Multiplicity of Nontrivial Solutions for a Class of Nonlocal Elliptic Operators Systems of Kirchhoff Type

Abstract

We investigate the existence and multiplicity of nontrivial solutions for a Kirchhoff type problem involving the nonlocal integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tool used for obtaining our result is Morse theory.

1. Introduction

This paper is concerned with the multiplicity of solutions to the following elliptic systems of Kirchhoff type involving the nonlocal integrodifferential operators: −𝑀_1ℝ2𝑛𝑢(𝑥)−𝑢𝑦²𝐾₁𝑥−𝑦𝑑𝑥𝑑𝑦ℒ_𝐾1𝑢=𝑓(𝑥,𝑣)inΩ,−𝑀_2ℝ2𝑛𝑣(𝑥)−𝑣𝑦²𝐾₂𝑥−𝑦𝑑𝑥𝑑𝑦ℒ_𝐾2𝑣=𝑔(𝑥,𝑢)inΩ,𝑢=𝑣=0inℝ^𝑛⧵Ω, (1) where Ω⊂ℝ^𝑛 (𝑛≥2) is a bounded domain with smooth boundary 𝜕Ω and 𝑓,𝑔∶Ω×ℝ→ℝ are two continuous functions. 𝑀_𝑖∶ℝ⁺→ℝ⁺ (𝑖=1,2) are two continuous functions whose properties will be introduced later. ℒ_𝐾𝑖 (𝑖=1,2) are the nonlocal operators defined byℒ_𝐾𝑖𝑢(𝑥)=12_ℝ^𝑛𝑢𝑥+𝑦+𝑢𝑥−𝑦−2𝑢(𝑥)𝐾_𝑖𝑦𝑑𝑦,𝑥∈ℝ^𝑛, (2)𝑖=1,2; here 𝐾_𝑖∶ℝ^𝑛⧵{0}→(0,+∞) is a function such that 𝑚𝐾_𝑖∈𝐿¹ℝ^𝑛,𝑖=1,2,where𝑚(𝑥)=min|𝑥|²,1; (3) there exist 𝜃_𝑖 and 𝑠_𝑖∈(0,1) (𝑖=1,2) such that 𝐾_𝑖(𝑥)≥𝜃_𝑖|𝑥|^{−(𝑛+2𝑠𝑖)},forany𝑥∈ℝ^𝑛⧵{0};𝐾_𝑖(𝑥)=𝐾_𝑖(−𝑥),∀𝑥∈ℝ^𝑛⧵{0}. (4)(5) A typical example for 𝐾_𝑖 is given by 𝐾_𝑖(𝑥)=|𝑥|^{−(𝑛+2𝑠𝑖)} (𝑖=1,2). In this case ℒ_𝐾𝑖 is the fractional Laplace operator −(−Δ)^𝑠𝑖, where −(−Δ)^𝑠𝑖 is defined by −(−Δ)^𝑠𝑖𝑢(𝑥)=12_ℝ^𝑛𝑢𝑥+𝑦+𝑢𝑥−𝑦−2𝑢(𝑥)𝑦^{𝑛+2𝑠𝑖}𝑑𝑦,𝑥∈ℝ^𝑛; (6) here 𝑠_𝑖∈(0,1) and 𝑛>2𝑠_𝑖 (𝑖=1,2). The fractional Laplacian −(−Δ)^𝑠𝑖 is a classical linear integrodifferential operator of order 2𝑠_𝑖 which gives the standard Laplacian when 𝑠_𝑖=1 (see [1]).

Denote by 𝑋_𝑖 the linear space of Lebesgue measurable functions 𝑢∶ℝ^𝑛→ℝ such that themap𝑥,𝑦↣𝑢(𝑥)−𝑢𝑦²𝐾_𝑖𝑥−𝑦isin𝐿¹𝑄,𝑑𝑥𝑑𝑦, (7) where 𝑄=(ℝ^𝑛×ℝ^𝑛)⧵𝒪 and 𝒪=(𝒞Ω)×(𝒞Ω)⊂ℝ^𝑛×ℝ^𝑛. The space 𝑋_𝑖 is endowed with the norm ‖𝑢‖_𝑋𝑖=‖𝑢‖_𝐿²_(ℝ^𝑛₎+_{ℝ𝑛×ℝ𝑛}𝑢(𝑥)−𝑢𝑦²𝐾_𝑖𝑥−𝑦𝑑𝑥𝑑𝑦^1/2,𝑖=1,2. (8)

