Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 98, pp. 1–29. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NEHARI MANIFOLD APPROACH FOR FRACTIONAL p(·)-LAPLACIAN SYSTEM INVOLVING CONCAVE-CONVEX NONLINEARITIES RESHMI BISWAS, SWETA TIWARI Communicated by Hongjie Dong Abstract. In this article, using Nehari manifold method we study the multi- plicity of solutions of the nonlocal elliptic system involving variable exponents and concave-convex nonlinearities, (−∆)sp(·)u = λa(x)|u|q(x)−2u+ α(x) α(x) + β(x) c(x)|u|α(x)−2u|v|β(x), x ∈ Ω; (−∆)sp(·)v = µb(x)|v|q(x)−2v + α(x) α(x) + β(x) c(x)|v|α(x)−2v|u|β(x), x ∈ Ω; u = v = 0, x ∈ Ωc := RN \ Ω, where Ω ⊂ RN , N ≥ 2 is a smooth bounded domain, λ, µ > 0 are parameters, and s ∈ (0, 1). We show that there exists Λ > 0 such that for all λ + µ < Λ, this system admits at least two non-trivial and non-negative solutions under some assumptions on q, α, β, a, b, c. 1. Introduction In this article, we consider the nonlocal elliptic system with variable exponents, (−∆)sp(·)u = λa(x)|u|q(x)−2u+ α(x) α(x) + β(x) c(x)|u|α(x)−2u|v|β(x), x ∈ Ω, (−∆)sp(·)v = µb(x)|v|q(x)−2v + α(x) α(x) + β(x) c(x)|v|α(x)−2v|u|β(x), x ∈ Ω, u = v = 0, x ∈ Ωc := RN \ Ω, (1.1) where Ω ⊂ RN , N ≥ 2 is a smooth bounded domain, λ, µ > 0 are the parameters, s ∈ (0, 1), p ∈ C(RN × RN , (1,∞)) with sp+ < N . Here q, α, β ∈ C(Ω, (1,∞)) are the variable exponents and a, b, c : Ω → [0,∞) are the non-negative weight functions. The nonlocal operator (−∆)sp(·) is defined as (−∆)sp(·)u(x) := P.V. ∫ RN |u(x)− u(y)|p(x,y)−2(u(x)− u(y)) |x− y|N+s(x,y)p(x,y) dy, x ∈ RN , (1.2) 2010 Mathematics Subject Classification. 35J48, 35J50, 35R11. Key words and phrases. Nonlocal problem with variable exponents; elliptic system; Nehari manifold; fibering map; concave-convex nonlinearities. c©2020 Texas State University. Submitted November 2, 2019. Published September 23, 2020. 1 2 R. BISWAS, S. TIWARI EJDE-2020/98 where P.V. stands for Cauchy’s principal value. Problems involving nonlocal op- erators have gained a lot of interest for research in recent years. Mathematical modeling of the problems in many areas like mechanics, population dynamics, thin obstacle problem, optimization and finance involve fractional Laplacian (−∆)s or fractional p-Laplacian (−∆)sp. We refer to [12, 24] for the basic results on problems involving nonlocal operators. Also, one can refer to [6, 11, 23, 26, 27] and the references therein for the existence, multiplicity, and regularity of the solutions of these problems. In this work, our objective is to study the nonlocal elliptic problems with variable exponents. Operators involving variable growth are extensively studied due to the precision in the modeling of various phenomenon where the property of the subject under consideration depends on the point of the observation, for example, in image restoration, study of electrorheological fluid flow, non-Newtonian processes, etc. We refer to [2, 13, 15, 16, 28] and references therein for the study of the problems in- volving the local p(x)-Laplace operator, defined as ∆p(x)u := div (|∇u|p(x)−2∇u). The fractional Sobolev spaces with variable exponents and the corresponding fractional p(·)-Laplace operator (−∆)sp(·) were recently introduced by Kaufmann et al in [22]. Also, in [3, 4, 5, 21], the authors have established the basic properties of such spaces and studied the problems involving fractional p(·)-Laplacian. Using the Nehari manifold and the fibering map, in the case of local p-Laplacian, Brown and Wu [8] have obtained multiple solutions of an elliptic system with sign changing weight functions and concave-convex nonlinearities. In the nonlocal set- up, Sreenadh and Goyal [20] have studied the same for the single fractional p- Laplacian equation . Also, we cite [10] where the authors have studied the fractional p-Laplacian system involving concave-convex nonlinearities via Nehari manifold and fibering map. In [18], Pucci et al. have modified the definition of Nehari manifold and fibering map for the fractional (p, q)-Laplacian system and studied the corresponding Dirichlet problem. Recently Alves et al [1] have used this Nehari manifold method to prove the multiplicity of solutions for p(x)-Laplacian problems in the whole of RN . Motivated by the above works, in this article, we address the multiplicity of the solutions of the nonlocal elliptic system with variable exponents involving con- cave and convex nonlinearities using the analysis of the fibering map and Nehari manifold. We note that the Nehari manifold approach through the fibering map analysis for the functional involving variable exponents is interesting due to the non-homogeneity that arises from the variable exponents. It is also worth mention- ing that due to the presence of the variable exponents, most of the estimates do not hold immediately, unlike in the constant exponent set-up. Hence, in our present work, we need to carry out some extra careful analysis to overcome this issue. To the best of our knowledge, this is the first work dealing with fractional p(·)-Laplacian system involving concave and convex nonlinearities using fibering-map approach. Next, we set some notation. Let D be a domain. For any function Φ : D → R, we set Φ− := inf D Φ(x), Φ+ := sup D Φ(x). (1.3) We also define the function space C+(D) := {g ∈ C(D,R) : 1 < g− ≤ g+ <∞}. EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 3 To state our result, we assume that the variable exponents p, q, α, β and the weight functions a, b, c satisfy the following hypotheses: (A1) The variable exponent p ∈ C+(RN × RN ). (A2) The function p is symmetric, i.e., p(x, y) = p(y, x) for all (x, y) ∈ RN ×RN . (A3) The variable exponents q, α, β ∈ C+(Ω) and p ∈ C+(RN × RN ) satisfy 1 < q− ≤ q+ < p− ≤ p+ < α− + β− ≤ α+ + β+ < p∗−s , where p∗s(x) = Np(x,x) N−sp(x,x) is the critical exponent. (A4) It holds that p− α+ + β+ < ( p− − q+ α+ + β+ − q+ )(α− + β− − q− p+ − q− ) . (A5) The non-negative weight functions a, b ∈ Lq∗(x)(Ω), where q∗(x) = α(x) + β(x) α(x) + β(x)− q(x) . (A6) The non-negative weight function c ∈ L∞(Ω). Observe that, when all the exponents are constants, (A4) is equivalent to the condition 0 < p < α+β . Now we define the weak solution of (1.1) in the functional space E (defined in Section 2) as follows. Definition 1.1. We say that (u, v) ∈ E is a weak solution of (1.1), if we have∫ RN×RN |u(x)− u(y)|p(x,y)−2(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp(x,y) dx dy + ∫ RN×RN |v(x)− v(y)|p(x,y)−2(v(x)− v(y))(ψ(x)− ψ(y)) |x− y|N+sp(x,y) dx dy = ∫ Ω ( λa(x)|u|q(x)−2uφ+ µb(x)|v|q(x)−2vψ ) dx + ∫ Ω α(x) α(x) + β(x) c(x)|u|α(x)−2u|v|β(x)φdx + ∫ Ω β(x) α(x) + β(x) c(x)|v|α(x)−2v|u|β(x)ψdx for all (φ, ψ) ∈ E. (1.4) The main result in this article is stated as follows. Theorem 1.2. Let Ω ⊂ RN , N ≥ 2 be a smooth bounded domain, s ∈ (0, 1) and p(·, ·) satisfy (A1)–(A2) with sp+ < N . Assume that the hypotheses (A3)–(A6) hold. Then there exists a positive constant Λ = Λ(N, s, p, q, α, β, a, b, c,Ω) such that for any pair of positive parameters (λ, µ) with λ+ µ < Λ, (1.1) admits at least two non-trivial, non-negative weak solutions. 