Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 83, pp. 1–23. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.83 MULTIPLE SOLUTIONS FOR p(x)-KIRCHHOFF TYPE PROBLEMS WITH EXTENDED ROBIN BOUNDARY CONDITIONS HOUSSAM BALADI, ABDELLATIF AGLZIM, MOHAMMED FILALI, NAJIB TSOULI Abstract. This article considers p(x)-Kirchhoff type problems with extended Robin boundary conditions. Using the mountain pass theorem, Ekeland’s vari- ational principle, and Krasnoselskii’s genus theory, we prove the existence at least two, and infinitely many non-trivial weak solutions under some suitable conditions on the non-linearities. The main results improve and generalize the results introduced in [1]. 1. Introduction In this article, we study the existence of weak solutions for p(x)-Kirchhoff type problems with extended Robin boundary conditions −M (∫ Ω 1 p(x) |∇u|p(x) dx + ∫ ∂Ω (β(x) p(x) |u|p(x) +G(x, u) ) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λh(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u+ g(x, u) = 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 ∈ L1(∂Ω), β− := infx∈∂Ω β(x) > 0, g : ∂Ω × R → R is a measurable function, with G(x, t) :=∫ t 0 g(x, s) ds, p ∈ C+(Ω̄), 1 < p− := inf x∈Ω̄ p(x) ≤ p+ := max x∈Ω̄ p(x) < N, λ is a non-negative parameter, f : Ω × R → R and M : R+ := [0,+∞) → R+ are two continuous functions, h : Ω → R is a measurable function. 2020 Mathematics Subject Classification. 35F20, 35F25, 35F30. Key words and phrases. p(x)-laplacian operator; variational method; Kirchhoff problem; Krasnoselskii’s genus; Robin boundary condition. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 23, 2024. Published December 11, 2024. 1 2 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 We can prove the same results if β(x) ≡ 0. Then the problem (1.1) becomes −M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω G(x, u) dσx ) div ( |∇u|p(x)−2∇u ) = f(x, u) + λh(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + g(x, u) = 0, x ∈ ∂Ω, (1.2) 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 (1.3) presented by Kirchhoff in 1883 as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings, see [24]. The parameters in (1.3) have the following meaning: 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 (1.3) is often called a non-local problem be- cause it contains an integral over Ω. This causes some mathematical difficulties which make the study of such a problem particularly interesting. The non-local problem models several physical and biological systems, where u describes a pro- cess which depends on the average of itself, such as the population density, see [9]. Kirchhoff type problems have been studied in many papers in the previous decades. In [8, 14, 23, 26, 29, 30], 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 [25, 27]. For this reason, ordinary differential and partial differential equations with non-standard 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 con- ditions [6, 10, 13, 15, 16, 22]. Relatively speaking, Kirchhoff type problems with Robin boundary conditions have rarely been considered. Robin boundary condi- tions are a weighted combination of Dirichlet and Neumann boundary conditions and it is also called impedance boundary conditions, from their application in elec- tromagnetic problems or convective boundary conditions from their application in heat transfer problems. Moreover, Robin conditions are commonly used in solving Sturm-Liouville problems which appear in many contexts in sciences and engineer- ing, see [17]. To the best of our knowledge, Allaoui [3] 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 [12, 4, 17, 21, 28], 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 [8], the purpose of this article is to consider Robin problem (1.1) with perturbation h 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 EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 3 if λ > 0 small enough, see Theorem 3.1. In the case when λ = 0, we prove problem (1.1) with subcritical growth condition has infinitely many solutions, see Theorem 3.8. Our methodology relies fundamentally on the utilization of the mountain pass theorem [5], the Ekeland variational principle [20], and Krasnoselskii’s genus theory [12], complemented by the approach outlined in the article by Afrouzi [1]. We high- light that the findings presented in this work are novel, extending to cases where p(·) is a constant. Furthermore, we do not require the non-degenerate condition on the Kirchhoff function M , as stipulated in [3, 8]; refer to the assumptions (A1) and (A2). This article si structured as follows: In the preliminaries we define the functional space and give results that we need for the proofs. In the main result section, we prove the existence and the multiplicity of the solution. In the application section, we apply the results of section 2. 