Three Weak Solutions for Nonlocal Fractional Laplacian Equations

Abstract

The existence of three weak solutions for the following nonlocal fractional equation (−Δ)^𝑠𝑢−𝜆𝑢=𝜇𝑓(𝑥,𝑢) in Ω,𝑢=0 in ℝ^𝑛⧵Ω, is investigated, where 𝑠∈(0,1) is fixed, (−Δ)^𝑠 is the fractional Laplace operator, 𝜆 and 𝜇 are real parameters, Ω is an open bounded subset of ℝ^𝑛, 𝑛>2𝑠, and the function 𝑓 satisfies some regularity and natural growth conditions. The approach is based on a three-critical-point theorem for differential functionals.

1. Introduction

In this work we investigate the existence of three weak solutions to the nonlocal counterpart of perturbed semilinear elliptic partial differential equations of the type−Δ𝑢−𝜆𝑢=𝜇𝑓(𝑥,𝑢)inΩ,𝑢=0inℝ^𝑛⧵Ω,(1) namely, (−Δ)^𝑠𝑢−𝜆𝑢=𝜇𝑓(𝑥,𝑢)inΩ,𝑢=0inℝ^𝑛⧵Ω,(2) where 𝑠∈(0,1) is fixed, Ω is a nonempty bounded open subset of ℝ^𝑛, 𝑛>2𝑠, 𝜆 and 𝜇 are positive real parameters, 𝑓∶Ω×ℝ→ℝ is a function satisfying suitable regularity and growth conditions, and (−Δ)^𝑠 is the fractional Laplace operator defined as −(−Δ)^𝑠𝑢(𝑥)∶=P.V._ℝ𝑛𝑢𝑥+𝑦+𝑢𝑥−𝑦−2𝑢(𝑥)𝑦^𝑛+2𝑠𝑑𝑦,𝑥∈ℝ^𝑛.(3)

Fractional Laplace operators have been proved to be valuable tools in the modeling of many phenomena in various fields, such as minimal surfaces, quasi-geostrophic flows, conservation laws, optimization, multiple scattering, anomalous diffusion, ultrarelativistic limits of quantum mechanics, finance, phase transitions, stratified materials, crystal dislocation, semipermeable membranes, flame propagation, soft thin films, and materials science. Recently, there has been significant development in fractional Laplace operators; for examples, see [1–13] and the references therein.

Motivated and inspired by the papers [13–15], in this paper, a variational approach is provided to investigate the existence of three weak solutions to a perturbed nonlocal fractional Laplacian equation (2), by using a three-critical-point theorem obtained by Bonanno and Marano in [14].

2. Preliminaries

Let 𝑠∈(0,1) such that 2𝑠<𝑛, Ω⊂ℝ^𝑛. The classical fractional Sobolev space 𝐻^𝑠(ℝ^𝑛) is defined by 𝐻^𝑠ℝ^𝑛=𝑣∈𝐿²ℝ^𝑛∶𝑣(𝑥)−𝑣𝑦𝑥−𝑦^{(𝑛+2𝑠)/2}∈𝐿²ℝ^𝑛×ℝ^𝑛,(4) endowed with the norm (the so-called Gagliardo norm) ‖𝑣‖_𝐻^𝑠_(ℝ^𝑛₎=‖𝑣‖_𝐿²_(ℝ^𝑛₎+_{ℝ𝑛×ℝ𝑛}𝑣(𝑥)−𝑣𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦^1/2.(5) Let 𝑋₀=𝑔∈𝐻^𝑠ℝ^𝑛∶𝑔=0a.e.inℝ^𝑛⧵Ω.(6) By [6] in the sequel we can take the function 𝑋₀∋𝑣↦‖𝑣‖_𝑋0=_{(ℝ𝑛×ℝ𝑛)⧵𝒪}𝑣(𝑥)−𝑣𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦^1/2(7) as norm on 𝑋₀, where 𝒪=(𝒞Ω)×(𝒞Ω)⊂ℝ^𝑛×ℝ^𝑛. It is easily seen that (𝑋₀,‖⋅‖_𝑋0) is a Hilbert space, with scalar product ⟨𝑢,𝑣⟩_𝑋0=_{(ℝ𝑛×ℝ𝑛)⧵𝒪}𝑢(𝑥)−𝑢𝑦𝑣(𝑥)−𝑣𝑦𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦.(8) Since 𝑣∈𝑋₀, we have that the integral in (7) (and in the related scalar product) can be extended to all ℝ^𝑛×ℝ^𝑛.