2. Preliminary results Here we recall the definition and some important properties of the Lebesgue spaces with variable exponents. For more details regarding these spaces, one can refer to [13, 16] and the references therein. For γ ∈ C+(Ω), we define the variable exponent Lebesgue space Lγ(x)(Ω) = { u : Ω→ R : u is measurable, ∫ Ω |u|γ(x) < +∞ } , 4 R. BISWAS, S. TIWARI EJDE-2020/98 This space is a separable, reflexive Banach space equipped with the Luxemburg norm ‖u‖Lγ(x)(Ω) = inf { η > 0 : ∫ Ω ∣∣∣u η ∣∣∣γ(x) ≤ 1 } . We have the following Hölder-type inequality (see [13]) for variable exponents Lebesgue spaces. Lemma 2.1. Let γ′ ∈ C+(Ω) such that 1 γ(x) + 1 γ′(x) = 1. Then for any u ∈ Lγ(x)(Ω) and v ∈ Lγ′(x)(Ω) we have∣∣ ∫ Ω uvdx ∣∣ ≤ ( 1 γ− + 1 γ′− ) ‖u‖Lγ(x)(Ω)‖v‖Lγ′(x)(Ω). Next, we recall [19, Lemma A.1]. Lemma 2.2. Let ν1(x) ∈ L∞(Ω) such that ν1 ≥ 0, ν1 6≡ 0. Let ν2 : Ω → R be a measurable function such that ν1(x)ν2(x) ≥ 1 a.e. in Ω. Then for every u ∈ Lν1(x)ν2(x)(Ω), ‖|u|ν1(·)‖Lν2(x)(Ω) ≤ ‖u‖ ν−1 Lν1(x)ν2(x)(Ω) + ‖u‖ν + 1 Lν1(x)ν2(x)(Ω) . The modular function ργ : Lγ(x)(Ω)→ R is defined as ργ(u) = ∫ Ω |u|γ(x)dx. Now we state the following two lemmas from [16], which establish the relationship between the norm ‖ · ‖Lγ(x)(Ω) and the corresponding modular function ργ(·). Lemma 2.3. Let u ∈ Lγ(x)(Ω), then (i) ‖u‖Lγ(x)(Ω) < 1 (= 1;> 1) if and only if ργ(u) < 1 (= 1;> 1); (ii) If ‖u‖Lγ(x)(Ω) > 1, then ‖u‖γ − Lγ(x)(Ω) ≤ ργ(u) ≤ ‖u‖γ + Lγ(x)(Ω) ; (iii) If ‖u‖Lγ(x)(Ω) < 1, then ‖u‖γ + Lγ(x)(Ω) ≤ ργ(u) ≤ ‖u‖γ − Lγ(x)(Ω) . Lemma 2.4. Let u, um ∈ Lγ(x)(Ω),m = 1, 2, 3, · · · . Then the following statements are equivalent. (i) limm→∞ ‖um − u‖Lγ(x) = 0; (ii) limm→∞ ργ(um − u) = 0; (iii) um converges to u in Ω in measure and limm→∞ ργ(um) = ργ(u). 2.1. Fractional Sobolev spaces with variable exponents. In this section, we discuss the properties of the fractional Sobolev spaces with variable exponents. These spaces have been introduced for the first time in [22]. Also, in [4, 5, 21], the authors have established some important properties of these spaces. Let Ω ⊂ RN be a smooth bounded domain and p(·, ·) satisfy (A1) and (A2). For any x ∈ RN , we denote p(x) := p(x, x). Thus, p ∈ C+(Ω). Now we define the fractional Sobolev space with variable expo- nents as follows. W = W s,p(x),p(x,y)(Ω) := { u ∈ Lp(x)(Ω) : ∫ Ω×Ω |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy <∞, for some η > 0 } . EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 5 We set the seminorm as [u] s,p(x,y) Ω := inf { η > 0 : ∫ Ω×Ω |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy < 1 } . Then (W, ‖ · ‖W ) is a separable reflexive Banach space (see [5]) equipped with the norm ‖u‖W := ‖u‖Lp(x)(Ω) + [u] s,p(x,y) Ω . Now we state the following continuous and compact embedding theorem (see [21]). Theorem 2.5. Let Ω be a smooth bounded domain in RN , s ∈ (0, 1) and p(·, ·) satisfy (A1), (A2) with sp+ < N . Let r ∈ C+(Ω) such that 1 < r− ≤ r(x) < p∗s(x) = Np(x) N−sp(x) for x ∈ Ω. Then, there exits a constant C = C(N, s, p, r,Ω) > 0 such that, for any u ∈W , ‖u‖Lr(x)(Ω) ≤ K‖u‖W . Moreover, this embedding is compact. For studying nonlocal problems involving the operator (−∆)sp(·) with Dirichlet boundary datum via variational methods, we define another fractional type Sobolev spaces with variable exponents. One can refer to [24] and references therein for this type of spaces in fractional Laplacian framework. We set Q := R2N \ (Ωc×Ωc) and define the new fractional Sobolev space with variable exponent as X = Xs,p(x),p(x,y)(Ω) := { u : RN → R : u|Ω ∈ L p(x)(Ω),∫ Q |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy <∞, for some η > 0 } . The space X is equipped with the norm ‖u‖X := ‖u‖Lp(x)(Ω) + inf { η > 0 : ∫ Q |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy < 1 } , where [u]X is the seminorm [u]X = inf { η > 0 : ∫ Q |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy < 1 } . Then (X, ‖·‖X) is a separable reflexive Banach space. Next, we define the subspace X0 of X as X0 = X s,p(x),p(x,y) 0 (Ω) := {u ∈ X : u = 0 a.e. in Ωc}. We define the norm on X0 as follows ‖u‖X0 := inf { η > 0 : ∫ Q |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy < 1 } . Remark 2.6. For u ∈ X0, we obtain∫ Q |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy = ∫ RN×RN |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy. 6 R. BISWAS, S. TIWARI EJDE-2020/98 Thus, we have ‖u‖X0 := inf { η > 0 : ∫ RN×RN |u(x)− u(y)|p(x,y) ηp(x,y)|x− y|N+sp(x,y) dx dy < 1 } . Now we state the following continuous and compact embedding result for the space X0. The proof follows from [3, Theorem 2.2, Remark 2.2] and [4, Lemma 2.1]. Theorem 2.7. Let Ω be a smooth bounded domain in RN and let s ∈ (0, 1). Let p(·, ·) satisfy (A1) and (A2) with sp+ < N . Then for any r ∈ C+(Ω) such that 1 < r(x) < p∗s(x) for all x ∈ Ω, there exits a constant C = C(N, s, p, r,Ω) > 0 such that for every u ∈ X0, ‖u‖Lr(x)(Ω) ≤ C‖u‖X0 . Moreover, this embedding is compact. Definition 2.8. For u ∈ X0, we define the modular ρX0 : X0 → R as ρX0 (u) := ∫ RN×RN |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy. (2.1) The interplay between the norm in X0 and the modular function ρX0 can be studied in the following lemma. Lemma 2.9. Let u ∈ X0 and ρX0 be defined as in (2.1). Then we have the following results: (i) ‖u‖X0 < 1 (= 1;> 1) if and only if ρX0 (u) < 1(= 1;> 1). (ii) If ‖u‖X0 > 1, then ‖u‖p − X0 ≤ ρX0 (u) ≤ ‖u‖p+X0 . (iii) If ‖u‖X0 < 1, then ‖u‖p + X0 ≤ ρX0 (u) ≤ ‖u‖p−X0 . The next lemma can easily be obtained using the properties of the modular function ρX0 from Lemma 2.9. Lemma 2.10. Let u, um ∈ X0, m ∈ N. Then the following two statements are equivalent: (i) limm→∞‖um − u‖X0 = 0, (ii) limm→∞ρX0(um − u) = 0. Lemma 2.11 ([3, Lemma 2.3]). (X0, ‖·‖X0 ) is a separable, reflexive and uniformly convex Banach space. We define E := X0 ×X0 as the solution space corresponding to (1.1), equipped with the norm ‖(u, v)‖ = max{‖u‖X0 , ‖v‖X0}. Clearly (E, ‖(·, ·)‖) is a reflexive, separable Banach space. 