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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 389530, 15 pages
http://dx.doi.org/10.1155/2012/389530
Research Article

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

Department of Mathematics, Faculty of Sciences, Razi University, Kermanshash 67149, Iran

Received 20 November 2011; Accepted 11 December 2011

Academic Editor: KanishkaΒ Perera

Copyright Β© 2012 Shapour Heidarkhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 βˆ’(|π‘’ξ…žπ‘–|π‘π‘–βˆ’2π‘’ξ…žπ‘–)β€²=πœ†πΉπ‘’π‘–(π‘₯,𝑒1,…,𝑒𝑛) in (0,1), π‘’π‘–βˆ‘(0)=π‘šπ‘—=1π‘Žπ‘—π‘’π‘–(π‘₯𝑗), π‘’π‘–βˆ‘(1)=π‘šπ‘—=1𝑏𝑗𝑒𝑖(π‘₯𝑗) for 1≀𝑖≀𝑛, is established.

1. Introduction

In this work, we consider the following multipoint boundary value system βˆ’ξ‚€||π‘’ξ…žπ‘–||π‘π‘–βˆ’2π‘’ξ…žπ‘–ξ‚ξ…ž=πœ†πΉπ‘’π‘–ξ€·π‘₯,𝑒1,…,𝑒𝑛in𝑒(0,1),𝑖(0)=π‘šξ“π‘—=1π‘Žπ‘—π‘’π‘–ξ€·π‘₯𝑗,𝑒𝑖(1)=π‘šξ“π‘—=1𝑏𝑗𝑒𝑖π‘₯𝑗,(1.1) for 1≀𝑖≀𝑛, where 𝑝𝑖>1 for 1≀𝑖≀𝑛, πœ†>0, π‘š,𝑛β‰₯1, 𝐹∢[0,1]×ℝ𝑛→ℝ is a function such that 𝐹(β‹…,𝑑1,…,𝑑𝑛) is continuous in [0,1] for all (𝑑1,…,𝑑𝑛)βˆˆβ„π‘›, 𝐹(π‘₯,β‹…,…,β‹…) is 𝐢1 in ℝ𝑛 for every π‘₯∈[0,1] and 𝐹(π‘₯,0,…,0)=0 for all π‘₯∈[0,1], π‘Žπ‘—,π‘π‘—βˆˆβ„ for 𝑗=1,…,π‘š and 0<π‘₯1<π‘₯2<π‘₯3<β‹―<π‘₯π‘š<1, and 𝐹𝑒𝑖 denotes the partial derivative of 𝐹 with respect to 𝑒𝑖 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 π‘’ξ…žξ…ž+πœ†π‘“(𝑒)=0in(0,1),𝑒(0)=𝑒(1)=0.(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 πœ† for which the functional has three critical points is established. Candito [7] extended the main result of [4] to the nonautonomous case π‘’ξ…žξ…ž+πœ†π‘“(π‘₯,𝑒)=0in(π‘Ž,𝑏),𝑒(π‘Ž)=𝑒(𝑏)=0.(1.3) In [8], He and Ge extended the main results of [4, 7] to the quasilinear differential equation ξ‚€πœ‘π‘ξ‚€π‘’β€²ξ‚ξ‚ξ…ž+πœ†π‘“(π‘₯,𝑒)=0in(π‘Ž,𝑏),𝑒(π‘Ž)=𝑒(𝑏)=0.(1.4) In [9], the authors extended the main results of [4, 7, 9] to the quasilinear differential equation with Sturm-Liouville boundary conditions ξ‚΅|||𝑒′|||π‘βˆ’2π‘’ξ…žξ‚Άξ…ž+πœ†π‘“(π‘₯,𝑒)=0in𝛼(π‘Ž,𝑏),1𝑒(π‘Ž)βˆ’π›Ό2𝑒′(π‘Ž)=0,𝛽1𝑒(𝑏)βˆ’π›½2π‘’ξ…ž(𝑏)=0,(1.5) where 𝑝>1 is a constant, πœ† is a positive parameter, π‘Ž,π‘βˆˆβ„;π‘Ž<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 𝑋𝑖=ξƒ―πœ‰βˆˆπ‘Š1,𝑝𝑖([]0,1);πœ‰(0)=π‘šξ“π‘—=1π‘Žπ‘—πœ‰ξ€·π‘₯𝑗,πœ‰(1)=π‘šξ“π‘—=1π‘π‘—πœ‰ξ€·π‘₯𝑗,(1.6) for 𝑖=1,…,𝑛, that is, 𝑋=𝑋1×⋯×𝑋𝑛 equipped with the norm ‖‖𝑒1,…,𝑒𝑛‖‖=𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–,(1.7) where β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–=ξ‚΅ξ€œ10||π‘’ξ…žπ‘–||(π‘₯)𝑝𝑖𝑑π‘₯1/𝑝𝑖(1.8) for 1≀𝑖≀𝑛.

We say that 𝑒=(𝑒1,…,𝑒𝑛) is a weak solution to (1.1) if 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘‹ andξ€œ10𝑛𝑖=1||π‘’ξ…žπ‘–||(π‘₯)π‘π‘–βˆ’2π‘’ξ…žπ‘–(π‘₯)π‘£ξ…žπ‘–ξ€œ(π‘₯)𝑑π‘₯βˆ’πœ†10𝑛𝑖=1𝐹𝑒𝑖π‘₯,𝑒1(π‘₯),…,𝑒𝑛𝑣(π‘₯)𝑖(π‘₯)𝑑π‘₯=0,(1.9)

for every (𝑣1,…,𝑣𝑛)βˆˆπ‘‹.