The space 𝑍_𝑖 denotes the closure of 𝐶^∞₀(Ω) in 𝑋_𝑖. By Lemmas 6 and 7 in [2], the space 𝑍_𝑖 is a Hilbert space which can be endowed with the norm defined as ‖𝑢‖_𝑍𝑖=_𝑄𝑢(𝑥)−𝑢𝑦²𝐾_𝑖𝑥−𝑦𝑑𝑥𝑑𝑦^1/2,𝑖=1,2. (9) Since 𝑢=0 a.e. in ℝ^𝑛⧵Ω, we have that the integral in (8) and (9) can be extended to all ℝ^2𝑛.

Let 𝐸=𝑍₁×𝑍₂ be the Cartesian product of two Hilbert spaces, which is a reflexive Banach space endowed with the norm ‖(𝑢,𝑣)‖=‖𝑢‖_𝑍1+‖𝑣‖_𝑍2. (10)

Denote by 0<𝜆₁<𝜆₂≤⋯≤𝜆_𝑘≤⋯ the eigenvalues of the following nonlocal operator eigenvalue problem: −ℒ_𝐾1𝑢=𝜆𝑢inΩ𝑢=0inℝ^𝑛⧵Ω. (11) Similarly, denote by 0<𝜇₁<𝜇₂≤⋯≤𝜇_𝑘≤⋯ the eigenvalues of the following nonlocal operator eigenvalue problem: −ℒ_𝐾2𝑣=𝜇𝑣inΩ,𝑣=0inℝ^𝑛⧵Ω. (12)

We say that (𝑢,𝑣)∈𝐸 is a weak solution of system (1) if, for every (𝜙,𝜓)∈𝐸, one has 𝑀₁‖𝑢‖²_{𝑍1ℝ2𝑛}𝑢(𝑥)−𝑢𝑦𝜙(𝑥)−𝜙𝑦×𝐾₁𝑥−𝑦𝑑𝑥𝑑𝑦+𝑀₂‖𝑣‖²_𝑍2×_ℝ2𝑛𝑣(𝑥)−𝑣𝑦𝜓(𝑥)−𝜓𝑦𝐾₂−𝑥−𝑦𝑑𝑥𝑑𝑦_Ω𝑓(𝑥,𝑣)𝜙(𝑥)𝑑𝑥−_Ω𝑔(𝑥,𝑢)𝜓(𝑥)𝑑𝑥=0. (13)

The fractional Laplacian and nonlocal operators of elliptic type arise in both pure mathematical research and concrete applications, since these operators occur in a quite natural way in many different contexts. For an elementary introduction to this topic, see [2] and the references therein. Recently, some elliptic boundary problems driven by the nonlocal integrodifferential operator ℒ_𝐾 have been studied in the works [3–8].

Recently, problems involving Kirchhoff type operators have been studied in many papers; we refer to [9–13] in which the authors have used the variational method and topological method to get the existence of solutions.

In this paper, motivated by the above mentioned works, we will use Morse theory to investigate the multiplicity of solutions of problem (1). To the best of our knowledge, there is no effort being made in the literature to study the existence of solutions for problem (1). This paper will make some contribution to this research field.

In order to establish solutions for problem (1), we make the following assumptions.(H1)𝑀_𝑖∶ℝ⁺→ℝ⁺ (𝑖=1,2) are two continuous functions, and there exist constants 𝑚₁, 𝑚₂, 𝑀₁, 𝑀₂>0 such that 𝑚_𝑖≤𝑀_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,∀𝑡≥0. (14)(H2)𝑓(𝑥,𝑣) and 𝑔(𝑥,𝑢) are two continuous functions with the subcritical growth; that is, there exist some positive constants 𝐶₁, 𝐶₂ such that 𝑓(𝑥,𝑣)≤𝐶₁1+|𝑣|^𝑝−1,𝑔(𝑥,𝑢)≤𝐶₂1+|𝑢|^𝑞−1,∀𝑥∈Ω,𝑢,𝑣∈ℝ (15) hold, where 1<𝑝<2^∗_𝑠1=2𝑛/(𝑛−2𝑠₁), 1<𝑞<2^∗_𝑠2=2𝑛/(𝑛−𝑠₂).(H3)There exists 𝑟>0, 𝜆∈(𝜆₁,𝜆₂) and 𝜇∈(𝜇₁,𝜇₂) such that 𝑀₁𝜆₁<𝑚₁𝜆, 𝑀₂𝜇₁<𝑚₂𝜇, and |𝑢|, |𝑣|≤𝑟 implies 12𝑀₂𝜇₁𝑣²()≤1≤𝐹𝑥,𝑣2𝑚₂𝜇𝑣²,12𝑀₁𝜆₁𝑢²()≤1≤𝐺𝑥,𝑢2𝑚₁𝜆𝑢²,a.e.𝑥∈Ω. (16)(H4)lim_|𝑣|→∞(𝐹(𝑥,𝑣)/𝑣²)<(1/2)𝑚₂𝜇₁, lim_|𝑢|→∞(𝐺(𝑥,𝑢)/𝑢²)<(1/2)𝑚₁𝜆₁, uniformly for all a.e. 𝑥∈Ω.