By a weak solution 𝑢 of (2) we mean a function 𝑢∈𝑋₀ such that _{ℝ𝑛×ℝ𝑛}(𝑢(𝑥)−𝑢𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦−𝜆_Ω𝑢(𝑥)𝜑(𝑥)𝑑𝑥=𝜇_Ω𝑓(𝑥,𝑢(𝑥))𝜑(𝑥)𝑑𝑥(9) for all 𝜑∈𝑋₀.

Denote by 𝜆₁>0 the first eigenvalue of the operator (−Δ)^𝑠 with homogeneous Dirichlet boundary data (−Δ)^𝑠𝑢=𝜆𝑢inΩ,𝑢=0inℝ^𝑛⧵Ω.(10)

For the existence and the basic properties of 𝜆₁ we may refer to [7]. From [7, 16], we know that if 𝜆<𝜆₁ then we can take a norm on 𝑋₀ as follows: ‖𝑣‖_𝑋0,𝜆=_{(ℝ𝑛×ℝ𝑛)⧵𝒪}𝑣(𝑥)−𝑣𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦−𝜆_Ω|𝑣(𝑥)|²𝑑𝑥^1/2.(11) Moreover, we have 𝑚_𝜆‖𝑣‖_𝑋0≤‖𝑣‖_𝑋0,𝜆≤𝑀_𝜆‖𝑣‖_𝑋0,(12) where 𝑚_𝜆∶=min𝜆₁−𝜆𝜆₁,𝑀,1_𝜆∶=max𝜆₁−𝜆𝜆₁,1.(13)

Remark 1. If 0<𝜆<𝜆₁, then 𝑚_𝜆=𝜆₁−𝜆𝜆₁,𝑀_𝜆=1.(14)
Taking into account Lemma 8 in [6], we know that the embedding 𝑗∶𝑋₀↪𝐿^𝜈(ℝ^𝑛) is continuous for any 𝜈∈[1,2^∗], while it is compact whenever 𝜈∈[1,2^∗). Thus, form any 𝜈∈[1,2^∗) there exists a positive constant 𝑐_𝜈 such that ‖𝑣‖_𝐿^𝜈_(ℝ^𝑛₎≤𝑐_𝜈‖𝑣‖_𝑋0≤𝑐_𝜈𝑚_𝜆⁻¹‖𝑣‖_𝑋0,𝜆(15) for any 𝑣∈𝑋₀.
Let 𝑅∶=sup_𝑥∈Ωdist(𝑥,𝜕Ω); simple calculations show that there is 𝑥₀∈Ω such that 𝐵(𝑥₀,𝑅)⊂Ω.
Set 𝑢_𝛿(𝑥)𝑥∶=𝛿if𝑥∈𝐵₀,𝑅2,2𝛿𝑅𝑅−𝑥−𝑥₀if𝑥∈𝐵𝑥₀,𝑅⧵𝐵𝑥₀,𝑅2,0if𝑥∈ℝ^𝑛⧵𝐵𝑥₀,𝑅.(16)

Lemma 2. Let 𝛿,𝑅>0, 𝑠∈(0,1), 0<𝜆<𝜆₁, and let 𝑢_𝛿 be defined by (16). Then 𝑢_𝛿∈𝑋₀, and there exist 𝐶_∗=𝐶_∗(𝑛,𝑠,𝑅)>0 and 𝐶^∗=𝐶^∗(𝑛,𝑠,𝑅) such that 𝐶_∗𝑚²_𝜆𝛿²≤𝑢_𝛿²_𝑋0,𝜆≤𝐶^∗𝛿²,(17) where 𝑚_𝜆 is as in (14).