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

Theorem 1.1. Let 𝑓,π‘”βˆΆβ„2→ℝ be two positive continuous functions such that the differential 1-form π‘€βˆΆ=𝑓(πœ‰,πœ‚)π‘‘πœ‰+𝑔(πœ‰,πœ‚)π‘‘πœ‚ is integrable, and let 𝐹 be a primitive of 𝑀 such that 𝐹(0,0)=0. Fix 𝑝,π‘ž>2, 0<π‘₯1<π‘₯2<1, and assume that liminf(πœ‰,πœ‚)β†’(0,0)𝐹(πœ‰,πœ‚)||πœ‰||𝑝||πœ‚||/𝑝+π‘ž/π‘ž=limsup||πœ‰||||πœ‚||β†’+∞,β†’+∞𝐹(πœ‰,πœ‚)||πœ‰||𝑝||πœ‚||/𝑝+π‘ž/π‘ž=0,(1.10) then there is πœ†βˆ—>0 such that for each πœ†>πœ†βˆ—, the problem βˆ’ξ‚€||π‘’ξ…ž1||π‘βˆ’2π‘’ξ…ž1ξ‚ξ…žξ€·π‘’=πœ†π‘“1,𝑒2ξ€Έinβˆ’ξ‚΅|||𝑒(0,1),β€²2|||π‘žβˆ’2π‘’ξ…ž2ξ‚Άξ…žξ€·π‘’=πœ†π‘”1,𝑒2ξ€Έin𝑒(0,1),𝑖(0)=π‘Ž1𝑒𝑖π‘₯1ξ€Έ+π‘Ž2𝑒𝑖π‘₯2ξ€Έ,𝑒𝑖(1)=𝑏1𝑒𝑖π‘₯1ξ€Έ+𝑏2𝑒𝑖π‘₯2ξ€Έ(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, Ξ¦βˆΆπ‘‹β†’β„ be a coercive continuously GΓ’teaux differentiable and sequentially weakly lower semicontinuous functional whose GΓ’teaux derivative admits a continuous inverse on π‘‹βˆ—, Ξ¨βˆΆπ‘‹β†’β„ be a continuously GΓ’teaux differentiable functional whose GΓ’teaux derivative is compact such that Ξ¦(0)=Ξ¨(0)=0. Assume that there exist π‘Ÿ>0 and π‘₯βˆˆπ‘‹, with π‘Ÿ<Ξ¦(π‘₯) such that(πœ…1)supΞ¦(π‘₯)β‰€π‘ŸΞ¨(π‘₯)/π‘Ÿ<Ξ¨(π‘₯)/Ξ¦(π‘₯), (πœ…2) for each πœ†βˆˆΞ›π‘ŸβˆΆ=]Ξ¦(π‘₯)/Ξ¨(π‘₯),π‘Ÿ/supΞ¦(π‘₯)β‰€π‘ŸΞ¨(π‘₯)[, the functional Ξ¦βˆ’πœ†Ξ¨ is coercive.
then, for each πœ†βˆˆΞ›π‘Ÿ, the functional Ξ¦βˆ’πœ†Ξ¨ has at least three distinct critical points in 𝑋.

Put ξƒ―π‘˜=maxsupπ‘’π‘–βˆˆπ‘‹π‘–β§΅{0}maxπ‘₯∈[0,1]||𝑒𝑖||(π‘₯)𝑝𝑖‖‖𝑒𝑖‖‖𝑝𝑖𝑝𝑖;forξƒ°.1≀𝑖≀𝑛(2.1) Since 𝑝𝑖>1 for 1≀𝑖≀𝑛, and the embedding 𝑋=𝑋1×⋯×𝑋𝑛β†ͺ(𝐢0([0,1]))𝑛 is compact, one has π‘˜<+∞. Moreover, from [10, Lemma3.1], one has supπ‘’βˆˆπ‘‹π‘–β§΅{0}maxπ‘₯∈[0,1]||||𝑒(π‘₯)‖‖𝑒′‖‖𝑝𝑖≀12ξƒ©βˆ‘1+π‘šπ‘—=1||π‘Žπ‘—||||βˆ‘1βˆ’π‘šπ‘—=1π‘Žπ‘—||+βˆ‘π‘šπ‘—=1||𝑏𝑗||||βˆ‘1βˆ’π‘šπ‘—=1𝑏𝑗||ξƒͺ.(2.2)

Put πœ™π‘π‘–(𝑠)=|𝑠|π‘π‘–βˆ’1𝑠 for 1≀𝑖≀𝑛. Let πœ™π‘βˆ’1𝑖 denotes the inverse of πœ™π‘π‘– for 1≀𝑖≀𝑛. then, πœ™π‘βˆ’1𝑖(𝑑)=πœ™π‘žπ‘–(𝑑) where 1/𝑝𝑖+1/π‘žπ‘–=1. It is clear that πœ™π‘π‘– is increasing on ℝ, limπ‘‘β†’βˆ’βˆžπœ™π‘π‘–(𝑑)=βˆ’βˆž,lim𝑑→+βˆžπœ™π‘π‘–(𝑑)=+∞.(2.3)

Lemma 2.2 (see [10, Lemma 3.3]). For fixed πœ†βˆˆβ„ and 𝑒=(𝑒1,…,𝑒𝑛)∈(𝐢([0,1]))𝑛, define 𝛼𝑖(𝑑;𝑒)βˆΆβ„β†’β„ by π›Όπ‘–ξ€œ(𝑑;𝑒)=10πœ™π‘βˆ’1π‘–ξ‚΅ξ€œπ‘‘βˆ’πœ†π›Ώ0πΉπ‘’π‘–ξ€·πœ‰,𝑒1(πœ‰),…,𝑒𝑛(πœ‰)π‘‘πœ‰π‘‘π›Ώ+π‘šξ“π‘—=1π‘Žπ‘—π‘’π‘–ξ€·π‘₯π‘—ξ€Έβˆ’π‘šξ“π‘—=1𝑏𝑗𝑒𝑖π‘₯𝑗.(2.4) then the equation 𝛼𝑖(𝑑;𝑒)=0(2.5)has a unique solution 𝑑𝑒,𝑖.

Direct computations show the following.

Lemma 2.3 (see [10, Lemma3.4]). The function 𝑒=(𝑒1,…,𝑒𝑛) is a solution of the system (1.1) if and only if 𝑒𝑖(π‘₯) is a solution of the equation 𝑒𝑖(π‘₯)=π‘šξ“π‘—=1π‘Žπ‘—π‘’π‘–ξ€·π‘₯𝑗+ξ€œπ‘₯0πœ™π‘βˆ’1𝑖𝑑𝑒,π‘–ξ€œβˆ’πœ†π›Ώ0πΉπ‘’π‘–ξ€·πœ‰,𝑒1(πœ‰),…,𝑒𝑛(πœ‰)π‘‘πœ‰π‘‘π›Ώ,(2.6) for 1≀𝑖≀𝑛, where 𝑑𝑒,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 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘‹ is a weak solution to (1.1), so ξ€œ10𝑛𝑖=1πœ™π‘π‘–ξ‚€π‘’β€²π‘–ξ‚π‘£(π‘₯)ξ…žπ‘–ξ€œ(π‘₯)𝑑π‘₯βˆ’πœ†10𝑛𝑖=1𝐹𝑒𝑖π‘₯,𝑒1(π‘₯),…,𝑒𝑛𝑣(π‘₯)𝑖(π‘₯)𝑑π‘₯,(2.7) for every (𝑣1,…,𝑣𝑛)βˆˆπ‘‹. 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=1ξ€œ10ξ‚ƒξ€·πœ™π‘π‘–ξ€·π‘’ξ…žπ‘–(π‘₯)ξ€Έξ€Έξ…ž+πœ†πΉπ‘’π‘–ξ€·π‘₯,𝑒1(π‘₯),…,𝑒𝑛𝑣(π‘₯)𝑖(π‘₯)𝑑π‘₯=0,(2.8) and so for 1≀𝑖≀𝑛, ξ€·πœ™π‘π‘–ξ€·π‘’ξ…žπ‘–(π‘₯)ξ€Έξ€Έξ…ž+πœ†πΉπ‘’π‘–ξ€·π‘₯,𝑒1(π‘₯),…,𝑒𝑛(π‘₯)=0,(2.9) for almost every π‘₯∈(0,1). Then, by Lemmas 2.2 and 2.3, we observe 𝑒𝑖(π‘₯)=π‘šξ“π‘—=1π‘Žπ‘—π‘’π‘–ξ€·π‘₯𝑗+ξ€œπ‘₯0πœ™π‘βˆ’1𝑖𝑑𝑒,π‘–ξ€œβˆ’πœ†π›Ώ0𝐹𝑒𝑖𝑠,𝑒1(𝑠),…,𝑒𝑛(𝑠)𝑑𝑠𝑑𝛿,(2.10) for 1≀𝑖≀𝑛, where 𝑑𝑒,𝑖 is the unique solution of (2.5). Hence, π‘’π‘–βˆˆπΆ1([0,1]) and πœ™π‘π‘–(π‘’ξ…žπ‘–(π‘₯))∈𝐢1([0,1]) for 1≀𝑖≀𝑛, namely 𝑒=(𝑒1,…,𝑒𝑛) is a classical solution to the system (1.1).