The main result of this paper is as follows.

Theorem 1. If (H1)–(H4) hold, then the problem (1) has at least two nontrivial weak solutions in 𝐸.

2. Preliminaries

For each (𝑢,𝑣)∈𝐸, we define the functional 𝒥∶𝐸→ℝ as follows: 𝒥()=1𝑢,𝑣2𝑀₁‖𝑢‖²_𝑍1+12𝑀₂‖𝑣‖²_𝑍2−_Ω𝐹(𝑥,𝑣)𝑑𝑥−_Ω𝐺(𝑥,𝑢)𝑑𝑥, (17) where 𝑀_𝑖(𝑡)=^𝑡₀𝑀_𝑖(𝜏)𝐹()=𝑑𝜏,𝑖=1,2,𝑡≥0,𝑥,𝑣^𝑣₀𝑓(𝑥,𝑠)𝑑𝑠,𝐺(𝑥,𝑢)=^𝑢₀𝑔(𝑥,𝑠)𝑑𝑠. (18)

It is easy to check that (𝑢,𝑣) is a weak solution of problem (1) which is equivalent to being a critical point of the functional 𝒥.

First let us recall the definition of the local linking which plays an important role in our paper.

Definition 2. Let 𝑋 be a Banach space with a direct sum decomposition 𝑋=𝑋¹⊕𝑋². The functional 𝑓∈𝐶¹(𝑋,ℝ) has a local linking at 0 with respect to (𝑋¹,𝑋²) if there is 𝑟>0 such that 𝑓(𝑢)≥0,∀𝑢∈𝑋¹‖𝑢‖𝑓(𝑢)with≤𝑟≤0,∀𝑢∈𝑋²with‖𝑢‖≤𝑟. (19)

Lemma 3. Assume that (H1) and (H4) hold; then the functional 𝒥 is coercive in 𝐸; that is, 𝒥(𝑢,𝑣)→+∞ as ‖(𝑢,𝑣)‖→∞.

Proof. From (H4) and the continuity of the potentials 𝐹 and 𝐺 we have that, for some 𝜖>0, there exists a positive constant 𝐶₃ such that 𝐹()≤𝑚𝑥,𝑡²2𝜇₁−𝜖|𝑡|²+𝐶₃,𝐺()≤𝑚𝑥,𝑡¹2𝜆₁−𝜖|𝑡|²+𝐶₃,∀𝑡∈ℝ,a.e.𝑥∈Ω. (20) Thus, by the Sobolev inequality [1] and (H1), for (𝑢,𝑣)∈𝐸, we obtain 𝒥()≥𝑚𝑢,𝑣¹2‖𝑢‖²_𝑍1+𝑚₂2‖𝑣‖²_𝑍2−𝑚₁𝜆₁−𝜖2_Ω𝑢²−𝑚𝑑𝑥₂𝜇₁−𝜖2_Ω𝑣²𝑑𝑥−2𝐶₃|Ω|≥𝑚₁2𝜆1−¹−𝜖𝜆₁‖𝑢‖²_𝑍1+𝑚₂2𝜇1−¹−𝜖𝜇₁‖𝑣‖²_𝑍2−2𝐶₃|Ω|⟶+∞, (21) as ‖(𝑢,𝑣)‖→∞. Hence, we have that 𝒥 is coercive in 𝐸.

Lemma 4. If (H1), (H2), and (H4) hold, then 𝒥 satisfies the (𝑃.𝑆.) condition.