2. Preliminaries 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 . For this context, we refer to the books [18, 27] and the papers [2, 17, 21, 25]. We define the set C+(Ω̄) := {h;h ∈ C(Ω̄), h(x) > 1 for all x ∈ Ω̄} For each h ∈ C+(Ω̄) we define h+ = sup x∈Ω̄ h(x), h− = inf x∈Ω̄ h(x). For each p(x) ∈ C+(Ω̄), 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 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 4 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 If u ∈ Lp(x)(Ω) and p+ < +∞ then the following relations hold ∥u∥p − p(x) ≤ ρp(x)(u) ≤ ∥u∥p + p(x), provided |u|p(x) > 1, while ∥u∥p + p(x) ≤ ρp(x)(u) ≤ ∥u∥p − p(x), provided ∥u∥p(x) < 1, and ∥un − u∥p(x) → 0 ⇔ ρp(x)(un − u) → 0 . If p ∈ C+(Ω̄) 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. Proposition 2.1 ([31]). Let ρ(u) = ∫ Ω |∇u|p(x) dx . Then ∥u∥p − ≤ ρ(u) ≤ ∥u∥p + provided ∥u∥ > 1, while ∥u∥p + ≤ ρ(u) ≤ ∥u∥p − provided ∥u∥ < 1, and ∥un − u∥ → 0 ⇔ ρ(un − u) → 0. The space of smooth functions are in general not dense in W 1,p(x)(Ω), but if the exponent p ∈ C+(Ω̄) is logarithmic Hölder continuous, that is, |p(x)− p(y)| ≤ − M log(|x− y|) , ∀x, y ∈ Ω, |x− y| ≤ 1 2 , 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+(Ω̄) and s(x) < p∗(x) for all x ∈ Ω̄ 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 . Proposition 2.2 ([31]). Let us define the functional L :W 1,p(x)(Ω) → R by L(u) = ∫ Ω 1 p(x) |∇u|p(x) dx (2.1) for all u ∈W 1,p(x)(Ω). Then L ∈ C1 ( W 1,p(x)(Ω),R ) and its derivative is L′(u)(v) = ∫ Ω |∇u|p(x)−2∇u∇v dx (2.2) EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 5 Moreover, we have the following assertions (i) L′ :W 1,p(x)(Ω) →W−1,p(x)(Ω) is a continuous, bounded and strictly mono- tone operator; (ii) L′ : 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′(un)(un − u) ≤ 0, then {un} converges strongly to u in W 1,p(x)(Ω). 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), we 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 [17, Theorem 2.1]. Now, let us introduce a norm which will be used later. Let β ∈ L1(∂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 ∥ · ∥∂ . Proposition 2.3 ([1]). Let ρβ(x)(u) = ∫ Ω |∇u|p(x) dx+ ∫ ∂Ω β(x)|u|p(x) dσx, we have ∥u∥p − β(x) ≤ ρβ(x)(u) ≤ ∥u∥p + β(x) (2.3) provided ∥u∥β(x) > 1, while ∥u∥p + β(x) ≤ ρβ(x)(u) ≤ ∥u∥p − β(x) (2.4) provided ∥u∥β(x) < 1, and ∥un − u∥β(x) → 0 ⇔ ρβ(x)(un − u) → 0. (2.5) Proposition 2.4 ([21]). For β ∈ L1(∂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 (2.6) for all u ∈W 1,p(x)(Ω). Then Lβ(x) ∈ C1 ( W 1,p(x)(Ω),R ) and its derivative is L′ β(x)(u)(v) = ∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx (2.7) 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; 6 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 (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 inW 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). In the rest of this section, we introduce some notion and results on Krasnoselskii’s genus theory, the readers can consult [7, 12]. 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 2.5 ([1]). 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 that 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 2.6. [1] 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 2.6 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 [12]. Proposition 2.7 ([1]). 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). 3. Main results 3.1. 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 functionM , we prove that (1.1) has at least two nontrivial weak solutions in the space X = W 1,p(x)(Ω). Our idea is to apply the mountain pass theorem in [5] combined with Ekeland’s variational principle in [20] to the energy functional Jλ associated qith 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 asumptions: (A1) There exists α ∈ (1, q−/p+) such that tM(t) ≤ αM̂(t) = α ∫ t 0 M(τ) dτ for all t ∈ R+ 0 , where q − = infx∈Ω̄ q(x), q ∈ C+(Ω̄) is given by (A3); (A2) For each τ > 0 there exists κ = κ(τ) > 0 such that M(t) ≥ κ for all t ≥ τ ; EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 7 (A3) There exists a positive constant C1 such that |f(x, t)| ≤ C1(1 + |t|q(x)−1) ∀(x, t) ∈ Ω× R, where q ∈ C+(Ω̄), p(x) < q(x) < p∗(x) = Np(x) N−p(x) for all x ∈ Ω̄; (A4) f(x, t) = o ( |t|αp+−1 ) , t→ 0, uniformly a.e. x ∈ Ω; (A5) There exists a constant µ > αp+ such that µF (x, t) := µ ∫ t 0 f(x, s) ds ≤ f(x, t)t, ∀(x, t) ∈ Ω× R; (A6) inf{x∈Ω;|t|=1} F (x, t) > 0. (A7) g(x, t) = o(|t|r1(x)−1) uniformly a.e. x ∈ ∂Ω, as t→ 0, where r1 ∈ C+(∂Ω), supx∈∂Ω r1(x) = r+1 < p− ≤ p(x) for all x ∈ ∂Ω; (A8) g(x, t) = o ( |t|r2(x)−1 ) , t→ +∞, uniformly a.e. x ∈ ∂Ω, where r2 ∈ C+(∂Ω), supx∈∂Ω r2(x) = r+2 < p− ≤ p(x) for all x ∈ ∂Ω; (A9) G(x, t) := ∫ t 0 g(x, s) ds ≥ 0, ∀(x, t) ∈ ∂Ω× R, where α and µ are given in (A1) and (A5). 