Multiple Solutions for a Class of Dirichlet Double Eigenvalue Quasilinear Elliptic Systems Involving the (𝑝₁,…,𝑝_𝑛)-Laplacian Operator

Abstract

Existence results of three weak solutions for a Dirichlet double eigenvalue quasilinear elliptic system involving the (𝑝₁,…,𝑝_𝑛)-Laplacian operator, under suitable assumptions, are established. Our main tool is based on a recent three-critical-point theorem obtained by Ricceri. We also give some examples to illustrate the obtained results.

1. Introduction and Preliminaries

The aim of this paper is to investigate the existence of at least three weak solutions for the following Dirichlet double eigenvalue quasilinear elliptic system: −Δ_𝑝𝑖𝑢_𝑖=𝜆𝐹_𝑢𝑖𝑥,𝑢₁,…,𝑢_𝑛+𝜇𝐺_𝑢𝑖𝑥,𝑢₁,…,𝑢_𝑛in𝑢Ω,_𝑖=0on𝜕Ω, (1) for 1≤𝑖≤𝑛, where Δ_𝑝𝑖𝑢_𝑖∶=div(|∇𝑢_𝑖|^𝑝𝑖−2∇𝑢_𝑖)is the 𝑝_𝑖-Laplacian operator, Ω⊂ℝ^𝑁(𝑁≥1) is a nonempty bounded open set with a boundary 𝜕Ω of class 𝐶¹, 𝜆 and 𝜇 are positive parameters, and 𝑝_𝑖>𝑁 for 1≤𝑖≤𝑛. Here, 𝐹,𝐺∶Ω×ℝ^𝑛→ℝ are measurable functions with respect to 𝑥∈Ω for every (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛 and are 𝐶¹ with respect to (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛 for a.e. 𝑥∈Ω, and 𝐹_𝑢𝑖 and 𝐺_𝑢𝑖 denote the partial derivatives of 𝐹 and 𝐺 with respect to 𝑢_𝑖, respectively.

Moreover, 𝐹 and 𝐺 satisfy the following additional assumptions:(F₁) for every 𝑀>0 and every 1≤𝑖≤𝑛, sup_{|(𝑡1,…,𝑡𝑛)|≤𝑀}𝐹_𝑢𝑖𝑥,𝑡₁,…,𝑡_𝑛∈𝐿¹(Ω); (2)(F₂)𝐹(𝑥,0,…,0)=0 for a.e. 𝑥∈Ω; (G) for every 𝑀>0 and every 1≤𝑖≤𝑛, sup_{|(𝑡1,…,𝑡𝑛)|≤𝑀}𝐺_𝑢𝑖𝑥,𝑡₁,…,𝑡_𝑛∈𝐿¹(Ω). (3)

Here and in the following, we let 𝑋 be the Cartesian product of the 𝑛 Sobolev spaces 𝑊^1,𝑝𝑖₀(Ω) for 1≤𝑖≤𝑛; that is, 𝑋=𝑊^1,𝑝1₀(Ω)×𝑊^1,𝑝2₀(Ω)×⋯×𝑊^1,𝑝𝑛₀(Ω) equipped with the norm ‖𝑢‖∶=^𝑛_𝑖=1𝑢_𝑖𝑝𝑖,𝑢=𝑢₁,𝑢₂,…,𝑢_𝑛, (4) where for 1≤𝑖≤𝑛, 𝑢_𝑖𝑝𝑖∶=_Ω∇𝑢_𝑖(𝑥)^𝑝𝑖𝑑𝑥^1/𝑝𝑖. (5) Put 𝑐∶=maxsup_{𝑢𝑖∈𝑊𝑖01,𝑝(Ω)⧵{0}}max_𝑥∈Ω𝑢_𝑖(𝑥)^𝑝𝑖𝑢_𝑖^𝑝𝑖_𝑝𝑖∶for1≤𝑖≤𝑛. (6) Since 𝑝_𝑖>𝑁 for 1≤𝑖≤𝑛, 𝑋 is compactly embedded in (𝐶⁰(Ω))^𝑛, so that 𝑐<+∞. In addition, it is known (see [1, formula (6b)]) that sup_{𝑢𝑖∈𝑊𝑖01,𝑝(Ω)⧵{0}}max_𝑥∈Ω𝑢_𝑖(𝑥)𝑢_𝑖𝑝𝑖≤𝑁^{−1/𝑝𝑖}√𝜋Γ𝑁1+2^1/𝑁𝑝_𝑖−1𝑝_𝑖−𝑁^{1−1/𝑝𝑖}[𝑚(Ω)]^{1/𝑁−1/𝑝𝑖} (7) for 1≤𝑖≤𝑛, where Γ denotes the Gamma function defined by Γ(𝑡)∶=₀^+∞𝑧^𝑡−1𝑒^−𝑧𝑑𝑧,∀𝑡>0, (8) and 𝑚(Ω) is the Lebesgue measure of the set Ω, and equality occurs when Ω is a ball.

