Multiple Solutions for a Class of New p ( x ) -Kirchhoff Problem without the Ambrosetti-Rabinowitz Conditions

: In this paper, we consider a nonlocal p ( x ) -Kirchhoff problem with a p + -superlinear subcritical Caratheodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Under some certain assumptions, we prove the existence of nontrivial solutions and many solutions. Our results are an improvement and generalization of the corresponding results obtained by Hamdani et al. (2020).


Introduction
This paper is concerned with the following nonlocal p(x)-Kirchhoff problem    − a − b Ω |∇u| p(x) dx div(|∇u| p(x)−2 ∇u) = λ|u| p(x)−2 u + g(x, u), in Ω, where Ω is a smooth bounded domain in R N , a ≥ b > 0, p ∈ C(Ω) with 1 < p(x) < N, λ > 0 is a real number, and g : Ω × R → R is a Carathéodory function whose potential satisfies some conditions which will be stated later on. The Kirchhoff type equations involving variable exponent growth conditions have been a very interesting topic in recent years, and we have seen the publication of a great number of manuscripts dealing with this subject (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references therein). Problems of this type arise in mathematical models of various physical and biological phenomena. We mention the works of Shahruz et al. [16] (in physics systems), Chipotv and Rodrigues [17] (in biological systems). Since the left-hand side in (1) contains an integral over Ω, it is no longer a pointwise identity, and therefore, it is often called a nonlocal problem. It was proposed by Kirchhoff in 1883 as a generalization of the well-known D'Alembert wave equation For free vibrations of elastic strings, see [18]. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. The parameters in (2) have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, ρ is the mass density, and p 0 is the initial tension.
Recently, Hamdani, Harrabi, Mtiri and Repovs established in [19] the existence of nontrivial solutions for problem 1 by assuming the following conditions: (g 1 ) g ∈ C(Ω × R), and positive constant C, such that |g(x, t)| ≤ C(1 + |t| q(x)−1 ), ∀(x, t) ∈ Ω × R, p + ) such that for all |t| ≥ s A and x ∈ Ω, for all x ∈ Ω and t ∈ R. Then by the Mountain Pass theorem and Fountain Theorem, the following result was presented.
It is well known that, the condition (g 3 ) is originally due to Ambrosetti and Rabinowitz [20]. This is a tool to study superlinear problems, it is a natural and useful condition not only to ensure that the Euler-Lagrange functional associated to problem (1) has a mountain pass geometry, but also to guarantee that the Palais-Smale sequence of the Euler-Lagrange functional is bounded. However, condition (g 3 ) is too restrictive and eliminates many nonlinearities. Clearly, the condition where c 1 , c 2 are two positive constants. However, there are many functions which are superlinear at infinity, but do not satisfy the condition (g 3 ), for example, At this purpose, we would note that from (3) and the fact that θ > p + , it follows that (g 5 ) lim |t|→∞ G(x,t) |t| p + = +∞, uniformly a.e. x ∈ Ω. Moreover, condition (g 5 ) characterizes the nonlinearity g to be p + -superlinear at infinity.
In this paper, we consider problem (1) in the case when the nonlinear term g(x, t) is p + -superlinear at infinity but does not satisfy condition (g 3 ). More precisely, we shall study the existence and multiplicity of weak solutions of problem (1) under the suitable conditions. To state our results, we make the following assumption on g: We remark that the condition (g 6 ) is a consequence of the following condition (g 6 ) , which was firstly introduced by Miyagaki and Souto [21] and developed by G. Li et al. [22] and C. Ji [23]: (g 6 ) There exists t 0 > 0 such that for ∀x ∈ Ω, is increasing in t ≥ t 0 and decreasing in t ≤ −t 0 .
The readers may consult the proof and comments on this assertion in the papers [21][22][23] and the references cited there. Now, we give an example to illustrate the feasibility of assumptions (g 1 ) − (g 2 ) and (g 4 ) − (g 6 ). Let by a straightforward computation, we deduce that So, it is easy to check that g(x, t) satisfies our conditions (g 1 ) (when (q(x) ≡ p + + 1), (g 2 ) and (g 4 ) − (g 6 ), but it does not satisfy the condition (g 3 ).
We are now in the position to state our main results.
The rest of this paper is organized as follows. In Section 2, we present some necessary preliminary (Ω). In Section 3, we establish the variational framework associated with problem (1), and we also state the critical point theorems needed for the proofs of our main results. We complete the proofs of Theorems 3 and 4 in Sections 4 and 5, respectively.

