Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 24, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MULTIPLE SOLUTIONS FOR p(x)-KIRCHHOFF TYPE PROBLEMS WITH ROBIN BOUNDARY CONDITIONS GHASEM A. AFROUZI, NGUYEN THANH CHUNG, ZOHREH NAGHIZADEH Abstract. This article considers p(x)-Kirchhoff type problems with Robin boundary conditions. Using the mountain pass theorem, the Ekeland’s varia- tional principle, and Krasnoselskii’s genus theory, we prove that the problem has at least two nontrivial weak solutions or infinitely many nontrivial weak so- lutions under some suitable conditions on the nonlinearities. The main results improve and generalize the previous ones introduced in [2, 7]. 1. Introduction In this article, we study the existence of weak solutions for p(x)-Kirchhoff type problems with Robin boundary conditions −M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λg(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u = 0, x ∈ ∂Ω, (1.1) where Ω is a bounded domain in RN with smooth boundary ∂Ω, ∂u ∂ν is the outer normal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L∞(∂Ω), β− := infx∈∂Ω β(x) > 0, p ∈ C+(Ω), 1 < p− := infx∈Ω p(x) ≤ p+ := maxx∈Ω p(x) < N , λ is a nonnegative parameter, f : Ω× R→ R and M : R+ := [0,+∞)→ R+ are two continuous functions, g : Ω→ R is a measurable function. Problem (1.1) is related to the stationary version of the Kirchhoff equation ρ ∂2u ∂t2 − (P0 h + E 2L ∫ L 0 |∂u ∂x |2 dx )∂2u ∂x2 = 0, (1.2) presented by Kirchhoff in 1883 as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings, see [22]. The parameters in (1.2) 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 P0 is the initial tension. Problem (1.2) is often called a nonlocal problem because it contains an integral over Ω. This causes some mathematical difficulties which make the study of such a problem particularly interesting. The nonlocal problem 2020 Mathematics Subject Classification. 35J70, 35J60, 35D05. Key words and phrases. p(x)-Kirchhoff type problems; Robin boundary conditions; variational methods. ©2022. This work is licensed under a CC BY 4.0 license. Submitted May 11, 2021. Published March 25, 2022. 1 2 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 models several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density, see [8]. Kirchhoff type problems have been studied in many papers in the previous decades. In [7, 13, 21, 24, 27, 28], using various methods the authors study the existence and multiplicity of solutions for Kirchhoff type problems involving the p-Laplacian operator −∆p(·) = − div(|∇ · |p−2∇·). The p(x)-Laplacian operator where p(·) is a continuous function possesses more complicated properties than the p-Laplacian operator, mainly due to the fact that it is not homogeneous. The study of various mathematical problems with variable exponent are interesting in applications and raise many difficult mathematical problems, see [23, 25]. For this reason, ordinary differential and partial differential equations with nonstan- dard growth conditions have received specific attention in recent years, we refer to some results on p(x)-Kirchhoff type problems with Dirichlet or Neumann boundary conditions [5, 9, 10, 12, 14, 15, 20]. Relatively speaking, Kirchhoff type problems with Robin boundary conditions have rarely been considered. Robin boundary conditions are a weighted combination of Dirichlet and Neuman boundary condi- tions and it is also called impedance boundary conditions, from their application in electromagnetic problems or convective boundary conditions from their applica- tion in heat transfer problems. Moreover, Robin conditions are commonly used in solving Sturm-Liouville problems which appear