Multiple Solutions for a Class of Multipoint Boundary Value Systems Driven by a One-Dimensional (𝑝₁,…,𝑝_𝑛)-Laplacian Operator

Abstract

Employing a recent three critical points theorem due to Bonanno and Marano (2010), the existence of at least three solutions for the following multipoint boundary value system −(|𝑢^𝑖|^𝑝𝑖−2𝑢^𝑖)′=𝜆𝐹_𝑢𝑖(𝑥,𝑢₁,…,𝑢_𝑛) in (0,1), 𝑢_𝑖(0)=∑^𝑚𝑗=1𝑎_𝑗𝑢_𝑖(𝑥_𝑗), 𝑢_𝑖(1)=∑^𝑚𝑗=1𝑏_𝑗𝑢_𝑖(𝑥_𝑗) for 1≤𝑖≤𝑛, is established.

1. Introduction

In this work, we consider the following multipoint boundary value system −𝑢||𝑢^𝑖||^𝑝𝑖−2𝑢^𝑖^=𝜆𝐹_𝑢𝑖𝑥,𝑢₁,…,𝑢_𝑛in(0,1),_𝑖(0)=^𝑚_𝑗=1𝑎_𝑗𝑢_𝑖𝑥_𝑗,𝑢_𝑖(1)=^𝑚_𝑗=1𝑏_𝑗𝑢_𝑖𝑥_𝑗,(1.1) for 1≤𝑖≤𝑛, where 𝑝_𝑖>1 for 1≤𝑖≤𝑛, 𝜆>0, 𝑚,𝑛≥1, 𝐹∶[0,1]×ℝ^𝑛→ℝ is a function such that 𝐹(⋅,𝑡₁,…,𝑡_𝑛) is continuous in [0,1] for all (𝑡₁,…,𝑡_𝑛)∈ℝ^𝑛, 𝐹(𝑥,⋅,…,⋅) is 𝐶¹ in ℝ^𝑛 for every 𝑥∈[0,1] and 𝐹(𝑥,0,…,0)=0 for all 𝑥∈[0,1], 𝑎_𝑗,𝑏_𝑗∈ℝ for 𝑗=1,…,𝑚 and 0<𝑥₁<𝑥₂<𝑥₃<⋯<𝑥_𝑚<1, and 𝐹_𝑢𝑖 denotes the partial derivative of 𝐹 with respect to 𝑢_𝑖 for 1≤𝑖≤𝑛.

The study of multiplicity of solutions is an important mathematical subject which is also interesting from the practical point of view because the physical processes described by boundary value problems for differential equations exhibit, generally, more than one solution. In [1–3], Ricceri proposed and developed an innovative minimal method for the study of nonlinear eigenvalue problems. Following that, Bonanno [4] gave an application of the method to the two-point problem 𝑢^+𝑢𝜆𝑓(0()𝑢=)=𝑢(01)i=n0(.0,1),(1.2) Bonanno also gave more precise versions of the three critical points of Ricceri in [5, 6]. In particular, in [5], an upper bound of the interval of parameters 𝜆 for which the functional has three critical points is established. Candito [7] extended the main result of [4] to the nonautonomous case 𝑢^+𝜆𝑢𝑓((𝑎𝑥),=𝑢)𝑢=(𝑏0)=in0.(𝑎,𝑏),(1.3) In [8], He and Ge extended the main results of [4, 7] to the quasilinear differential equation 𝜑^𝑝𝑢^′^𝑢+(𝑎𝜆)𝑓=(𝑢𝑥,(𝑏𝑢))==00.in(𝑎,𝑏),(1.4) In [9], the authors extended the main results of [4, 7, 9] to the quasilinear differential equation with Sturm-Liouville boundary conditions |||𝑢^′|||^𝑝−2𝑢^^𝛼+𝜆𝑓(𝑥,𝑢)=0in(𝑎,𝑏),₁𝑢(𝑎)−𝛼₂𝑢^′(𝑎)=0,𝛽₁𝑢(𝑏)−𝛽₂𝑢^(𝑏)=0,(1.5) where 𝑝>1 is a constant, 𝜆 is a positive parameter, 𝑎,𝑏∈ℝ;𝑎<𝑏. In particular, in [10], the authors motivated by these works, established some criteria for the existence of three classical solutions of the system (1.1), while in [11], based on Ricceri’s three critical points theorem [3], the existence of at least three classical solutions to doubly eigenvalue multipoint boundary value systems was established.

In the present paper, based on a three critical points theorem due to Bonanno and Marano [12], we ensure the existence of least three classical solutions for the system (1.1).

Several results are known concerning the existence of multiple solutions for multipoint boundary value problems, and we refer the reader to the papers [13–16] and the references cited therein.

Here and in the sequel, 𝑋 will denote the Cartesian product of 𝑛 space 𝑋_𝑖=𝜉∈𝑊^1,𝑝𝑖([0,1]);𝜉(0)=^𝑚_𝑗=1𝑎_𝑗𝜉𝑥_𝑗,𝜉(1)=^𝑚_𝑗=1𝑏_𝑗𝜉𝑥_𝑗,(1.6) for 𝑖=1,…,𝑛, that is, 𝑋=𝑋₁×⋯×𝑋_𝑛 equipped with the norm ‖‖𝑢₁,…,𝑢_𝑛‖‖=^𝑛_𝑖=1‖‖𝑢^𝑖‖‖_𝑝𝑖,(1.7) where ‖‖𝑢^𝑖‖‖_𝑝𝑖=¹⁰||𝑢^𝑖(𝑥)||^𝑝𝑖𝑑𝑥^1/𝑝𝑖(1.8) for 1≤𝑖≤𝑛.