Moreover, let 𝐷∶=sup_𝑥∈Ωdist(𝑥,𝜕Ω). (9) Simple calculations show that there is 𝑥⁰∈Ω such that 𝐵(𝑥⁰,𝐷)⊆Ω, where 𝐵(𝑥,𝑟) stands for the open ball in ℝ^𝑁 of radius 𝑟 centered at 𝑥.

Put 𝜅_𝑖2∶=𝐷𝑐𝜋^𝑁/2𝐷^𝑁−(𝐷/2)^𝑁Γ(1+𝑁/2)^1/𝑝𝑖 (10) for 1≤𝑖≤𝑛.

By a (weak) solution of system (1), we mean any 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋 such that _Ω^𝑛_𝑖=1∇𝑢_𝑖(𝑥)^𝑝𝑖−2∇𝑢_𝑖(𝑥)∇𝑣_𝑖(𝑥)𝑑𝑥−𝜆_Ω^𝑛_𝑖=1𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥−𝜇_Ω^𝑛_𝑖=1𝐺_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥=0 (11) for all 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑋.

In the literature many papers (see, e.g., the papers [2–9] and references therein) deal with nonlinear elliptic problems. Motivated by the fact that such kind of problems is used to describe a large class of physical phenomena, many authors have studied the existence and multiplicity of solutions for (1).

The goal of this work is to establish some new criteria for system (1) to have at least three weak solutions in 𝑋, by means of a very recent abstract critical point result of Ricceri [10]. We first recall the following three-critical-point theorem that follows from a combination of [11, Theorem  3.6] and [10, Theorem  1].

Lemma 1. Let 𝑋 be a reflexive real Banach space; let Φ∶𝑋→ℝ be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗, bounded on bounded subsets of 𝑋; Ψ∶𝑋→ℝ a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Φ(0)=Ψ(0)=0. (12) Assume that there exists 𝑟>0 and 𝑥∈𝑋, with 𝑟<Φ(𝑥), such that (a₁)(sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)/𝑟)<Ψ(𝑥)/Φ(𝑥); (a₂)for each𝜆∈Λ_𝑟∶=]Φ(𝑥)/Ψ(𝑥),𝑟/sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)[, the functional Φ−𝜆Ψ is coercive.
Then, for each compact interval [𝑎,𝑏]⊆Λ_𝑟, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏] and every 𝐶¹ functional Γ∶𝑋→ℝ with compact derivative, there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the equation Φ^′(𝑥)−𝜆Ψ^′(𝑥)−𝜇Γ^′(𝑥)=0 (13) has at least three solutions in 𝑋 whose norms are less than 𝜌.

For other basic notations and definitions, we refer the reader to [12–14].

2. Main Result

In the present section we discuss the existence of multiple solutions for system (1). For any 𝛾>0, we denote by 𝐾(𝛾) the set 𝑡₁,…,𝑡_𝑛∈ℝ^𝑛∶^𝑛_𝑖=1𝑡_𝑖^𝑝𝑖𝑝_𝑖≤𝛾. (14) This set will be used in some of our hypotheses with appropriate choices of 𝛾.

We formulate our main result as follows.