in many contexts in sciences and engineering, see [16]. To the best of our knowledge, Allaoui [2] first introduced the p(x)-Kirchhoff type problems involving Robin boundary conditions and studied problem (1.1) in the case λ = 0 by using the mountain pass theorem, the fountain theorem and some properties of (S)+ type operator. Regarding the p(x)-Laplacian problems with the Robin boundary conditions in the local case when M(t) ≡ 1, we refer to some papers [1, 3, 16, 19, 26], in which some existence and multiplicity results were obtained by using variational methods. Motivated by above mentioned papers and the results on the Kirchhoff type problem involving Laplace operator −∆(·) in [7], the purpose of this article is to consider Robin problem (1.1) with perturbation g and parameter λ. More precisely, under some suitable conditions on the nonlinear term f and the Kirchhoff function M , we prove that problem (1.1) has at least two weak solutions if λ > 0 small enough, see Theorem 2.2. In the case when λ = 0, we prove problem (1.1) with subcritical growth condition has infinitely many solutions, see Theorem 2.7. Our proofs are essentially based on the mountain pass theorem [4], the Ekeland variational principle [18] and Krasnoselskii’s genus theory [11]. We emphasize that the results introduced here are new even in the case when p(·) is a constant and we do not need the non-degenerate condition on the Kirchhoff function M as in [2, 7], see assumption (A1). Next we recall some definitions and basic properties of the generalized Lebesgue- Sobolev spaces Lp(x)(Ω) and W 1,p(x)(Ω) where Ω is an open subset of RN . In that context, we refer to the books [17, 25] and the papers [1, 16, 19, 23]. Set C+(Ω) := {h; h ∈ C(Ω), h(x) > 1 for all x ∈ Ω}. For any h ∈ C+(Ω) we define h+ = sup x∈Ω h(x), h− = inf x∈Ω h(x). EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 3 For any p(x) ∈ C+(Ω), we define the variable exponent Lebesgue space Lp(x)(Ω) = { u measurable real-valued functions such that ∫ Ω |u(x)|p(x) dx <∞ } . We recall the following so-called Luxemburg norm on this space defined by the formula |u|Lp(x)(Ω) = |u|p(x) := inf { λ > 0 : ∫ Ω ∣∣u(x) λ ∣∣p(x) dx ≤ 1 } . 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 p1, p2 are variable exponents so that p1(x) ≤ p2(x) a.e. x ∈ Ω then there exists the continuous embedding Lp2(x)(Ω) ↪→ Lp1(x)(Ω). We denote by Lp ′(x)(Ω) the conjugate space of Lp(x)(Ω), where 1 p(x) + 1 p′(x) = 1. For any u ∈ Lp(x)(Ω) and v ∈ Lp′(x)(Ω) the Hölder inequalities∣∣ ∫ Ω uv dx ∣∣ ≤ ( 1 p− + 1 (p′)− ) |u|p(x)|v|p′(x) ≤ 2|u|p(x)|v|p′(x) hold. An important role in manipulating the generalized Lebesgue-Sobolev spaces is played by the modular of the Lp(x)(Ω) space, which is the mapping ρp(x) : Lp(x)(Ω)→ R defined by ρp(x)(u) = ∫ Ω |u|p(x) dx. If u ∈ Lp(x)(Ω) and p+ < +∞ then the following relations hold |u|p − p(x) ≤ ρp(x)(u) ≤ |u|p + p(x), (1.3) provided |u|p(x) > 1 while |u|p + p(x) ≤ ρp(x)(u) ≤ |u|p − p(x), (1.4) provided |u|p(x) < 1 and |un − u|p(x) → 0 ⇔ ρp(x)(un − u)→ 0. (1.5) If p ∈ C+(Ω) the variable exponent Sobolev space W 1,p(x)(Ω), consisting of functions u ∈ Lp(x)(Ω) whose distributional gradient ∇u exists almost everywhere and belongs to [Lp(x)(Ω)]N , endowed with the norm ‖u‖ := inf { λ > 0 : ∫ Ω [∣∣∇u(x) λ ∣∣p(x) + ∣∣u(x) λ ∣∣p(x) ] dx ≤ 1 } or ‖u‖ = |u|p(x) + |∇u|p(x), is a separable and reflexive Banach space. The space of smooth functions are in general not dense in W 1,p(x)(Ω), but if the exponent p ∈ C+(Ω) is logarithmic Hölder continuous, that is, |p(x)− p(y)| ≤ − M log(|x− y|) , ∀x, y ∈ Ω, |x− y| ≤ 1 2 , 4 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 then the smooth functions are dense in W 1,p(x)(Ω). The