We say that 𝑢=(𝑢₁,…,𝑢_𝑛) is a weak solution to (1.1) if 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋 and^10𝑛_𝑖=1||𝑢^𝑖(𝑥)||^𝑝𝑖−2𝑢^𝑖(𝑥)𝑣^𝑖(𝑥)𝑑𝑥−𝜆^10𝑛_𝑖=1𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥=0,(1.9)

for every (𝑣₁,…,𝑣_𝑛)∈𝑋.

A special case of our main result is the following theorem.

Theorem 1.1. Let 𝑓,𝑔∶ℝ²→ℝ be two positive continuous functions such that the differential 1-form 𝑤∶=𝑓(𝜉,𝜂)𝑑𝜉+𝑔(𝜉,𝜂)𝑑𝜂 is integrable, and let 𝐹 be a primitive of 𝑤 such that 𝐹(0,0)=0. Fix 𝑝,𝑞>2, 0<𝑥₁<𝑥₂<1, and assume that liminf_{(𝜉,𝜂)→(0,0)}𝐹(𝜉,𝜂)||𝜉||^𝑝/𝑝+||𝜂||^𝑞/𝑞=limsup_{||𝜉||||𝜂||→+∞,→+∞}𝐹(𝜉,𝜂)||𝜉||^𝑝/𝑝+||𝜂||^𝑞/𝑞=0,(1.10) then there is 𝜆^∗>0 such that for each 𝜆>𝜆^∗, the problem −−|||||𝑢𝑢^1||^𝑝−2𝑢^1^=𝜆𝑓𝑢₁,𝑢₂in(0,1),_′2|||^𝑞−2𝑢^2^𝑢=𝜆𝑔𝑢₁,𝑢₂in(0,1),_𝑖(0)=𝑎₁𝑢_𝑖𝑥₁+𝑎₂𝑢_𝑖𝑥₂,𝑢_𝑖(1)=𝑏₁𝑢_𝑖𝑥₁+𝑏₂𝑢_𝑖𝑥₂(1.11) admits at least two positive classical solutions.

The main aim of the present paper is to obtain further applications of [12, Theorem 2.6] (see Theorem 2.1 in the next section) to the system (1.1), and the obtained results are strictly comparable with those of [9–11], and here we wil give the exact collocation of the interval of positive parameters.

For other basic notations and definitions, we refer the reader to [17–26]. We note that some of the ideas used here were motivated by corresponding ones in [10].

2. Main Results

Our main tool is a three critical points theorem obtained in [12] (see also [1, 2, 5, 27] for related results), which is a more precise version of Theorem 3.2 of [28], to transfer the existence of three solutions of the system (1.1) into the existence of critical points of the Euler functional. We recall it here in a convenient form (see [23]).

Theorem 2.1 (see [12, Theorem 2.6]). Let 𝑋 be a reflexive real Banach space, Φ∶𝑋→ℝ be a coercive continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗, Ψ∶𝑋→ℝ be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Φ(0)=Ψ(0)=0. Assume that there exist 𝑟>0 and 𝑥∈𝑋, with 𝑟<Φ(𝑥) such that(𝜅₁)sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)/𝑟<Ψ(𝑥)/Φ(𝑥), (𝜅₂) for each 𝜆∈Λ_𝑟∶=]Φ(𝑥)/Ψ(𝑥),𝑟/sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)[, the functional Φ−𝜆Ψ is coercive.
then, for each 𝜆∈Λ_𝑟, the functional Φ−𝜆Ψ has at least three distinct critical points in 𝑋.

Put 𝑘=maxsup_{𝑢𝑖∈𝑋𝑖⧵{0}}max_𝑥∈[0,1]||𝑢_𝑖(𝑥)||^𝑝𝑖‖‖𝑢_𝑖‖‖^𝑝𝑖_𝑝𝑖;for1≤𝑖≤𝑛.(2.1) Since 𝑝_𝑖>1 for 1≤𝑖≤𝑛, and the embedding 𝑋=𝑋₁×⋯×𝑋_𝑛↪(𝐶⁰([0,1]))^𝑛 is compact, one has 𝑘<+∞. Moreover, from [10, Lemma3.1], one has sup_{𝑢∈𝑋𝑖⧵{0}}max_𝑥∈[0,1]||𝑢(𝑥)||‖‖𝑢^′‖‖_𝑝𝑖≤121+∑^𝑚𝑗=1||𝑎_𝑗||||1−∑^𝑚𝑗=1𝑎_𝑗||+∑^𝑚𝑗=1||𝑏_𝑗||||1−∑^𝑚𝑗=1𝑏_𝑗||.(2.2)

Put 𝜙_𝑝𝑖(𝑠)=|𝑠|^𝑝𝑖−1𝑠 for 1≤𝑖≤𝑛. Let 𝜙^𝑝−1_𝑖 denotes the inverse of 𝜙_𝑝𝑖 for 1≤𝑖≤𝑛. then, 𝜙^𝑝−1_𝑖(𝑡)=𝜙_𝑞𝑖(𝑡) where 1/𝑝_𝑖+1/𝑞_𝑖=1. It is clear that 𝜙_𝑝𝑖 is increasing on ℝ, lim_{𝑡→−∞}𝜙_𝑝𝑖(𝑡)=−∞,lim_𝑡→+∞𝜙_𝑝𝑖(𝑡)=+∞.(2.3)