For all 𝛾>0, we denote by 𝐾(𝛾) the set 𝑑1,…,π‘‘π‘›ξ€Έβˆˆπ‘…π‘›βˆΆπ‘›ξ“π‘–=1||𝑑𝑖||𝑝𝑖𝑝𝑖.≀𝛾(2.11) Now, we formulate our main result as follows.

Theorem 2.5. Assume that there exist 2π‘š constants π‘Žπ‘—,𝑏𝑗 for 1β‰€π‘—β‰€π‘š with βˆ‘π‘šπ‘—=1π‘Žπ‘—β‰ 1 and βˆ‘π‘šπ‘—=1𝑏𝑗≠1, a positive constant π‘Ÿ and a function 𝑀=(𝑀1,…,𝑀𝑛)βˆˆπ‘‹ such that (A1) βˆ‘π‘›π‘–=1(β€–π‘€ξ…žπ‘–β€–π‘π‘–π‘π‘–/𝑝𝑖)>π‘Ÿ, (A2)∫10sup(𝑑1,…,𝑑𝑛)∈𝐾(π‘˜π‘Ÿ)𝐹(π‘₯,𝑑1,…,π‘‘π‘›βˆ)𝑑π‘₯<(π‘Ÿπ‘›π‘–=1𝑝𝑖)∫10𝐹(π‘₯,𝑀1(π‘₯),…,π‘€π‘›βˆ‘(π‘₯))𝑑π‘₯/𝑛𝑖=1βˆπ‘›π‘—=1,π‘—β‰ π‘–π‘π‘—β€–π‘€ξ…žπ‘–β€–π‘π‘–π‘π‘–where 𝐾(π‘˜π‘Ÿ)={(𝑑1,…,π‘‘π‘›βˆ‘)|𝑛𝑖=1(|𝑑𝑖|𝑝𝑖/𝑝𝑖)≀kr}(see (2.11)),(A3)limsup|𝑑1|β†’+∞,…,|𝑑𝑛|β†’+∞𝐹(π‘₯,𝑑1,…,π‘‘π‘›βˆ‘)/𝑛𝑖=1(|𝑑𝑖|𝑝𝑖/π‘π‘–βˆ«)<10sup(𝑑1,…,𝑑𝑛)∈𝐾(π‘˜π‘Ÿ)×𝐹(π‘₯,𝑑1,…,𝑑𝑛)𝑑π‘₯/π‘˜π‘Ÿ uniformly with respect to π‘₯∈[0,1].Then, for each πœ†βˆˆΞ›π‘Ÿβˆ‘βˆΆ=]𝑛𝑖=1(‖𝑀′𝑖𝑝𝑖𝑝𝑖‖/π‘π‘–βˆ«)/10𝐹(π‘₯,𝑀1(π‘₯),…,π‘€π‘›βˆ«(π‘₯))𝑑π‘₯,π‘Ÿ/11sup(𝑑1,…,𝑑𝑛)∈𝐾(π‘˜π‘Ÿ)𝐹(π‘₯,𝑑1,…,𝑑𝑛)𝑑π‘₯[, 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 Ξ¦,Ξ¨βˆΆπ‘‹β†’β„ for each 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘‹, as follows: Ξ¦(𝑒)=𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–,ξ€œΞ¨(𝑒)=10𝐹π‘₯,𝑒1(π‘₯),…,𝑒𝑛(π‘₯)𝑑π‘₯.(2.12) Since 𝑝𝑖>1 for 1≀𝑖≀𝑛, 𝑋 is compactly embedded in (𝐢0([0,1]))𝑛 and it is well known that Ξ¦ and Ξ¨ are well defined and continuously differentiable functionals whose derivatives at the point 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘‹ are the functionals Ξ¦ξ…ž(𝑒), Ξ¨ξ…ž(𝑒)βˆˆπ‘‹βˆ—, given by Ξ¦ξ…žξ€œ(𝑒)(𝑣)=10𝑛𝑖=1||π‘’ξ…žπ‘–||(π‘₯)π‘π‘–βˆ’2π‘’ξ…žπ‘–(π‘₯)π‘£ξ…žπ‘–Ξ¨(π‘₯)𝑑π‘₯,ξ…žξ€œ(𝑒)(𝑣)=10𝑛𝑖=1𝐹𝑒𝑖π‘₯,𝑒1(π‘₯),…,𝑒𝑛𝑣(π‘₯)𝑖(π‘₯)𝑑π‘₯(2.13) for every 𝑣=(𝑣1,…,𝑣𝑛)βˆˆπ‘‹, respectively, as well as Ξ¨ is sequentially weakly upper semicontinuous. Furthermore, Lemma 2.6 of [11] gives that Ξ¦ξ…ž admits a continuous inverse on π‘‹βˆ—, and since Ξ¦β€² is monotone, we obtain that Ξ¦ is sequentially weakly lower semiacontinuous (see [29, Proposition 25.20]). Moreover, Ξ¨ξ…žβˆΆπ‘‹β†’π‘‹βˆ— is a compact operator. From assumption (A1), we get 0<π‘Ÿ<Ξ¦(𝑀). Since from (2.1) for each π‘’π‘–βˆˆπ‘‹π‘–, supπ‘₯∈[0,1]||𝑒𝑖||(π‘₯)π‘π‘–β€–β€–π‘’β‰€π‘˜ξ…žπ‘–β€–β€–π‘π‘–(2.14) for 𝑖=1,…,𝑛, we have sup𝑛π‘₯∈[0,1]𝑖=1||𝑒𝑖||(π‘₯)π‘π‘–π‘π‘–β‰€π‘˜π‘›ξ“π‘–=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–,(2.15) for each 𝑒=(𝑒1,…,𝑒𝑛)βˆˆπ‘‹, and so using (2.15), we observe