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) +G(x, u)) dσx ) × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx + ∫ ∂Ω g(x, u)v dσx ) − ∫ Ω f(x, u)v dx− λ ∫ Ω h(x)v dx = 0 for all v ∈W 1,p(x)(Ω). The first results of this article reads as follows. Theorem 3.1. Suppose that β ∈ L1(∂Ω), β− := infx∈∂Ω β(x) > 0, h ∈ L αp+ αp+−1 (Ω) and g ̸≡ 0. Let M(0) = 0 and the conditions (A1)–(A9) hold. Then there exists λ∗ > 0 such that (1.1) has at least two non-trivial weak solutions when λ ∈ (0, λ∗). Theorem 3.2. Suppose that β(x) ≡ 0, h ∈ L αp+ αp+−1 (Ω) and g ̸≡ 0. Let M(0) = 0 and the conditions (A1)–(A9) hold. Then there exists λ∗ > 0 such that the problem (1.2) has at least two non-trivial weak solutions when λ ∈ (0, λ∗). Let us denote by X the variable exponent Sobolev spaceW 1,p(x)(Ω) and consider the energy functional Iλ : X → R given by Iλ(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx + ∫ ∂Ω G(x, u) dσx ) − ∫ Ω F (x, u) dx− λ ∫ Ω h(x)u dx. Then by (A3) and the continuous embeddings, we can show that the functional Iλ is well-defined on X and Iλ ∈ C1(X,R) with the derivative I ′λ(u)(v) =M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) 8 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx + ∫ ∂Ω g(x, u)v dσx ) − ∫ Ω f(x, u)v dx− λ ∫ Ω h(x)v dx for all u, v ∈ X. Hence, we can find weak solutions of (1.1) as the critical points of the functional Iλ in the space X. In what follows, we study the degenerate problem (1.1), and in passing we recall that (A1) is assumed throughout the paper. Remark 3.3. Obviously, (A2) implies thatM(t) > 0 as t > 0. Hence, ifM(t0) > 0 for some t0 > 0, then (A1) yields that tα0 M̂(t) ≥ M̂(t0)t α ∀t ∈ [0, t0], (3.1) tα0 M̂(t) ≤ M̂(t0)t α ∀t ≥ t0. (3.2) assuming that {un}n ⊂ X. If inf n∈N (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) > 0 Then from condition (A2) we can choose τ = inf n∈N (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) > 0. Then there exists κ = κ(τ) > 0 such that M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) ⩾ κ ∀n ∈ N. We will discuss the other case later. Lemma 3.4. Assume that (A1)–(A5), (A9) hold and that h ∈ L αp+ αp+−1 (Ω). Then there exist constants ρ, r, λ∗ > 0 such that Iλ(u) ≥ r for all u ∈ X with ∥u∥β(x) = ρ, when λ ∈ (0, λ∗). Proof. Since 1 < r−1 < p− ≤ p(x) ≤ p+ < αp+ < q− ≤ q(x) < p∗(x) for all x ∈ Ω̄, 1 < r−1 < p− and 1 < r−2 < p−, the embeddings X ↪→ Lp(x)(Ω), X ↪→ Lαp+ (Ω), X ↪→ Lq(x)(Ω), X ↪→ Lp(x)(∂Ω) are continuous, and there exists positive constants C2, C3, C4, C5 such that ∥u∥p(x) ≤ C2∥u∥β(x), ∥u∥αp+ ≤ C3∥u∥β(x), ∥u∥q(x) ≤ C4∥u∥β(x), ∥u∥∂Ω,p(x) ≤ C5∥u∥β(x). (3.3) Let 0 < ϵ1. From the assumptions (A1) and (A2), there exists a constant C(ϵ1) depending on ϵ1 such that |F (x, t)| ≤ ϵ1|t|αp + + C(ϵ1)|t|q(x), ∀(x, t) ∈ Ω× R (3.4) Let u ∈ X with ∥u∥β(x) < 1 sufficiently small. From (A1), (A2), (2.4), (3.3), (3.4), and (A9) applying the Hölder inequality we have Iλ(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 9 − λ ∫ Ω h(x)u dx ≥ 1 α M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) × (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx − λ ∫ Ω h(x)u dx ≥ κ αp+ ( ρβ(x)(u) + p+ ∫ ∂Ω G(x, u)dσx ) − ∫ Ω F (x, u) dx− λ ∫ Ω h(x)u dx ≥ κ αp+ ∥u∥p + β(x) − ϵ1 ∫ Ω |u|αp + dx− C(ϵ1) ∫ Ω |u|q(x)dx− λ ∫ Ω h(x)u dx ≥ κ αp+ ∥u∥p + β(x) − ϵ1C αp+ 3 ∥u∥αp + β(x) − C(ϵ1)C q− 4 ∥u∥q − β(x) − λC3∥h∥ αp+ αp+−1 ∥u∥β(x) ≥ ( κ αp+ ∥u∥p +−1 β(x) − ϵ1C αp+ 3 ∥u∥αp +−1 β(x) − C(ϵ1)C q− 4 ∥u∥q −−1 β(x) − λC3∥h∥ αp+ αp+−1 ) ∥u∥β(x), for ϵ1 sufficiently small. Then Iλ(u) ≥ ( κ 2αp+ ∥u∥p +−1 β(x) − C(ϵ1)C q− 4 ∥u∥q −−1 β(x) − λC3∥h∥ αp+ αp+−1 ) ∥u∥β(x) where C3, C4 > 0 are given by (3.3). Consider the functions γ1 : [0,+∞) → R is given by γ1(τ) = κ 2αp+ τp +−1 − C(ϵ1)C q− 4 τ q −−1 Since q− > p+, there exists a constant τ = ρ > 0 obeying the relationship γ1(ρ) = maxτ∈[0,+∞) γ1(τ) > 0. Taking λ∗ = γ1(ρ) 2C3∥h∥ αp+ αp+−1 > 0, it then follows that, if λ ∈ (0, λ∗), we can choose r and ρ > 0 such that Iλ(u) ≥ r > 0 for all u ∈ X with ∥u∥β(x) = ρ. □ Lemma 3.5. Assume