Lemma 2.2 (see [10, Lemma 3.3]). For fixed 𝜆∈ℝ and 𝑢=(𝑢₁,…,𝑢_𝑛)∈(𝐶([0,1]))^𝑛, define 𝛼_𝑖(𝑡;𝑢)∶ℝ→ℝ by 𝛼_𝑖(𝑡;𝑢)=¹⁰𝜙^𝑝−1_𝑖𝑡−𝜆^𝛿0𝐹_𝑢𝑖𝜉,𝑢₁(𝜉),…,𝑢_𝑛(𝜉)𝑑𝜉𝑑𝛿+^𝑚_𝑗=1𝑎_𝑗𝑢_𝑖𝑥_𝑗−^𝑚_𝑗=1𝑏_𝑗𝑢_𝑖𝑥_𝑗.(2.4) then the equation 𝛼_𝑖(𝑡;𝑢)=0(2.5)has a unique solution 𝑡_𝑢,𝑖.

Direct computations show the following.

Lemma 2.3 (see [10, Lemma3.4]). The function 𝑢=(𝑢₁,…,𝑢_𝑛) is a solution of the system (1.1) if and only if 𝑢_𝑖(𝑥) is a solution of the equation 𝑢_𝑖(𝑥)=^𝑚_𝑗=1𝑎_𝑗𝑢_𝑖𝑥_𝑗+^𝑥0𝜙^𝑝−1_𝑖𝑡^𝑢,𝑖−𝜆^𝛿0𝐹_𝑢𝑖𝜉,𝑢₁(𝜉),…,𝑢_𝑛(𝜉)𝑑𝜉𝑑𝛿,(2.6) for 1≤𝑖≤𝑛, where 𝑡_𝑢,𝑖 is the unique solution of (2.5).

Lemma 2.4. A weak solution to the systems (1.1) coincides with classical solution one. Proof. Suppose that 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋 is a weak solution to (1.1), so ^10𝑛_𝑖=1𝜙_𝑝𝑖𝑢^′𝑖(𝑥)𝑣^𝑖(𝑥)𝑑𝑥−𝜆^10𝑛_𝑖=1𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥,(2.7) for every (𝑣₁,…,𝑣_𝑛)∈𝑋. Note that, in one dimension, any weakly differentiable function is absolutely continuous, so that its classical derivative exists almost everywhere, and that the classical derivative coincides with the weak derivative. Now, using integration by part, from (2.7), we obtain ^𝑛_𝑖=1¹⁰𝜙^𝑝_𝑖𝑢^𝑖(𝑥)^+𝜆𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥=0,(2.8) and so for 1≤𝑖≤𝑛, 𝜙_𝑝𝑖𝑢^𝑖(𝑥)^+𝜆𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)=0,(2.9) for almost every 𝑥∈(0,1). Then, by Lemmas 2.2 and 2.3, we observe 𝑢_𝑖(𝑥)=^𝑚_𝑗=1𝑎_𝑗𝑢_𝑖𝑥_𝑗+^𝑥0𝜙^𝑝−1_𝑖𝑡^𝑢,𝑖−𝜆^𝛿0𝐹_𝑢𝑖𝑠,𝑢₁(𝑠),…,𝑢_𝑛(𝑠)𝑑𝑠𝑑𝛿,(2.10) for 1≤𝑖≤𝑛, where 𝑡_𝑢,𝑖 is the unique solution of (2.5). Hence, 𝑢_𝑖∈𝐶¹([0,1]) and 𝜙_𝑝𝑖(𝑢^𝑖(𝑥))∈𝐶¹([0,1]) for 1≤𝑖≤𝑛, namely 𝑢=(𝑢₁,…,𝑢_𝑛) is a classical solution to the system (1.1).

For all 𝛾>0, we denote by 𝐾(𝛾) the set 𝑡¹,…,𝑡_𝑛∈𝑅^𝑛∶^𝑛_𝑖=1||𝑡_𝑖||^𝑝𝑖𝑝_𝑖≤𝛾.(2.11) Now, we formulate our main result as follows.