Proof. Let {𝑧_𝑛=(𝑢_𝑛,𝑣_𝑛)} be a (PS) sequence of 𝒥; then {(𝑢_𝑛,𝑣_𝑛)} must be bounded by Lemma 3. Passing to a subsequence if necessary, there exists 𝑧=(𝑢,𝑣)∈𝐸 such that (𝑢_𝑛,𝑣_𝑛)⇀(𝑢,𝑣) weakly in 𝐸. Thus, there exists a strictly decreasing subsequence 𝜖_𝑛, lim_𝑛→∞𝜖_𝑛=0, such that 𝒥^′𝑢_𝑛,𝑣_𝑛𝑢_𝑛−𝑢,0≤𝜖_𝑛𝑢_𝑛−𝑢,0. (22) In particular, 𝑀₁𝑢_𝑛²_𝑍1×_ℝ2𝑛𝑢_𝑛(𝑥)−𝑢_𝑛𝑦×𝑢_𝑛−𝑢(𝑥)−𝑢_𝑛−𝑢𝑦𝐾₁−𝑥−𝑦𝑑𝑥𝑑𝑦_Ω𝑓𝑥,𝑣_𝑛𝑢_𝑛−𝑢𝑑𝑥≤𝜖_𝑛𝑢_𝑛−𝑢,0. (23) Since the potential 𝐹 satisfies (H2) and by remark (3.2.24) in [14] we have _Ω𝑓𝑥,𝑣_𝑛𝑢_𝑛−𝑢𝑑𝑥⟶0. (24) Combining (23) with (24), we obtain 𝑚_1ℝ2𝑛𝑢_𝑛(𝑥)−𝑢_𝑛𝑦𝑢_𝑛−𝑢(𝑥)−𝑢_𝑛−𝑢𝑦×𝐾₁≤𝑀𝑥−𝑦𝑑𝑥𝑑𝑦₁𝑢_𝑛²_{𝑍1ℝ2𝑛}𝑢_𝑛(𝑥)−𝑢_𝑛𝑦×𝑢_𝑛−𝑢(𝑥)−𝑢_𝑛−𝑢𝑦×𝐾₁𝑥−𝑦𝑑𝑥𝑑𝑦⟶0. (25) On the other hand, we have lim_{𝑛→∞ℝ2𝑛}𝑢(𝑥)−𝑢𝑦𝑢_𝑛−𝑢(𝑥)−𝑢_𝑛−𝑢𝑦×𝐾₁𝑥−𝑦𝑑𝑥𝑑𝑦=0. (26) Adding (25) to (26), we conclude that 0=lim_{𝑛→∞ℝ2𝑛}𝑢_𝑛(𝑥)−𝑢_𝑛𝑦²𝐾₁−𝑥−𝑦𝑑𝑥𝑑𝑦_ℝ2𝑛𝑢(𝑥)−𝑢𝑦²𝐾₁𝑥−𝑦𝑑𝑥𝑑𝑦, (27) which implies ‖𝑢_𝑛‖²_𝑍1→‖𝑢‖²_𝑍1. So, ‖𝑢_𝑛‖_𝑍1→‖𝑢‖_𝑍1.
Similarly, we can obtain that ‖𝑣_𝑛‖_𝑍2→‖𝑣‖_𝑍2. The uniform convexity of 𝐸 yields that {𝑧_𝑛} converges strongly to 𝑧 in 𝐸.
Thanks to the fact that 𝐿^2∗𝑠1(Ω)↪𝐿^𝑝1(Ω) (2<𝑝₁<2^∗_𝑠1) continuously, we get by Lemma 6 in [2] and (4) that ‖𝑢‖_𝐿^𝑝_1(Ω)≤|Ω|^{(2∗𝑠1−𝑝1)/(𝑝12∗𝑠1)}‖𝑢‖_𝐿2^∗_𝑠1(Ω)≤|Ω|^{(2∗𝑠1−𝑝1)/(𝑝12∗𝑠1)}×√𝑐_1ℝ2𝑛𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑛+2𝑠1𝑑𝑥𝑑𝑦^1/2≤𝐶₄‖𝑢‖_𝑍1, (28) where 𝐶₄=|Ω|^{(2∗𝑠1−𝑝1)/(𝑝12∗𝑠1)}𝑐₁/𝜃₁. Similarly, for 2<𝑞₁<2^∗_𝑠2, there exists a constant 𝐶₅>0 such that ‖𝑣‖_𝐿^𝑞_1(Ω)≤𝐶₅‖𝑣‖_𝑍2. (29)
In the following, set 𝑈=span{𝜑₁}×span{𝜓₁}∶=⟨𝜑₁⟩×⟨𝜓₁⟩, where 𝜑₁>0 with ‖𝜑₁‖_𝑍1=1 is the corresponding eigenfunction of 𝜆₁ and 𝜓₁>0 with ‖𝜓₁‖_𝑍2=1 is the corresponding eigenfunction of 𝜇₁. Eigenvalues 𝜆₁ and 𝜇₁ are as in (11) and (12), respectively. Taking 𝑉=(𝑢,𝑣)∈𝐸∶𝑢∈𝜑₁^⟂,𝑣∈𝜓₁^⟂, (30) we can easily know that 𝑉 is complementary subspace of 𝑈. Hence we have the following direct sum: 𝐸=𝑈⊕𝑉. (31)
If (𝑢,𝑣)∈𝑈, from Proposition 9 in [4], we get ‖𝑢‖²_𝑍1=𝜆_1Ω|𝑢(𝑥)|²𝑑𝑥,‖𝑣‖²_𝑍2=𝜇_1Ω|𝑣(𝑥)|²𝑑𝑥. (32) Moreover, if (𝑢,𝑣)∈𝑉, by Proposition 9 in [4], we have ‖𝑢‖²_𝑍1≥𝜆_2Ω|𝑢(𝑥)|²𝑑𝑥,‖𝑣‖²_𝑍2≥𝜇_2Ω|𝑣(𝑥)|²𝑑𝑥. (33)