3. Nehari manifold and fibering map analysis Here first we discuss certain technical results regarding the Nehari manifold and the fibering map and the behavior of the energy functional corresponding to (1.1). EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 7 The energy functional Jλ,µ : E → R associated with (1.1) is defined as Jλ,µ(u, v) = ∫ RN×RN 1 p(x, y) |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN 1 p(x, y) |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω 1 q(x) ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx − ∫ Ω 1 α(x) + β(x) c(x)|u|α(x)|v|β(x)dx. (3.1) By a direct computation, it can be checked that Jλ,µ ∈ C1(E,R) and for any (φ, ψ) ∈ E, we have 〈J ′λ,µ(u, v), (φ, ψ)〉 = ∫ RN×RN |u(x)− u(y)|p(x,y)−2(u(x)− u(y))(φ(x)− φ(y)) |x− y|N+sp(x,y) dx dy + ∫ RN×RN |v(x)− v(y)|p(x,y)−2(v(x)− v(y))(ψ(x)− ψ(y)) |x− y|N+sp(x,y) dx dy − ∫ Ω ( λa(x)|u|q(x)−2uφ+ µb(x)|v|q(x)−2vψ ) dx − ∫ Ω α(x) α(x) + β(x) c(x)|u|α(x)−2u|v|β(x)φdx − ∫ Ω β(x) α(x) + β(x) c(x)|v|α(x)−2v|u|β(x)ψdx. Therefore, the weak solutions of (1.1) are the critical points of the functional Jλ,µ. One can note that Jλ,µ is not bounded below on E, but it is bounded below on the following subset of E. We define the Nehari manifold as Nλ,µ := {(u, v) ∈ E \ {(0, 0)} : 〈J ′λ,µ(u, v), (u, v)〉 = 0}. Therefore, (u, v) ∈ Nλ,µ if and only if ∫ RN×RN |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx− ∫ Ω c(x)|u|α(x)|v|β(x)dx = 0. (3.2) The Nehari manifold is closely associated with the behavior of the fibering maps ϕu,v : R+ → R, defined as ϕu,v(t) = Jλ,µ(tu, tv), where (u, v) ∈ E. These maps are first given by Drabek and Pohozaev in [14] and are discussed in detail in [9] and [20]. 8 R. BISWAS, S. TIWARI EJDE-2020/98 For (u, v) ∈ E, we have ϕu,v(t) = Jλ,µ(tu, tv) = ∫ RN×RN tp(x,y) p(x, y) { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy − ∫ Ω tq(x) q(x) ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx − ∫ Ω tα(x)+β(x) α(x) + β(x) c(x)|u|α(x)|v|β(x)dx. (3.3) ϕ′u,v(t) = 〈J ′λ,µ(tu, tv), (u, v)〉 = ∫ RN×RN tp(x,y)−1 { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy − ∫ Ω tq(x)−1 ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx − ∫ Ω tα(x)+β(x)−1c(x)|u|α(x)|v|β(x)dx. (3.4) ϕ′′u,v(t) = ∫ RN×RN (p(x, y)− 1)tp(x,y)−2 { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy − ∫ Ω (q(x)− 1)tq(x)−2 ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx − ∫ Ω (α(x) + β(x)− 1)tα(x)+β(x)−2c(x)|u|α(x)|v|β(x)dx. (3.5) Then using that ϕ′u,v(t) = 〈J ′λ,µ(tu, tv), (u, v)〉, we can see that (tu, tv) ∈ Nλ,µ if and only if ϕ′u,v(t) = 0. In particular, (u, v) ∈ Nλ,µ if and only if ϕ′u,v(1) = 0. Thus, it is natural to split Nλ,µ into three parts corresponding to the points of local maxima, local minima and inflection of the function ϕu,v as follows: N + λ,µ :={(u, v) ∈ Nλ,µ : ϕ′′u,v(1) > 0} ={(tu, tv) ∈ E \ {(0, 0)} : ϕ′u,v(t) = 0, ϕ′′u,v(1) > 0}, N − λ,µ :={(u, v) ∈ Nλ,µ : ϕ′′u,v(1) < 0} ={(tu, tv) ∈ E \ {(0, 0)} : ϕ′u,v(t) = 0, ϕ′′u,v(1) < 0}, N 0 λ,µ :={(u, v) ∈ Nλ,µ : ϕ′′u,v(1) = 0} ={(tu, tv) ∈ E \ {(0, 0)} : ϕ′u,v(t) = 0, ϕ′′u,v(1) = 0}. EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 9 Hence, for any (u, v) ∈ Nλ,µ, from (3.2), (3.4) and (3.5), we deduce ϕ′′u,v(1) = ∫ RN×RN p(x, y) { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy − ∫ Ω q(x) ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx − ∫ Ω (α(x) + β(x))c(x)|u|α(x)|v|β(x)dx. (3.6) For a given pair of functions (u, v) ∈ E, we set P (u, v) := ∫ RN×RN { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy, Q(u, v) := ∫ Ω ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx, R(u, v) := ∫ Ω c(x)|u|α(x)|v|β(x)dx. In the next lemma, we obtain some estimations on P,Q and R. Lemma 3.1. Let (u, v) ∈ E. Then we have the following: Is this what you had in mind(i) P (u, v) ≥ { ‖(u, v)‖p+ , if ‖(u, v)‖ < 1 ‖(u, v)‖p− , if ‖(u, v)‖ > 1, P (u, v) ≤ { 2‖(u, v)‖p− , if ‖(u, v)‖ < 1 2‖(u, v)‖p+ , if ‖(u, v)‖ > 1. (ii) There exists a constant C1 = C1(N, s, p, q, α, β, a, b,Ω) > 0 such that Q(u, v) ≤ C1(λ+ µ) max{‖(u, v)‖q − , ‖(u, v)‖q + }. (iii) There exists a constant C2 = C2(N, s, p, α, β, c,Ω) > 1 such that R(u, v) ≤ C2 max{‖(u, v)‖r − , ‖(u, v)‖r + }. Proof. (i) Clearly P (u, v) = ρX0 (u) + ρX0 (v). Hence, we have max{ρX0 (u), ρX0 (v)} ≤ P (u, v) ≤ 2 max{ρX0 (u), ρX0 (v)} (3.7) For ‖(u, v)‖ > 1, there are two cases. Case I: ‖u‖X0 > 1 and ‖v‖X0 > 1. Then from Lemma 2.9 , we obtain ‖u‖p − X0 < ρX0 (u) < ‖u‖p + X0 and ‖v‖p − X0 < ρX0 (v) < ‖v‖p + X0 . (3.8) Thus, from (3.7) and (3.8), we obtain P (u, v) ≤ 2 max{‖u‖p + X0 , ‖v‖p + X0 } = 2‖(u, v)‖p + ; P (u, v) ≥ max{‖u‖p − X0 , ‖v‖p − X0 } = ‖(u, v)‖p − . Case II: ‖v‖X0 < 1 < ‖u‖X0 : Then ‖(u, v)‖ = ‖u‖X0 . Now Lemma 2.9 implies that ‖u‖p − X0 < ρX0(u) < ‖u‖p + X0 and ‖v‖p + X0 < ρX0 (v) < ‖v‖p − X0 . (3.9) 10 R. BISWAS, S. TIWARI EJDE-2020/98 Combining (3.7) and (3.9), we deduce that P (u, v) ≤ 2 max{‖u‖p + X0 , ‖v‖p + X0 } = 2‖(u, v)‖p + , P (u, v) ≥ max{‖u‖p − X0 , ‖v‖p − X0 } = ‖(u, v)‖p − . Next, for ‖(u, v)‖ < 1, we have ‖u‖X0 < 1 and ‖v‖X0 < 1. By Lemma 2.9, we obtain ‖u‖p + X0 < ρX0 (u) < ‖u‖p − X0 and ‖v‖p + X0 < ρX0 (v) < ‖v‖p − X0 . (3.10) Hence, from (3.7) and (3.10), it follows that P (u, v) ≤ 2 max{‖u‖p − X0 , ‖v‖p − X0 } = 2‖(u, v)‖p − , P (u, v) ≥ max{‖u‖p + X0 , ‖v‖p + X0 } = ‖(u, v)‖p + . Thus, we obtain (i). (ii) Using Hölder’s inequality (Lemma 2.1), Sobolev-type embedding (Lemma 2.7) and Lemma 2.2, we obtain Q(u, v) = ∫ Ω ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx ≤ 2λ‖a‖Lq∗(x)(Ω)‖|u|q(·)‖ L α(x)+β(x) q(x) (Ω) + 2µ‖b‖Lq∗(x)(Ω)‖|v|q(·)‖ L α(x)+β(x) q(x) (Ω) ≤ 2λ‖a‖Lq∗(x)(Ω) { ‖u‖q − Lα(x)+β(x)(Ω) + ‖u‖q + Lα(x)+β(x)(Ω) } + 2µ‖b‖Lq∗(x)(Ω) { ‖v‖q − Lα(x)+β(x)(Ω) + ‖v‖q + Lα(x)+β(x)(Ω) } ≤ K1 [ λ { ‖u‖q − X0 + ‖u‖q + X0 } + µ { ‖v‖q − X0 + ‖v‖q + X0 }] ≤ C1(λ+ µ) max { ‖u‖q − X0 , ‖u‖q + X0 , ‖v‖q − X0 , ‖v‖q + X0 } = C1(λ+ µ) max { max { ‖u‖q − X0 , ‖v‖q − X0 } ,max { ‖u‖q + X0 , ‖v‖q + X0 }} = C1(λ+ µ) max { ‖(u, v)‖q − , ‖(u, v)‖q + } , where K1 = 2 ( ‖a‖Lq∗(x)(Ω) + ‖b‖Lq∗(x)(Ω) ) ×max {( C(N, s, p, α, β,Ω) )q− , ( C(N, s, p, α, β,Ω) )q+} and C1 = 4K1. (iii) Using Young’s inequality, Lemma 2.2 and Lemma 2.7, we deduce R(u, v) = ∫ Ω c(x)|u|α(x)|v|β(x)dx ≤ ‖c‖L∞(Ω) ∫ Ω |u|α(x)|v|β(x)dx ≤ ‖c‖L∞(Ω) ∫ Ω { α(x) α(x) + β(x) |u|α(x)+β(x) + β(x) α(x) + β(x) |v|α(x)+β(x) } dx ≤ ‖c‖L∞(Ω) [{ ‖u‖α ++β+ Lα(x)+β(x)(Ω) + ‖u‖α −+β− Lα(x)+β(x)(Ω) } + { ‖v‖α ++β+ Lα(x)+β(x)(Ω) + ‖v‖α −+β− Lα(x)+β(x)(Ω) }] EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 11 ≤ K2 [{ ‖u‖α ++β+ X0 + ‖u‖α −+β− X0 } + { ‖v‖α ++β+ X0 + ‖v‖α −+β− X0 }] ≤ C2 max { ‖u‖α −+β− X0 , ‖u‖α ++β+ X0 , ‖v‖α −+β− X0 , ‖v‖α ++β+ X0 } = C2 max { max { ‖u‖α −+β− X0 , ‖v‖α −+β− X0 } ,max { ‖u‖α ++β+ X0 , ‖v‖α ++β+ X0 }} = C2 max { ‖(u, v)‖α −+β− , ‖(u, v)‖α ++β+ } , where K2 = ‖c‖L∞(Ω) max {( C(N, s, p, α, β,Ω) )α−+β− , ( C(N, s, p, α, β,Ω) )α++β+} and C2 = 4K2 + 1. � In the following lemma, we characterize the critical points of Jλ,µ as the local minimizers of Jλ,µ on N + λ,µ (or N − λ,µ). Lemma 3.2. Let (u∗, v∗) ∈ N + λ,µ (or N − λ,µ) be a local minimizer for Jλ,µ on N + λ,µ (or N − λ,µ). Then (u∗, v∗) is a critical point of Jλ,µ. Proof. First assume that (u∗, v∗) ∈ N + λ,µ is a local minimizer for Jλ,µ on N + λ,µ. Let Iλ,µ(u, v) = 〈J ′λ,µ(u, v), (u, v)〉. Note that for (u, v) ∈ E \ {0} with Iλ,µ(u, v) = 0, we have ϕ′′u,v(1) > 0 if and only if 〈I ′λ,µ(u, v), (u, v)〉 > 0. Since (u∗, v∗) is a local minimizer for Jλ,µ on N + λ,µ, using Lagrange’s multiplier theorem