Theorem 2.5. Assume that there exist 2𝑚 constants 𝑎_𝑗,𝑏_𝑗 for 1≤𝑗≤𝑚 with ∑^𝑚𝑗=1𝑎_𝑗≠1 and ∑^𝑚𝑗=1𝑏_𝑗≠1, a positive constant 𝑟 and a function 𝑤=(𝑤₁,…,𝑤_𝑛)∈𝑋 such that (A1) ∑^𝑛𝑖=1(‖𝑤^𝑖‖^𝑝𝑖_𝑝𝑖/𝑝_𝑖)>𝑟, (A2)∫¹⁰sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝑘𝑟)}𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥<(𝑟∏^𝑛𝑖=1𝑝_𝑖)∫¹⁰𝐹(𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥))𝑑𝑥/∑^𝑛𝑖=1∏^𝑛𝑗=1,_𝑗≠𝑖𝑝_𝑗‖𝑤^𝑖‖^𝑝𝑖_𝑝𝑖where 𝐾(𝑘𝑟)={(𝑡₁,…,𝑡_𝑛)|∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖)≤kr}(see (2.11)),(A3)limsup_{|𝑡1|→+∞,…,|𝑡𝑛|→+∞}𝐹(𝑥,𝑡₁,…,𝑡_𝑛)/∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖)<∫¹⁰sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝑘𝑟)}×𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥/𝑘𝑟 uniformly with respect to 𝑥∈[0,1].Then, for each 𝜆∈Λ_𝑟∶=]∑^𝑛𝑖=1(‖𝑤^′_𝑖𝑝^𝑖_𝑝𝑖‖/𝑝_𝑖)/∫¹⁰𝐹(𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥))𝑑𝑥,𝑟/∫¹¹sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝑘𝑟)}𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥[, the system (1.1) admits at least three distinct classical solutions in 𝑋.Proof. In order to apply Theorem 2.1 to our problem, we introduce the functionals Φ,Ψ∶𝑋→ℝ for each 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋, as follows: Φ(𝑢)=^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖Ψ(𝑢)=₁₀,𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥.(2.12) Since 𝑝_𝑖>1 for 1≤𝑖≤𝑛, 𝑋 is compactly embedded in (𝐶⁰([0,1]))^𝑛 and it is well known that Φ and Ψ are well defined and continuously differentiable functionals whose derivatives at the point 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋 are the functionals Φ^(𝑢), Ψ^(𝑢)∈𝑋^∗, given by Φ^(𝑢)(𝑣)=^10𝑛_𝑖=1Ψ||𝑢^𝑖(𝑥)||^𝑝𝑖−2𝑢^𝑖(𝑥)𝑣^𝑖(𝑥)𝑑𝑥,_(𝑢)(𝑣)=^10𝑛_𝑖=1𝐹_𝑢𝑖𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑣_𝑖(𝑥)𝑑𝑥(2.13) for every 𝑣=(𝑣₁,…,𝑣_𝑛)∈𝑋, respectively, as well as Ψ is sequentially weakly upper semicontinuous. Furthermore, Lemma 2.6 of [11] gives that Φ^ admits a continuous inverse on 𝑋^∗, and since Φ′ is monotone, we obtain that Φ is sequentially weakly lower semiacontinuous (see [29, Proposition 25.20]). Moreover, Ψ^∶𝑋→𝑋^∗ is a compact operator. From assumption (A1), we get 0<𝑟<Φ(𝑤). Since from (2.1) for each 𝑢_𝑖∈𝑋_𝑖, sup_𝑥∈[0,1]||𝑢_𝑖(𝑥)||^𝑝𝑖≤𝑘‖‖𝑢^𝑖‖‖^𝑝𝑖(2.14) for 𝑖=1,…,𝑛, we have sup_{𝑛𝑥∈[0,1]}_𝑖=1||𝑢_𝑖(𝑥)||^𝑝𝑖𝑝_𝑖≤𝑘^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖,(2.15) for each 𝑢=(𝑢₁,…,𝑢_𝑛)∈𝑋, and so using (2.15), we observe Φ⁻¹(]−∞,𝑟])==𝑢₁𝑢,…,𝑢_𝑛∈𝑋;Φ𝑢₁,…,𝑢_𝑛≤𝑟₁,…,𝑢_𝑛∈𝑋;^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖⊆𝑢≤𝑟₁,…,𝑢_𝑛∈𝑋;^𝑛_𝑖=1||𝑢_𝑖(𝑥)||^𝑝𝑖𝑝_𝑖≤𝑘𝑟∀𝑥∈[0,1],(2.16) and it follows that sup_{𝑢1,…,𝑢𝑛∈Φ−1(]])−∞,𝑟}Ψ𝑢₁,…,𝑢_𝑛=sup_{𝑢1,…,𝑢𝑛∈Φ−1(]])−∞,𝑟}¹⁰≤𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥₁₀sup_{𝑡1,…,𝑡𝑛∈𝐾(𝑘𝑟)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥.(2.17) Therefore, owing to assumption (A2), we have sup_{𝑢∈Φ−1(]])−∞,𝑟}Ψ𝑢₁,…,𝑢_𝑛=sup_{𝑢1,…,𝑢𝑛∈Φ−1(]])−∞,𝑟}¹⁰≤𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥₁₀<𝑟sup_{𝑡1,…,𝑡𝑛∈𝐾(𝑘𝑟)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥_𝑛_𝑖=1𝑝_𝑖∫¹⁰𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛𝑖=1∏^𝑛𝑗=1,_𝑗≠𝑖𝑝_𝑗‖‖𝑤^𝑖‖‖^𝑝𝑖_𝑝𝑖<𝑟∫¹⁰𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥=𝑟Ψ(𝑤)∑^𝑛𝑖=1‖‖𝑤^𝑖‖‖^𝑝𝑖/𝑝_𝑖Φ(𝑤).(2.18) Furthermore, from (A3), there exist two constants 𝛾,𝜐∈ℝ with 0<𝛾<∫¹⁰sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝑘𝑟)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥𝑟,(2.19) such that 𝑘𝐹𝑥,𝑡₁,…,𝑡_𝑛≤𝛾^𝑛_𝑖=1||𝑡_𝑖||^𝑝𝑖𝑝_𝑖+𝜐,∀𝑥∈[0,1],∀𝑡₁,…,𝑡_𝑛∈ℝ^𝑛.(2.20) Fix (𝑢₁,…,𝑢_𝑛)∈𝑋, Then 𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)≤1𝑘𝛾^𝑛_𝑖=1||𝑢_𝑖(𝑥)||^𝑝𝑖𝑝_𝑖+𝜐∀𝑥∈[0,1].(2.21) So, for any fixed 𝜆∈Λ_𝑟, from (2.15) and (2.21), we have Φ(𝑢)−𝜆Ψ(𝑢)=^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖≥−𝜆¹⁰𝐹𝑥,𝑢₁(𝑥),…,𝑢_𝑛(𝑥)𝑑𝑥_𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝛾𝑘^𝑛_𝑖=11𝑝_𝑖¹⁰||𝑢_𝑖(𝑥)||^𝑝𝑖𝑑𝑥−𝜆𝜐≥_𝑛𝑘_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝛾𝑘𝑘^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖=_𝑛𝑝_𝑖−𝜆𝑘𝜐_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝛾^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝜐≥1−𝛾𝑟𝑘∫¹⁰sup_{(𝑡1,…,𝑡𝑛)∈𝐾(𝑘𝑟)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥^𝑛_𝑖=1‖‖𝑢^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖−𝜆𝜐𝑘,(2.22) and thus, lim_{‖‖𝑢1,…,𝑢𝑛‖‖→+∞}Φ𝑢₁,…,𝑢_𝑛−𝜆Ψ𝑢₁,…,𝑢_𝑛=+∞,(2.23) which means that the functional Φ−𝜆Ψ is coercive. So, all assumptions of Theorem 2.1 are satisfied. Hence, from Theorem 2.1 with 𝑥=𝑤, taking into account that the weak solutions of the system (1.1) are exactly the solutions of the equation Φ′(𝑢₁,…,𝑢_𝑛)−𝜆Ψ′(𝑢₁,…,𝑢_𝑛)=0 and using Lemma 2.4, we have the conclusion.