Proof. By Proposition 3.4 in [5], we have 𝑢_𝛿²_𝑋0=2𝐶(𝑛,𝑠)⁻¹_ℝ𝑛𝜉^2𝑠ℱ𝑢_𝛿(𝜉)²𝑑𝜉,(18) where 𝐶(𝑛,𝑠)=_ℝ𝑛1−cos𝜁₁𝜁^𝑛+2𝑠𝑑𝜁⁻¹.(19) Here 𝜁=(𝜁₁,…,𝜁_𝑛). From the trivial inequality |𝜉|^2𝑠≤1+|𝜉|², 𝑠∈(0,1], and (18), we obtain 𝑢_𝛿²_𝑋0≤2𝐶(𝑛,𝑠)⁻¹_ℝ𝑛1+𝜉²ℱ𝑢_𝛿(𝜉)²()𝑑𝜉=2𝐶𝑛,𝑠⁻¹𝑢_𝛿²_{𝐻1(ℝ𝑛)}.(20) Moreover, according to the definition of norm for 𝐻¹(ℝ^𝑛), we get 𝑢_𝛿²_{𝐻1(ℝ𝑛)}=_ℝ𝑛∇𝑢_𝛿(𝑥)²=𝑑𝑥_{𝐵(𝑥0,𝑅)⧵𝐵(𝑥0,𝑅/2)}(2𝛿)²𝑅²=𝑑𝑥4𝛿²𝑅²meas𝐵𝑥₀,𝑅−meas𝐵𝑥₀,𝑅2=4𝛿²𝑅²𝜋^𝑛/2Γ()𝑅1+𝑛/2^𝑛−𝑅2^𝑛=4𝑅^𝑛−2𝜋^𝑛/21−1/2^𝑛Γ()𝛿1+𝑛/2².(21) By [5], we have 𝐶(𝑛,𝑠)⁻¹=_ℝ𝑛1−cos𝜁₁𝜁^𝑛+2𝑠𝑑𝜁=𝐴(𝑛,𝑠)𝐷(𝑠),(22) where 𝐴(𝑛,𝑠)=_ℝ𝑛−111+𝜂^{2(𝑛+2𝑠)/2}𝐷(𝑠)=𝑑𝜂,_ℝ1−cos𝑡|𝑡|^1+2𝑠𝑑𝑡.(23) By polar coordinates, for any 𝑠∈(0,1), we obtain 𝐴(𝑛,𝑠)=𝜔_𝑛−20^+∞𝜌^𝑛−21+𝜌^{2(𝑛+2𝑠)/2}𝑑𝜌=𝜔_𝑛−2¹₀𝜌^𝑛−21+𝜌^{2(𝑛+2𝑠)/2}𝑑𝜌+₁^+∞𝜌^𝑛−21+𝜌^{2(𝑛+2𝑠)/2}𝑑𝜌<𝜔_𝑛−2¹₀1+𝜌^{2(𝑛−2)/2}1+𝜌^{2(𝑛+2𝑠)/2}+𝑑𝜌_1+∞𝜌^𝑛−2𝜌^{2(𝑛+2𝑠)/2}𝑑𝜌<𝜔_𝑛−2¹₀𝑑𝜌1+𝜌²+₁^+∞𝑑𝜌𝜌^2(1+𝑠)=𝜔_𝑛−2𝜋4+11+2𝑠∶=𝐶₁,(24) where 𝜔_𝑛−2 is the Lebesgue measure of the unit sphere in ℝ^𝑛−1. Furthermore, we have 𝐷(𝑠)=2₀^+∞1−cos𝑡𝑡^1+2𝑠𝑑𝑡=2¹₀1−cos𝑡𝑡^1+2𝑠𝑑𝑡+₁^+∞1−cos𝑡𝑡^1+2𝑠𝑑𝑡<2¹₀𝑡²/2𝑡^1+2𝑠𝑑𝑡+2₁^+∞1𝑡^1+2𝑠=1𝑑𝑡2()+21−𝑠𝑠∶=𝐶₂.(25) Thanks to (20)–(25), we conclude that 𝑢_𝛿²_𝑋0<8𝑅^𝑛−2𝜋^𝑛/21−1/2^𝑛Γ()𝐶1+𝑛/2¹𝐶₂𝛿²<∞,(26) which implies that 𝑢_𝛿∈𝑋₀. By (12), (14), and (26), we obtain 𝑢_𝛿²_𝑋0,𝜆≤𝑢_𝛿²_𝑋0≤𝐶^∗𝛿²,(27) where 𝐶^∗∶=8𝜋^𝑛/21−1/2^𝑛Γ()𝐶1+𝑛/2¹𝐶₂𝑅^𝑛−2.(28) Hence, the conclusion of right-hand side of (17) holds.
On the other hand, we have 