Ξ¦βˆ’1(]]π‘’βˆ’βˆž,π‘Ÿ)=ξ€½ξ€·1,…,π‘’π‘›ξ€Έξ€·π‘’βˆˆπ‘‹;Ξ¦1,…,𝑒𝑛=ξƒ―ξ€·π‘’β‰€π‘Ÿ1,…,π‘’π‘›ξ€Έβˆˆπ‘‹;𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–ξƒ°βŠ†ξƒ―ξ€·π‘’β‰€π‘Ÿ1,…,π‘’π‘›ξ€Έβˆˆπ‘‹;𝑛𝑖=1||𝑒𝑖||(π‘₯)𝑝𝑖𝑝𝑖[]ξƒ°,β‰€π‘˜π‘Ÿβˆ€π‘₯∈0,1(2.16) and it follows that sup𝑒1,…,π‘’π‘›ξ€ΈβˆˆΞ¦βˆ’1(]])βˆ’βˆž,π‘ŸΞ¨ξ€·π‘’1,…,𝑒𝑛=sup𝑒1,…,π‘’π‘›ξ€ΈβˆˆΞ¦βˆ’1(]])βˆ’βˆž,π‘Ÿξ€œ10𝐹π‘₯,𝑒1(π‘₯),…,π‘’π‘›ξ€Έβ‰€ξ€œ(π‘₯)𝑑π‘₯10sup𝑑1,…,π‘‘π‘›ξ€ΈβˆˆπΎ(π‘˜π‘Ÿ)𝐹π‘₯,𝑑1,…,𝑑𝑛𝑑π‘₯.(2.17) Therefore, owing to assumption (A2), we have supπ‘’βˆˆΞ¦βˆ’1(]])βˆ’βˆž,π‘ŸΞ¨ξ€·π‘’1,…,𝑒𝑛=sup𝑒1,…,π‘’π‘›ξ€ΈβˆˆΞ¦βˆ’1(]])βˆ’βˆž,π‘Ÿξ€œ10𝐹π‘₯,𝑒1(π‘₯),…,π‘’π‘›ξ€Έβ‰€ξ€œ(π‘₯)𝑑π‘₯10sup𝑑1,…,π‘‘π‘›ξ€ΈβˆˆπΎ(π‘˜π‘Ÿ)𝐹π‘₯,𝑑1,…,𝑑𝑛<ξƒ©π‘Ÿπ‘‘π‘₯𝑛𝑖=1𝑝𝑖ξƒͺ∫10𝐹π‘₯,𝑀1(π‘₯),…,𝑀𝑛(π‘₯)𝑑π‘₯βˆ‘π‘›π‘–=1βˆπ‘›π‘—=1,π‘—β‰ π‘–π‘π‘—β€–β€–π‘€ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–βˆ«<π‘Ÿ10𝐹π‘₯,𝑀1(π‘₯),…,𝑀𝑛(π‘₯)𝑑π‘₯βˆ‘π‘›π‘–=1ξ€·β€–β€–π‘€ξ…žπ‘–β€–β€–π‘π‘–/𝑝𝑖=π‘ŸΞ¨(𝑀)Ξ¦.(𝑀)(2.18) Furthermore, from (A3), there exist two constants 𝛾,πœβˆˆβ„ with ∫0<𝛾<10sup(𝑑1,…,𝑑𝑛)∈𝐾(π‘˜π‘Ÿ)𝐹π‘₯,𝑑1,…,𝑑𝑛𝑑π‘₯π‘Ÿ,(2.19) such that ξ€·π‘˜πΉπ‘₯,𝑑1,…,𝑑𝑛≀𝛾𝑛𝑖=1||𝑑𝑖||𝑝𝑖𝑝𝑖[]𝑑+𝜐,βˆ€π‘₯∈0,1,βˆ€1,…,π‘‘π‘›ξ€Έβˆˆβ„π‘›.(2.20) Fix (𝑒1,…,𝑒𝑛)βˆˆπ‘‹, Then 𝐹π‘₯,𝑒1(π‘₯),…,𝑒𝑛≀1(π‘₯)π‘˜ξƒ©π›Ύπ‘›ξ“π‘–=1||𝑒𝑖||(π‘₯)𝑝𝑖𝑝𝑖ξƒͺ[].+πœβˆ€π‘₯∈0,1(2.21) So, for any fixed πœ†βˆˆΞ›π‘Ÿ, from (2.15) and (2.21), we have Ξ¦(𝑒)βˆ’πœ†Ξ¨(𝑒)=𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–ξ€œβˆ’πœ†10𝐹π‘₯,𝑒1(π‘₯),…,𝑒𝑛β‰₯(π‘₯)𝑑π‘₯𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–βˆ’πœ†π›Ύπ‘˜ξƒ©π‘›ξ“π‘–=11π‘π‘–ξ€œ10||𝑒𝑖||(π‘₯)𝑝𝑖ξƒͺβˆ’π‘‘π‘₯πœ†πœπ‘˜β‰₯𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–βˆ’πœ†π›Ύπ‘˜ξƒ©π‘˜π‘›ξ“π‘–=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–ξƒͺβˆ’πœ†πœπ‘˜=𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–βˆ’πœ†π›Ύπ‘›ξ“π‘–=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–βˆ’πœ†πœπ‘˜β‰₯ξƒ©π‘Ÿ1βˆ’π›Ύβˆ«10sup(𝑑1,…,𝑑𝑛)∈𝐾(π‘˜π‘Ÿ)𝐹π‘₯,𝑑1,…,𝑑𝑛ξƒͺ𝑑π‘₯𝑛𝑖=1β€–β€–π‘’ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–βˆ’πœ†πœπ‘˜,(2.22) and thus, lim‖‖𝑒1,…,𝑒𝑛‖‖→+βˆžξ€·Ξ¦ξ€·π‘’1,…,π‘’π‘›ξ€Έξ€·π‘’βˆ’πœ†Ξ¨1,…,𝑒𝑛=+∞,(2.23) which means that the functional Ξ¦βˆ’πœ†Ξ¨ is coercive. So, all assumptions of Theorem 2.1 are satisfied. Hence, from Theorem 2.1 with π‘₯=𝑀, taking into account that the weak solutions of the system (1.1) are exactly the solutions of the equation Ξ¦β€²(𝑒1,…,𝑒𝑛)βˆ’πœ†Ξ¨β€²(𝑒1,…,𝑒𝑛)=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 𝑀 is specified.