Now we want to present a verifiable consequence of the main result where the test function 𝑤 is specified.

Put𝜎_𝑖=⎡⎢⎢⎣2^𝑝𝑖−1⎛⎜⎜⎝𝑥_{1−𝑝𝑖1}|||||1−_𝑚_𝑗=1𝑎_𝑗|||||^𝑝𝑖+1−𝑥_𝑚^1−𝑝𝑖|||||1−_𝑚_𝑗=1𝑏_𝑗|||||^𝑝𝑖⎞⎟⎟⎠⎤⎥⎥⎦^1/𝑝𝑖for1≤𝑖≤𝑛.(2.24)

Define 𝐵_1,𝑛(𝑥)=⎧⎪⎪⎨⎪⎪⎩𝑥^𝑚_𝑗=1𝑎_𝑗,𝑥^𝑛if^𝑚_𝑗=1𝑥,𝑥_𝑚𝑎_𝑗<1,_𝑗=1𝑎_𝑗^𝑛if^𝑚_𝑗=1𝐵𝑎_𝑗>1,_2,𝑛(𝑥)=⎧⎪⎪⎨⎪⎪⎩𝑥^𝑚_𝑗=1𝑏_𝑗,𝑥^𝑛if^𝑚_𝑗=1𝑥,𝑥_𝑚𝑏_𝑗<1,_𝑗=1𝑏_𝑗^𝑛if^𝑚_𝑗=1𝑏_𝑗>1,(2.25) where [⋅,⋅]^𝑛=[⋅,⋅]×⋯×[⋅,⋅], then we have the following consequence of Theorem 2.5.

Corollary 2.6. Assume that there exist 2𝑚 constants 𝑎_𝑗,𝑏_𝑗 for 1≤𝑗≤𝑚 with ∑^𝑚𝑗=1𝑎_𝑗≠1 and ∑^𝑚𝑗=1𝑏_𝑗≠1 and two positive constants 𝜃 and 𝜏 with ∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)>𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖 such that (B1)𝐹(𝑥,𝑡₁,…,𝑡_𝑛)≥0 for each 𝑥∈[0,𝑥₁/2]∪[(1+𝑥_𝑚)/2,1] and (𝑡₁,…,𝑡_𝑛)∈𝐵_1,𝑛(𝜏)∪𝐵_2,𝑛(𝜏),(B2)∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)∫¹⁰sup_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥<  (𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖)∫^{(1+𝑥𝑚𝑥)/2}_1/2𝐹(𝑥,𝜏,…,𝜏)𝑑𝑥, where 𝜎_𝑖 is given by (2.24) and 𝐾(𝜃/∏^𝑛𝑖=1𝑝_𝑖)={(𝑡₁,…,𝑡_𝑛)∣∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖)≤𝑐/∏^𝑛𝑖=1𝑝_𝑖}(see (2.11));(B3)limsup_{|𝑡1|→+∞,…,|𝑡𝑛|→+∞}(𝐹(𝑥,𝑡₁,…,𝑡_𝑛)/∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖))<  ∏^𝑛𝑖=1𝑝_𝑖/𝜃×∫¹⁰sup_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥 uniformly with respect to 𝑥∈[0,1].then, for each 𝜆∈]∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)/∫^{(1+𝑥𝑚𝑥)/2}_1/2𝐹(𝑥,𝜏,…,𝜏)𝑑𝑥,(𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖)/∫¹⁰sup_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑥,𝑡₁,…,𝑡_𝑛)𝑑𝑥[ the systems (1.1) admits at least three distinct classical solutions.