Theorem 2. Assume that there exist two positive constants 𝜃 and 𝛿 with ∑^𝑛_𝑖=1((𝛿𝜅_𝑖)^𝑝𝑖/𝑝_𝑖)>(𝜃/∏^𝑛_𝑖=1𝑝_𝑖) such that (b₁)𝐹(𝑥,𝑡₁,…,𝑡_𝑛)≥0 for a.e. 𝑥∈Ω⧵𝐵(𝑥⁰,𝐷/2) and all 𝑡_𝑖∈[0,𝛿] for 1≤𝑖≤𝑛; (b₂)𝜃∏^𝑛_𝑖=1𝑝_{𝑖𝐵(𝑥}⁰_,𝐷/2)𝐹()−𝑥,𝛿,…,𝛿𝑑𝑥_𝑛𝑖=1𝛿𝜅_𝑖^𝑝𝑖𝑝_𝑖Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥>0; (15)(b₃)limsup_{(|𝑡1|,…,|𝑡𝑛|)→(+∞,…,+∞)}𝐹𝑥,𝑡₁,…,𝑡_𝑛∑^𝑛_𝑖=1𝑡_𝑖^𝑝𝑖/𝑝_𝑖<∏^𝑛_𝑖=1𝑝_𝑖𝑚(Ω)𝜃_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥, (16)uniformly with respect to 𝑥∈Ω.
Then, setting∑Λ∶=^𝑛𝑖=1𝛿𝜅_𝑖^𝑝𝑖/𝑝_𝑖𝑐∫_{𝐵(𝑥0,𝐷/2)}𝐹(),𝜃𝑥,𝛿,…,𝛿𝑑𝑥𝑐∏^𝑛_𝑖=1𝑝_𝑖∫_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏}^𝑛_{𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛,𝑑𝑥 (17) for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], system (1) admits at least three weak solutions in 𝑋 whose norms are less than 𝜌.