we obtain a real number τ such that J ′λ,µ(u∗, v∗) = τI ′λ,µ(u∗, v∗). Therefore, 0 = 〈J ′λ,µ(u∗, v∗), (u∗, v∗)〉 = τ〈I ′λ,µ(u∗, v∗), (u∗, v∗)〉 = τφ′′(u∗,v∗)(1). Since (u∗, v∗) ∈ N + λ,µ, we obtain that φ′′(u∗,v∗)(1) > 0 and hence τ = 0. This completes the proof. Similarly we can prove the result when (u∗, v∗) ∈ N − λ,µ is a local minimizer for Jλ,µ on N − λ,µ. � Next we show that the set of points of inflection of the function ϕu,v is empty for certain values of the parameters λ and µ. Lemma 3.3. There exists δ > 0, given by δ = 1 C1 (α− + β− − p+ α− + β− − q− )( p− − q+ C2(α+ + β+ − q+) ) p+−q− α−+β−−p+ such that for any pair of (λ, µ) ∈ R+ × R+ with λ + µ < δ, we have N 0 λ,µ = ∅, where the positive constants C1, C2 are given as in Lemma 3.1. Proof. We prove this lemma by contradiction. Let us assume that there exist λ, µ > 0 with λ+ µ < δ such that N 0 λ,µ 6= ∅. Hence, there is (u, v) ∈ N 0 λ,µ. Now, if ‖(u, v)‖ < 1, then using (3.2),(3.6) and Lemma 3.1 (i), (ii), we obtain 0 = ϕ′′(u,v)(1) ≤ p+P (u, v)− q−Q(u, v)− (α− + β−)R(u, v) = (p+ − (α− + β−))P (u, v) + (α− + β− − q−)Q(u, v) ≤ (p+ − (α− + β−))‖(u, v)‖p + + (α− + β− − q−)C1(λ+ µ)‖(u, v)‖q − . 12 R. BISWAS, S. TIWARI EJDE-2020/98 This implies ‖(u, v)‖p +−q− ≤ (α− + β− − q−) (α− + β− − p+) C1(λ+ µ). (3.11) Again using (3.2), (3.6) and Lemma 3.1 (i), (iii), we deduce 0 = ϕ′′(u,v)(1) ≥ p−P (u, v)− q+Q(u, v)− (α+ + β+)R(u, v) = (p− − q+)P (u, v)− (α+ + β+ − q+)R(u, v) ≥ (p− − q+)‖(u, v)‖p + − (α+ + β+ − q+)C2‖(u, v)‖α −+β− . This yields 1 ≥ ‖(u, v)‖α −+β−−p+ ≥ (p− − q+) C2(α+ + β+ − q+) . (3.12) Combining (3.11) and (3.12), we obtain λ+ µ ≥ 1 C1 (α− + β− − p+ α− + β− − q− )( p− − q+ C2(α+ + β+ − q+) ) p+−q− α−+β−−p+ , which is a contradiction. Next, if ‖(u, v)‖ > 1, again using (3.2), (3.6) and Lemma 3.1 (i), (ii), we find that 0 = ϕ′′u,v(1) ≤ (p+ − (α− + β−))‖(u, v)‖p − + (α− + β− − q−)C1(λ+ µ)‖(u, v)‖q + , that is, ‖(u, v)‖p −−q+ ≤ (α− + β− − q−) (α− + β− − p+) C1(λ+ µ). (3.13) On the other hand, by taking into account (3.2), (3.6) and Lemma 3.1 (i), (iii), it follows that 0 = ϕ′′u,v(1) ≥ (p− − q+)‖(u, v)‖p − − (α+ + β+ − q+)C2‖(u, v)‖α ++β+ , that is, ‖(u, v)‖α ++β+−p− ≥ (p− − q+) C2(α+ + β+ − q+) . (3.14) Thus, combining (3.13) and (3.14), we obtain λ+ µ ≥ 1 C1 (α− + β− − p+ α− + β− − q− )( p− − q+ C2(α+ + β+ − q+) ) p−−q+ α++β+−p− . (3.15) Since 0 < ( p−−q+ C2(α++β+−q+) ) < 1 and p−−q+ α++β+−p− < p+−q− α−+β−−p+ , from (3.15) we infer that λ+ µ ≥ 1 C1 (α− + β− − p+ α− + β− − q− )( p− − q+ C2(α+ + β+ − q+) ) p+−q− α−+β−−p+ , which is a contradiction. The proof is complete. � In the next result, we discuss the behavior of the functional Jλ,µ on Nλ,µ. Lemma 3.4. For λ+ µ < δ, Jλ,µ is coercive and bounded below on Nλ,µ. EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 13 Proof. Let (u, v) ∈ Nλ,µ. Then for ‖(u, v)‖ > 1, from (3.1) and (3.2) and Lemma 3.1 (ii), we deduce Jλ,µ(u, v) ≥ 1 p+ P (u, v)− 1 q− Q(u, v)− 1 α− + β− R(u, v) = ( 1 p+ − 1 α− + β− ) P (u, v)− ( 1 q− − 1 α− + β− ) Q(u, v) ≥ ( 1 p+ − 1 α− + β− ) ‖(u, v)‖p − − C1(λ+ µ) ( 1 q− − 1 α− + β− ) ‖(u, v)‖q + . (3.16) Since from (A3), we have 1 < q− ≤ q+ < p− ≤ p+ < α− + β−, (3.16) yields that Jλ,µ(u, v)→ +∞ as ‖(u, v)‖ → +∞. Therefore, Jλ,µ is coercive and bounded below on Nλ,µ. � Lemma 3.5. We have the following results: (i) If (u, v) ∈ N + λ,µ, then Q(u, v) > 0. (ii) If (u, v) ∈ N − λ,µ, then R(u, v) > 0. Proof. (i). Since (u, v) ∈ N + λ,µ, we have φ′′(u,v)(1) > 0. Thus, using (3.2) and (3.6), we obtain 0 < ϕ′′(u,v)(1) ≤ p+P (u, v)− q−Q(u, v)− (α− + β−)R(u, v) = {p+ − (α− + β−)}P (u, v) + (α− + β− − q−)Q(u, v). This implies that Q(u, v) ≥ (α− + β− − p+) (α− + β− − q−) P (u, v) > 0. (ii). Since (u, v) ∈ N − λ,µ, we have φ′′(u,v)(1) < 0. Thus, taking into account (3.2) and (3.6), we obtain 0 > ϕ′′(u,v)(1) ≥ p−P (u, v)− q+Q(u, v)− (α+ + β+)R(u, v) = (p− − q+)P (u, v)− (α+ + β+ − q+)R(u, v), that is, R(u, v) ≥ (α+ + β+ − p−) (α+ + β+ − q+) P (u, v) > 0. � From Lemma 3.3 and Lemma 3.4, we conclude that for any pair of parameters (λ, µ) ∈ R+ × R+ with λ + µ < δ, Nλ,µ = N − λ,µ ∪N + λ,µ and Jλ,µ is coercive and bounded below on N − λ,µ and N + λ,µ. Therefore, we can define θλ,µ = inf (u,v)∈Nλ,µ Jλ,µ(u, v), θ+ λ,µ = inf (u,v)∈N + λ,µ Jλ,µ(u, v), θ−λ,µ = inf (u,v)∈N − λ,µ Jλ,µ(u, v). The following two lemmas give the signs of θ+ λ,µ and θ−λ,µ, respectively. Lemma 3.6. If λ+ µ < δ, then θλ,µ ≤ θ+ λ,µ < 0. 14 R. BISWAS, S. TIWARI EJDE-2020/98 Proof. Let (u, v) ∈ N + λ,µ. Then ϕ′′u,v(1) > 0. Now combining (3.2) and (3.6), we obtain 0 < ϕ′′u,v(1) < p+P (u, v)− q−Q(u, v)− (α− + β−)R(u, v) = (p+ − q−)P (u, v)− (α− + β− − q−)R(u, v), that is, R(u, v) < (p+ − q−) (α− + β− − q−) P (u, v). (3.17) Using (3.1), (3.2) and (3.17), we deduce Jλ,µ(u, v) ≤ 1 p− P (u, v)− 1 q+ Q(u, v)− 1 α+ + β+ R(u, v) = ( 1 p− − 1 q+ ) P (u, v) + ( 1 q+ − 1 α+ + β+ ) R(u, v) ≤ {( 1 p− − 1 q+ ) + ( 1 q+ − 1 α+ + β+ ) (p+ − q−) (α− + β− − q−) } P (u, v) = { (q+ − p−)(α+ + β+) + p−(α+ + β+ − q+) (p+−q−) (α−+β−−q−) p−q+(α+ + β+) } P (u, v). (3.18) From (A4), we have (q+ − p−)(α+ + β+) + p−(α+ + β+ − q+) (p+−q−) (α−+β−−q−) < 0. Hence, (3.18) implies that Jλ,µ(u, v) < 0. Therefore, from the definition of θλ,µ and θ+ λ,µ, it follows that θλ,µ ≤ θ+ λ,µ < 0. � Lemma 3.7. If λ+µ < ( q− p+ ) δ, then θ−λ,µ > K, where K is some positive constant depending on N, s, p, q, α, β, a, b, λ, µ,Ω. Proof. Let (u, v) ∈ N − λ,µ. Then ϕ′′u,v(1) < 0. Therefore, from (3.12) and (3.13), we obtain ‖(u, v)‖ ≥  { (p−−q+) C2(α++β+−q+) }1/(α−+β−−p+) , if ‖(u, v)‖ < 1{ (p−−q+) C2(α++β+−q+) }1/(α++β+−p−) , if ‖(u, v)‖ > 1. (3.19) Now for ‖(u, v)‖ < 1, plugging (3.2) into (3.1) and using Lemma 3.1 (i), (ii) and (3.19), we deduce that Jλ,µ(u, v) ≥ 1 p+ P (u, v)− 1 q− Q(u, v)− 1 α− + β− R(u, v) = ( 1 p+ − 1 α− + β− ) P (u, v)− ( 1 q− − 1 α− + β− ) Q(u, v) ≥ ( 1 p+ − 1 α− + β− ) ‖(u, v)‖p + − ( 1 q− − 1 α− + β− ) C1(λ+ µ)‖(u, v)‖q − = ‖(u, v)‖q − [( 1 p+ − 1 α− + β− ) ‖(u, v)‖p +−q− − ( 1 q− − 1 α− + β− ) C1(λ+ µ) ] EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 15 ≥ { (p− − q+) C2(α+ + β+ − q+) } q− (α−+β−−p+) [( 1 p+ − 1 α− + β− ) × { (p− − q+) C2(α+ + β+ − q+) } (p+−q−) (α−+β−−p+) − ( 1 q− − 1 α− + β− ) C1(λ+ µ) ] := d1 (3.20) If λ+ µ < (q− p+ ) δ = (q− p+ ) 1 C1 (α− + β− − p+ α− + β− − q− ){ (p− − q+) C2(α+ + β+ − q+) } (p+−q−) (α−+β−−p+) , then λ+ µ < α− + β− − p+ p+(α− + β−) { (p− − q+) C2(α+ + β+ − q+) } (p+−q−) (α−+β−−p+) (α− + β−)q− α− + β− − q− . 