.Lemma 3.12.Assume that the conditions (M 1 ), (M 2 ), (F 0 ) and (F 1 ) are satisfied.Then the functional J satisfies the (PS) * c condition.
Proof.Let {u n j } ⊂ X be such that u n j ∈ Y n j and J(u n j ) → 0 and (J| Y n j ) (u n j ) → 0 as n j → ∞.Similar to the process of verifying the (PS) condition in the proof of Lemma 3.4, we can get the boundedness of { u n j }.Going, if necessary, to a subsequence, we can assume that {u n j } converges weakly to u in X.As X = ∪ n j Y n j , we can choose v n j ∈ Y n j such that v n j → u.Hence, lim From the proof of Lemma 3.4, J is of (S + ) type, so we can conclude that u n j → u as n j → ∞, furthermore we have J (u n j ) → J (u).
Let us prove J (u) = 0, i.e., u is a critical point of J. Indeed, taking arbitrarily w k ∈ Y k , notice that when n j ≥ k we have J (u)(w k ) = (J (u) − J (u n j ))(w k ) + J (u n j )(w k ) = (J (u) − J (u n j ))(w k ) + (J| Y n j ) (u n j )(w k ).
Going to limit in the right hand-side of above equation reaches J (u)(w k ) = 0 for all w k ∈ Y k .Thus, J (u) = 0 and the functional J satisfies the (PS) * c condition for every c ∈ R.
Proof of Theorem 3.11.From (F 0 ), (F 1 ), (F 3 ) and Lemma 3.12, we know that J is an even functional and satisfies the (PS) * c condition, the assertion of conclusion can be obtained from Dual Fountain Theorem, see Proposition 3.9.
Hence, d k → 0 as k → ∞, i.e., (B 3 ) is satisfied.Conclusion of Theorem 3.11 is reached by the Dual Fountain Theorem.
from the definition of the functional J and relations (3.5)-(3.7),we have 0 inf