PutπœŽπ‘–=⎑⎒⎒⎣2π‘π‘–βˆ’1βŽ›βŽœβŽœβŽπ‘₯1βˆ’π‘π‘–1|||||1βˆ’π‘šξ“π‘—=1π‘Žπ‘—|||||𝑝𝑖+ξ€·1βˆ’π‘₯π‘šξ€Έ1βˆ’π‘π‘–|||||1βˆ’π‘šξ“π‘—=1𝑏𝑗|||||π‘π‘–βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦1/𝑝𝑖for1≀𝑖≀𝑛.(2.24)

Define 𝐡1,π‘›βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξƒ¬π‘₯(π‘₯)=π‘šξ“π‘—=1π‘Žπ‘—ξƒ­,π‘₯𝑛ifπ‘šξ“π‘—=1π‘Žπ‘—ξƒ¬<1,π‘₯,π‘₯π‘šξ“π‘—=1π‘Žπ‘—ξƒ­π‘›ifπ‘šξ“π‘—=1π‘Žπ‘—π΅>1,2,π‘›βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©ξƒ¬π‘₯(π‘₯)=π‘šξ“π‘—=1𝑏𝑗,π‘₯𝑛ifπ‘šξ“π‘—=1𝑏𝑗<1,π‘₯,π‘₯π‘šξ“π‘—=1𝑏𝑗𝑛ifπ‘šξ“π‘—=1𝑏𝑗>1,(2.25) where [β‹…,β‹…]𝑛=[β‹…,β‹…]Γ—β‹―Γ—[β‹…,β‹…], then we have the following consequence of Theorem 2.5.

Corollary 2.6. Assume that there exist 2π‘š constants π‘Žπ‘—,𝑏𝑗 for 1β‰€π‘—β‰€π‘š with βˆ‘π‘šπ‘—=1π‘Žπ‘—β‰ 1 and βˆ‘π‘šπ‘—=1𝑏𝑗≠1 and two positive constants πœƒ and 𝜏 with βˆ‘π‘›π‘–=1((πœŽπ‘–πœ)𝑝𝑖/π‘π‘–βˆ)>πœƒ/π‘˜π‘›π‘–=1𝑝𝑖 such that (B1)𝐹(π‘₯,𝑑1,…,𝑑𝑛)β‰₯0 for each π‘₯∈[0,π‘₯1/2]βˆͺ[(1+π‘₯π‘š)/2,1] and (𝑑1,…,𝑑𝑛)∈𝐡1,𝑛(𝜏)βˆͺ𝐡2,𝑛(𝜏),(B2)βˆ‘π‘›π‘–=1((πœŽπ‘–πœ)𝑝𝑖/𝑝𝑖)∫10sup(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(π‘₯,𝑑1,…,𝑑𝑛)𝑑π‘₯<β€‰β€‰βˆ(πœƒ/π‘˜π‘›π‘–=1𝑝𝑖)∫(1+π‘₯π‘šπ‘₯)/21/2𝐹(π‘₯,𝜏,…,𝜏)𝑑π‘₯, where πœŽπ‘– is given by (2.24) and ∏𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)={(𝑑1,…,π‘‘π‘›βˆ‘)βˆ£π‘›π‘–=1(|𝑑𝑖|𝑝𝑖/π‘π‘–βˆ)≀𝑐/𝑛𝑖=1𝑝𝑖}(see (2.11));(B3)limsup|𝑑1|β†’+∞,…,|𝑑𝑛|β†’+∞(𝐹(π‘₯,𝑑1,…,π‘‘π‘›βˆ‘)/𝑛𝑖=1(|𝑑𝑖|𝑝𝑖/𝑝𝑖))<β€‰β€‰βˆπ‘›π‘–=1𝑝𝑖/πœƒΓ—βˆ«10sup(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(π‘₯,𝑑1,…,𝑑𝑛)𝑑π‘₯ uniformly with respect to π‘₯∈[0,1].then, for each βˆ‘πœ†βˆˆ]𝑛𝑖=1((πœŽπ‘–πœ)𝑝𝑖/π‘π‘–βˆ«)/(1+π‘₯π‘šπ‘₯)/21/2∏𝐹(π‘₯,𝜏,…,𝜏)𝑑π‘₯,(πœƒ/π‘˜π‘›π‘–=1π‘π‘–βˆ«)/10sup(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(π‘₯,𝑑1,…,𝑑𝑛)𝑑π‘₯[ the systems (1.1) admits at least three distinct classical solutions.