space ( W 1,p(x) 0 (Ω), ‖ · ‖ ) is a separable and Banach space. We note that if s ∈ C+(Ω) and s(x) < p∗(x) for all x ∈ Ω then the embedding W 1,p(x)(Ω) ↪→ Ls(x)(Ω) is compact and continuous, where p∗(x) = Np(x) N−p(x) if p(x) < N or p∗(x) = +∞ if p(x) > N . If s ∈ C+(∂Ω) and s(x) < p∗(x) for all x ∈ ∂Ω then the trace embedding W 1,p(x)(Ω) ↪→ Ls(x)(∂Ω) is compact and continuous, where p∗(x) = (N−1)p(x) N−p(x) if p(x) < N or p∗(x) = +∞ if p(x) > N . Moreover, for any u ∈W 1,p(x)(Ω), let us define ‖u‖∂ := |∇u|Lp(x)(Ω) + |u|Lp(x)(∂Ω), then ‖u‖∂ is a norm on W 1,p(x)(Ω) which is equivalent to the norm ‖u‖, see [16, Theorem 2.1]. Now, let us introduce a norm which will be used later. Let β ∈ L∞(∂Ω) with β− = infx∈∂Ω β(x) > 0, and for any u ∈W 1,p(x)(Ω), define ‖u‖β(x) := inf { λ > 0 : ∫ Ω ∣∣∇u(x) λ ∣∣p(x) dx+ ∫ ∂Ω β(x) ∣∣u(x) λ ∣∣p(x) dσx dx ≤ 1 } , where dσx is the measure on the boundary ∂Ω. Then ‖u‖β(x) is also a norm on W 1,p(x)(Ω) which is equivalent to ‖ · ‖ and ‖ · ‖∂ . Let Iβ(x)(u) = ∫ Ω |∇u|p(x) dx+ ∫ ∂Ω β(x)|u|p(x) dσx, we have ‖u‖p − β(x) ≤ Iβ(x)(u) ≤ ‖u‖p + β(x) (1.6) provided ‖u‖β(x) > 1 while ‖u‖p + β(x) ≤ Iβ(x)(u) ≤ ‖u‖p − β(x) (1.7) provided ‖u‖β(x) < 1 and ‖un − u‖β(x) → 0 ⇔ Iβ(x)(un − u)→ 0. (1.8) Proposition 1.1 (see [19]). For β ∈ L∞(∂Ω) with β− := infx∈∂Ω β(x) > 0, let us define the functional Lβ(x) : W 1,p(x)(Ω)→ R by Lβ(x)(u) = ∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx (1.9) for all u ∈W 1,p(x)(Ω). Then Lβ(x) ∈ C1(W 1,p(x)(Ω),R) and its derivative is given by L′β(x)(u)(v) = ∫ Ω |∇u|p(x)−2uv dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx. (1.10) Moreover, we have the following assertions (i) L′β(x) : W 1,p(x)(Ω) → W−1,p(x)(Ω) is a continuous, bounded and strictly monotone operator; (ii) L′β(x) : W 1,p(x)(Ω) → W−1,p(x)(Ω) is a mapping of type (S)+, i.e. if {un} converges weakly to u in W 1,p(x)(Ω) and lim supn→∞ L′β(x)(un)(un−u) ≤ 0, then {un} converges strongly to u in W 1,p(x)(Ω). EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 5 In the rest of this section, we introduce some notions and results on Krasnosel- skii’s genus theory, the readers can consult [6, 11]. Let Y be a real Banach space. Let us denote by R the class of all closed subsets A ⊂ X\{0} that are symmetric with respect to the origin, that is, u ∈ A implies −u ∈ A, i.e. R = {A ⊂ Y \{0} : A is compact and A = −A}. Definition 1.2. Let A ∈ R and Y = RN . The genus γ(A) of A is defined by γ(A) = min{k ≥ 1 : there exists an odd continuous mapping φ : A→ Rk\{0}}. If such a mapping φ does not exist for any k > 0, we set γ(A) = +∞. Note that if A is a subset, which consists of finitely many pairs of points, then γ(A) = 1. Moreover, from the above definition, γ(∅) = 0. A typical example of a set of genus k is a set, which is homeomorphic to a (k − 1) dimensional sphere via an odd map. Proposition 1.3. Let Y = RN and ∂Ω be the boundary of an open, symmetric and bounded subset Ω ⊂ RN with 0 ∈ Ω. Then we have γ(∂Ω) = N . Let us denote by S the unit sphere in Y . It follows from Proposition 1.3 that γ(SN−1) = N . If Y is of infinite dimension and separable then γ(S) = +∞. We now recall an application of Palais-Smale “compactness” criterion, which was introduced by Clark [11]. Proposition 1.4. Let J ∈ C1(Y,R) be a functional satisfying the Palais-Smale condition. Furthermore, let us suppose that (i) J is bounded from below and even; (ii) There is a compact set K ∈ R such that γ(K) = k and supx∈K J(x) < J(0). Then J possesses at least k pairs of distinct critical points, and their corresponding critical values are less than J(0). 