that (A1), (A2), (A5), (A6), (A8) hold. Then there exists a function e ∈ X with ∥e∥β(x) > ρ such that Jλ(e) < 0, where ρ is given by Lemma 3.4. Proof. By (A8) for ϵ > 0 there is a constant Mϵ > 0 such that |G(x, t)| ≤ ϵ|t|r2(x) +Mϵ ∀(x, t) ∈ ∂Ω× R. For each x ∈ Ω and t ∈ R, we define the function γ2(τ) = τ−µF (x, τt)− F (x, t) for all τ ≥ 1. Then we deduce from (A5) that γ′2(τ) = τ−µ−1(f(x, τt)τt− µF (x, τt)) ≥ 0, ∀τ ≥ 1 thus 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. (3.5) Let φ ∈ C∞ 0 (Ω) and φ ̸≡ 0 such that ∫ Ω F (x, φ) dx > 0, by (3.2), (3.7) and (3.5) we have Iλ(τφ) = M̂ (∫ Ω 1 p(x) |∇τφ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τφ|p(x) +G(x, τφ)) dσx ) 10 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 − ∫ Ω F (x, τφ) dx− λ ∫ Ω h(x)τφ dx ≤ mταp + (p−)α (∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω (β(x)|φ|p(x) + p−τ−p+ G(x, τφ)) dσx )α − τµ ∫ Ω F (x, φ) dx− λτ ∫ Ω h(x)φdx ≤ mταp + (p−)α (∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω β(x)|φ|p(x) dσx + p−ϵ ∫ ∂Ω (|u|r2(x) dσx + p−τ−p+ Mϵ|∂Ω|) )α − τµ ∫ Ω F (x, φ) dx− λτ ∫ Ω h(x)φdx→ −∞, as τ → +∞, where m = M̂(t0)t −α 0 since µ > αp+. Therefore, there exists a constant τ0 > 0 such that ∥τ0φ∥β(x) > ρ and Iλ(τ0φ) < 0. Letting e = τ0φ we complete the proof. □ Lemma 3.6. Assume that (A1)–(A5), (A7), (A8) hold. Then the functional Iλ satisfies the Palais-Smale condition. Proof. Since 1 < r+2 < r+1 < p− ≤ p(x) for all x ∈ ∂Ω, the embeddings X ↪→ Lr1(x)(Ω), X ↪→ Lr2(x)(Ω), are continuous, and there existspositive constants Cr1 , Cr2 such that ∥u∥∂Ω,r1(x) ≤ Cr1∥u∥β(x), ∥u∥∂Ω,r2(x) ≤ Cr2∥u∥β(x) Let {un} ⊂ X be such that Iλ(un) → c ∈ R, I ′λ(un) → 0 in X∗ (3.6) 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 (A7) and (A8) for 0 < ϵ2, ϵ3 there exist a constant C(ϵ2) depending on ϵ2 and a constant C(ϵ3) depending on ϵ3 such that |G(x, t)| ≤ ϵ2|t|r1(x) + C(ϵ2)|t|r2(x), ∀(x, t) ∈ ∂Ω× R, (3.7) |g(x, t)| ≤ ϵ3|t|r1(x)−1 + C(ϵ3)|t|r2(x)−1, ∀(x, t) ∈ ∂Ω× R (3.8) By assumptions (A1), (A2), (A5), (A7), (A8) and (2.5), (3.6), applying the Hölder inequality we deduce for n large enough that ∥un∥β(x) > 1, we have c+ 1 + ∥un∥β(x) ≥ Iλ(un)− 1 µ ⟨I ′λun, un⟩ = M̂ (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) − ∫ Ω F (x, un) dx− λ ∫ Ω h(x)un dx− 1 µ M (∫ Ω 1 p(x) |∇un|p(x) dx + ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx )(∫ Ω |∇un|p(x) dx EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 11 + ∫ ∂Ω (β(x)|un|p(x) + g(x, un)un) dσx ) + 1 µ ∫ Ω f(x, un)un dx + λ µ ∫ Ω h(x)un dx ≥ 1 α M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) × (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) − ∫ Ω F (x, un) dx − λ ∫ Ω h(x)un dx− 1 µ M (∫ Ω 1 p(x) |∇un|p(x) d+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx )(∫ Ω |∇un|p(x) dx+ ∫ ∂Ω (β(x)|un|p(x) + g(x, un)un) dσx ) + 1 µ ∫ Ω f(x, un)un dx+ λ µ ∫ Ω h(x)un dx ≥M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) × ( 1 α (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) − 1 µ (∫ Ω |∇un|p(x) dx+ ∫ ∂Ω (β(x)|un|p(x) + g(x, un)un) dσx )) − ∫ Ω F (x, un) dx − λ ∫ Ω h(x)un dx+ 1 µ ∫ Ω f(x, un)un dx+ λ µ ∫ Ω h(x)un dx ≥M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) × ( 1 α (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |un|p(x) +G(x, un)) dσx ) − 1 µ (∫ Ω |∇un|p(x) dx+ ∫ ∂Ω (β(x)|un|p(x) + g(x, un)un) dσx )) + ∫ Ω ( 1 µ f(x, un)un − F (x, un) ) dx− λ ( 1− 1 µ ) ∫ Ω h(x)un dx ≥ κ ( 1 αp+ ρβ(x)(un) + 1 α ∫ ∂Ω G(x, un) dσx − 1 µ ρβ(x)(un)− 1 µ ∫ ∂Ω g(x, un)un dσx ) + ∫ Ω ( 1 µ f(x, un)un − F (x, un) ) dx− λ ( 1− 1 µ ) ∫ Ω h(x)un dx ≥ κ [( 1 αp+ − 1 µ ) ρβ(x)(un)− 1 α ϵ2 ∫ ∂Ω |un|r1(x)dσx − 1 α C(ϵ2) ∫ ∂Ω |un|r2(x)dσx − 1 µ ϵ3 ∫ ∂Ω |un|r2(x)−1dσx − 1 µ C(ϵ3) ∫ ∂Ω |un|r2(x)dσx ] + ∫ Ω ( 1 µ f(x, un)un − F (x, un) ) dx− λ ( 1− 1 µ ) ∫ Ω h(x)un dx ≥ κ ( 1 αp+ − κ µ ) ∥un∥p − β(x) − κ α ϵ2C r−1 r1 ∥un∥ r−1 β(x) − κ α C(ϵ2)C r−2 r2 ∥un∥ r−1 β(x) 12 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 − κ µ ϵ3C r−1 r1 ∥un∥ r−1 β(x) − κ µ C r−2 r2 (ϵ3)C∥un∥ r−2 β(x) − λC3 ( 1− 1 µ ) ∥h∥ αp+ αp+−1 ∥un∥β(x), where µ > αp+ and κ > 0. Dividing by ∥u∥p − 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 inX and converges strongly to u in the spaces Lq(x)(Ω). Using the condition (A3) and Hölder inequality, we have∣∣ ∫ Ω f(x, un)(un − u) dx ∣∣ ≤ ∫ Ω |f(x, un)| |un − u| dx ≤ C1 ∫ Ω ( 1 + |un|q(x)−1 ) |un − u| dx ≤ 2C1 ( 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 . (3.9) Moreover,∣∣ ∫ Ω h(x)(un − u) dx ∣∣ ≤ ∫ Ω |h(x)| |un − u| dx ≤ 2∥h∥ αp+ αp+−1 ∥un − u∥αp+ → 0 as n→ ∞. (3.10) Since {un} converges weakly to u in