Proof. Our aim is to apply Lemma 1 to our problem. To this end, for each 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋, we let the functionals Φ,Ψ∶𝑋→ℝ be defined by Φ(𝑢)∶=^𝑛_𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖,Ψ(𝑢)∶=_Ω𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥. (18) Clearly, Φ is bounded on each bounded subset of 𝑋 and it is known that Φ and Ψ are well-defined and continuously Gâteaux differentiable functionals whose derivatives at the point 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋 are the functionals Φ^′(𝑢) and Ψ^′(𝑢) given by Φ^′(𝑢)(𝑣)=_Ω^𝑛_𝑖=1∇𝑢_𝑖(𝑥)^𝑝𝑖−2∇𝑢_𝑖(𝑥)∇𝑣_𝑖(𝑥)Ψ𝑑𝑥,_′(𝑢)(𝑣)=_Ω^𝑛_𝑖=1𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥 (19) for every 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑋, and Φ is sequentially weakly lower semicontinuous (see Proposition  25.20 of [14]). Also, Φ^′∶𝑋→𝑋^∗ is a uniformly monotone operator in 𝑋 (for more details, see (2.2) of [15]), and since Φ^′ is coercive and hemicontinuous in 𝑋, by applying Minty-Browder theorem (Theorem  26.A of [14]), Φ^′ admits a continuous inverse on 𝑋^∗.
We claim that Ψ^′∶𝑋→𝑋^∗ is a compact operator. To this end, it is enough to show that Ψ^′ is strongly continuous on 𝑋. For this, for fixed (𝑢₁,…,𝑢_𝑛)∈𝑋, let(𝑢_1𝑚,…,𝑢_𝑛𝑚)→(𝑢₁,…,𝑢_𝑛) weakly in 𝑋 as 𝑚→+∞. Then we have that (𝑢_1𝑚,…,𝑢_𝑛𝑚) converges uniformly to (𝑢₁,…,𝑢_𝑛) on Ω as 𝑚→+∞ (see [14]). Since 𝐹(𝑥,⋅,…,⋅) is 𝐶¹ in ℝ^𝑛 for every 𝑥∈Ω, the derivatives of 𝐹 are continuous in ℝ^𝑛 for every 𝑥∈Ω, so fo 1≤𝑖≤𝑛, 𝐹_𝑢𝑖(𝑥,𝑢_1𝑚,…,𝑢_𝑛𝑚)→𝐹_𝑢𝑖(𝑥,𝑢₁,…,𝑢_𝑛) strongly as 𝑚→+∞. By the Lebesgue control convergence theorem, Ψ^′(𝑢_1𝑚,…,𝑢_𝑛𝑚)→Ψ^′(𝑢₁,…,𝑢_𝑛) strongly as 𝑚→+∞. Thus we proved that Ψ^′ is strongly continuous on 𝑋, which implies that Ψ^′ is a compact operator by [14, Proposition  26.2]. Hence the claim is true.
Moreover, we have Φ(0)=Ψ(0)=0. (20) Next, put 𝑤(𝑥)=(𝑤₁(𝑥),…,𝑤_𝑛(𝑥)) such that for 1≤𝑖≤𝑛, 𝑤_𝑖(𝑥)𝑥∶=0𝑥∈Ω⧵𝐵⁰,𝐷,2𝛿𝐷𝐷−^𝑁_𝑗=1𝑥_𝑗−𝑥⁰_𝑗²𝑥∈𝐵𝑥⁰,𝐷⧵𝐵𝑥⁰,𝑥,𝐷/2𝛿𝑥∈𝐵⁰,𝐷/2, (21) and 𝑟∶=(𝜃/𝑐∏^𝑛_𝑖=1𝑝_𝑖). Clearly 𝑤=(𝑤₁,…,𝑤_𝑛)∈𝑋 and, in particular, one has for 1≤𝑖≤𝑛,𝑤_𝑖^𝑝𝑖_𝑝𝑖=𝜋^𝑁/2Γ()𝐷1+𝑁/2^𝑁−(𝐷/2)^𝑁2𝛿𝐷^𝑝𝑖. (22) So Φ(𝑤)=𝜋^𝑁/2Γ()𝐷1+𝑁/2^𝑁−(𝐷/2)^𝑁𝑛_𝑖=11𝑝_𝑖2𝛿𝐷^𝑝𝑖. (23) Bearing in mind that ∑^𝑛_𝑖=1((𝛿𝜅_𝑖)^𝑝𝑖/𝑝_𝑖)>(𝜃/∏^𝑛_𝑖=1𝑝_𝑖) and that ‖𝑤_𝑖‖^𝑝𝑖_𝑝𝑖=((𝛿𝜅_𝑖)^𝑝𝑖/𝑐) for 1≤𝑖≤𝑛, one has Φ(𝑤)>𝑟.
Since 0≤𝑤_𝑖(𝑥)≤𝛿 for each 𝑥∈Ω for 1≤𝑖≤𝑛, condition (b₁) ensures that _{Ω⧵𝐵(𝑥0,𝐷)}𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)+𝑑𝑥_{𝐵(𝑥0,𝐷)⧵𝐵(𝑥0,𝐷/2)}𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥≥0. (24) Hence _Ω𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)≥𝑑𝑥_{𝐵(𝑥0,𝐷/2)}𝐹(𝑥,𝛿,…,𝛿)𝑑𝑥. (25) Moreover, owing to assumption (b₂), we have _Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛<𝜃𝑑𝑥∑^𝑛_𝑖=1𝛿𝜅_𝑖^𝑝𝑖/𝑝_𝑖∏^𝑛_𝑖=1𝑝_𝑖^{𝐵(𝑥0,𝐷/2)}𝐹()≤𝜃𝑥,𝛿,…,𝛿𝑑𝑥𝑐∫_Ω𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛_𝑖=1∏^𝑛_{𝑗=1𝑗≠𝑖}𝑝_𝑗𝑤_𝑖^𝑝𝑖_𝑝𝑖. (26) Taking