1 C1 , that is, ( 1 p+ − 1 α− + β− ){ (p− − q+) C2(α+ + β+ − q+) } (p+−q−) (α−+β−−p+) − ( 1 q− − 1 α− + β− ) C1(λ+ µ) > 0, and thus, from (3.20), we obtain that d1 > 0. Similarly for ‖(u, v)‖ > 1, again plugging (3.2) in (3.1) and using Lemma 3.1 (i), (ii) and (3.19), we obtain Jλ,µ(u, v) ≥ 1 p+ P (u, v)− 1 q− Q(u, v)− 1 α− + β− R(u, v) = ( 1 p+ − 1 α− + β− ) P (u, v)− ( 1 q− − 1 α− + β− ) Q(u, v) ≥ ( 1 p+ − 1 α− + β− ) ‖(u, v)‖p − − ( 1 q− − 1 α− + β− ) C1(λ+ µ)‖(u, v)‖q + = ‖(u, v)‖q + {( 1 p+ − 1 α− + β− ) ‖(u, v)‖p −−q+ − ( 1 q− − 1 α− + β− ) C1(λ+ µ) } ≥ { (p− − q+) C2(α+ + β+ − q+) } q+ (α++β+−p−) [( 1 p+ − 1 α− + β− ) { (p− − q+) C2(α+ + β+ − q+) } (p−−q+) (α++β+−p−) − ( 1 q− − 1 α− + β− ) C1(λ+ µ) ] (3.21) Combining that (p−−q+) (α++β+−p−) < (p+−q−) (α−+β−−p+) and (p−−q+) C2(α++β+−q+) < 1, and taking into account (3.20) and (3.21), we deduce that Jλ,µ(u, v) 16 R. BISWAS, S. TIWARI EJDE-2020/98 ≥ { (p− − q+) C2(α+ + β+ − q+) } q+ (α−+β−−p+) [( 1 p+ − 1 α− + β− ) × { (p− − q+) C2(α+ + β+ − q+) } (p+−q−) (α−+β−−p+) − ( 1 q− − 1 α− + β− ) C1(λ+ µ) ] ≥ { (p− − q+) C2(α+ + β+ − q+) } (q+−q−) (α−+β−−p+) d1 := d2 > 0. Finally by choosing K = min{d1, d2} > 0, the proof is complete. � The next lemma describes the nature of the map ϕu,v. We refer to [8] and [10] for the similar results in the case of local p-Laplacian and nonlocal p-Laplacian, respectively, and [1, 15] for variable exponent Laplacian. Lemma 3.8. For (u, v) ∈ E\{(0, 0)}, there exists δ′ > 0 such that for all λ+µ < δ′, we have the following: (i) If Q(u, v) = 0, then there exists a unique t− = t−(u, v) such that (t−u, t−v) ∈ N − λ,µ and Jλ,µ(t−u, t−v) = supt≥0 Jλ,µ(tu, tv). (ii) If Q(u, v) > 0, then there exist t∗ > 0 and unique positive numbers t+ = t+(u, v) < t− = t−(u, v) such that (t−u, t−v) ∈ N − λ,µ, (t+u, t+v) ∈ N + λ,µ and Jλ,µ(t+u, t+v) = inf 0≤t≤t∗ Jλ,µ(tu, tv), Jλ,µ(t−u, t−v) = sup t≥0 Jλ,µ(tu, tv). Proof. (i) Using the given assumption, for 0 < t < 1 sufficiently small, we obtain ϕu,v(t) > tp + p+ P (u, v)− tα ++β+ α+ + β+ R(u, v) > 0 and for t > 1 sufficiently large, we obtain ϕu,v(t) < tp + p− P (u, v)− tα ++β+ α− + β− R(u, v) < 0. Hence, ϕu,v achieves its maximum at some point t−(u, v) on [0,∞). Thus, ϕ′u,v(t −) = 〈J ′λ,µ(t−u, t−v), (u, v)〉 = 0. Set (t−u, t−v) := (u, v). Then 〈J ′λ,µ(u, v), (u, v)〉 = 0, which implies (u, v) ∈ Nλ,µ. Thus, from (3.2), we obtain P (u, v) = R(u, v). (3.22) Now we define the function Θu,v : [0,∞) → R as Θu,v(t) = Jλ,µ(tu, tv). We know that Θu,v(1) = Jλ,µ(u, v) = maxt∈[0,∞) Θu,v(t) and Θ′u,v(1) = 〈J ′λ,µ(u, v), (u, v)〉 = 0. For t > 1, by (3.22), we deduce that Θ′u,v(t) = 〈J ′λ,µ(tu, tv), (u, v)〉 ≤ tp +−1P (u, v)− tα −+β−−1R(u, v) < 0, and on the other hand, for t ∈ (0, 1), again using (3.22), we obtain Θ′u,v(t) = 〈J ′λ,µ(tu, tv), (u, v)〉 ≥ tp −−1P (u, v)− tα ++β+−1R(u, v) > 0. This shows that the point t− is unique. Hence, the result follows. EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 17 (ii) To prove this lemma, first we set f1(t) := ∫ RN×RN tp(x,y) { |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) + |v(x)− v(y)|p(x,y) |x− y|N+sp(x,y) } dx dy, f2(t) := ∫ Ω tq(x) ( λa(x)|u|q(x) + µb(x)|v|q(x) ) dx, f3(t) := ∫ Ω tα(x)+β(x)c(x)|u|α(x)|v|β(x)dx. The fi’s are continuous and strictly increasing functions with fi(0) = 0 for i = 1, 2, 3. Also, we have the following observations: (I) limt→0+ f3(t)/f1(t) = 0. (II) limt→+∞ f2(t) = +∞. (III) limt→+∞(f1 − f3)(t))/f2(t) = 0. (IV) f1 − f3 has unique point of maximum, say tmax and (f1 − f3)(t)→ −∞ as t→ +∞. (V) There exists t̃ ∈ (0, tmax) such that f1−f3 f2 is strictly increasing on (0, t̃). From (I), we note that (f1− f3)(t) > 0 for t→ 0+ sufficiently small. Hence, by (V) and intermediate value theorem, for each choice of the pair (λ, µ) ∈ R+ ×R+ with f2(t̃) < (f1 − f3)(t̃), there exists a unique t+ = t+(λ, µ) ∈ (0, t̃) such that (f1 − f3)(t+) f2(t+) = 1. (3.23) Since (f1−f3) f2 is strictly monotone increasing in (t+, t̃), from (3.23), we obtain 1 = (f1 − f3)(t+) f2(t+) < (f1 − f3)(t) f2(t) for all t ∈ (t+, t̃), that is, f2(t) < (f1 − f3)(t) for all t ∈ (t+, t̃). (3.24) Now we can fix (λ∗, µ∗) ∈ R+ ×R+ such that for all λ ∈ (0, λ∗), µ ∈ (0, µ∗), taking into account (3.24), we have f2(t) < (f1 − f3)(t) for all t ∈ (t+, tmax). (3.25) Since f1 − f3 is strictly decreasing in (tmax,∞) and f2 is monotonically increasing in (0,∞), by (II) and (3.25), there exists a unique positive real number t− > tmax such that f2(t−) = (f1 − f3)(t−) for all (λ, µ) ∈ (0, λ∗)× (0, µ∗). (3.26) Hence, combining (3.23) and (3.26), we yield that the function ϕ′u,v(t) = f1−f2−f3 has exactly two nontrivial zeroes, t+ < t−, that is, t+ and t− are critical points of ϕu,v(t). For δ′ := λ∗ + µ∗, we can choose λ∗, µ∗ > 0 sufficiently small such that δ′ < δ, where δ is given as in Lemma 3.3. Since ϕu,v(0) = 0 and ϕu,v(t) < 0 for t → 0+ sufficiently small, we obtain that ϕ′u,v(t) < 0 for all t ∈ (0, t+) and ϕ′u,v(t) > 0 for all t ∈ (t+, tmax) and ϕ′u,v(t +) = 0. Again, by the Lemma 3.3, we have N 0 λ,µ = ∅ and thus, we infer that ϕu,v attains a local minimum at t+ and consequently ϕ′′u,v(t +) > 0. Hence, (t+u, t+v) ∈ N +. Similarly, since ϕ′u,v(t) > 0 for all t ∈ [tmax, t −), ϕ′u,v(t) < 0 for all t > t−, and ϕ′u,v(t −) = 0, using the fact that N 0 λ,µ = ∅ from Lemma 3.3, it follows that t− is the point of global maximum for ϕu,v and consequently ϕ′′u,v(t −) < 0. Hence, 18 R. BISWAS, S. TIWARI EJDE-2020/98 (t−u, t−v) ∈ N −. Now by appealing to Lemma 3.6 and Lemma 3.7, we obtain ϕu,v(t +) < 0 and ϕu,v(t −) > 0. Also, by the above discussions, ϕu,v is strictly increasing on [t+, t−] and strictly decreasing for all t > t− with ϕu,v(t) → −∞ as t → +∞. Thus, there exists a unique t∗ ∈ (t+, t−) such that ϕu,v(t ∗) = 0. Therefore, Jλ,µ(t+u, t+v) = ϕu,v(t +) = inf 0≤t≤t∗ φu,v(t) = inf 0≤t≤t∗ Jλ,µ(tu, tv), Jλ,µ(t−u, t−v) = ϕu,v(t −) = sup t≥0 φu,v(t) = sup t≥0 Jλ,µ(tu, tv). This completes the proof. � 4. Existence of multiple solutions In this section, we will prove the existence of at least two distinct non-trivial and non-negative weak solutions of (1.1). The next two propositions ensure the existence of minimizers for the functional Jλ,µ in N + λ,µ and N − λ,µ, respectively, which serve as the weak solutions of (1.1). We set δ0 := min {( q− p+ ) δ, δ′ } , where δ and δ′ are given as in Lemma 3.7 and Lemma 3.8, respectively. Proposition 4.1. For λ+ µ < δ0, the functional Jλ,µ has a minimizer (u0, v0) in N + λ,µ, which satisfies the following assertions: (i) Jλ,µ(u0, v0) = θ+ λ,µ < 0, (ii) (u0, v0) is a solution of (1.1) Proof. (i) Since Jλ,µ is bounded below on Nλ,µ and hence on N + λ,µ, there exists a minimizing sequence {(um, vm)} ⊂ N + λ,µ, such that lim m→∞ Jλ,µ(um, vm) = inf (u,v)∈N + λ,µ Jλ,µ(u, v). By Lemma 3.4, we have Jλ,µ is coercive on N + λ,µ, which implies that the sequence {(um, vm)} is bounded on E. Therefore, there exists (u0, v0) ∈ E such that, passing to a subsequence, um ⇀ u0, vm ⇀ v0 in X0 as m→∞ and hence, using Sobolev-type embedding result (Lemma 2.7), we have um → u0 strongly in Lq(x)(Ω), and Lα(x)+β(x)(Ω), vm → v0 strongly in Lq(x)(Ω) and Lα(x)+β(x)(Ω), um(x)→ u0(x) and vm(x)→ v0(x) a.e. in Ω as m → ∞. Now by applying Lemma 2.4 and Lebesgue dominated convergence theorem, one can check that lim m→∞ ∫ Ω a(x)|um|q(x)dx = ∫ Ω a(x)|u0|q(x)dx, lim m→∞ ∫ Ω b(x)|vm|q(x)dx = ∫ Ω