𝑢_𝛿²_𝑋0=_{ℝ𝑛×ℝ𝑛}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠=𝑑𝑥𝑑𝑦_{𝐵(𝑥0,𝑅)×𝐵(𝑥0,𝑅)}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦+2_{𝐵(𝑥0,𝑅)×(ℝ𝑛⧵𝐵(𝑥0,𝑅))}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦=2_{𝐵𝑥0×𝐵𝑥,𝑅/20𝑥,𝑅⧵𝐵0,𝑅/2}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠+𝑑𝑥𝑑𝑦_{𝐵𝑥0𝑥,𝑅⧵𝐵0×𝐵𝑥,𝑅/20𝑥,𝑅⧵𝐵0,𝑅/2}×𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦+2_{𝐵(𝑥0,𝑅)×(ℝ𝑛⧵𝐵(𝑥0,𝑅))}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠=𝑑𝑥𝑑𝑦8𝛿²𝑅²×_{𝐵𝑥0,𝑅/2𝐵(𝑥0,𝑅)⧵𝐵(𝑥0,𝑅/2)}𝑥−𝑥₀−𝑅/2²𝑥−𝑦^𝑛+2𝑠+𝑑𝑥𝑑𝑦4𝛿²𝑅²×_{𝐵𝑥0𝑥,𝑅⧵𝐵0×𝐵𝑥,𝑅/20𝑥,𝑅⧵𝐵0,𝑅/2}×𝑥−𝑥₀−𝑦−𝑥₀²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦+2_{𝐵𝑥0,𝑅/2ℝ𝑛⧵𝐵(𝑥0,𝑅)}𝛿²𝑥−𝑦^𝑛+2𝑠+𝑑𝑥𝑑𝑦8𝛿²𝑅²×_{𝐵𝑥0𝑥,𝑅⧵𝐵0,𝑅/2ℝ𝑛⧵𝐵(𝑥0,𝑅)}𝑦−𝑥₀−𝑅²𝑥−𝑦^𝑛+2𝑠≥𝑑𝑥𝑑𝑦8𝛿²𝑅²×_{𝐵𝑥0,𝑅/2𝐵(𝑥0,𝑅)⧵𝐵(𝑥0,𝑅/2)}𝑥−𝑥₀−𝑅/2²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦+2_{𝐵(𝑥0,𝑅/2)ℝ𝑛⧵𝐵(𝑥0,𝑅)}𝛿²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦. For 𝑥∈ℝ^𝑛⧵𝐵(𝑥₀,𝑅) and 𝑦∈𝐵(𝑥₀,𝑅/2), we have 𝑥−𝑦≤𝑥−𝑥₀+𝑦−𝑥₀≤𝑥−𝑥₀+𝑅2≤32𝑥−𝑥₀.(30) Thus, _{𝐵(𝑥0,𝑅/2)ℝ𝑛⧵𝐵(𝑥0,𝑅)}1𝑥−𝑦^𝑛+2𝑠≥2𝑑𝑥𝑑𝑦3^𝑛+2𝑠_{𝐵(𝑥0,𝑅/2)ℝ}^𝑛_{⧵𝐵(𝑥0,𝑅)}1𝑥−𝑥₀^𝑛+2𝑠≥2𝑑𝑥𝑑𝑦3^𝑛+2𝑠⋅𝜔_𝑛−1𝑛𝑅2^𝑛⋅𝜔_𝑛−1^∞_𝑅𝜌^𝑛−1𝜌^𝑛+2𝑠=2𝑑𝜌3^𝑛+2𝑠𝜔²_𝑛−1𝑛𝑅2^𝑛⋅1𝑅2𝑠^−2𝑠=𝐶₃𝑅^𝑛−2𝑠,(31) where 𝐶₃∶=(2^2𝑠−1/3^𝑛+2𝑠𝑛𝑠)𝜔²_𝑛−1. Moreover, we obtain _{𝐵(𝑥0,𝑅/2)𝐵(𝑥0,𝑅)⧵𝐵(𝑥0,𝑅/2)}𝑥−𝑥₀−𝑅/2²𝑥−𝑦^𝑛+2𝑠≥1𝑑𝑥𝑑𝑦((3/2)𝑅)^𝑛+2𝑠⋅𝜔_𝑛−1𝑛𝑅2^𝑛⋅_{𝐵(𝑥0,𝑅)⧵𝐵(𝑥0,𝑅/2)}𝑥−𝑥₀−𝑅2²=2𝑑𝑥_2𝑠𝜔_𝑛−1𝑛3^𝑛+2𝑠𝑅^2𝑠×_{𝐵(𝑥0,𝑅)}𝑥−𝑥₀²−𝑅𝑥−𝑥₀+𝑅²4−𝑑𝑥_{𝐵(𝑥0,𝑅/2)}𝑥−𝑥₀²−𝑅𝑥−𝑥₀+𝑅²4=2𝑑𝑥_2𝑠𝜔_𝑛−1𝑛3^𝑛+2𝑠𝑅^2𝑠⋅𝜔_𝑛−1𝑅^𝑛+2×𝑛²−𝑛+2()+14𝑛𝑛+1)(𝑛+22^𝑛+1𝑛(𝑛+1)(𝑛+2)=𝐶₄𝑅^{𝑛+2−2𝑠},(32) where 𝐶₄2∶=^2𝑠𝜔²_𝑛−1𝑛²(𝑛+1)(𝑛+2)3^𝑛+2𝑠𝑛²−𝑛+24+12^𝑛+1.(33) Substitute (31) and (32) into (29), we get _{ℝ𝑛×ℝ𝑛}𝑢_𝛿(𝑥)−𝑢_𝛿𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦≥2𝐶₃+4𝐶₄𝑅^𝑛−2𝑠𝛿².(34) From (34) and (12), we obtain 𝑢_𝛿²_𝑋0,𝜆≥𝑚²_𝜆𝑢_𝛿²_𝑋0≥𝐶_∗𝑚²_𝜆𝛿²,(35) where 𝐶_∗∶=2𝐶₃+4𝐶₄𝑅^𝑛−2𝑠.(36) Thus, the conclusion of left-hand side of (17) holds.