Proof. Set 𝑤(𝑥)=(𝑤₁(𝑥),…,𝑤_𝑛(𝑥)) such that for 1≤𝑖≤𝑛, 𝑤_𝑖(𝑥)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝜏^𝑚_𝑗=1𝑎_𝑗+21−∑^𝑚𝑗=1𝑎_𝑗𝑥₁𝑥if𝑥∈0,𝑥¹𝜏2,if𝑥∈𝑥¹𝜏2−∑_𝑚𝑗=12,1+2𝑥^𝑚,𝑏_𝑗−𝑥_𝑚∑^𝑚𝑗=1𝑏_𝑗1−𝑥_𝑚−21−∑^𝑚𝑗=1𝑏_𝑗1−𝑥_𝑚𝑥if𝑥∈1+𝑥^𝑚2,1,(2.26) and 𝑟=𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖. It is easy to see that 𝑤=(𝑤₁,…,𝑤_𝑛)∈𝑋, and, in particular, one has ‖‖𝑤^′_𝑖‖‖^𝑝𝑖_𝑝𝑖=𝜎_𝑖𝜏^𝑝𝑖,(2.27) for 1≤𝑖≤𝑛, which, employing the condition ∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)>𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖, gives ^𝑛_𝑖=1‖‖𝑤^𝑖‖‖^𝑝𝑖_𝑝𝑖𝑝_𝑖>𝑟.(2.28) Since for 1≤𝑖≤𝑛, 𝜏^𝑚_𝑗=1𝑎_𝑗≤𝑤_𝑖(𝑥)≤𝜏foreach𝑥∈0,𝑥¹2if^𝑚_𝑗=1𝜏≤𝑤_𝑖𝑎_𝑗<1,(𝑥)≤𝜏^𝑚_𝑗=1𝑎_𝑗foreach𝑥∈0,𝑥¹2if^𝑚_𝑗=1𝜏𝑎_𝑗>1,_𝑚_𝑗=1𝑏_𝑗≤𝑤_𝑖(𝑥)≤𝜏foreach𝑥∈1+𝑥^𝑚2,1if^𝑚_𝑗=1𝜏≤𝑤_𝑖𝑏_𝑗<1,(𝑥)≤𝜏^𝑚_𝑗=1𝑏_𝑗foreach𝑥∈1+𝑥^𝑚2,1if^𝑚_𝑗=1𝑏_𝑗>1,(2.29) the condition (B1) ensures that ^𝑥10/2𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥+^11+𝑥_𝑚/2𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥≥0.(2.30) Therefore, owing to assumption (B2), we have ¹⁰sup_{𝑡1,…,𝑡𝑛∈𝐾(𝑘𝑟)}𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥<𝜃∑^𝑛𝑖=1𝜎_𝑖𝜏^𝑝𝑖/𝑝_𝑖𝑘∏^𝑛𝑖=1𝑝_𝑖^{(1+𝑥𝑚𝑥)/2}₁^/2≤𝜃𝐹(𝑥,𝜏,…,𝜏)𝑑𝑥𝑘∫¹⁰𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛𝑖=1∏^𝑛𝑗=1,_𝑗≠𝑖𝑝_𝑗‖‖𝑤^′_𝑖‖‖^𝑝𝑖_𝑝𝑖=𝑟^𝑛_𝑖=1𝑝_𝑖∫¹⁰𝐹𝑥,𝑤₁(𝑥),…,𝑤_𝑛(𝑥)𝑑𝑥∑^𝑛𝑖=1∏^𝑛𝑗=1,_𝑗≠𝑖𝑝_𝑗‖‖𝑤^𝑖‖‖^𝑝𝑖_𝑝𝑖,(2.31) where 𝐾(𝜃/∏^𝑛𝑖=1𝑝_𝑖)={(𝑡₁,…,𝑡_𝑛)∣∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖)≤𝜃/∏^𝑛𝑖=1𝑝_𝑖}. Moreover, from assumption (B3) it follows that assumption (A3) is fulfilled. Hence, taking into account that ⎤⎥⎥⎦∑^𝑛𝑖=1𝜎_𝑖𝜏^𝑝𝑖/𝑝_𝑖∫^{(1+𝑥𝑚𝑥)/2}_1/2𝐹(𝑥,𝜏,…,𝜏)𝑑𝑥,𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖∫¹⁰sup_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹𝑥,𝑡₁,…,𝑡_𝑛𝑑𝑥⎡⎢⎢⎣⊆Λ^𝑟,(2.32) using Theorem 2.5, we have the desired conclusion.

Let us present an application of Corollary 2.6.