b(x)|v0|q(x)dx; (4.1) EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 19 and lim m→∞ ∫ Ω a(x) |um|q(x) q(x) dx = ∫ Ω a(x) |u0|q(x) q(x) dx, lim m→∞ ∫ Ω b(x) |vm|q(x) q(x) dx = ∫ Ω b(x) |v0|q(x) q(x) dx. (4.2) Also, by Lemma 5.1 and Lemma 5.2 (see Appendix), we have lim m→∞ R(um, vm) = R(u0, v0), lim m→∞ ∫ Ω c(x)|um|α(x)|vm|β(x) α(x) + β(x) dx = ∫ Ω c(x)|u0|α(x)|v0|β(x) α(x) + β(x) dx, (4.3) respectively. We claim that (u0, v0) 6≡ (0, 0). Note that Q(u0, v0) > 0. Indeed, if not then from (4.1), Q(um, vm)→ Q(u0, v0) = 0 as m→∞. (4.4) Since (um, vm) ∈ N + λ,µ, using (3.1) and (3.2), we obtain Jλ,µ(um, vm) ≥ ( 1 p+ − 1 α− + β− ) P (um, vm)− ( 1 q− − 1 α− + β− ) Q(um, vm). Now letting m → ∞ in the both side of the last expression and using (4.4), we obtain lim m→∞ Jλ,µ(um, vm) ≥ 0. (4.5) But Lemma 3.6 gives limm→∞ Jλ,µ(um, vm) = inf(u,v)∈N + λ,µ Jλ,µ(u, v) < 0, which contradicts (4.5). Thus, the claim is proved and we obtain that (u0, v0) ∈ E \ {(0, 0)}. Next, we claim that um → u0 and vm → v0 strongly in X0 as m → ∞. If not, then um 6→ u0 or vm 6→ v0 in X0 as m → ∞. Therefore, using Lemma 2.10 and Brezis-Lieb lemma (see [7]), it follows that either ∫ RN×RN 1 p(x, y) |u0(x)− u0(y)|p(x,y) |x− y|N+sp(x,y) dx dy < lim inf m→∞ ∫ RN×RN 1 p(x, y) |um(x)− um(y)|p(x,y) |x− y|N+sp(x,y) dx dy or∫ RN×RN 1 p(x, y) |v0(x)− v0(y)|p(x,y) |x− y|N+sp(x,y) dx dy < lim inf m→∞ ∫ RN×RN 1 p(x, y) |vm(x)− vm(y)|p(x,y) |x− y|N+sp(x,y) dx dy. (4.6) 20 R. BISWAS, S. TIWARI EJDE-2020/98 Thus, combining (3.1), (4.2), (4.3) and (4.6), we obtain lim m→∞ Jλ,µ(um, vm) = lim inf m→∞ [ ∫ RN×RN 1 p(x, y) { |um(x)− um(y)|p(x,y) |x− y|N+sp(x,y) + |vm(x)− vm(y)|p(x,y) |x− y|N+sp(x,y) } dx dy − ∫ Ω 1 q(x) ( λa(x)|um|q(x) + µb(x)|vm|q(x) ) dx − ∫ Ω 1 α(x) + β(x) c(x)|um|α(x)|vm|β(x)dx ] ≥ lim inf m→∞ ∫ RN×RN 1 p(x, y) |um(x)− um(y)|p(x,y) |x− y|N+sp(x,y) dx dy + lim inf m→∞ ∫ RN×RN 1 p(x, y) |vm(x)− vm(y)|p(x,y) |x− y|N+sp(x,y) dx dy − lim m→∞ ∫ Ω 1 q(x) ( λa(x)|um|q(x) + µb(x)|vm|q(x) ) dx − lim m→∞ ∫ Ω 1 α(x) + β(x) c(x)|um|α(x)|vm|β(x)dx > ∫ RN×RN 1 p(x, y) |u0(x)− u0(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN 1 p(x, y) |v0(x)− v0(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω 1 q(x) ( λa(x)|u0|q(x) + µb(x)|v0|q(x) ) dx − ∫ Ω 1 α(x) + β(x) c(x)|u0|α(x)|v0|β(x)dx = Jλ,µ(u0, v0) (4.7) By Lemma 3.8 (ii), for (u0, v0) ∈ E \ {(0, 0)}, there exists a positive real number t+0 (u0, v0) such that (t+0 u0, t + 0 v0) ∈ N + λ,µ. Again, considering the assumption um 6→ u0 or vm 6→ v0 in X0, we have ρX0(t+0 u0) < lim inf m→∞ ρX0(t+0 um) or ρX0(t+0 v0) < lim inf m→∞ ρX0(t+0 vm). (4.8) Furthermore, appealing Lemma 2.4 and Lebesgue dominated convergence theorem, we obtain Q(t+0 u0, t + 0 v0) = lim m→∞ Q(t+0 um, t + 0 vm) (4.9) and by Lemma 5.1 (see Appendix), we obtain R(t+0 u0, t + 0 v0) = lim m→∞ R(t+0 um, t + 0 vm). (4.10) EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 21 Taking into account (3.4), (4.9), (4.10) and (4.8), we deduce that lim m→∞ ϕ′um,vm(t+0 ) = lim inf m→∞ [ ∫ RN×RN (t+0 )p(x,y)−1 |um(x)− um(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN (t+0 )p(x,y)−1 |vm(x)− vm(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω (t+0 )q(x)−1 ( λa(x)|um|q(x) + µb(x)|vm|q(x) ) dx − ∫ Ω (t+0 )α(x)+β(x)−1c(x)|um|α(x)|vm|β(x)dx ] ≥ 1 t+0 [ lim inf m→∞ ρX0 (t+0 um) + lim inf m→∞ ρX0 (t+0 vm)− lim m→∞ Q(t+0 um, t + 0 vm) − lim m→∞ R(t+0 um, t + 0 vm) ] > 1 t+0 [ ρX0 (t+0 u0) + ρX0 (t+0 vm)−Q(t+0 u0, t + 0 v0)−R(t+0 u0, t + 0 v0) ] = ϕ′u0,v0 (t+0 ) = 0. (4.11) Thus, for m large enough, ϕ′um,vm(t+0 ) > 0. Since (um, vm) ∈ N + λ,µ for all m ∈ N, we have ϕ′um,vm(1) = 0 and ϕ′′um,vm(1) > 0. Then using Lemma 3.8 (ii), we obtain ϕ′um,vm(t) < 0 for all t ∈ (0, 1) and therefore, from (4.11), we must have t+0 > 1. Since (t+0 u0, t + 0 v0) ∈ N + λ,µ, again by Lemma 3.8 (ii), we obtain that ϕu0,v0(t) is monotone decreasing on (0, t+0 ). Hence using (4.7), we obtain Jλ,µ(t+0 u0, t + 0 v0) ≤ Jλ,µ(u0, v0) < lim m→∞ Jλ,µ(um, vm) = inf (u,v)∈N + λ,µ Jλ,µ(u, v). This is a contradiction to the fact that (t+0 u0, t + 0 v0) ∈ N + λ,µ. So, (um, vm)→ (u0, v0) strongly in E as m → ∞ and thus, (u0, v0) ∈ Nλ,µ. Since Lemma 3.3 gives that N 0 λ,µ = ∅ and from Lemma 3.6, we have Jλ,µ(u0, v0) = limm→∞ Jλ,µ(um, vm) < 0, we infer that (u0, v0) ∈ N + λ,µ. (ii) Using Lemma 3.2, we conclude that (u0, v0) is a solution of (1.1). � Proposition 4.2. If λ+ µ < δ0, then Jλ,µ has a minimizer (w0, z0) in N − λ,µ such that the following assertions hold: (i) Jλ,µ(w0, z0) = θ−λ,µ > 0. (ii) (w0, z0) is a non-semi trivial solution of (1.1). Proof. (i) From Lemma 3.4, Jλ,µ is bounded below on N − λ,µ. Hence, there exists a minimizing sequence {(wm, zm)} ⊂ N − λ,µ such that lim m→∞ Jλ,µ(wm, zm) = inf (u,v)∈N − λ,µ Jλ,µ(u, v). Again from Lemma 3.4, we have Jλ,µ is coercive, which implies that the sequence {(wm, zm)} is bounded on E and thus, there exists (w0, z0) ∈ E such that up to a subsequence, (wm, zm) ⇀ (w0, z0) weakly and by Sobolev-type embedding result 22 R. BISWAS, S. TIWARI EJDE-2020/98 (Theorem 2.7), we obtain wm → w0, zm → z0 strongly in Lq(x)(Ω) and Lα(x)+β(x)(Ω), wm(x)→ w0(x) and zm(x)→ z0(x) a.e. in Ω as m→∞. Using Lemma 2.4 and Dominated convergence theorem, we have lim m→∞ ∫ Ω a(x)|wm|q(x)dx = ∫ Ω a(x)|w0|q(x)dx and lim m→∞ ∫ Ω b(x)|zm|q(x)dx = ∫ Ω b(x)|z0|q(x)dx. (4.12) Also, Lemma 5.1 (see Appendix) gives R(wm, zm)→ R(w0, z0) as m→∞. (4.13) Next, we have (w0, z0) 6≡ (0, 0). Indeed, if (w0, z0) = (0, 0), from (4.13), we obtain R(wm, zm)→ R(w0, z0) = 0 as m→∞. (4.14) Since (wm, zm) ∈ N − λ,µ, using (3.1), (3.2) and Lemma 3.6, we have 0 < K < Jλ,µ(wm, zm) ≤ ( 1 p+ − 1 q− ) P (wm, zm) + ( 1 q− − 1 α− + β− ) R(wm, zm) + om(1). Now letting m→∞, from the last expression and (4.14), we obtain 0 < K < lim m→∞ Jλ,µ(wm, zm) ≤ 0, which is a contradiction. Thus, (w0, z0) ∈ E\{(0, 0)}. If Q(w0, z0) = 0, then we use Lemma 3.8 (i) and if Q(w0, z0) > 0, then we use Lemma 3.8 (ii). In both the cases, there exists a positive real number t−0 = t−0 (w0, z0) such that (t−0 w0, t − 0 z0) ∈ N − λ,µ. Next, we claim that wm → w0 strongly in X0 and zm → z0 strongly in X0 as m→∞. Suppose the claim does not hold. Then t−0 wm 6→ t−0 w0 or t−0 zm 6→ t−0 z0 in X0 as m→∞. This implies that either ρX0(t−0 w0) < lim inf m→∞ ρX0(t−0 wm) (4.15) or ρX0 (t−0 z0) < lim inf m→∞ ρX0 (t−0 zm). (4.16) Furthermore, using the same assumption, we can have the following as in Proposi- tion 4.1: either∫ RN×RN 1 p(x, y) |t−0 w0(x)− t−0 w0(y)|p(x,y) |x− y|N+sp(x,y) dx dy < lim inf m→∞ ∫ RN×RN 1 p(x, y) |t−0 wm(x)− t−0 wm(y)|p(x,y) |x− y|N+sp(x,y) dx dy or∫ RN×RN 1 p(x, y) |t−0 z0(x)− t−0 z0(y)|p(x,y) |x− y|N+sp(x,y) dx dy < lim inf m→∞ ∫ RN×RN 1 p(x, y) |t−0 zm(x)− t−0 zm(y)|p(x,y) |x− y|N+sp(x,y) dx dy. (4.17) EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 23 Note that, using Lemma 2.4 and Lebesgue dominated converges theorem, we deduce that lim m→∞ ∫ Ω 1 