X, by (3.6) we have I ′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) +G(x, un)) dσx ) × (∫ Ω |∇un|p(x)−2∇un (∇un −∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un − u) dσx + ∫ ∂Ω g(x, un)(un − u) dσx ) − ∫ Ω f(x, un)(un − u) dx− λ ∫ Ω h(x)(un − u) dx → 0, which from (3.9) and (3.10) leads to M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) × (∫ Ω |∇un|p(x)−2∇un (∇un −∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un − u) dσx + ∫ ∂Ω g(x, un)(un − u) dσx ) → 0 If (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) → 0, EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 13 then by (A9), (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x)σx ) → 0. This implies that ∫ 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 + ∫ ∂Ω G(x, un) dσx ) → t1 > 0 as n→ ∞ then from the continuity of M it follows that M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx+ ∫ ∂Ω G(x, un) dσx ) →M (t1) > 0, so that M (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx+ ∫ ∂Ω G(x, un) dσx ) ≥ 1 2 M (t1) > 0 for all n large enough. Therefore,(∫ Ω |∇un|p(x)−2∇un (∇un −∇u) dx+ ∫ ∂Ω β(x)|un|p(x)−2un(un − u) dσx + ∫ ∂Ω g(x, un)(un − u) dσx ) → 0 by (3.8) and Hölder inequaly, we have∣∣ ∫ ∂Ω g(x, un)(un − u) dx ∣∣ ≤ ∫ ∂Ω |g(x, un)| |un − u| dx ≤ ∫ ∂Ω ( ϵ3|t|r1(x)−1 + C(ϵ3)|t|r2(x)−1 ) |un − u| dx ≤ 2ϵ3∥|un|r1(x)−1∥ r1(x) r1(x)−1 ∥un − u∥r1(x) + 2C(ϵ3)∥|un|r2(x)−1∥ r2(x) r2(x)−1 ∥un − u∥r2(x) → 0 as n→ ∞ . Then 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 (2.6) and (2.7). From Proposition 2.4 the sequence {un} converges strongly to u as n→ ∞. Thus, the functional Iλ satisfies the Palais-Smale condition. □ Lemma 3.7. Assume that h ∈ L αp+ αp+−1 (Ω) with h ̸≡ 0, and that (A1)–(A8) hold. Then there exists a function ψ ∈ X,ψ ̸≡ 0 such that Iλ(τψ) < 0 for all τ > 0 small enough. 14 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 Proof. For (x, t) ∈ Ω× R, set γ3(τ) = F ( x, τ−1t ) τµ, τ ≥ 1. By (A5), 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 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 ≥ C6|t|µ (3.11) where C6 = infx∈Ω,|t|=1 F (x, t) > 0 by (A6). From (A4), there exists a constant η > 0 such that ∣∣f(x, t)t |t|αp+ ∣∣ = ∣∣ f(x, t) |t|αp+−1 ∣∣ ≤ 1 (3.12) for all x ∈ Ω and all 0 < |t| ≤ η. By (A3), for all x ∈ Ω and all η ≤ |t| ≤ 1, there exists C7 > 0 such that∣∣f(x, t)t |t|αp+ ∣∣ ≤ C ( 1 + |t|q(x)−1 ) |t| |t|αp+ ≤ C7 (3.13) From (3.12) and (3.13), we deduce that f(x, t)t ≥ − (C7 + 1) |t|αp + for all x ∈ Ω and all |t| ∈ [0, 1]. Using the equality F (x, t) = ∫ 1 0 f(x, τt)tdτ , we obtain F (x, t) ≥ − 1 αp+ (C7 + 1) |t|αp + (3.14) for all x ∈ Ω and all |t| ∈ [0, 1]. Taking C8 = 1 αp+ (C7 + 1) + C6, then from (3.11) and (3.14) we obtain that F (x, t) ≥ C6|t|µ − C8|t|αp + (3.15) 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 h ∈ L αp+ αp+−1 (Ω) and h ̸≡ 0, we can choose a function ψ ∈ X be such that ∫ Ω h(x)ψ(x) dx > 0 . If ∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx ≥ t0 with t0 defined in (3.1) and (3.2), then by (3.15) we have Iλ(τψ) = M̂ (∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx ) − ∫ Ω F (x, τψ) dx− λ ∫ Ω h(x)τψ dx ≤ m (∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx )α − C8τ µ ∫ Ω |ψ|µ dx+ C6τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω h(x)ψ dx EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 15 ≤ mταp + (p−)α (∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω (β(x)|φ|p(x) + p−τ−p+ G(x, τφ)) dσx )α − C8τ µ ∫ Ω |ψ|µ dx+ C6τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω h(x)ψ dx ≤ mταp + (p−)α [ ∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω β(x)|φ|p(x) dσx + p−τ−p+ ϵ2 ∫ ∂Ω |u|r1(x) dσx + p−τ−p+ C(ϵ2) ∫ ∂Ω |u|r2(x) dσx ]α − C8τ µ ∫ Ω |ψ|µ dx + C6τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω h(x)ψ dx < 0 for all τ > 0 and ϵ2 small enough. If∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx ≤ t0 by M(0) = 0 and the continuity of M , there exists m0 > 0 such that M(t) ≤ m0 for all t ∈ [0, t0]. Then M̂(t) ≤ m0t ∀t ≤ t0. (3.16) Then by (3.15) and (3.16), we have Iλ(τψ) = M̂ (∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx ) − ∫ Ω F (x, τψ) dx− λ ∫ Ω h(x)τψ dx ≤ m0 (∫ Ω 1 p(x) |∇τψ|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |τψ|p(x) +G(x, τψ)) dσx ) − C8τ µ ∫ Ω |ψ|µ dx+ C6τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω h(x)ψ dx ≤ m0τ p+ p− (∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω (β(x)|φ|p(x) + p−τ−p+ G(x, τφ)) dσx ) − C8τ µ ∫ Ω |ψ|µ dx+ C6τ αp+ ∫ Ω |ψ|αp + dx− λτ ∫ Ω h(x)ψ dx ≤ m0τ p+ p− (∫ Ω |∇φ|p(x) dx+ ∫ ∂Ω β(x)|φ|p(x) dσx + p−τ−p+ ϵ2 ∫ ∂Ω |u|r1(x) dσx + p−τ−p+ C(ϵ2) ∫ ∂Ω |u|r2(x) dσx ) − C8τ µ ∫ Ω |ψ|µ dx+ C6τ αp+ ∫ Ω |ψ|αp + dx − λτ ∫ Ω h(x)ψ dx < 0 for all τ > 0 and ϵ2 smalls enough. □ Proof of Theorem 3.1. By Lemmas 3.4–3.7, there exists λ∗ > 0 such that for if λ ∈ (0, λ∗), all assumptions of the mountain pass theorem by Ambrosetti-Rabinowitz [5] hold. Then, there exists a critical point u1 ∈ X of the functional Iλ, i.e. I ′ λ(u1) = 0 and thus, problem (1.1) has a nontrivial weak solution u1 ∈ X with positive energy Iλ(u1) = c̄ := inf γ∈Γ max t∈[0,1] Iλ(γ(t)) > 0 16 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 where Γ := {γ ∈ C([0, 1], X) : γ(0) = 0, γ(1) = e} and the function e is given by Lemma 3.5. In the case inf n∈N (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, un) dσx ) = 0, by (A9), inf n∈N (∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx ) = 0 . Here, either 0 is an accumulation point for the real sequence{∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx } n = {vn}n and so there is a subsequence of {vn}n strongly converging to v = 0 and by (2.4) there exists a subcequence of {∥un∥β(x)}n strongly converging to u = 0, or 0 is an isolated point of{∫ Ω 1 p(x) |∇un|p(x) dx+ ∫ ∂Ω β(x) p(x) |un|p(x) dσx + ∫ ∂Ω G(x, unk ) dσx } n . The first case can not occur since it implies that the trivial solution is a critical point at level c. This is impossible, being 0 = Iλ(0) = c > 0. Hence only the latter case can occur, so that there is a subsequence, denoted by{∫ Ω 1 p(x) |∇unk |p(x) dx+ ∫ ∂Ω β(x) p(x) |unk |p(x) dσx + ∫ ∂Ω G(x, unk ) dσx } k , such that inf k∈N {∫ Ω 1 p(x) |∇unk |p(x) dx+ ∫ ∂Ω β(x) p(x) |unk |p(x) dσx + ∫ ∂Ω G(x, unk ) dσx } > 0 and we can proceed as before. □ Proof of Theorem 3.2. If β(x) ≡ 0, then we can proceed as before using the usual norm on W 1,p(x)(Ω) instead of ∥ · ∥β(x), and the propositions 2.1 and 2.2 instead of the proposition 2.3 and 2.4. □ Next we show the existence of the second nontrivial weak solution u2 ∈ X and u2 ̸= u1 by using the Ekeland variational principle. Indeed, by Lemma 3.4 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) Iλ(u) > 0 On the other hand, by Lemma 3.4 again, the functional Iλ is bounded from below on Bρ(0). Moreover, by Lemma 3.7 there exists φ ∈ X such that Jλ(τφ) < 0 for all τ small enough. It follows that −∞ < c = inf u∈B̄ρ(0) Iλ(u) < 0 Let us choose ϵ > 0 such that 0 < ϵ < inf u∈∂Bρ(0) Iλ(u)− inf u∈B̄ρ(0) Iλ(u) EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 17 Applying the Ekeland variational principle in [20] to the functional Iλ : B̄ρ(0) → R, it follows that there exists uϵ ∈ B̄ρ(0) such that Iλ (uϵ) < inf u∈B̄ρ(0) Iλ(u) + ϵ Iλ (uϵ) < Iλ(u) + ϵ∥u− uϵ∥β(x), u ̸= uϵ. Then, we have Iλ (uϵ) < infu∈∂B(0) Iλ(u) and thus, uϵ ∈ Bρ(0). Now, we define the functional Jλ : B̄ρ(0) → R by Jλ(u) = Iλ(u) + ϵ∥u− uϵ∥β(x). It is clear that uϵ is a minimum point of Jλ and thus Jλ (uϵ + τv)− Jλ (uϵ) t ≥ 0 for all τ > 0 small enough and all v ∈ Bρ(0). The above information shows that Iλ (uϵ + τv)− Iλ (uϵ) τ + ϵ∥v∥β(x) ≥ 0 Letting τ → 0+, we deduce that ⟨I ′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 ⟨I ′λ (uϵ) ,−v⟩ ≥ −ϵ∥ − v∥β(x) or ⟨I ′λ (uϵ) , v⟩ ≤ ϵ∥v∥β(x) which helps us to deduce that ∥I ′λ (uϵ) ∥X∗ ≤ ϵ. Therefore, there exists a sequence {un} ⊂ Bρ(0) such that Iλ(un) → c = inf u∈B̄ρ(0) Iλ(u) < 0 and I ′λ(un) → 0 in X∗ as n→ ∞ . (3.17) Based on Lemma 3.7 the sequence {un} converges strongly to some u2 as n→ ∞. Moreover, since Iλ ∈ C1(X,R), by (3.17) it follows that I ′λ (u2) = 0. Thus, u2 is a non-trivial weak solution of problem (1.1) with negative energy Iλ (u2) = c < 0. Finally, we point out that u1 ̸= u2 since Iλ(u1) = c̄ > 0 > c = Iλ (u2). The proof is complete. We can do the same for the problem (1.2). 3.2. 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 (1.1) by using the Krasnoselskii’s genus theory [12, Prop 1.4]. We introduce the following assumptions: (A10) f : Ω → R is a continuous function such that A1k(x)|t|s(x)−1 ≤ f(x, t) ≤ A2k(x)|t|s(x)−1, ∀(x, t) ∈ Ω× R+, where A1, A2 > 0 are positive constants and s ∈ C+(Ω̄) such that 1 < s(x) < p∗(x) for all x ∈ Ω̄, the function k ≡ 1 if p(x) ≤ s(x) < p∗(x) for all x ∈ Ω̄ while k ∈ L s0(x) + (Ω) with s0(x) = p(x) p(x)−s(x) if 1 < s(x) < p(x) for all x ∈ Ω̄ (A11) f(x,−t) = −f(x, t) for all (x, t) ∈ Ω× R; (A12) g(x,−t) = −g(x, t) for all (x, t) ∈ ∂Ω× R; Then we have the following result. 