into account that for each 𝑢_𝑖∈𝑊^1,𝑝𝑖₀(Ω)sup_𝑥∈Ω𝑢_𝑖(𝑥)^𝑝𝑖≤𝑐𝑢_𝑖^𝑝𝑖_𝑝𝑖 (27) for 1≤𝑖≤𝑛 (see (6)), we have that sup^𝑛_{𝑥∈Ω𝑖=1}𝑢_𝑖(𝑥)^𝑝𝑖𝑝_𝑖≤𝑐^𝑛_𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖=𝑐Φ(𝑢) (28) for every 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋, and taking into account (26) and (28), it follows that sup_{𝑢∈Φ−1(]−∞,𝑟])}Ψ(𝑢)=sup_{Φ(𝑢)≤𝑟Ω}𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)≤𝑑𝑥_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛<𝜃𝑑𝑥𝑐∫_Ω𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛_𝑖=1∏^𝑛_{𝑗=1𝑗≠𝑖}𝑝_𝑗𝑤_𝑖^𝑝𝑖_𝑝𝑖=𝜃𝑐∏^𝑛_𝑖=1𝑝_𝑖∫_Ω𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛_𝑖=1𝑤_𝑖^𝑝𝑖_𝑝𝑖/𝑝_𝑖Ψ(𝑤)=𝑟Φ(𝑤). (29) Therefore, assumption (a₁) of Lemma 1 is satisfied.
Now, fixed 𝜆∈Λ, due to (b₃), there exist two constants 𝛾,𝜗∈ℝ with ∏0<𝛾<^𝑛𝑖=1𝑝_𝑖𝑚(Ω)𝜃_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥 (30) such that 𝐹𝑥,𝑡₁,…,𝑡_𝑛≤𝛾^𝑛_𝑖=1𝑡_𝑖^𝑝𝑖𝑝_𝑖+𝜗 (31) for all 𝑥∈Ω and for all (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛. Fix 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋. Then 𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)≤𝛾^𝑛_𝑖=1𝑢_𝑖(𝑥)^𝑝𝑖𝑝_𝑖+𝜗 (32) for all 𝑥∈Ω. So, for any fixed 𝜆∈Λ, from (28) and (32), we have Φ(𝑢)(𝑢)=−𝜆Ψ_𝑛𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆_Ω𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥≥)𝑑𝑥_𝑛𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝛾_Ω^𝑛_𝑖=1𝑢_𝑖(𝑥)^𝑝𝑖𝑝_𝑖(Ω)≥𝑑𝑥−𝜆𝜗𝑚_𝑛𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝛾𝑐𝑚(Ω)^𝑛_𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖(Ω)≥𝑚(Ω)−𝜆𝜗𝑚1−𝛾𝜃∏^𝑛_𝑖=1𝑝_𝑖∫_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏}^𝑛_{𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛×𝑑𝑥_𝑛𝑖=1𝑢_𝑖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝑚(Ω)𝜗𝜃𝑐∏^𝑛_𝑖=1𝑝_𝑖∫_Ωsup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏}^𝑛_{𝑖=1𝑝𝑖)}𝐹𝑥,𝑡₁,…,𝑡_𝑛,𝑑𝑥 (33) and thus lim_{‖𝑢‖→+∞}(Φ(𝑢)−𝜆Ψ(𝑢))=+∞, (34) which means that the functional Φ−𝜆Ψ is coercive. Then, also condition (a₂) of Lemma 1 holds.
In addition, since 𝐺∶Ω×ℝ^𝑛→ℝ is a measurable function with respect to 𝑥∈Ω for every (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛 and is 𝐶¹ with respect to (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛 for a.e. 𝑥∈Ω, satisfying condition (G), the functional Γ(𝑢)=_Ω𝐺𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥 (35) is well defined and continuously Gâteaux differentiable on 𝑋 with a compact derivative, and Γ^′(𝑢)(𝑣)=_Ω^𝑛_𝑖=1𝐺_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥 (36) for all 𝑢=(𝑢₁,…,𝑢_𝑛), 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑋. Thus, all the hypotheses of Lemma 1 are satisfied. Also note that the solutions of the equation Φ^′(𝑢)−𝜆Ψ^′(𝑢)−𝜇Γ^′(𝑢)=0 (37) are exactly the weak solutions of (1). So, the conclusion follows from Lemma 1.