q(x) ( λa(x)|t−0 wm|q(x) + µb(x)|t−0 zm|q(x) ) dx = ∫ Ω 1 q(x) ( λa(x)|t−0 w0|q(x) + µb(x)|t−0 z0|q(x) ) dx. (4.18) Also, by Lemma 5.2 (see Appendix), we have lim m→∞ ∫ Ω 1 α(x) + β(x) c(x)|t−0 wm|α(x)|t−0 zm|β(x)dx = ∫ Ω 1 α(x) + β(x) c(x)|t−0 w0|α(x)|t−0 z0|β(x)dx. (4.19) Thus, combining (3.1), (4.17), (4.18) and (4.19), we obtain lim m→∞ Jλ,µ(t−0 wm, t − 0 zm) = lim inf m→∞ [ ∫ RN×RN 1 p(x, y) |t−0 wm(x)− t−0 wm(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN 1 p(x, y) |t−0 zm(x)− t−0 zm(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω 1 q(x) ( λa(x)|t−0 wm|q(x) + µb(x)|t−0 zm|q(x) ) dx − ∫ Ω 1 α(x) + β(x) c(x)|t−0 wm|α(x)|t−0 zm|β(x)dx ] ≥ lim inf m→∞ ∫ RN×RN 1 p(x, y) |t−0 wm(x)− t−0 wm(y)|p(x,y) |x− y|N+sp(x,y) dx dy + lim inf m→∞ ∫ RN×RN 1 p(x, y) |t−0 zm(x)− t−0 zm(y)|p(x,y) |x− y|N+sp(x,y) dx dy − lim m→∞ ∫ Ω 1 q(x) ( λa(x)|t−0 wm|q(x) + µb(x)|t−0 zm|q(x) ) dx − lim m→∞ ∫ Ω 1 α(x) + β(x) c(x)|t−0 wm|α(x)|t−0 zm|β(x)dx > ∫ RN×RN 1 p(x, y) |t−0 w0(x)− t−0 w0(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN 1 p(x, y) |t−0 z0(x)− t−0 z0(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω 1 q(x) ( λa(x)|t−0 w0|q(x) + µb(x)|t−0 z0|q(x) ) dx − ∫ Ω 1 α(x) + β(x) c(x)|t−0 w0|α(x)|t−0 z0|β(x)dx = Jλ,µ(t−0 w0, t − 0 z0). (4.20) Again, using that wm → w0 and zm → z0 in Lq(x)(Ω), we obtain lim m→∞ Q(t−0 wm, t − 0 zm) = Q(t−0 w0, t − 0 z0) (4.21) 24 R. BISWAS, S. TIWARI EJDE-2020/98 and Lemma 5.1 (see Appendix) gives us lim m→∞ R(t−0 wm, t − 0 zm) = R(t−0 w0, t − 0 z0). (4.22) Considering (3.4), (4.15), (4.21) and (4.22), we obtain lim m→∞ ϕ′wm,zm(t−0 ) = lim inf m→∞ [ ∫ RN×RN (t−0 )p(x,y)−1 |wm(x)− wm(y)|p(x,y) |x− y|N+sp(x,y) dx dy + ∫ RN×RN (t−0 )p(x,y)−1 |zm(x)− zm(y)|p(x,y) |x− y|N+sp(x,y) dx dy − ∫ Ω (t−0 )q(x)−1 ( λa(x)|wm|q(x) + µb(x)|zm|q(x) ) dx − ∫ Ω (t−0 )α(x)+β(x)−1c(x)|wm|α(x)|zm|β(x)dx ] ≥ 1 t−0 [ lim inf m→∞ ρX0(t−0 wm) + lim inf m→∞ ρX0(t−0 zm)− lim m→∞ Q(t−0 wm, t − 0 zm) − lim m→∞ R(t−0 wm, t − 0 zm) ] > 1 t−0 [ ρX0 (t−0 w0) + ρX0 (t−0 zm)−Q(t−0 w0, t − 0 z0)−R(t−0 w0, t − 0 z0) ] = ϕ′u0,v0 (t−0 ) = 0. (4.23) For m large enough, ϕ′wm,zm(t−0 ) > 0. Since (wm, zm) ∈ N − λ,µ for all m ∈ N, we have ϕ′wm,zm(1) = 0 and ϕ′′wm,zm(1) < 0 for all m ∈ N. Now using the Lemma 3.8, we obtain ϕ′wm,zm(t) < 0 for all t > 1. Then from (4.23), we must have t−0 < 1. Since (t−0 w0, t − 0 z0) ∈ N − λ,µ, again using Lemma 3.8, we obtain that 1 is the global maximum point for ϕwm,zm(t). Therefore, from (4.20), it follows that Jλ,µ(t−0 w0, t − 0 z0) < lim m→∞ Jλ,µ(t−0 wm, t − 0 zm) ≤ lim m→∞ Jλ,µ(wm, zm) = inf (u,v)∈N − λ,µ Jλ,µ(u, v). This is a contradiction to the fact that (t−0 w0, t − 0 z0) ∈ N − λ,µ. Hence, (wm, zm) → (w0, z0) strongly in E as m→∞ and (w0, z0) ∈ N . Now using the fact that N 0 λ,µ = ∅ from Lemma 3.3 and noticing that Jλ,µ(w0, z0) = inf(u,v)∈N − λ,µ Jλ,µ(u, v) > 0, we conclude that (w0, z0) ∈ N − λ,µ. (ii) Using Lemma 3.2, we infer that (w0, z0) is a solution of (1.1). Now we will show that (w0, z0) is not semi-trivial, that is, (w0, z0) is not of the form (u, 0) (or (0, v)). The proof follows adapting the similar approach as in [10]. If (u, 0) (or (0, v)) is a semi-trivial solution of (1.1), then from (1.4) we obtain ρX0 (u) = ∫ RN×RN |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy = λ ∫ Ω a(x)|u|q(x)dx. Therefore, Jλ,µ(u, 0) = ∫ RN×RN 1 p(x, y) |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy − λ ∫ Ω 1 q(x) a(x)|u|q(x)dx EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 25 ≤ 1 p− ∫ RN×RN |u(x)− u(y)|p(x,y) |x− y|N+sp(x,y) dx dy − λ q+ ∫ Ω a(x)|u|q(x)dx = ( 1 p− − 1 q+ ) ρX0 (u) < 0, since by Lemma 3.7, Jλ,µ(w0, z0) > 0, it follows that (w0, z0) is not semi-trivial. � Proof of Theorem 1.2. Define Λ = δ0 (given as in Section 4). Let (u0, v0) be ob- tained as in Proposition 4.1. Now using Lemma 3.5 and the fact that (u0, v0) ∈ N + λ,µ, for (|u0|, |v0|) ∈ E \ {(0, 0)}, we have Q(|u0|, |v0|) = Q(u0, v0) > 0, and thus by Lemma 3.8 (ii), there exists t1 > 0 such that (t1|u0|, t1|v0|) ∈ N + λ,µ. This implies 0 = ϕ′|u0|,|v0|(t1) ≤ ϕ′u0,v0 (t1). (4.24) Combining (4.24) with the facts that (u0, v0) ∈ N + λ,µ, ϕ′u0,v0 (1) = 0, and again using Lemma 3.8 (ii), we obtain t1 ≥ 1. This yields Jλ,µ(t1|u0|, t1|v0|) ≤ Jλ,µ(|u0|, |v0|) ≤ Jλ,µ(u0, v0) = inf (u,v)∈N + λ,µ Jλ,µ(u, v). Therefore, there exists a non-negative minimizer for Jλ,µ on N + λ,µ, which is a solu- tion of (1.1) by Lemma 3.2. Next, we assert that there exists a non-negative minimizer for Jλ,µ(w, z) on N − λ,µ. Indeed, for (|w0|, |z0|) ∈ E \ {(0, 0)}, by Lemma 3.8, there exists t2 > 0 such that (t2|w0|, t2|z0|) ∈ N − λ,µ, where (w0, z0) is as in Proposition 4.2. Since (w0, z0) ∈ N − λ,µ, again by Lemma 3.8, we obtain Jλ,µ(t2|w0|, t2|z0|) ≤ Jλ,µ(t2w0, t2z0) ≤ Jλ,µ(w0, z0) = inf (u,v)∈N − λ,µ Jλ,µ(u, v). Thus, we obtain a non-negative minimizer for Jλ,µ on N − λ,µ, which is a solution of (1.1), thanks to Lemma 3.2. Hence for all 0 < λ + µ < Λ, (1.1) admits two non-trivial and non-negative solutions in N + λ,µ and N − λ,µ, respectively. Since N + λ,µ ∩N − λ,µ = ∅, these solutions are distinct. This completes the proof. � 5. Appendix Lemma 5.1. Let {um}, {vm} be any two bounded sequences in X0 and c, α, β be as in Theorem 1.2. Then lim m→∞ ∫ Ω c(x)|um|α(x)|vm|β(x)dx = ∫ Ω c(x)|u|α(x)|v|β(x)dx. Proof. Since {um} and {vm} are bounded sequences in X0 and X0 is reflexive, up to subsequences, um ⇀ u and vm ⇀ v weakly in X0 as m → ∞. First we claim that lim m→∞ ∫ Ω |um − u|α(x)|vm − v|β(x)dx = lim m→∞ ∫ Ω |um|α(x)|vm|β(x)dx− ∫ Ω |u|α(x)|v|β(x)dx (5.1) 26 R. BISWAS, S. TIWARI EJDE-2020/98 For t ∈ (0, 1), we note that∫ Ω ∫ 1 0 α(x)|um − tu|α(x)−2(um − tu)u|vm|β(x) dx dt − ∫ Ω ∫ 1 0 β(x)|um − u|α(x)|vm − tv|β(x)−2v(vm − tv) dx dt = ∫ Ω |um|α(x)|vm|β(x)dx− ∫ Ω |um − u|α(x)|vm − v|β(x)dx. (5.2) Set fm(x, t) := |um − tu|α(x)−2(um − tu)|vm|β(x) gm(x, t) := |um − u|α(x)|vm − tv|β(x)−2(vm − tv). Now from the given assumptions, we have fm(x, t)→ (1− t)α(x)−1|u|α(x)−2u|v|β(x) a.e. in RN × (0, 1) as m→∞, gm(x, t)→ 0 a.e. in RN × (0, 1) as m→∞. (5.3) Next, using Hölder’s inequality and Sobolev-type embedding result (Theorem 2.7), we obtain ∫ Ω ∫ 1 0 |fm| α(x)+β(x) α(x)+β(x)−1 dx dt ≤ ‖ |um − tu|{(α−1)( α+β α+β−1 )}(·)‖ L α(x)+β(x)−1 α(x)−1 (Ω×(0,1)) × ‖ |vm|β(·)‖ L α(x)+β(x)−1 β(x) (Ω×(0,1)) < M1, (5.4) and ∫ Ω ∫ 1 0 |gm| α(x)+β(x) α(x)+β(x)−1 dx dt ≤ ‖ |um − u|α( α+β α+β−1 )(·)‖ L α(x)+β(x)−1 α(x) (Ω×(0,1)) × ‖ |vm|(β−1) α+β α+β−1 (·)‖ L α(x)+β(x)−1 β(x)−1 (Ω×(0,1)) < M2, (5.5) where M1,M2 are positive constants independent of m. Hence, the sequences {fm} and {gm} are uniformly bounded in L α(x)+β(x) α(x)+β(x)−1 (Ω × (0, 1)) and thus we have, up to subsequences, fm ⇀ (1− t)α(x)−1|u|α(x)−2u|v|β(x) weakly in L α(x)+β(x) α(x)+β(x)−1 (Ω× (0, 1)), gm ⇀ 0 weakly in L α(x)+β(x) α(x)+β(x)−1 (Ω× (0, 1)). (5.6) as m→∞. Using (5.6), we deduce that lim m→∞ ∫ Ω ∫ 1 0 α(x)fmudxdt = lim m→∞ ∫ Ω ∫ 1 