18 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 Theorem 3.8. Let M(0) = 0 and (A1)–(A12) hold. Then (1.1) with λ = 0 has infinitely many weak solutions. With similar arguments as those used in the proof of Theorem 3.1, by assumption (A10), we can show that the functional I0 : X → R defined by I0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx is of class C1 on X and its derivative is I ′0(u)(v) =M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u))dσx ) × (∫ Ω |∇u|p(x)−2∇u∇v dx+ ∫ ∂Ω β(x)|u|p(x)−2uv dσx + ∫ ∂Ω g(x, u)u dσx ) − ∫ Ω f(x, u)v dx for all u, v ∈ X. Thus, weak solutions of (1.1) with λ = 0 are exactly the critical points of I0. Lemma 3.9. Assume that (A1)–(A6), (A9), (A10) hold. Then the functional I0 is bounded from below on X and satisfies the Palais-Smale condition. Proof. Since 1 < s(x) < p+(x) for all x ∈ Ω̄, the embedding X ↪→ Ls(x)(Ω) is continuous and compact, then there exists C9 > 0 such that ∥u∥s(x) ≤ C9∥u∥β(x), ∀u ∈ X By (A1), (A2), (A9), (A10) and the Hölder iniquality, it follows from the definition of the functional I0, (λ = 0) that I0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx ≥ 1 α M (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) × (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx ≥ κ α (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω β(x) p(x) |u|p(x) dσx ) − ∫ Ω F (x, u) dx ≥ κ αp+ ∥u∥p − β(x) − A2C s+ 9 s− ∥k∥s0∥u∥s + β(x), for all u ∈ X with ∥u∥β(x) > 1 large enough. Since we always have that s+ < p−, I0 is coercive, i.e. I0(u) → +∞ as ∥u∥β(x) → +∞ and bounded from below on X. From these statements, if {un} is a Palais-Smale sequence for the functional I0, i.e. I(un) → c̄, I ′(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 ↪→ Ls(x)(Ω) is continuous and compact, using (A10) and the Hölder inequality, we have∣∣ ∫ Ω f(x, un)(un − u) dx ∣∣ ≤ ∫ Ω |f(x, un)| |un − u| dx EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 19 ≤ A2 ∫ Ω h(x)|un|s(x)−1 |un − u| dx ≤ 2A2∥h∥s0(x)∥|un| s(x)−1∥ s(x) s(x)−1 ∥un − u∥s(x) → 0 as n→ ∞. Then lim m→∞ ∫ Ω f(x, un)(un − u) dx = 0 . (3.18) From (3.18), with similar arguments as those presented in the proof of Lemma 3.7, we can show that {un} converges strongly to u ∈ X and thus, the functional I0 satisfies the Palais-Smale condition. □ Proof of Theorem 3.8. We known that for p ∈ C+(Ω̄) and 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 ̸= j , X = span{en : n = 1, 2, . . .}, X∗ = span{e∗n : n = 1, 2, . . .}. For each k ∈ N, consider Xj = span{e1, e2, . . . , ej}, the subspace if X spanned by the vectors e1, e2, . . . , ej . Let k ∈ Ls0(x)(Ω) for all x ∈ Ω̄, we define a norm ∥ · ∥Ls(x)(Ω,k(x)) on the space Xj sa follows ∥u∥Ls(x)(Ω,k(x)) := inf{λ > 0; ∫ Ω k(x) ∣∣u(x) λ ∣∣s(x) dx ≤ 1}. Note that the embedding Xj ↪→ Ls(x)(Ω), 1 < s(x) < p(x) is continuous. Since all norms on the finite dimensional space Xj are equivalent, so are the norms ∥ · ∥β(x) and ∥ · ∥Ls(x)(Ω,k(x)). Moreover, for any u ∈ Xj , it follows that if∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ≥ t0 with t0 defined in (3.1) and (3.2), by (3.2) and (3.7) for ϵ2 small enough we have I0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx ≤ m (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx )α − A1 s+ ∫ Ω k(x)|u|s(x) dx ≤ m (p−)α (∫ Ω |∇u|p(x) dx+ ∫ ∂Ω β(x)|u|p(x) dσx + p−C(ϵ2) ∫ ∂Ω |u|r2(x)dσx )α × A1 s+ ∫ Ω k(x)|u|s(x) dx ≤ m (p−)α (∥u∥p − β(x) + C10(ϵ2)∥u∥r + 2 )α − A1 s+ C(j)∥u∥s + β(x) = m (p−) α ∥u∥ αr+2 β(x) ( ∥u∥p −−r+2 β(x) + C10(ϵ2) )α − A1 s+ C(j)∥u∥s + β(x) 20 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 = ∥u∥s + β(x) ( m (p−) α ∥u∥ αr+2 −s+ β(x) (∥u∥p −−r+2 β(x) + C10(ϵ2)) α − A1 s+ C(j) ) , where C(j) is a positive constant depending on j. For each j ∈ N as before, let us denote by Rj the positive constant such that m (p−) α ρ αr+2 −s+ j (ρ p−−r+2 j − C10(ϵ2)) α < A1 s+ C(j) then, for all 0 < ρj < Rj , and u ∈ Sρj := {u ∈ Xj : ∥u∥β(x) = ρj}, Sρj is a closed subset of X\{0} that is symmetric with respect to the origin. We obtain I0(u) ≤ ρs + j ( m (p−) α ρ αr+2 −s+ j (ρ p−−r+2 j − C10(ϵ2)) α − A1 s+ C(j) ) ≤ Rs+ j ( m (p−) αR αr+2 −s+ j (R p−−r+2 j − C10(ϵ2)) α − A1 s+ C(j) ) < 0 = I0(0). If ∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ≤ t0, then by (3.16), I0(u) = M̂ (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − ∫ Ω F (x, u) dx ≤ m0 (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω ( β(x) p(x) |u|p(x) +G(x, u)) dσx ) − A1 s+ ∫ Ω k(x)|u|s(x) dx ≤ m (p−) α (∥u∥p − β(x) + C10(ϵ2)∥u∥r + 2 )− A1 s+ C(j)∥u∥s + β(x) ≤ ∥u∥s + β(x) ( m0 (p−) α (∥u∥p −−s+ β(x) + C10(ϵ2)∥u∥r + 2 −s+)− A1 s+ C(j) ) where C(j) is a positive constant depending on j. For each j ∈ N as before, let us denote