Preliminaries
Firstly, we introduce some definitions and basic properties of the Lebesgue-Sobolev spaces with variable exponents. The detailed result can be found in [24][25][26][27][28][29]. Let Ω be a bounded domain of R N . Set For any h ∈ C + (Ω), we define For any p ∈ C + (Ω), we define the variable exponent Lebesgue space: and define the variable exponent Sobolev space . With these norms, the spaces L p(x) (Ω) and W 1,p(x) (Ω) are separable and reflexive Banach spaces: see [27] for details.
(2) If h ∈ C(Ω) and 1 ≤ h(x) ≤ p * (x) for any x ∈ Ω, then the embedding from W By (1) of Proposition 2, we know that |∇u| p(x) and u are equivalent norms on W 1,p(x) 0 (Ω). We will use |∇u| p(x) to replace u in the following discussions.
To the best of our knowledge, necessary and sufficient conditions in order to ensure that has not been obtained yet, except in the case N = 1, (Theorem 3.2 [31]). To overcome this difficulty, the following definition is given.

Definition 1.
We say that p(·) belongs to the modular Poincaré inequality, MPI(Ω), if there exists necessary conditions to ensure that (7) holds.

Variational Setting and Some Preliminary Lemmas
To prove our theorems, we recall the variational setting corresponding to the problem (1). Firstly, we introduce the energy functional ϕ λ (u) : W

1.p(x) 0
(Ω) → R associated with problem (1), defined by From the hypotheses on g, it is standard to check that ϕ λ ∈ C 1 (W
Proof. (a) From (g 5 ), it follows that, ∀M > 0, ∃C M > 0, such that Therefore, ϕ λ is unbounded from below. (b) Firstly, from (g 1 ) and (g 2 ), it follows that, for any given ε > 0, there exists C ε > 0, such that Thus, for u ∈ W 1,p(x) 0 (Ω) with u ≤ 1, using Proposition 2, (6) and (7), we have From this, and the fact that p + < 2p − < q − , we can choose r > 0 and (Ω) and u = r. This proves (b). So far, we complete the proof. Definition 2. Let (X, · ) be a real Banach space, I ∈ C 1 (X, R). We say that I ∈ C 1 (E, R) satisfies (C) c -condition if any sequence {u n } ⊂ E satisfying I(u n ) → c and I (u n ) E * (1 + u n ) → 0 contains a convergent subsequence. If this condition is satisfied at every level c ∈ R, then we say that I satisfies (C)-condition. Now, we present the following Lemmas which will play a crucial role in the proof of Main Theorems. First of all, let us recall the mountain pass theorem, which we use in the proof of Theorem 3.
Lemma 2 (Theorem 1 [35]). Let X be a real Banach space, let I : X → R be a functional of class C 1 (X, R) that satisfies the (C) c condition for any c ∈ R, I(0) = 0, and the following conditions hold: (1) There exist positive constants ρ and α such that I(u) ≥ α for any u ∈ X with u = ρ.
(2) There exists a function e ∈ X such that e > ρ and I(e) ≤ 0. Then, the functional I has a critical value c ≥ α, that is, there exists u ∈ X such that I(u) = c and I (u) = 0 in X * .
In order to prove the Theorem 4, we will use the following symmetric mountain pass theorem of Rabinowitz [36]. It is remarked that the symmetric mountain pass theorem is established under the (PS) condition. Since the deformation theorem is still valid under the (C) c -condition ( [37]), we see that the symmetric mountain pass theorem also holds under the (C) c -condition (see [38]).