0 α(x)fu dx dt = lim m→∞ ∫ Ω |u|α(x)|v|β(x)dx, (5.7) lim m→∞ ∫ Ω ∫ 1 0 β(x)gmv dx dt = 0. (5.8) EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 27 Thus, plugging (5.7) and (5.8) into (5.2), we obtain (5.1). Note that, from Theorem 2.7 and Lemma 2.4, we have∫ Ω |um − u|α(x)+β(x)dx→ 0 and ∫ Ω |vm − v|α(x)+β(x)dx→ 0 as m→∞. Now using the above and Young’s inequality, we have∫ Ω |um − u|α(x)|vm − v|β(x)dx ≤ ∫ Ω { α(x) α(x) + β(x) |um − u|α(x)+β(x) + α(x) α(x) + β(x) |vm − v|α(x)+β(x) } dx ≤ α+ α− + β− ∫ Ω |um − u|α(x)+β(x)dx+ β+ α− + β− ∫ Ω |vm − v|α(x)+β(x)dx → 0 as m→∞. (5.9) Thus, inserting (5.9) into (5.1), we obtain lim m→∞ ∫ Ω |um|α(x)|vm|β(x)dx = ∫ Ω |u|α(x)|v|β(x)dx. (5.10) Now ∣∣∣ ∫ Ω c(x)|um|α(x)|vm|β(x)dx− ∫ Ω c(x)|u|α(x)|v|β(x)dx ∣∣∣ ≤ ‖c‖L∞(Ω) ∫ Ω ∣∣|um|α(x)|vm|β(x) − |u|α(x)|v|β(x) ∣∣dx. (5.11) We define wm := |um|α(x)|vm|β(x) + |u|α(x)|v|β(x) − ∣∣|um|α(x)|vm|β(x) − |u|α(x)|v|β(x) ∣∣ ≥ 0. Since um(x)→ u(x) and vm(x)→ v(x) a.e. in RN as m→∞, we have wm(x)→ 2|u(x)|α(x)|v(x)|β(x) a.e. in RN as m→∞. Thus, by Fatou’s Lemma, lim inf m→∞ ∫ Ω wm(x)dx ≥ 2 ∫ Ω |u|α(x)|v|β(x)dx. (5.12) Again from (5.10), we find that lim sup m→∞ ∫ Ω wm(x)dx ≤ lim m→∞ ∫ Ω |um|α(x)|vm|β(x)dx+ lim m→∞ ∫ Ω |u|α(x)|v|β(x)dx − lim sup m→∞ ∫ Ω ∣∣|um|α(x)|vm|β(x)dx− |u|α(x)|v|β(x) ∣∣dx = 2 ∫ Ω |u|α(x)|v|β(x)dx− lim sup m→∞ ∫ Ω ∣∣|um|α(x)|vm|β(x) − |u|α(x)|v|β(x) ∣∣dx (5.13) Combining (5.12) and (5.13), we have lim sup m→∞ ∫ Ω ∣∣|um|α(x)|vm|β(x) − |u|α(x)|v|β(x) ∣∣dx ≤ 0, that is, lim m→∞ ∫ Ω ∣∣|um|α(x)|vm|β(x) − |u|α(x)|v|β(x) ∣∣dx = 0. 28 R. BISWAS, S. TIWARI EJDE-2020/98 Thus, combining the above together with (5.11), we obtain our final result. � The proof of the next lemma is similarly to the one for Lemma 5.1, using that α, β ∈ C+(Ω). Lemma 5.2. Let {um}, {vm} be any two bounded sequences in X0 and c, α, β be as in Theorem 1.2. Then lim m→∞ ∫ Ω 1 α(x) + β(x) c(x)|um|α(x)|vm|β(x)dx = ∫ Ω 1 α(x) + β(x) c(x)|u|α(x)|v|β(x)dx. Acknowledgements. R. Biswas was supported by research grant from Indian Institute of Technology Guwahati. S. Tiwari was supported by MATRICS grant MTR/2018/000010 funded by Science and Engineering Research Board (SERB) India. References [1] C. O. Alves, L. P. Barreiro, V. A. Goncalves; Multiplicity of solutions of some quasilinear equations in RN with variable exponents and concave-convex nonlinearities, Topol. Methods Nonlinear Anal., 47 (2016), no. 2, 529–559. [2] C. O. Alves, M. A. S. Souto; Existence of solutions for a class of problems in RN involving p(x)-Laplacian, Progr. Nonlinear Differential Equations Appl., 66 (2005), 17–32. [3] E. Azroul, A. Benkirane, M. Shimi, M. Srati; On a class of fractional p(x)-Kirchhoff type, Appl. Anal., (2019), DOI: 10.1080/00036811.2019.1603372 [4] A. Bahrouni; Comparison and sub-super solution principles for the fractional p(x)-Laplacian, J. Math. Anal. Appl., 458 (2018), no. 2, 1363–1372. [5] A. Bahrouni, V. D. Rǎdulescu; On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), no. 3, 379–389. [6] L. Brasco, E. Lindgren, E. Parini; The fractional Cheeger problem, Interfaces Free Bound., 16 (2014), no. 3, 419–458. [7] H. Brézis, E. Lieb; A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), no. 3, 486–490. [8] K. J. Brown, T. F. Wu; The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), no. 6, 1733–1745. [9] K. J. Brown and Y. Zhang; The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), no. 2, 481-499. [10] W. Chen, S. Deng; The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Anal. Real World Appl., 27 (2016), 80–92. [11] M. Chhetri, P. Girg, E. Hollifield; Existence of positive solutions for fractional Laplacian equations: theory and numerical experiments, Electron. J. Differential Equations, 2020 (2020), no. 81, 1–31. [12] E. Di Nezza, G. Palatucci, E. Valdinoci; Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), no. 5, 521–573. [13] L. Diening, P. Harjulehto, P. Hästö, M. Ružicka; Lebesgue and Sobolev spaces with variable exponents, 2017, Springer-Verlag, Heidelberg, 2011. [14] P. Drábek, S. I. Pohozaev; Positive solutions for the p-Laplacian: Application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), no. 4, 703–726. [15] X. Fan; p(x)-Laplacian equations in RN with periodic data and nonperiodic perturbations, J. Math. Anal. Appl., 341 (2008), no. 1, 103–119. [16] X. Fan, D. Zhao; On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl., 263 (2001), no. 2, 424–446. [17] G. Franzina, G. Palatucci; Fractional p-eigenvalues, Riv. Math. Univ. Parma (N.S.), 5 (2014), no. 2, 373–386. [18] Y. Fu, H. Li, P. Pucci; Existence of Nonnegative Solutions for a Class of Systems Involving Fractional (p, q)-Laplacian Operators, Chin. Ann. Math. Ser. B, 39 (2018), no. 2, 357–372. EJDE-2020/98 FRACTIONAL p(·)-LAPLACIAN SYSTEM 29 [19] J. Giacomoni, S. Tiwari, G. Warnault; Quasilinear parabolic problem with p(x)-Laplacian: ex- istence, uniqueness of weak solutions and stabilization, NoDEA Nonlinear Differential Equa- tions Appl., 23 (2016), no. 3, Art. 24. [20] S. Goyal, K. Sreenadh; Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign changing wight function, Proc. Indian Acad. Sci. Math. Sci., 125 (2015), no. 4, 545-558. [21] K. Ho, Y. H. Kim; A-priori bounds and multiplicity of solutions for nonlinear elliptic prob- lems involving the fractional p(·)-Laplacian, Nonlinear Anal., 188 (2019), 179–201. [22] U. Kaufmann, J. D. Rossi, R. Vidal; Fractional Sobolev spaces with variable exponents and fractional p(x)- Laplacians, Electron. J. Qual. Theory Differ. Equ., 76 (2017), 1–10. [23] G. Molica Bisci, V. D. Rădulescu; Multiplicity results for elliptic fractional equations with subcritical term, NoDEA Nonlinear Differential Equations Appl., 22 (2015), no. 4, 721–739. [24] G. Molica Bisci, V. Radulescu, R. Servadei; Variational methods for nonlocal fractional prob- lems, Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. [25] S. Mosconi, M. Squassina; Nonlocal problems at nearly critical growth, Nonlinear Anal., 136 (2016), 84–101. [26] S. Mosconi, M. Squassina; Recent progresses in the theory of nonlinear nonlocal problems, Bruno Pini Math. Anal. Semin., 7 (2017), no. 1, 147–164. [27] R. Servadei and E. Valdinoci; Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), no. 5, 2105–2137. [28] K. Sreenadh, S. Tiwari; Global multiplicity results for p(x)-Laplacian equations with non- linear Neumann boundary condition, Differential Integral Equations, 26 (2013), no. 7/8, 815–836. Reshmi Biswas Department of Mathematics, IIT Guwahati, Assam 781039, India Email address: b.reshmi@iitg.ac.in Sweta Tiwari Department of Mathematics, IIT Guwahati, Assam 781039, India Email address: swetatiwari@iitg.ac.in 1. Introduction 2. Preliminary results 2.1. Fractional Sobolev spaces with variable exponents 3. Nehari manifold and fibering map analysis 4. Existence of multiple solutions 5. Appendix Acknowledgements References