by R′ j the positive constant such that m0 (p−) α (ρ ′ j) p−−s+ − C10(ϵ2)(ρ ′ j) r+−s+) < A1 s+ C(j). Then, for all 0 < ρ′j < R′ j , and u ∈ Sρ′ j := {u ∈ Xj : ∥u∥β(x) = ρ′j}, Sρ′ j is a closed subset of X\{0} that is symmetric with respect to the origin. We obtain I0(u) ≤ (ρ′j) s+ ( m0 (p−) α (ρ ′ j) p−−s+ − C10(ϵ2)(ρ ′ j) r+2 −s+ − A1 s+ C(j) ) ≤ (R′ j) s+ ( m0 (p−) α (R ′ j) p−−s+ − C10(ϵ2)(R ′ j) r+2 −s+ − A1 s+ C(j) ) < 0 = I0(0). Then in both cases there exists a sphere Sj such that sup u∈Sj I0(u) < I0(0). BecauseXj and Rj are isomorphic and Sj and S j−1 are homeomorphic, we conclude that γ (Sj) = j. Moreover, by (A11) and (A12), I0 is even. By Proposition 2.7 the functional I0 has at least j pair of different critical points. Since j is arbitrary, we EJDE-2024/83 p(x)-KIRCHHOFF TYPE PROBLEMS 21 obtain infinitely many critical points of I0 and thus problem (1.1) with λ = 0 has infinitely many weak solutions. We can prove the same result for β(x) ≡ 0. □ 4. Applications In this section we give two examples of problems that satisfy the hypotheses of Theorem 3.1 and Theorem 3.8. The first is an example of a non-linear problem with variable exponent and the second is an example of the linear case where p(x) is a constant equal to 2. 4.1. Non-linear case. Let us consider the problem − aα (∫ Ω 1 p(x) |∇u|p(x) dx+ ∫ ∂Ω (β(x) p(x) |u|p(x) + γ(x) r(x) |u|r(x) ) dσx )α−1 ∆p(x)u = k(x)|u|s(x)−2u+ λh(x), x ∈ Ω, |∇u|p(x)−2 ∂u ∂ν + β(x)|u|p(x)−2u+ γ(x)|u|r(x)−2u = 0, x ∈ ∂Ω, (4.1) where Ω is a bounded domain in RN with smooth boundary ∂Ω, ∂u ∂ν is the outer nor- mal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L1(∂Ω), infx∈∂Ω β(x) > 0, γ ∈ L1(∂Ω), infx∈∂Ω γ(x) > 0, p ∈ C+(Ω̄), 1 < p− := infx∈Ω̄ p(x) ≤ p+ := maxx∈Ω̄ p(x) < s(x) < p∗(x), a and λ are non-negative parameters, α ≥ 1 and h : Ω → R is a measurable function. Let M(t) = aαtα−1, f(x, t) = |t|s(x)−2t, g(x, t) = γ(x)|t|r(x)−2t, with G(x, t) = γ(x) r(x) |u| r(x). It is clear that M satisfies (A1) and (A2); f satisfies (A3)–(A6), (A10), (A11); and g satisfies (A7) and (A8) for every r(x) such that r1(x) < r(x) < r2(x) < p(x), and g satisfies (A9) and (A12). Then we can deduce the next propositions Proposition 4.1. There exists λ∗ > 0 such that problem (4.1) has at least two non-trivial weak solutions when λ ∈ (0, λ∗). Proposition 4.2. Problem (4.1) with λ = 0 has infinitely many weak solutions. 4.2. Linear case. Let us consider the problem −a ( ∥u∥2β(x) + ∫ ∂Ω e−u2 dσx ) ∆u = |u|u+ λh(x), x ∈ Ω, ∂u ∂ν + β(x)u− ue−u2 = 0, x ∈ ∂Ω, (4.2) where Ω is a bounded domain in R3 with smooth boundary ∂Ω, ∂u ∂ν is the outer nor- mal derivative, dσx is the measure on the boundary ∂Ω, β ∈ L1(∂Ω), infx∈∂Ω β(x) > 0, a and λ are non-negative parameters, α ≥ 1, and h : Ω → R is a measurable function. Let M(t) = 2at, f(x, t) = |t|t t(k ≡ 1 and s(x) = 3) and g(x, t) = −te−t2 , with G(x, t) = 1 2e −t2 . Iti s clear that M satisfies (A1) and (A2); f satisfies (A3)–(A6), (A10), (A11); and g satisfies (A7) and (A8) for every r1(x) = r2(x) = r = 3 2 , and g satisfies (A9) and (A12). 22 H. BALADI, A. AGLZIM, M. FILALI, N. TSOULI EJDE-2024/83 Proposition 4.3. There exists λ∗ > 0 such that problem (4.2) has at least two non-trivial weak solutions when λ ∈ (0, λ∗). Proposition 4.4. Problem (4.2) with λ = 0 has infinitely many weak solutions. Acknowledgments. The authors would like to thank the anonymous referees for their valuable comments and constructive suggestions. All authors contributed equally to the writing of this article. References [1] Afrouzi, G. A.; Chung, T. N.; Naghizadeh, Z.; Muliple solutions for p(x)-Kirchhoff type problems with Robin Boundary conditions, Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 24, pp. 1-16. 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Houssam Baladi University of Mohamed Premier, Mathematics Department, Faculty of Science, Oujda, Morocco Email address: houssam.baladi@ump.ac.ma Abdellatif Aglzim University of Mohamed Premier, Mathematics Department, Faculty of Science, Oujda, Morocco Email address: abdellatif.aglzim@ump.ac.ma Mohammed Filali University of Mohamed Premier, Mathematics Department, Faculty of Science, Oujda, Morocco Email address: filali1959@yahoo.fr Najib Tsouli University of Mohamed Premier, Mathematics Department, Faculty of Science, Oujda, Morocco Email address: n.tsouli@ump.ac.ma 1. Introduction 2. Preliminaries 3. Main results 3.1. Existence of at least two solutions 3.2. Existence of infinitely many solutions 4. Applications 4.1. Non-linear case 4.2. Linear case Acknowledgments References