Example 2.7. Let 𝐹∶[0,1]×ℝ³→ℝ be the function defined as 𝐹𝑥,𝑡₁,𝑡₂,𝑡₃=⎧⎪⎪⎪⎨⎪⎪⎪⎩0∀𝑡^𝑖𝑥<0,𝑖=1,2,₂𝑥𝑡¹¹⁰⁰𝑒^−𝑡1for𝑡₁≥0,𝑡₂<0,𝑡₃<0,₂𝑥𝑡²¹⁰⁰𝑒^−𝑡2for𝑡₁<0,𝑡₂≥0,𝑡₃<0,₂𝑥𝑡³¹⁰⁰𝑒^−𝑡3for𝑡₁<0,𝑡₂<0,𝑡₃≥0.₂₃_𝑖=1𝑡^𝑖100𝑒^−𝑡𝑖for𝑡_𝑖≥0,𝑖=1,2,3,(2.33) for each (𝑥,𝑡₁,𝑡₂,𝑡₃)∈[0,1]×ℝ³. In fact, by choosing 𝑝₁=𝑝₂=𝑝₃=3 and 𝑎₁=𝑏₁=𝑥₁=1/2, by a simple calculation, we obtain that 𝑘=27/8 and 𝜎₁=𝜎₂=𝜎₃=4^1/3, and so with 𝜃=9 and 𝜏=100, we observe that the assumptions (B1) and (B3) in Corollary 2.6 are satisfied. For (B2), ³_𝑖=1𝜎_𝑖𝜏^𝑝𝑖𝑝_𝑖¹⁰=4(100)₃sup_{𝑡1,𝑡2,𝑡3∏∈𝐾𝜃/3𝑖=1𝑝𝑖}𝐹𝑥,𝑡₁,𝑡₂,𝑡₃𝑑𝑥¹⁰≤4(100)₃sup_{𝑡1,𝑡2,𝑡3∈𝐾(1/3)}𝐹𝑥,𝑡₁,𝑡₂,𝑡₃𝑑𝑥¹⁰sup_{𝑡1,𝑡2,𝑡3∈𝐾(1/3)}𝑥²³_𝑖=1=4(100)₃𝑡^𝑖100𝑒^−𝑡𝑖𝑑𝑥max_{(𝑡1,𝑡2,𝑡33)∈𝐾(1/3)}_𝑖=1𝑡^𝑖100𝑒^−𝑡𝑖¹⁰≤4𝑥²𝑑𝑥3(100)³3max_|𝑡|≤1𝑡¹⁰⁰𝑒^−𝑡=4(100)₃<13𝑒4×3⁴(100)¹⁰⁰𝑒⁻¹⁰⁰=𝜃𝑘∏^3𝑖=1𝑝_𝑖^{(1+𝑥1𝑥)/2}_1/2𝐹(𝑥,𝜏,𝜏,𝜏)𝑑𝑥.(2.34) So, for every 𝜆∈4⁴(100)³26(100)¹⁰⁰𝑒⁻¹⁰⁰,83⁵(100)³𝑒,(2.35) Corollary 2.6 is applicable to the system −|||𝑢^′₁|||𝑢_1−|||𝑢′=𝜆𝑥²𝑢⁺¹⁹⁹𝑒^−𝑢+1100−𝑢⁺¹in(0,1),_′2|||𝑢_2−|||𝑢′=𝜆𝑥²𝑢⁺²⁹⁹𝑒^−𝑢+2100−𝑢⁺²in(0,1),_′3|||𝑢_3𝑢′=𝜆𝑥²𝑢⁺³⁹⁹𝑒^−𝑢+3100−𝑢⁺³in(0,1),_𝑖(0)=𝑢_𝑖(1)=12𝑢^𝑖21for𝑖=1,2,3,(2.36) where 𝑡⁺=max{𝑡,0}.

Here is a remarkable consequence of Corollary 2.6.

Corollary 2.8. Let 𝐹∶ℝ^𝑛→ℝ be a 𝐶¹ function in 𝑅^𝑛 such that 𝐹(0,…,0)=0. Assume that there exist 2𝑚 constants 𝑎_𝑗,𝑏_𝑗 for 1≤𝑗≤𝑚 with ∑^𝑚𝑗=1𝑎_𝑗≠1 and ∑^𝑚𝑗=1𝑏_𝑗≠1 and two positive constants 𝜃 and 𝜏 with ∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)>𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖 such that (C1)𝐹(𝑡₁,…,𝑡_𝑛)≥0 for each (𝑡₁,…,𝑡_𝑛)∈𝐵_1,𝑛(𝜏)∪𝐵_2,𝑛(𝜏),(C2)∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)max_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑡₁,…,𝑡_𝑛)<(𝜃(1+𝑥_𝑚−𝑥₁)/2𝑘∏^𝑛𝑖=1𝑝_𝑖)𝐹(𝜏,…,𝜏)where 𝜎_𝑖 is given by (2.24),(C3)limsup_{|𝑡1|→+∞,…,|𝑡𝑛|→+∞}(𝐹(𝑡₁,…,𝑡_𝑛)/∑^𝑛𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖))<(∏^𝑛𝑖=1𝑝_𝑖/𝜃)×max_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑡₁,…,𝑡_𝑛),Then, for each 𝜆∈]∑^𝑛𝑖=1((𝜎_𝑖𝜏)^𝑝𝑖/𝑝_𝑖)/((1+𝑥_𝑚−𝑥₁)/2)𝐹(𝜏,…,𝜏),(𝜃/𝑘∏^𝑛𝑖=1𝑝_𝑖)/max_{(𝑡1,…,𝑡𝑛∏)∈𝐾(𝜃/}^𝑛𝑖=1_𝑝𝑖)𝐹(𝑡₁,…,𝑡_𝑛)[, the systems −|||𝑢^′_𝑖|||^𝑝𝑖−2𝑢^𝑖^𝑢=𝜆𝐹_𝑢𝑖𝑢₁,…,𝑢_𝑛in(0,1),_𝑖(0)=^𝑚_𝑗=1𝑎_𝑗𝑢_𝑖𝑥_𝑗,𝑢_𝑖(1)=^𝑚_𝑗=1𝑏_𝑗𝑢_𝑖𝑥_𝑗,(2.37) for 1≤𝑖≤𝑛, admits at least three distinct classical solutions.

Proof. Set 𝐹(𝑥,𝑡₁,…,𝑡_𝑛)=𝐹(𝑡₁,…,𝑡_𝑛) for all 𝑥∈[0,1] and 𝑡_𝑖∈ℝ for 1≤𝑖≤𝑛. From the hypotheses, we see that all assumptions of Corollary 2.6 are satisfied. So, we have the conclusion by using Corollary 2.6.