Proof. Set 𝑀(π‘₯)=(𝑀1(π‘₯),…,𝑀𝑛(π‘₯)) such that for 1≀𝑖≀𝑛, π‘€π‘–βŽ§βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©πœξƒ©(π‘₯)=π‘šξ“π‘—=1π‘Žπ‘—+2ξ€·βˆ‘1βˆ’π‘šπ‘—=1π‘Žπ‘—ξ€Έπ‘₯1π‘₯ξƒͺifπ‘₯π‘₯∈0,12ξ‚„,𝜏ifξ‚Έπ‘₯π‘₯∈12,1+π‘₯π‘š2ξ‚Ή,πœξƒ©βˆ‘2βˆ’π‘šπ‘—=1π‘π‘—βˆ’π‘₯π‘šβˆ‘π‘šπ‘—=1𝑏𝑗1βˆ’π‘₯π‘šβˆ’2ξ€·βˆ‘1βˆ’π‘šπ‘—=1𝑏𝑗1βˆ’π‘₯π‘šπ‘₯ξƒͺifξ‚Έπ‘₯∈1+π‘₯π‘š2ξ‚Ή,,1(2.26) and βˆπ‘Ÿ=πœƒ/π‘˜π‘›π‘–=1𝑝𝑖. It is easy to see that 𝑀=(𝑀1,…,𝑀𝑛)βˆˆπ‘‹, and, in particular, one has ‖‖𝑀′𝑖‖‖𝑝𝑖𝑝𝑖=ξ€·πœŽπ‘–πœξ€Έπ‘π‘–,(2.27) for 1≀𝑖≀𝑛, which, employing the condition βˆ‘π‘›π‘–=1((πœŽπ‘–πœ)𝑝𝑖/π‘π‘–βˆ)>πœƒ/π‘˜π‘›π‘–=1𝑝𝑖, gives 𝑛𝑖=1β€–β€–π‘€ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–π‘π‘–>π‘Ÿ.(2.28) Since for 1≀𝑖≀𝑛, πœπ‘šξ“π‘—=1π‘Žπ‘—β‰€π‘€π‘–(π‘₯)β‰€πœforeachπ‘₯π‘₯∈0,12ξ‚„ifπ‘šξ“π‘—=1π‘Žπ‘—<1,πœβ‰€π‘€π‘–(π‘₯)β‰€πœπ‘šξ“π‘—=1π‘Žπ‘—foreachπ‘₯π‘₯∈0,12ξ‚„ifπ‘šξ“π‘—=1π‘Žπ‘—πœ>1,π‘šξ“π‘—=1𝑏𝑗≀𝑀𝑖(π‘₯)β‰€πœforeachξ‚Έπ‘₯∈1+π‘₯π‘š2ξ‚Ή,1ifπ‘šξ“π‘—=1𝑏𝑗<1,πœβ‰€π‘€π‘–(π‘₯)β‰€πœπ‘šξ“π‘—=1𝑏𝑗foreachξ‚Έπ‘₯∈1+π‘₯π‘š2ξ‚Ή,1ifπ‘šξ“π‘—=1𝑏𝑗>1,(2.29) the condition (B1) ensures that ξ€œπ‘₯10/2𝐹π‘₯,𝑀1(π‘₯),…,π‘€π‘›ξ€Έξ€œ(π‘₯)𝑑π‘₯+1ξ€·1+π‘₯π‘šξ€Έ/2𝐹π‘₯,𝑀1(π‘₯),…,𝑀𝑛(π‘₯)𝑑π‘₯β‰₯0.(2.30) Therefore, owing to assumption (B2), we have ξ€œ10sup𝑑1,…,π‘‘π‘›ξ€ΈβˆˆπΎ(π‘˜π‘Ÿ)𝐹π‘₯,𝑑1,…,π‘‘π‘›ξ€Έπœƒπ‘‘π‘₯<ξ€·βˆ‘π‘›π‘–=1πœŽξ€·ξ€·π‘–πœξ€Έπ‘π‘–/π‘π‘–π‘˜βˆξ€Έξ€Έξ€·π‘›π‘–=1π‘π‘–ξ€Έξ€œ(1+π‘₯π‘šπ‘₯)/21/2β‰€πœƒπΉ(π‘₯,𝜏,…,𝜏)𝑑π‘₯π‘˜βˆ«10𝐹π‘₯,𝑀1(π‘₯),…,𝑀𝑛(π‘₯)𝑑π‘₯βˆ‘π‘›π‘–=1βˆπ‘›π‘—=1,𝑗≠𝑖𝑝𝑗‖‖𝑀′𝑖‖‖𝑝𝑖𝑝𝑖=ξƒ©π‘Ÿπ‘›ξ‘π‘–=1𝑝𝑖ξƒͺ∫10𝐹π‘₯,𝑀1(π‘₯),…,𝑀𝑛(π‘₯)𝑑π‘₯βˆ‘π‘›π‘–=1βˆπ‘›π‘—=1,π‘—β‰ π‘–π‘π‘—β€–β€–π‘€ξ…žπ‘–β€–β€–π‘π‘–π‘π‘–,(2.31) where ∏𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)={(𝑑1,…,π‘‘π‘›βˆ‘)βˆ£π‘›π‘–=1(|𝑑𝑖|𝑝𝑖/π‘π‘–βˆ)β‰€πœƒ/𝑛𝑖=1𝑝𝑖}. Moreover, from assumption (B3) it follows that assumption (A3) is fulfilled. Hence, taking into account that ⎀βŽ₯βŽ₯βŽ¦βˆ‘π‘›π‘–=1πœŽξ€·ξ€·π‘–πœξ€Έπ‘π‘–/π‘π‘–ξ€Έβˆ«(1+π‘₯π‘šπ‘₯)/21/2,∏𝐹(π‘₯,𝜏,…,𝜏)𝑑π‘₯πœƒ/π‘˜π‘›π‘–=1π‘π‘–βˆ«10sup(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹π‘₯,𝑑1,…,π‘‘π‘›ξ€ΈβŽ‘βŽ’βŽ’βŽ£π‘‘π‘₯βŠ†Ξ›π‘Ÿ,(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π‘₯,𝑑1,𝑑2,𝑑3ξ€Έ=⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩0βˆ€π‘‘π‘–π‘₯<0,𝑖=1,2,2𝑑1100π‘’βˆ’π‘‘1for𝑑1β‰₯0,𝑑2<0,𝑑3π‘₯<0,2𝑑2100π‘’βˆ’π‘‘2for𝑑1<0,𝑑2β‰₯0,𝑑3π‘₯<0,2𝑑3100π‘’βˆ’π‘‘3for𝑑1<0,𝑑2<0,𝑑3π‘₯β‰₯0.23𝑖=1𝑑𝑖100π‘’βˆ’π‘‘π‘–for𝑑𝑖β‰₯0,𝑖=1,2,3,(2.33) for each (π‘₯,𝑑1,𝑑2,𝑑3)∈[0,1]×ℝ3. In fact, by choosing 𝑝1=𝑝2=𝑝3=3 and π‘Ž1=𝑏1=π‘₯1=1/2, by a simple calculation, we obtain that π‘˜=27/8 and 𝜎1=𝜎2=𝜎3=41/3, and so with πœƒ=9 and 𝜏=100, we observe that the assumptions (B1) and (B3) in Corollary 2.6 are satisfied. For (B2), 3𝑖=1ξ€·πœŽπ‘–πœξ€Έπ‘π‘–π‘π‘–ξ€œ10sup𝑑1,𝑑2,𝑑3ξ€Έξ‚€βˆβˆˆπΎπœƒ/3𝑖=1𝑝𝑖𝐹π‘₯,𝑑1,𝑑2,𝑑3𝑑π‘₯=4(100)3ξ€œ10sup𝑑1,𝑑2,𝑑3ξ€ΈβˆˆπΎ(1/3)𝐹π‘₯,𝑑1,𝑑2,𝑑3𝑑π‘₯≀4(100)3ξ€œ10sup𝑑1,𝑑2,𝑑3ξ€ΈβˆˆπΎ(1/3)π‘₯23𝑖=1𝑑𝑖100π‘’βˆ’π‘‘π‘–π‘‘π‘₯=4(100)3max(𝑑1,𝑑2,𝑑33)∈𝐾(1/3)𝑖=1𝑑𝑖100π‘’βˆ’π‘‘π‘–ξ€œ10π‘₯2≀4𝑑π‘₯3(100)3ξ‚΅3max|𝑑|≀1𝑑100π‘’βˆ’π‘‘ξ‚Ά=4(100)3𝑒<134Γ—34(100)100π‘’βˆ’100=πœƒπ‘˜βˆ3𝑖=1π‘π‘–ξ€œ(1+π‘₯1π‘₯)/21/2𝐹(π‘₯,𝜏,𝜏,𝜏)𝑑π‘₯.(2.34) So, for every ξ‚Ή4πœ†βˆˆ4(100)326(100)100π‘’βˆ’100,835(100)3𝑒,(2.35) Corollary 2.6 is applicable to the system βˆ’ξ‚€|||𝑒′1|||π‘’ξ…ž1′=πœ†π‘₯2𝑒+1ξ€Έ99π‘’βˆ’π‘’+1ξ€·100βˆ’π‘’+1ξ€Έinβˆ’ξ‚€|||𝑒(0,1),β€²2|||π‘’ξ…ž2′=πœ†π‘₯2𝑒+2ξ€Έ99π‘’βˆ’π‘’+2ξ€·100βˆ’π‘’+2ξ€Έinβˆ’ξ‚€|||𝑒(0,1),β€²3|||π‘’ξ…ž3′=πœ†π‘₯2𝑒+3ξ€Έ99π‘’βˆ’π‘’+3ξ€·100βˆ’π‘’+3ξ€Έin𝑒(0,1),𝑖(0)=𝑒𝑖1(1)=2𝑒𝑖12for𝑖=1,2,3,(2.36) where 𝑑+=max{𝑑,0}.

Here is a remarkable consequence of Corollary 2.6.