We now point out the following special case of Theorem 2 when 𝐹 does not depend on 𝑥∈Ω.

Theorem 3. Let 𝐹∶ℝ^𝑛→ℝ be a 𝐶¹-function and assume that there exist two positive constants 𝜃 and 𝛿 with ∑^𝑛_𝑖=1((𝛿𝜅_𝑖)^𝑝𝑖/𝑝_𝑖)>(𝜃/∏^𝑛_𝑖=1𝑝_𝑖) such that (b₄)𝐹(𝑡₁,…,𝑡_𝑛)≥0 for all 𝑡_𝑖∈[0,𝛿] for 1≤𝑖≤𝑛; (b₅)𝜃𝜋^𝑁/2Γ(1+(𝑁/2))∏^𝑛_𝑖=1𝑝_𝑖𝐷2^𝑁𝐹()(Ω)𝛿,…,𝛿−𝑚^𝑛_𝑖=1𝛿𝜅_𝑖^𝑝𝑖𝑝_𝑖sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏𝑛𝑖=1𝑝𝑖)}𝐹𝑡₁,…,𝑡_𝑛>0; (38)(b₆)limsup_{(|𝑡1|,…,|𝑡𝑛|)→(+∞,…,+∞)}𝐹(𝑡₁,…,𝑡_𝑛)/∑^𝑛_𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖)≤0. Then, settingΓ()∑Λ∶=1+𝑁/2^𝑛_𝑖=1𝛿𝜅_𝑖^𝑝𝑖/𝑝_𝑖𝑐𝜋^𝑁/2𝐹()2𝛿,…,𝛿𝐷^𝑁,𝜃𝑐∏^𝑛_𝑖=1𝑝_𝑖𝑚(Ω)sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝜃/∏}^𝑛_{𝑖=1𝑝𝑖)}𝐹𝑡₁,…,𝑡_𝑛, (39)for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the system −Δ_𝑝𝑖𝑢_𝑖=𝜆𝐹_𝑢𝑖𝑢₁,…,𝑢_𝑛+𝜇𝐺_𝑢𝑖𝑥,𝑢₁,…,𝑢_𝑛𝑢𝑖𝑛Ω,_𝑖=0𝑜𝑛𝜕Ω, (40) for 1≤𝑖≤𝑛, admits at least three weak solutions in 𝑋 whose norms are less than 𝜌.

Proof. Set 𝐹(𝑥,𝑡₁,…,𝑡_𝑛)=𝐹(𝑡₁,…,𝑡_𝑛) for all 𝑥∈Ω and 𝑡_𝑖∈ℝ for 1≤𝑖≤𝑛. Since ∫_{𝐵(𝑥0,𝐷/2)}𝐹(𝛿,…,𝛿)𝑑𝑥=(𝜋^𝑁/2/Γ(1+𝑁/2))(𝐷/2)^𝑁𝐹(𝛿,…,𝛿), Theorem 2 ensures the conclusion.

Let 𝜅=𝜅₁ and 𝑝=𝑝₁. Then we have the following existence result.

Corollary 4. Let 𝑓∶ℝ→ℝ be a continuous function and let 𝑔∶Ω×ℝ→ℝ be an 𝐿¹-Carathéodory function. Put 𝐹(𝑡)=∫^𝑡₀𝑓(𝜉)𝑑𝜉 for each 𝑡∈ℝ and assume that there exist two positive constants 𝜃 and 𝛿 with (𝛿𝜅)^𝑝>𝜃 such that (b₇)𝐹(𝑡)≥0 for all 𝑡∈[0,𝛿];(b₈)(𝜃𝜋^𝑁/2/Γ(1+𝑁/2))(𝐷/2)^𝑁𝐹(𝛿)−𝑚(Ω)(𝛿𝜅)^𝑝sup_𝑡∈[−^𝑝_√𝜃,^𝑝_√𝜃]𝐹(𝑡)>0;(b₉)limsup_{|𝑡|→+∞}𝐹(𝑡)/|𝑡|^𝑝≤0. Then, setting ΛΓ()∶=1+𝑁/2)(𝛿𝜅^𝑝𝑝𝑐𝜋^𝑁/2𝐹(𝛿)2𝐷^𝑁,𝜃𝑚(Ω)𝑝𝑐sup_𝑡∈[−^𝑝_√𝜃,^𝑝_√𝜃]𝐹(𝑡), (41) for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the problem −Δ_𝑝𝑢=𝜆𝑓(𝑢)+𝜇𝑔(𝑥,𝑢)𝑖𝑛Ω,𝑢=0𝑜𝑛𝜕Ω (42) admits at least three weak solutions in 𝑊₀^1,𝑝(Ω) whose norms are less than 𝜌.