Lemma 5. Assume that (H1)–(H3) hold. Then the functional 𝒥 has a local linking at the origin with respect to 𝐸=𝑈⊕𝑉.

Proof. (i) Let (𝑢,𝑣)∈𝑈. Since ‖(𝑢,𝑣)‖⟶0⟹_Ω|𝑢(𝑥)|²𝑑𝑥⟶0,_Ω|𝑣(𝑥)|²⟶0 (34) by (32), we have that, for given 𝑟>0, there is some 𝜌>0 small enough such that (𝑢,𝑣)∈𝑈,‖(𝑢,𝑣)‖≤𝜌⟹|𝑢(𝑥)|≤𝑟,|𝑣(𝑥)|≤𝑟,a.e.𝑥∈Ω. (35) Now on 𝑈, we have by (H1) and (H3) that, for (𝑢,𝑣)∈𝑈 with ‖(𝑢,𝑣)‖≤𝜌, 𝒥()=1𝑢,𝑣2𝑀₁‖𝑢‖²_𝑍1+12𝑀₂‖𝑣‖²_𝑍2−_Ω𝐹(𝑥,𝑣)𝑑𝑥−_Ω𝐺()≤𝑀𝑥,𝑢𝑑𝑥₁2𝜆_1Ω|𝑢|²𝑀𝑑𝑥+²2𝜇_1Ω|𝑣|²−𝑑𝑥_Ω𝐹(𝑥,𝑣)𝑑𝑥−_Ω𝐺()=𝑥,𝑢𝑑𝑥_{|𝑢|≤𝑟}12𝑀₁𝜆₁|𝑢|²(+−𝐺𝑥,𝑢)𝑑𝑥_{|𝑣|≤𝑟}12𝑀₂𝜇₁|𝑣|²−𝐹(𝑥,𝑣)𝑑𝑥≤0. (36)
(ii) Let (𝑢,𝑣)∈𝑉. By (33), similar to (34) and (35), we obtain by (H1)–(H3) that, for (𝑢,𝑣)∈𝑉 with ‖(𝑢,𝑣)‖≤𝜌, 𝒥()=1𝑢,𝑣2𝑀₁‖𝑢‖²_𝑍1+12𝑀₂‖𝑣‖²_𝑍2−12𝑚₁𝜆_Ω𝑢²−1𝑑𝑥2𝑚₂𝜇_Ω𝑣²𝑑𝑥−_{{|𝑣|≤𝑟}}𝐹()−1𝑥,𝑣2𝑚₂𝜇|𝑣|²−𝑑𝑥_{{|𝑣|>𝑟}}𝐹()−1𝑥,𝑣2𝑚₂𝜇|𝑣|²−𝑑𝑥_{{|𝑢|≤𝑟}}𝐺()−1𝑥,𝑢2𝑚₁𝜆|𝑢|²−𝑑𝑥_{{|𝑢|>𝑟}}𝐺()−1𝑥,𝑢2𝑚₁𝜆|𝑢|²≥𝑚𝑑𝑥₁21−𝜆𝜆₂‖𝑢‖²_𝑍1+𝑚₂21−𝜇𝜇₂‖𝑣‖²_𝑍2−_{{|𝑣|>𝑟}}𝐹()−1𝑥,𝑣2𝑚₂𝜇|𝑣|²−𝑑𝑥_{{|𝑢|>𝑟}}𝐺()−1𝑥,𝑢2𝑚₁𝜆|𝑢|²≥𝑚𝑑𝑥₁21−𝜆𝜆₂‖𝑢‖²_𝑍1+𝑚₂21−𝜇𝜇₂‖𝑣‖²_𝑍2−𝐶_{6{|𝑣|>𝑟}}|𝑣|^𝑝2𝑑𝑥−𝐶_{7{|𝑢|>𝑟}}|𝑢|^𝑞2≥𝑚𝑑𝑥₁21−𝜆𝜆₂‖𝑢‖²_𝑍1+𝑚₂21−𝜇𝜇₂‖𝑣‖²_𝑍2−𝐶₈‖𝑢‖^𝑝2_𝑍1−𝐶₉‖𝑣‖^𝑞2_𝑍2,by(28)-(29), (37) where 𝐶_𝑖 (𝑖=6,…,9) are positive constants, 2<𝑝₂<2^∗_𝑠1, and 2<𝑞₂<2^∗_𝑠2. Thus, (37) implies that 𝒥(𝑢,𝑣)>0 for 0<‖(𝑢,𝑣)‖≤𝜌 with 𝜌>0 is small enough. The proof is complete.