Corollary 2.8. Let πΉβˆΆβ„π‘›β†’β„ be a 𝐢1 function in 𝑅𝑛 such that 𝐹(0,…,0)=0. Assume that there exist 2π‘š constants π‘Žπ‘—,𝑏𝑗 for 1β‰€π‘—β‰€π‘š with βˆ‘π‘šπ‘—=1π‘Žπ‘—β‰ 1 and βˆ‘π‘šπ‘—=1𝑏𝑗≠1 and two positive constants πœƒ and 𝜏 with βˆ‘π‘›π‘–=1((πœŽπ‘–πœ)𝑝𝑖/π‘π‘–βˆ)>πœƒ/π‘˜π‘›π‘–=1𝑝𝑖 such that (C1)𝐹(𝑑1,…,𝑑𝑛)β‰₯0 for each (𝑑1,…,𝑑𝑛)∈𝐡1,𝑛(𝜏)βˆͺ𝐡2,𝑛(𝜏),(C2)βˆ‘π‘›π‘–=1((πœŽπ‘–πœ)𝑝𝑖/𝑝𝑖)max(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(𝑑1,…,𝑑𝑛)<(πœƒ(1+π‘₯π‘šβˆ’π‘₯1∏)/2π‘˜π‘›π‘–=1𝑝𝑖)𝐹(𝜏,…,𝜏)where πœŽπ‘– is given by (2.24),(C3)limsup|𝑑1|β†’+∞,…,|𝑑𝑛|β†’+∞(𝐹(𝑑1,…,π‘‘π‘›βˆ‘)/𝑛𝑖=1(|𝑑𝑖|𝑝𝑖/π‘π‘–βˆ))<(𝑛𝑖=1𝑝𝑖/πœƒ)Γ—max(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(𝑑1,…,𝑑𝑛),Then, for each βˆ‘πœ†βˆˆ]𝑛𝑖=1((πœŽπ‘–πœ)𝑝𝑖/𝑝𝑖)/((1+π‘₯π‘šβˆ’π‘₯1∏)/2)𝐹(𝜏,…,𝜏),(πœƒ/π‘˜π‘›π‘–=1𝑝𝑖)/max(𝑑1,…,π‘‘π‘›βˆ)∈𝐾(πœƒ/𝑛𝑖=1𝑝𝑖)𝐹(𝑑1,…,𝑑𝑛)[, the systems βˆ’ξ‚΅|||𝑒′𝑖|||π‘π‘–βˆ’2π‘’ξ…žπ‘–ξ‚Άξ…ž=πœ†πΉπ‘’π‘–ξ€·π‘’1,…,𝑒𝑛in𝑒(0,1),𝑖(0)=π‘šξ“π‘—=1π‘Žπ‘—π‘’π‘–ξ€·π‘₯𝑗,𝑒𝑖(1)=π‘šξ“π‘—=1𝑏𝑗𝑒𝑖π‘₯𝑗,(2.37) for 1≀𝑖≀𝑛, admits at least three distinct classical solutions.

Proof. Set 𝐹(π‘₯,𝑑1,…,𝑑𝑛)=𝐹(𝑑1,…,𝑑𝑛) for all π‘₯∈[0,1] and π‘‘π‘–βˆˆβ„ 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,π‘š=2,π‘₯1=1/3,π‘₯2=2/3 and π‘Žπ‘–=𝑏𝑖=1/3, 𝑖=1,2. Consider the system βˆ’ξ‚€|||𝑒′1|||π‘’ξ…ž1ξ‚ξ…žξ€·π‘’=πœ†βˆ’π‘’1𝑒111ξ€·12βˆ’π‘’1,ξ€Έξ€Έinβˆ’ξ‚€|||𝑒(0,1),β€²2|||π‘’ξ…ž2ξ‚ξ…žξ€·π‘’=πœ†βˆ’π‘’2𝑒92ξ€·10βˆ’π‘’2,ξ€Έξ€Έin𝑒(0,1),1(0)=𝑒11(1)=3𝑒1ξ‚€13+13𝑒1ξ‚€23,𝑒2(0)=𝑒21(1)=3𝑒2ξ‚€13+13𝑒2ξ‚€23.(2.38) Clearly, (H1) and (H2) hold. A simple calculation shows that π‘˜=125/8 and 𝜎1=𝜎2=(8/3)1/3. So, by choosing πœƒ=3 and 𝜏=10, we observe that the assumptions (C1) and (C3) in Corollary 2.8 hold. For (C3), since 𝐹(𝑑1,𝑑2)=𝑑112π‘’βˆ’π‘‘1+𝑑210π‘’βˆ’π‘‘2 for every (𝑑1,𝑑2)βˆˆβ„2, we have max(𝑑1,𝑑2∏)∈𝐾(πœƒ/2𝑖=1𝑝𝑖)𝐹𝑑1,𝑑2ξ€Έ=max𝑑1,𝑑2ξ€ΈβˆˆπΎ(1/3)𝐹𝑑1,𝑑2ξ€Έ=max𝑑1,𝑑2ξ€ΈβˆˆπΎ(1/3)𝑑112π‘’βˆ’π‘‘1+𝑑210π‘’βˆ’π‘‘2≀max||𝑑1||≀1𝑑112π‘’βˆ’π‘‘1+max||𝑑2||≀1𝑑210π‘’βˆ’π‘‘2ξƒ­<1=2𝑒125β‹…103ξ€·1012π‘’βˆ’10+1010π‘’βˆ’10ξ€Έ=πœƒξ€·1+π‘₯2βˆ’π‘₯1ξ€Έξ‚€βˆ‘2𝑖=1ξ€·ξ€·πœπœŽπ‘–ξ€Έπ‘π‘–/π‘π‘–ξ€Έβˆξ‚ξ‚€2π‘˜2𝑖=1𝑝𝑖𝐹(𝜏,𝜏).(2.39) Note that lim|𝑑1|β†’βˆž,|𝑑2|β†’βˆž(𝐹(𝑑1,𝑑2βˆ‘)/2𝑖=1(|𝑑𝑖|𝑝𝑖/𝑝𝑖))=0. We see that for every ξƒ­8πœ†βˆˆ3ξ€·109π‘’βˆ’10+107π‘’βˆ’10ξ€Έ,4,375𝑒(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 πœ†>πœ†βˆ—βˆΆ=((𝜎1𝜏)𝑝/𝑝+(𝜎2𝜏)π‘ž/π‘ž)/((1+π‘₯2βˆ’π‘₯1)/2)𝐹(𝜏,𝜏) for some 𝜏>0. Note that liminf(πœ‰,πœ‚)β†’(0,0)(𝐹(πœ‰,πœ‚)/(|πœ‰|𝑝/𝑝+|πœ‚|π‘ž/π‘ž))=0, and there is {πœƒπ‘›}π‘›βˆˆπ‘βŠ†]0,+∞[such that lim𝑛→+βˆžπœƒπ‘›=0 and lim𝑛→+∞max(πœ‰,πœ‚)∈𝐾(πœƒπ‘›/π‘π‘ž)𝐹(πœ‰,πœ‚)πœƒπ‘›=0.(2.41) In fact, one has lim𝑛→+∞(max(πœ‰,πœ‚)∈𝐾(πœƒπ‘›/π‘π‘ž)𝐹(πœ‰,πœ‚)/πœƒπ‘›)=lim𝑛→+∞(𝐹(πœ‰πœƒπ‘›,πœ‚πœƒπ‘›)/(|πœ‰πœƒπ‘›|𝑝/𝑝+|πœ‚πœƒπ‘›|π‘ž/π‘ž))β‹…(|πœ‰πœƒπ‘›|𝑝/𝑝+|πœ‚πœƒπ‘›|π‘ž/π‘ž)/πœƒπ‘›=0, where 𝐹(πœ‰πœƒπ‘›,πœ‚πœƒπ‘›)=sup(πœ‰,πœ‚)∈𝐾(πœƒπ‘›/π‘π‘ž)𝐹(πœ‰,πœ‚). Hence, there is πœƒ>0 such that max(πœ‰,πœ‚)∈𝐾(πœƒ/π‘π‘ž)𝐹(πœ‰,πœ‚)πœƒξƒ―ξ€·<min1+π‘₯2βˆ’π‘₯1ξ€Έ2ξ€·π‘žξ€·πœŽ1πœξ€Έπ‘ξ€·πœŽ+𝑝2πœξ€Έπ‘žξ€Έπ‘˜1𝐹(𝜏,𝜏);ξƒ°,πœ†π‘π‘žπ‘˜(2.42) and πœƒ<π‘˜(π‘ž(𝜎1𝜏)𝑝+𝑝(𝜎2𝜏)π‘ž).
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.