Lemma 3 ([38]).
Assume that X is an infinite dimensional Banach space, and let I : X → R be an even functional of class C 1 (X, R) that satisfies the (C) c condition for any c ∈ R, I(0) = 0, and the following conditions hold: (1) There exist two constants ρ, α > 0 such that I(u) ≥ α for any u ∈ X with u = ρ; (2) for all finite dimensional subspace X ⊂ X, there exists R = R( X) > 0 such that I(u) ≤ 0 for any u ∈ X with u = ρ. Then, I possesses an unbounded sequence of critical values characterized by a minimax argument.

The Proof of Theorem 3
In this section, we will prove Theorem 3. Firstly, we show that (C)-condition holds. The proof idea is mainly due to Hamdani, Harrabi, Mtiri and Repovs [19], where the Palais-Smale compactness condition was obtained.
Proof. Let {u n } ⊂ E be a (C) c sequence. Firstly, we claim that the sequence {u n } is bounded in E. Indeed, if u n ≤ 1, we have done. If u n > 1, then from (g 6 ), (9) and 2p(x) ≥ 2p − > p + , we have that From this, we conclude that It follows from (10) and 2p − > p + > p − that {u n } is bounded in E. Therefore, going if necessary to a subsequence, we may assume that u n u 0 in E, It is easy to compute directly that and Ω |u n | p(x)−2 u n (u n − u 0 )dx Moreover, by (9), one yields lim n→∞ ϕ λ (u n ), u n − u 0 = 0.
Finally, the combination of (12)-(15) implies Similar to the proof of Lemma 3.1 in [19], we can deduce that the sequence This fact combined with (16) implies that lim n→∞ L(u n ), u n − u 0 = 0.
Since L is of type (S) + by Proposition 3, we obtain u n → u 0 in E. The proof is complete. Now, we are ready to prove Theorem 3.
Proof of Theorem 1. Let X = E and I = ϕ λ . Obviously, ϕ λ (0) = 0, and Lemma 4 implies that ϕ λ satisfies the (C)-condition for any c < a 2 2b . In view of Lemma 1, ϕ λ satisfies the mountain pass geometry for any λ < λ 0 . Therefore, all the assumptions of Lemma 2 are satisfied, so that, for each λ < λ 0 , the problem (1) admits at least one nontrivial solution in E. This completes the proof.

The Proof of Theorem 4
In this section, we will show that (1) has many pairs of solutions by using Lemma 3. To prove the Theorem 4, we will need the following Lemma 5.
Proof. Arguing indirectly, assume that there exists a sequence {u n } ⊂ E such that u n → +∞, n → +∞ and ϕ λ (u n ) ≥ M, ∀n ∈ N, where M ∈ R is a fixed constant not depending on n ∈ N.
Let v n = u n u n . Then, it is obvious that v n = 1. Since dim E < +∞, there exists v ∈ E \ {0} such that up to a subsequence, v n − v → 0 and v n (x) → v(x) a.e. x ∈ Ω as n → +∞.
Then, by (18) and (19), we deduce that there is a constant C 5 ∈ R, such that G(x, t) ≥ C 5 , ∀(x, t) ∈ Ω × R. (20) some weaker assumptions on f . To deal with the difficulty caused by the noncompactness due to the Kirchhoff function term, we must estimate precisely the value of c and give a threshold value (see Lemma 4) under which the Cerami condition at the level c for ϕ λ is satisfied. So, the variational technique for problem (1) becomes more delicate. Furthermore, under an additional assumption of symmetry, the infinitely many solutions are shown, formulated in the paper as Theorem 4. One example is given to show the effectiveness of our results.