2. Main results 2.1. Existence of at least two solutions. In this part, we consider problem (1.1) in the case when λ > 0. Under suitable conditions on the nonlinear term f and the Kirchhoff function M , we prove that (1.1) has at least two nontrivial weak solutions in the space X. Our idea is to apply the mountain pass theorem in [4] combined with Ekeland’s variational principle in [18] to the energy functional Jλ associated to problem (1.1) when λ > 0 small enough. For this purpose, let us assume that M : R+ → R+ and f : Ω × R → R are continuous functions, and introduce the following conditions: (A1) There exist constants m1,m2 > 0 and 1 < α < q−/p+ such that m1t α−1 ≤M(t) ≤ m2t α−1 for all t ∈ R+ := [0,+∞), where q− = infx∈Ω q(x), q ∈ C+(Ω) is given by assumption (A2); (A2) There exists a positive constant C such that |f(x, t)| ≤ C(1 + |t|q(x)−1), ∀(x, t) ∈ Ω× R, where q ∈ C+(Ω), p(x) < q(x) < p∗(x) = Np(x) N−p(x) for all x ∈ Ω; (A3) f(x, t) = o(|t|αp+−1), t→ 0, uniformly a.e. x ∈ Ω; 6 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 (A4) There exists a constant µ > m2α(p+)α m1(p−)α−1 such that µF (x, t) := µ ∫ t 0 f(x, s) ds ≤ f(x, t)t, ∀(x, t) ∈ Ω× R; (A5) inf{x∈Ω; |t|=1} F (x, t) > 0. Definition 2.1. We say that u ∈W 1,p(x)(Ω) is a weak solution of problem (1.1) if M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx ) − ∫ Ω f(x, u)v dx− λ ∫ Ω g(x)v dx = 0 for all v ∈W 1,p(x)(Ω). The first result of this article is stated as follows. Theorem 2.2. Suppose that g ∈ L αp+ αp+−1 (Ω) and g 6≡ 0. Let conditions (A1)– (A5) hold, then there exists λ∗ > 0 such that (1.1) has at least two nontrivial weak solutions when λ ∈ (0, λ∗). Let us denote by X the variable exponent Sobolev space W 1,p(x)(Ω) and consider the energy functional Jλ : X → R given by Jλ(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx− λ ∫ Ω g(x)u dx. Then by (A2) and the continuous embeddings, we can show that that the func- tional Jλ is well-defined on X and Jλ ∈ C1(X,R) with the derivative given by J ′λ(u)(v) = M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx ) − ∫ Ω f(x, u)v dx− λ ∫ Ω g(x)v dx for all u, v ∈ X. Hence, we can find weak solutions of (1.1) as the critical points of the functional Jλ in the space X. Lemma 2.3. Assume that (A2), (A4) hold and that g ∈ L αp+ αp+−1 (Ω). Then there exist constants ρ, r, λ∗ > 0 such that Jλ(u) ≥ r for all u ∈ X with ‖u‖β(x) = ρ, when λ ∈ (0, λ∗). Proof. Since αp+ < q− ≤ q(x) < p∗(x) for all x ∈ Ω, the embeddings X ↪→ Lαp + (Ω), X ↪→ Lq(x)(Ω) are continuous, and there exists two constants C1, C2 > 0 such that |u|αp+ ≤ C1‖u‖β(x), |u|q(x) ≤ C2‖u‖β(x). (2.1) EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 7 Let 0 < ε < m1 2α(p+)αCαp + 1 , where C1 is given by (2.1). From the assumptions (A3), (A4), there exists a constant C(ε) depending on ε such that F (x, t) ≤ ε|t|αp + + C(ε)|t|q(x), ∀(x, t) ∈ Ω× R, (2.2) Let u ∈ X with ‖u‖β(x) < 1 sufficiently small. From (1.6) and (2.1)-(2.2), applying the Hölder inequality we have Jλ(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx− λ ∫ Ω g(x)u dx ≥ m1 α(p+)α ‖u‖αp + β(x) − ε ∫ Ω |u|αp + dx− C(ε) ∫ Ω |u|q(x) dx− λ|g| αp+ αp+−1 |u|αp+ ≥ m1 α(p+)α ‖u‖αp + β(x) − εC αp+ 1 ‖u‖αp + β(x) − C(ε)Cq − 2 ‖u‖ q− β(x) − λC1|g| αp+ αp+−1 ‖u‖β(x) ≥ ( m1 2α(p+)α ‖u‖αp +−1 β(x) − C(ε)Cq − 2 ‖u‖ q−−1 β(x) − λC1|g| αp+ αp+−1 ) ‖u‖β(x), where C1, C2 > 0 are given by (2.1). Consider the functions γ1 : [0,+∞) → R is given by γ1(τ) = m1 2α(p+)α ταp +−1 − C(ε)Cq − 2 τ q −−1. Since q− > αp+, there exists a constant τ = ρ > 0 obeying the relationship γ1(ρ) = maxτ∈[0,+∞) γ1(τ) > 0. Taking λ∗ = γ1(ρ) 2C1|g| αp+ αp+−1 > 0, it then follows that, if λ ∈ (0, λ∗), we can choose r and ρ > 0 such that Jλ(u) ≥ r > 0 for all u ∈ X with ‖u‖β(x) = ρ. � Lemma 2.4. Assume that (A1), (A4), (A5) hold. Then there exists a function e ∈ X with ‖e‖β(x) > ρ such that Jλ(e) < 0, where ρ is given by Lemma 2.3. Proof. For each x ∈ Ω and t ∈ R, let us define the function γ2(τ) = τ−µF (x, τt)− F (x, t) for all τ ≥ 1. Then we deduce from (A4) that γ′2(τ) = τ−µ−1 (f(x, τt)τt− µF (x, τt)) ≥ 0, ∀τ ≥ 1 of the function γ2 is increasing on [1,+∞) and γ2(τ) ≥ γ2(1) = 0 for all τ ∈ [1,+∞). Hence, F (x, τt) ≥ τµF (x, t), ∀x ∈ Ω, t ∈ R, τ ≥ 1. (2.3) Let ϕ ∈ C∞0 (Ω) and ϕ 6≡ 0 such that ∫ Ω F (x, ϕ) dx > 0, by (A1) we have Jλ(τϕ) = M̂ (∫ Ω 1 p(x) |∇τϕ|p(x) dx+ ∫ ∂Ω β(x) p(x) |τϕ|p(x) dσx ) − ∫ Ω F (x, τϕ) dx− λ ∫ Ω g(x)τϕ dx ≤ m2τ αp+ α (∫ Ω 1 p(x) |∇ϕ|p(x) dx+ ∫ ∂Ω β(x) p(x) |ϕ|p(x) dσx )α − τµ ∫ Ω F (x, ϕ) dx− λτ ∫ Ω g(x)ϕdx→ −∞, as τ → +∞ since µ > m2α(p+)α m1(p−)α−1 ≥ αp+. Therefore, there exists a constant τ0 > 0 such that ‖τ0ϕ‖β(x) > ρ and Jλ(τ0ϕ) < 0. Let e = τ0ϕ the proof is complete. � 8 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 Lemma 2.5. Assume that (A1)–(A4) hold. Then the functional Jλ satisfies the Palais-Smale condition. Proof. Let {un} ⊂ X be such that Jλ(un)→ c ∈ R, J ′λ(un)→ 0 in X∗, (2.4) where X∗ is the dual space of X. We will prove that {un} is bounded in X. Indeed, assume by contradiction that ‖un‖β(x) → +∞ as n→∞. By the conditions (A1), (A4) and (1.5), (2.4), applying the Hölder inequality we deduce for n large enough that ‖un‖β(x) > 1 and c+ 1 + ‖un‖β(x) ≥ Jλ(un)− 1 µ J ′λ(un)(un) = M̂ (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) − ∫ Ω F (x, un) dx − λ ∫ Ω g(x)un dx−M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) × (∫ Ω |∇un|p(x) dx+ ∫ ∂Ω β(x)|un|p(x) dσx ) + 1 µ ∫ Ω f(x, un)un dx + λ µ ∫ Ω g(x)un dx ≥ ( m1 α(p+)α − m2 µ(p−)α−1 )(∫ Ω |∇un|p(x) dx+ ∫ ∂Ω β(x)|un|p(x) dσx )α + ∫ Ω ( 1 µ f(x, un)un − F (x, un) ) dx− λ ( 1− 1 µ ) ∫ Ω g(x)un dx ≥ ( m1 α(p+)α − m2 µ(p−)α−1 ) ‖un‖αp − β(x) − λC1 ( 1− 1 µ ) |g| αp+ αp+−1 ‖u‖β(x), where µ > m2α(p+)α/m1(p−)α−1. Dividing by ‖u‖αp − β(x) in the above inequality and passing to the limit as n → ∞, we obtain a contradiction. This follows that the sequence {un} is bounded in X. Now, since the Banach space X is reflexive, there exists u ∈ X such that passing to a subsequence, still denoted by {un}, it converges weakly to u in X and converges strongly to u in the spaces Lq(x)(Ω). Using the condition (A2) and Hölder inequality, we have ∣∣∣ ∫ Ω f(x, un)(un − u) dx ∣∣∣ ≤ ∫ Ω |f(x, un)||un − u| dx ≤ C ∫ Ω (1 + |un|q(x)−1)|un − u| dx ≤ 2C ( 1 + ||un|q(x)−1| q(x) q(x)−1 ) |un − u|q(x) → 0 as n→∞, which yields lim m→∞ ∫ Ω f(x, un)(un − u) dx = 0. (2.5) EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 9 Moreover, we have∣∣ ∫ Ω g(x)(un − u) dx ∣∣ ≤ ∫ Ω |g(x)||un − u| dx ≤ 2|g| αp+ αp+−1 |un − u|αp+ → 0 as n→∞. (2.6) Since {un} converges weakly to u in X, by (2.4) we have J ′λ(un)(un − u) → 0 as n→∞ or M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) × (∫ Ω |∇un|p(x)−2∇un(∇un −∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un − u) dσx ) − ∫ Ω f(x, un)(un − u) dx− λ ∫ Ω g(x)(un − u) dx→ 0, which leads from (2.5)-(2.6) to M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) × (∫ Ω |∇un|p(x)−2∇un(∇un −∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un − u) dσx ) → 0. If ∫ Ω 1 p(x) |∇un| p(x) dx + ∫ ∂Ω β(x) p(x) |un| p(x) σx → 0 then we have ∫ Ω |∇un|p(x) dx +∫ ∂Ω β(x)|un|p(x) dσx → 0 as n → ∞ and thus un → 0 strongly in X as n → ∞. If∫ Ω 1 p(x) |∇un| p(x) dx+ ∫ ∂Ω β(x) p(x) |un| p(x) dσx → t0 > 0 as n→∞ then it follows from the continuity of M that M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) →M(t0) > 0, so that M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) ≥ 1 2 M(t0) > 0 for all n large enough. Hence, lim n→∞ (∫ Ω |∇un|p(x)−2∇un(∇un−∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un− u) dσx ) = 0 or lim n→∞ L′β(x)(un)(un − u) = 0, where Lβ(x) and L′β(x) are given by formulas (1.9) and (1.10). From Proposition 1.1, the sequence {un} converges strongly to u as n → ∞. Thus, the functional Jλ satisfies the Palais-Smale condition. � Lemma 2.6. Assume that g ∈ L αp+ αp+−1 (Ω) with g 6≡ 0, and that (A2)-(A5) hold. Then there exists a function ψ ∈ X, ψ 6≡ 0 such that Jλ(τψ) < 0 for all τ > 0 small enough. Proof. For (x, t) ∈ Ω× R, set γ3(τ) = F (x, τ−1t)τµ, τ ≥ 1. By (A4), we have γ′3(τ) = f(x, τ−1t) ( − t τ2 ) τµ + F (x, τ−1t)µτµ−1 = τµ−1[µF (x, τ−1t)− τ−1tf(x, τ−1t)] ≤ 0 10 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 so, γ3(t) is non-increasing. Thus, for any |t| ≥ 1, we have γ3(1) ≥ γ3(|t|), that is F (x, t) ≥ F (x, |t|−1t)|t|µ ≥ C3|t|µ, (2.7) where C3 = infx∈Ω,|t|=1 F (x, t) > 0 by (A5). From (A3), there exists a constant η > 0 such that ∣∣∣f(x, t)t |t|αp+ ∣∣∣ = ∣∣∣ f(x, t) |t|αp+−1 ∣∣∣ ≤ 1, (2.8) for all x ∈ Ω and all 0 < |t| ≤ η. By (A2), for all x ∈ Ω and all η ≤ |t| ≤ 1, there exists C4 > 0 such that∣∣∣f(x, t)t |t|αp+ ∣∣∣ ≤ C(1 + |t|q(x)−1)|t| |t|αp+ ≤ C4. (2.9) From (2.8) and (2.9), we deduce that f(x, t)t ≥ −(C4 + 1)|t|αp + , for all x ∈ Ω and all |t| ∈ [0, 1]. Using the equality F (x, t) = ∫ 1 0 f(x, τt)t dτ , we obtain F (x, t) ≥ − 1 αp+ (C4 + 1)|t|αp + , (2.10) for all x ∈ Ω and all |t| ∈ [0, 1]. Taking C5 = 1 αp+ (C4 + 1) + C3, we then get from (2.7) and (2.10) that F (x, t) ≥ C3|t|µ − C5|t|αp + , (2.11) for all x ∈ Ω and all t ∈ R. We now prove that there exists a function ψ ∈ X such that Jλ(τψ) < 0 for all τ > 0 small enough. Since g ∈ L αp+ αp+−1 (Ω) and g 6≡ 0, we can choose a function ψ ∈ X be such that ∫ Ω g(x)ψ(x) dx > 0, then by (2.11) we have Jλ(τψ) = M̂ (∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω β(x) p(x) |τψ|p(x) dσx ) − ∫ Ω F (x, τψ) dx− λ ∫ Ω g(x)τψ dx ≤ m2 α(p−)α ταp − (∫ Ω |∇ψ|p(x) dx+ ∫ ∂Ω β(x)|ψ|p(x) dσx )α − C5τ µ ∫ Ω |ψ|µ dx+ C3τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω g(x)ψ dx < 0, for all τ > 0 small enough. � Proof of Theorem 2.2. By Lemmas 2.3–2.5, there exists λ∗ > 0 such that for if λ ∈ (0, λ∗), all assumptions of the mountain pass theorem by Ambrosetti-Rabinowitz [4] hold. Then, there exists a critical point u1 ∈ X of the functional Jλ, i.e. J ′λ(u1) = 0 and thus, problem (1.1) has a nontrivial weak solution u1 ∈ X with positive energy Jλ(u1) = c := inf γ∈Γ max t∈[0,1] Jλ(γ(t)) > 0, where Γ := {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e} and the function e is given by Lemma 2.4. EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 11 We will show the existence of the second nontrivial weak solution u2 ∈ X and u2 6= u1 by using the Ekeland variational principle. Indeed, by Lemma 2.3, it follows that on the boundary of the ball centered at the origin and of radius ρ in X, denoted by Bρ(0), we have inf u∈∂Bρ(0) Jλ(u) > 0. On the other hand, by Lemma 2.3 again, the functional Jλ is bounded from below on Bρ(0). Moreover, by Lemma 2.6, there exists ϕ ∈ X such that Jλ(τϕ) < 0 for all τ small enough. It follows that −∞ < c = inf u∈Bρ(0) Jλ(u) < 0. Let us choose ε > 0 such that 0 < ε < inf u∈∂Bρ(0) Jλ(u)− inf u∈Bρ(0) Jλ(u). Applying the Ekeland variational principle in [18] to the functional Jλ : Bρ(0)→ R, it follows that there exists uε ∈ Bρ(0) such that Jλ(uε) < inf u∈Bρ(0) Jλ(u) + ε, Jλ(uε) < Jλ(u) + ε‖u− uε‖β(x), u 6= uε, then, we have Jλ(uε) < infu∈∂B(0) Jλ(u) and thus, uε ∈ Bρ(0). Now, we define the functional Iλ : Bρ(0)→ R by Iλ(u) = Jλ(u) + ε‖u− uε‖β(x). It is clear that uε is a minimum point of Iλ and thus Iλ(uε + τv)− Iλ(uε) t ≥ 0 for all τ > 0 small enough and all v ∈ Bρ(0). The above information shows that Jλ(uε + τv)− Jλ(uε) τ + ε‖v‖β(x) ≥ 0. Letting τ → 0+, we deduce that 〈J ′λ(uε), v〉 ≥ −ε‖v‖β(x). It should be noticed that −v also belongs to Bρ(0), so replacing v by −v, we obtain 〈J ′λ(uε),−v〉 ≥ −ε‖ − v‖β(x) or 〈J ′λ(uε), v〉 ≤ ε‖v‖β(x), which helps us to deduce that ‖J ′λ(uε)‖X∗ ≤ ε. Therefore, there exists a sequence {un} ⊂ Bρ(0) such that Jλ(un)→ c = inf u∈Bρ(0) Jλ(u) < 0 and J ′λ(un)→ 0 in X∗ as n→∞. (2.12) Based on Lemma 2.6, the sequence {un} converges strongly to some u2 as n→∞. Moreover, since Jλ ∈ C1(X,R), by (2.12) it follows that J ′λ(u2) = 0. Thus, u2 is a non-trivial weak solution of problem (1.1) with negative energy Jλ(u2) = c < 0. Finally, we point out the fact that u1 6= u2 since Jλ(u1) = c > 0 > c = Jλ(u2). The proof is complete. � 12 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 2.2. Existence of infinitely many solutions. The purpose of this part is to consider problem (1.1) in the case λ = 0. Under some suitable conditions on M and f , we prove the existence of infinitely many solutions for problem (1.1) by using the Krasnoselskii’s genus theory [11], see Proposition 1.4. Let us introduce the following conditions: (A6) f : Ω→ R is a continuous function such that D1h(x)|t|r(x)−1 ≤ f(x, t) ≤ D2h(x)|t|r(x)−1, ∀(x, t) ∈ Ω× R+, where D1, D2 > 0 are positive constants and r ∈ C+(Ω) such that 1 < r(x) < p∗(x) = Np(x) N−p(x) for all x ∈ Ω, the function h ≡ 1 if p(x) ≤ r(x) < p∗(x) for all x ∈ Ω while h ∈ Lr0(x) + (Ω) with r0(x) = p(x) p(x)−r(x) if 1 < r(x) < p(x) for all x ∈ Ω; (A7) f(x,−t) = −f(x, t) for all (x, t) ∈ Ω× R; We have the following result. Theorem 2.7. Let (A1), (A6), (A7) hold. If p(x) ≤ r(x) < p∗(x) for all x ∈ Ω with r+ < αp− or 1 < r(x) < p(x) for all x ∈ Ω, then (1.1) with λ = 0 has infinitely many weak solutions. With similar arguments as those used in the proof of Theorem 2.2, by assumption (A6), we can show that the functional J0 : X → R defined by J0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx is of C1 on X and its derivative is J ′0(u)(v) = M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx ) − ∫ Ω f(x, u)v dx, for all u, v ∈ X. Thus, weak solutions of problem (1.1) with λ = 0 are exactly the critical points of J0. Lemma 2.8. Assume that (A1), (A6) hold. If p(x) ≤ r(x) < p∗(x) for all x ∈ Ω with r+ < αp− or 1 < r(x) < p(x) for all x ∈ Ω, then the functional J0 is bounded from below on X and satisfies the Palais-Smale condition. Proof. Since 1 < r(x) < p∗(x) for all x ∈ Ω, the embedding X ↪→ Lr(x)(Ω) is continuous and compact, then there exists C6 > 0 such that |u|r(x) ≤ C6‖u‖β(x), ∀u ∈ X. If p(x) ≤ r(x) < p∗(x) for all x ∈ Ω then h ≡ 1, by (A1) and (A6), it follows from the definition of the functional J0, (λ = 0) that J0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx ≥ m1 α (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx )α − D2 r− ∫ Ω |u|r(x) dx ≥ m1 α(p+)α ‖u‖αp − β(x) − D2C r+ 6 r− ‖u‖r + β(x), EJDE-2022/24 p(x)-KIRCHHOFF TYPE PROBLEMS 13 for all u ∈ X with ‖u‖β(x) > 1 large enough. Since we always have that r+ < αp−, J0 is coercive, i.e. J0(u)→ +∞ as ‖u‖β(x) → +∞ and bounded from below on X. Similarly, if 1 < r(x) < p(x) for all x ∈ Ω then h ∈ Lr0(x)(Ω) with r0(x) = p(x) p(x)−r(x) . Applying the Hölder inequality and embedding theorem, we also have J0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx ≥ m1 α (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx )α − D2 r− ∫ Ω h(x)|u|r(x) dx ≥ m1 α(p+)α ‖u‖αp − β(x) − 2D2C r+ 6 r− |h|r0(x)‖u‖r + β(x), for all u ∈ X with ‖u‖β(x) > 1 large enough. Since r+ < p− < αp−, J0 is coercive and bounded from below on X. From these statements, if {un} is a Palais-Smale sequence for the functional J0, i.e. J(un) → c, J ′(un) → 0 in X∗, then {un} is