References

  1. B. Ricceri, β€œExistence of three solutions for a class of elliptic eigenvalue problems,” Mathematical and Computer Modelling, vol. 32, no. 11–13, pp. 1485–1494, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  2. B. Ricceri, β€œOn a three critical points theorem,” Archiv der Mathematik, vol. 75, no. 3, pp. 220–226, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  3. B. Ricceri, β€œA three critical points theorem revisited,” Nonlinear Analysis, vol. 70, no. 9, pp. 3084–3089, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. G. Bonannao, β€œExistence of three solutions for a two point boundary value problem,” Applied Mathematics Letters, vol. 13, no. 5, pp. 53–57, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  5. G. Bonanno, β€œSome remarks on a three critical points theorem,” Nonlinear Analysis, vol. 54, no. 4, pp. 651–665, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  6. G. Bonanno, β€œA critical points theorem and nonlinear differential problems,” Journal of Global Optimization, vol. 28, no. 3-4, pp. 249–258, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  7. P. Candito, β€œExistence of three solutions for a nonautonomous two point boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 532–537, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  8. X. He and W. Ge, β€œExistence of three solutions for a quasilinear two-point boundary value problem,” Computers & Mathematics with Applications, vol. 45, no. 4-5, pp. 765–769, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  9. Z. Du, X. Lin, and C. C. Tisdell, β€œA multiplicity result for p-Lapacian boundary value problems via critical points theorem,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 231–237, 2008. View at Publisher Β· View at Google Scholar
  10. Z. Du and L. Kong, β€œExistence of three solutions for systems of multi-point boundary value problems,” Electronic Journal of Qualitative Theory of Differential Equations, no. 10, article 17, 2009.
  11. J. R. Graef, S. Heidarkhani, and L. Kong, β€œA critical points approach to multiplicity results for multi-point boundary value problems,” Applicable Analysis, vol. 90, pp. 1909–1925, 2011.
  12. G. Bonanno and S. A. Marano, β€œOn the structure of the critical set of non-differentiable functions with a weak compactness condition,” Applicable Analysis, vol. 89, no. 1, pp. 1–10, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  13. J. R. Graef and L. Kong, β€œExistence of solutions for nonlinear boundary value problems,” Communications on Applied Nonlinear Analysis, vol. 14, no. 1, pp. 39–60, 2007. View at Zentralblatt MATH
  14. J. R. Graef, L. Kong, and Q. Kong, β€œHigher order multi-point boundary value problems,” Mathematische Nachrichten, vol. 284, no. 1, pp. 39–52, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  15. J. R. Graef and B. Yang, β€œMultiple positive solutions to a three point third order boundary value problem,” Discrete and Continuous Dynamical Systems. Series A, supplement, pp. 337–344, 2005. View at Zentralblatt MATH
  16. J. Henderson, β€œSolutions of multipoint boundary value problems for second order equations,” Dynamic Systems and Applications, vol. 15, no. 1, pp. 111–117, 2006. View at Zentralblatt MATH
  17. G. A. Afrouzi and S. Heidarkhani, β€œThree solutions for a quasilinear boundary value problem,” Nonlinear Analysis, vol. 69, no. 10, pp. 3330–3336, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  18. G. Bonanno, S. Heidarkhani, and D. O'Regan, β€œMultiple solutions for a class of Dirichlet quasilinear elliptic systems driven by a (p,q)-Laplacian operator,” Dynamic Systems and Applications, vol. 20, no. 1, pp. 89–99, 2011.
  19. G. Bonanno and G. Molica Bisci, β€œThree weak solutions for elliptic Dirichlet problems,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 1–8, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. G. Bonanno, G. Molica Bisci, and D. O'Regan, β€œInfinitely many weak solutions for a class of quasilinear elliptic systems,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 152–160, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. G. Bonanno, G. Molica Bisci, and V. Rădulescu, β€œExistence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces,” Nonlinear Analysis, vol. 74, no. 14, pp. 4785–4795, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  22. G. Bonanno, G. Molica Bisci, and V. Rădulescu, β€œMultiple solutions of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems,” Nonlinear Analysis, vol. 12, no. 5, pp. 2656–2665, 2011. View at Publisher Β· View at Google Scholar
  23. G. Bonanno and G. Riccobono, β€œMultiplicity results for Sturm-Liouville boundary value problems,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 294–297, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. S. Heidarkhani and D. Motreanu, β€œMultiplicity results for a two-point boundary value problem,” Panamerican Mathematical Journal, vol. 19, no. 3, pp. 69–78, 2009. View at Zentralblatt MATH
  25. S. Heidarkhani and Y. Tian, β€œMultiplicity results for a class of gradient systems depending on two parameters,” Nonlinear Analysis, vol. 73, no. 2, pp. 547–554, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. R. Livrea, β€œExistence of three solutions for a quasilinear two point boundary value problem,” Archiv der Mathematik, vol. 79, no. 4, pp. 288–298, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  27. S. A. Marano and D. Motreanu, β€œOn a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems,” Nonlinear Analysis, vol. 48, pp. 37–52, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  28. G. Bonanno and P. Candito, β€œNon-differentiable functionals and applications to elliptic problems with discontinuous nonlinearities,” Journal of Differential Equations, vol. 244, no. 12, pp. 3031–3059, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  29. E. Zeidler, Nonlinear Functional Analysis and Its Applications, vol. 1–3, Springer, New York, NY, USA, 1985.