Let 𝑋 be a real Banach space and 𝑓∈𝐶¹(𝑋,ℝ). Suppose 𝑝 is an isolated critical point of 𝑓 with 𝑓(𝑝)=𝑐 and 𝑈 is a neighborhood of 𝑝, containing the unique critical point; the group 𝐶_𝑞𝑓,𝑝=𝐻_𝑞𝑓_𝑐∩𝑈,𝑓_𝑐∩𝑈⧵𝑝,𝑞=0,1,2,…, (38) is called the 𝑞th critical group of 𝑓 at 𝑝, where 𝑓_𝑐={𝑢∈𝑋∶𝑓(𝑢)≤𝑐} and 𝐻_𝑞(⋅,⋅) is the 𝑞th singular relative homology group with integer coefficients.

Lemma 6 (see [15]). Let 𝐸 be a Banach space and 𝑓∶𝐸→ℝ a 𝐶¹-functional satisfying the (P.S) condition. Assume that 𝑓 has a local linking to the decomposition 𝐸=𝑈⊕𝑉 near the origin, where dim𝑈=𝑚<∞. If 0∈𝐸 is the unique critical point of 𝑓 in 𝐵_𝜌, then 𝐶_𝑚𝑓,0=𝐻_𝑚𝑓_𝑐∩𝐵_𝜌,𝑓_𝑐∩𝐵_𝜌⧵{0}≠0. (39)

3. The Proof of Theorem 1

We say that 𝑢 is a homological nontrivial critical point of 𝑓 if at least one of its critical groups is nontrivial. By [16], we have the following abstract critical point theorem.

Lemma 7 (see [16]). Let 𝑋 be a real Banach space and let Φ∈𝐶¹(𝑋,ℝ) satisfy the (𝑃.𝑆) condition and be bounded from below. If Φ has a critical point that is homologically nontrivial and is not the minimizer of Φ, then Φ has at least three critical points.

From the proof of Lemma 3, we can conclude that (0,0)∈𝐸 is the unique critical point of our 𝒥 in a ball that is small enough. Since dim𝑈=dim⟨𝜑₁⟩×⟨𝜓₁⟩=2<∞, by Lemmas 5 and 6, we have the following lemma.

Lemma 8. Let (H1)–(H3) hold. Then (0,0) is a critical point of 𝒥 and 𝐶₂(𝒥,(0,0))≠0.

Proof of Theorem 1. By Lemmas 3 and 4, 𝒥 is coercive and satisfies the (P.S) condition. Hence 𝒥 is bounded below. By Lemma 8, (0,0)∈𝐸 is homologically nontrivial critical point of 𝒥 but not a minimizer. Then the conclusion follows from Lemma 7.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their valuable and helpful suggestions and comments that improved the paper. This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and the Natural Science Foundation of China (11271364 and 10771212).