bounded in X. Since X is a reflexive Banach space, {un} has a subsequence, still denoted by {un}, that converges weakly to some u ∈ X. Moreover, the embedding X ↪→ Lr(x)(Ω) is continuous and compact, using (A6) and the Hölder inequality, we have∣∣ ∫ Ω f(x, un)(un − u) dx ∣∣ ≤ ∫ Ω |f(x, un)||un − u| dx ≤ D2 ∫ Ω |un|r(x)−1|un − u| dx ≤ 2D2||un|r(x)−1| r(x) r(x)−1 |un − u|r(x) → 0 as n→∞, if p(x) ≤ r(x) < p∗(x) for all x ∈ Ω, or∣∣ ∫ Ω f(x, un)(un − u) dx ∣∣ ≤ ∫ Ω |f(x, un)||un − u| dx ≤ D2 ∫ Ω h(x)|un|r(x)−1|un − u| dx ≤ 3D2|h|r0(x)||un|r(x)−1| p(x) r(x)−1 |un − u|p(x) → 0 as n→∞, if 1 < r(x) < p(x) for all x ∈ Ω, which yields lim m→∞ ∫ Ω f(x, un)(un − u) dx = 0. (2.13) From (2.13), with similar arguments as those presented in the proof of Lemma 2.6, we can show that {un} converges strongly to u ∈ X and thus, the functional J0 satisfies the Palais-Smale condition. � Proof of Theorem 2.7. We have known that for p ∈ C+(Ω), 1 < p− ≤ p+ < N , X = W 1,p(x)(Ω) is a separable and reflexive Banach space, then there exist {en} ⊂ X and {e∗n} ⊂ X∗ such that 〈e∗i , ej〉 = { 1, i = j, 0, i 6= j, 14 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 X = span{en : n = 1, 2, . . . }, X∗ = span{e∗n : n = 1, 2, . . . }. For each k ∈ N, consider Xk = span{e1, e2, . . . , ek}, the subspace if X spanned by the vectors e1, e2, . . . , ek. Let h(x) ≡ 1 if p(x) ≤ r(x) < p∗(x) for all x ∈ Ω and h ∈ Lr0(x)(Ω) if 1 < r(x) < p(x) for all x ∈ Ω, we define a norm | · |Lr(x)(Ω,h(x)) on the space Xk sa follows |u|Lr(x)(Ω,h(x)) := inf { λ > 0; ∫ Ω h(x)|u(x) λ |r(x) dx ≤ 1 } . (2.14) Note that the embedding Xk ↪→ Ls(x)(Ω), 1 < s(x) < p∗(x) is continuous. Since all norms on the finite dimensional space Xk are equivalent, so are the norms ‖·‖β(x) and | · |Lr(x)(Ω,h(x)). Moreover, for any u ∈ Xk, it follows that J0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx ≤ m2 α (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx )α − D1 r+ ∫ Ω h(x)|u|r(x) dx ≤ m2 α(p−)α ‖u‖αp − β(x) − D1 r+ C(k)‖u‖r + β(x) = ‖u‖r + β(x) ( m2 α(p−)α ‖u‖αp −−r+ β(x) − D1 r+ C(k) ) , where C(k) is a positive constant depending on k. For each k ∈ N as before, let us denote by Rk the positive constant such that m2 α(p−)α rαp −−r+ k < D1 r+ C(k), then, for all 0 < ρk < Rk, and u ∈ Sρk := {u ∈ Xk : ‖u‖β(x) = ρk}, Srk is a closed subset of X\{0} that is symmetric with respect to the origin, we obtain J0(u) ≤ ρr + k ( m2 α(p−)α ραp −−r+ k − D1 r+ C(k) ) ≤ Rr + k ( m2 α(p−)α Rαp −−r+ k − D1 r+ C(k) ) < 0 = J0(0), which implies sup u∈Sρk J0(u) < J0(0). 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[26] Tsouli, N.; Darhouche, O.; Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator, Opuscula Math., 34 (3) (2014), 621–638. [27] Wang, W. B.; Tang, W.; Infinitely many solutions for Kirchhoff type problems with nonlinear Neumann boundary conditions, Electron. J. Diff. Equ., 2016 (188) (2016), 1–9. [28] Wang, L.; Xie, K.; Zhang, B.; Existence and multiplicity of solutions for critical Kirchhoff- type p-Laplacian problems, J. Math. Anal. Appl., 458 (1) (2018), 361–378. Ghasem A. Afrouzi Department of Mathematics, Faculty of Mathematical Sciences, University of Mazan- daran, Babolsar, Iran Email address: afrouzi@umz.ac.ir 16 G. A. AFROUZI, N. T. CHUNG, Z. NAGHIZADEH EJDE-2022/24 Nguyen Thanh Chung Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam Email address: ntchung82@yahoo.com Zohreh Naghizadeh Faculty of Sciences, Department of Mathematics, University of Science and Technol- ogy of Mazandaran, P.O. Box 48518-78195, Behshahr, Iran Email address: z.naghizadeh@mazust.ac.ir 1. Introduction 2. Main results 2.1. Existence of at least two solutions 2.2. Existence of infinitely many solutions References