Now, we want to point out a simple consequence of Corollary 4 in the case when 𝑁=1 and 𝑝=2. For simplicity, we fix Ω=(𝛼,𝛽) and note that in this situation, 𝑐=(𝛽−𝛼)/4 and 𝜅=((𝛽−𝛼)/𝐷)^1/2.

Corollary 5. Let 𝑓∶ℝ→ℝ be a continuous function and 𝑔∶(𝛼,𝛽)×ℝ→ℝ an 𝐿¹-Carathéodory function. Put 𝐹(𝑡)=∫^𝑡₀𝑓(𝜉)𝑑𝜉 for each 𝑡∈ℝ and assume that there exist two positive constants 𝜃 and 𝛿 with (𝛿²(𝛽−𝛼)/𝐷)>𝜃 such that assumption (b₇) in Corollary 4 holds, and (b₁₀)(𝜃𝐷)𝐹(𝛿)−(𝛿²(𝛽−𝛼)²/𝐷)sup_{√𝑡∈[−√𝜃,𝜃]}𝐹(𝑡)>0;(b₁₁)limsup_{|𝑡|→+∞}(𝐹(𝑡)/𝑡²)≤0. Then, setting Λ∶=2𝛿²𝐷²𝐹(𝛿),2𝜃𝛽−𝛼²sup_{√𝑡∈[−√𝜃,𝜃]}𝐹(𝑡), (43) for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], the problem −𝑢^′′(𝑢)(),𝑢(𝛼)𝛽=𝜆𝑓+𝜇𝑔𝑥,𝑢𝑖𝑛𝛼,𝛽=𝑢=0 (44) admits at least three classical solutions in 𝐶²([𝛼,𝛽]) whose norms are less than 𝜌.

3. Examples

First, we present an example of the application of Theorem 3.