Example 2.9. Let 𝑝₁=𝑝₂=3,𝑚=2,𝑥₁=1/3,𝑥₂=2/3 and 𝑎_𝑖=𝑏_𝑖=1/3, 𝑖=1,2. Consider the system −|||𝑢^′₁|||𝑢_1−|||𝑢^=𝜆𝑒^−𝑢1𝑢¹¹¹12−𝑢₁,in(0,1),_′2|||𝑢_2𝑢^=𝜆𝑒^−𝑢2𝑢⁹²10−𝑢₂,in(0,1),₁(0)=𝑢₁(1)=13𝑢¹31+31𝑢¹32,𝑢²(0)=𝑢₂(1)=13𝑢²31+31𝑢²32.(2.38) Clearly, (H1) and (H2) hold. A simple calculation shows that 𝑘=125/8 and 𝜎₁=𝜎₂=(8/3)^1/3. So, by choosing 𝜃=3 and 𝜏=10, we observe that the assumptions (C1) and (C3) in Corollary 2.8 hold. For (C3), since 𝐹(𝑡₁,𝑡₂)=𝑡¹¹²𝑒^−𝑡1+𝑡²¹⁰𝑒^−𝑡2 for every (𝑡₁,𝑡₂)∈ℝ², we have max_{(𝑡1,𝑡2∏)∈𝐾(𝜃/2𝑖=1𝑝𝑖)}𝐹𝑡₁,𝑡₂==mmaaxx_{𝑡1,𝑡2∈𝐾(1/3)𝑡1,𝑡2∈𝐾(1/3)}𝐹𝑡₁,𝑡₂≤max_{||𝑡1||≤1}𝑡¹¹²𝑒^−𝑡1+𝑡²¹⁰𝑒^−𝑡2𝑡¹¹²𝑒^−𝑡1+max_{||𝑡2||≤1}𝑡²¹⁰𝑒^−𝑡2<=2𝑒1125⋅10³=𝜃101+¹²𝑒𝑥⁻¹⁰₂+10¹⁰𝑒⁻¹⁰−𝑥₁∑^2𝑖=1𝜏𝜎_𝑖^𝑝𝑖/𝑝_𝑖2𝑘∏^2𝑖=1𝑝_𝑖𝐹(𝜏,𝜏).(2.39) Note that lim_{|𝑡1|→∞,|𝑡2|→∞}(𝐹(𝑡₁,𝑡₂)/∑^2𝑖=1(|𝑡_𝑖|^𝑝𝑖/𝑝_𝑖))=0. We see that for every 𝜆∈8310⁹𝑒⁻¹⁰+10⁷𝑒⁻¹⁰,3745𝑒,(2.40) Corollary 2.8 is applicable to the system (2.38).

Finally, we prove the theorem in the introduction.

Proof of Theorem 1.1. Since 𝑓 and 𝑔 are positive, then 𝐹 is nonnegative in ℝ². Fix 𝜆>𝜆^∗∶=((𝜎₁𝜏)^𝑝/𝑝+(𝜎₂𝜏)^𝑞/𝑞)/((1+𝑥₂−𝑥₁)/2)𝐹(𝜏,𝜏) for some 𝜏>0. Note that liminf_{(𝜉,𝜂)→(0,0)}(𝐹(𝜉,𝜂)/(|𝜉|^𝑝/𝑝+|𝜂|^𝑞/𝑞))=0, and there is {𝜃_𝑛}_𝑛∈𝑁⊆]0,+∞[such that lim_𝑛→+∞𝜃_𝑛=0 and lim_𝑛→+∞max_{(𝜉,𝜂)∈𝐾(𝜃𝑛/𝑝𝑞)}𝐹(𝜉,𝜂)𝜃_𝑛=0.(2.41) In fact, one has lim_𝑛→+∞(max_{(𝜉,𝜂)∈𝐾(𝜃𝑛/𝑝𝑞)}𝐹(𝜉,𝜂)/𝜃_𝑛)=lim_𝑛→+∞(𝐹(𝜉_𝜃𝑛,𝜂_𝜃𝑛)/(|𝜉_𝜃𝑛|^𝑝/𝑝+|𝜂_𝜃𝑛|^𝑞/𝑞))⋅(|𝜉_𝜃𝑛|^𝑝/𝑝+|𝜂_𝜃𝑛|^𝑞/𝑞)/𝜃_𝑛=0, where 𝐹(𝜉_𝜃𝑛,𝜂_𝜃𝑛)=sup_{(𝜉,𝜂)∈𝐾(𝜃𝑛/𝑝𝑞)}𝐹(𝜉,𝜂). Hence, there is 𝜃>0 such that max_{(𝜉,𝜂)∈𝐾(𝜃/𝑝𝑞)}𝐹(𝜉,𝜂)𝜃<min1+𝑥²−𝑥₁2𝑞𝜎₁𝜏^𝑝+𝑝𝜎₂𝜏^𝑞𝑘𝐹(𝜏,𝜏);𝜆𝑝1𝑞𝑘,(2.42) and 𝜃<𝑘(𝑞(𝜎₁𝜏)^𝑝+𝑝(𝜎₂𝜏)^𝑞).
from Corollary 2.8, with 𝑛=2 follows the conclusion.

Acknowledgment

The author would like to thank Professor Juan J. Nieto for valuable suggestions and reading this paper carefully.