Example 1. Let Ω={(𝑥,𝑦)∈ℝ²∶𝑥²+𝑦²<4}. Consider the system −Δ₃𝑢₁=𝜆𝑒^1−𝑢1𝑢₁¹¹12−𝑢₁+𝜇𝐺_𝑢1𝑥,𝑦,𝑢₁,𝑢₂inΩ,−Δ₃𝑢₂=𝜆𝑒^1−𝑢2𝑢₂¹³14−𝑢₂+𝜇𝐺_𝑢2𝑥,𝑦,𝑢₁,𝑢₂in𝑢Ω,₁=𝑢₂=0on𝜕Ω, (45) where 𝐺∶Ω×ℝ²→ℝ is an arbitrary function which is measurable with respect to (𝑥,𝑦)∈Ω for every (𝑡₁,𝑡₂)∈ℝ² and is 𝐶¹ with respect to (𝑡₁,𝑡₂)∈ℝ² for a.e. (𝑥,𝑦)∈Ω, satisfying sup_{|(𝑡1,𝑡2)|≤𝑀}𝐺_𝑢𝑖𝑥,𝑦,𝑡₁,𝑡₂∈𝐿¹(Ω) (46) for every 𝑀>0 and 𝑖=1,2. Taking into account 𝑐=4/𝜋, picking 𝑥⁰=(0,0) and 𝐹𝑡₁,𝑡₂=𝑒^1−𝑡1𝑡₁¹²+𝑒^1−𝑡2𝑡₂¹⁴ (47) for each (𝑡₁,𝑡₂)∈ℝ², so that 𝐷=2 and 𝜅₁=𝜅₂=√³12, by choosing 𝛿=𝜃=3, we have 𝐹(𝑡₁,𝑡₂)≥0 for all (𝑡₁,𝑡₂)∈[0,3]×[0,3] and limsup_{(|𝑡1|,|𝑡2|)→(+∞,+∞)}𝐹𝑡₁,𝑡₂(1/3)𝑡₁³+(1/3)𝑡₂³=0. (48) We see that sup_{|𝑡1|3+|𝑡2|3≤1}𝑒^1−𝑡1𝑡₁¹²+𝑒^1−𝑡2𝑡₂¹⁴≤sup_|𝑡1|≤1𝑒^1−𝑡1𝑡₁¹²+sup_|𝑡2|≤1𝑒^1−𝑡2𝑡₂¹⁴=2𝑒², (49) which gives that 𝜃𝜋𝑝₁𝑝₂𝐷2²𝐹()(Ω)𝛿,𝛿−𝑚²_𝑖=1𝛿𝜅_𝑖^𝑝𝑖𝑝_𝑖sup_{(𝑡1,𝑡2)∈𝐾(𝜃/𝑝1𝑝2)}𝐹𝑡₁,𝑡₂≥𝜋3𝐹(3,3)−(4𝜋)2³⋅3³×sup_|𝑡1|≤1𝑒^1−𝑡1𝑡₁¹²+sup_|𝑡2|≤1𝑒^1−𝑡2𝑡₂¹⁴=3¹¹⋅10𝜋𝑒²−2⁶⋅3³⋅𝜋𝑒²>0. (50) Hence, all the assumptions of Theorem 3, with 𝑝₁=𝑝₂=3, are satisfied. So, setting 𝑒Λ=²3⁹,1⋅52⁵⋅3𝑒², (51) for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], system (45) has at least three weak solutions in 𝑊₀^1,3(Ω)×𝑊₀^1,3(Ω) whose norms are less than 𝜌.

The next example follows directly by Corollary 5.

Example 2. Suppose 𝑓(𝑡)=(1/10)𝑡⁹𝑒^−𝑡(10−𝑡) for all 𝑡∈ℝ. Then one has 𝐹(𝑡)=(1/10)𝑡¹⁰𝑒^−𝑡 for all 𝑡∈ℝ. Now, consider the following two-point boundary value problem −𝑢^′′=𝜆𝑓(𝑢)+𝜇𝑔(𝑥,𝑢)in(),𝑢()(1)−1,1−1=𝑢=0, (52) where 𝑔∶(−1,1)×ℝ→ℝ is an arbitrary 𝐿¹-Carathéodory function. Note that, in this case, 𝑥⁰=0, 𝐷=1, 𝑐=1/2, and 𝜅=√2. In fact, by choosing 𝜃=1 and 𝛿=2, we have 𝐹(𝑡)≥0 for all 𝑡∈[0,2]. Also ()𝐹(𝛿)−𝛿𝜃𝐷²𝛽−𝛼²𝐷sup_{√𝑡∈[−√𝜃,𝜃]}𝐹(𝑡)(2)()=1=𝐹−16𝐹−152⁹𝑒⁻²−8𝑒>0, (53) which shows that assumption (b₁₀) is fulfilled. Furthermore, we have limsup_{|𝑡|→+∞}(𝐹(𝑡)/𝑡²)=0. Thus, all hypotheses of Corollary 5 are satisfied. So, setting Λ=](5𝑒²/2⁷),(5/𝑒)[, for each compact interval [𝑎,𝑏]⊆Λ, there exists 𝜌>0 with the following property: for every 𝜆∈[𝑎,𝑏], there exists 𝛿>0 such that, for each 𝜇∈[0,𝛿], problem (52) has at least three classical solutions in 𝐶²([−1,1]) whose norms are less than 𝜌.
In particular, there exist two positive constants 𝜌 and 𝛿 such that, for each 𝜇∈[0,𝛿], the problem −𝑢^′′=𝑓(𝑢)+𝜇𝑔(𝑥,𝑢)in(),𝑢()(1)−1,1−1=𝑢=0 (54) admits at least three classical solutions in 𝐶²([−1,1]) whose norms are less than 𝜌.