In this paper our main tool is a three-critical-point theorem of [14] which is recalled below.

Theorem 3 (see [14]). Let 𝑋 be a reflexive real Banach space; let Φ∶𝑋→𝐑 be a coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on 𝑋^∗, and let Ψ∶𝑋→𝐑 be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Φ(0)=Ψ(0)=0.(37) Assume that there exist 𝑟>0 and 𝑥∈𝑋, with 𝑟<Φ(𝑥), such that(i)sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)/𝑟<Ψ(𝑥)/Φ(𝑥);(ii)for each 𝜇∈Λ_𝑟∶=]Φ(𝑥)/Ψ(𝑥),𝑟/sup_{Φ(𝑥)≤𝑟}Ψ(𝑥)[ the functional Φ−𝜇Ψ is coercive.Then, for each 𝜇∈Λ_𝑟, the functional Φ−𝜇Ψ has at least three distinct critical points in 𝑋.

3. Main Result

Let 𝑓∶Ω×ℝ→ℝ be a Carathéodory function such that(H1)there exist 𝑎₁,𝑎₂≥0 and 𝑞∈(1,2^∗_𝑠), 2^∗_𝑠∶=2𝑛/(𝑛−2𝑠), such that 𝑓(𝑥,𝑣)≤𝑎₁+𝑎₂|𝑣|^𝑞−1,∀(𝑥,𝑣)∈Ω×ℝ.(38)

Theorem 4. Let function 𝑓 satisfy condition (H1). Assume that (H2)𝐹(𝑥,𝑣)∶=∫^𝑣₀𝑓(𝑥,𝜏)𝑑𝜏≥0 for all (𝑥,𝑣)∈Ω×ℝ⁺;(H3)there exist two positive constants 𝑏 and 𝑝<2 such that 𝐹(𝑥,𝑣)≤𝑏1+|𝑣|^𝑝,(39) for almost every 𝑥∈Ω and for every 𝑣∈ℝ;(H4)let 0<𝜆<𝜆₁ such that there exist two positive constants 𝛾 and 𝛿, with 𝛿>2/𝐶_∗(𝛾/𝑚_𝜆) such that inf_𝑥∈Ω𝐹(𝑥,𝛿)𝛿²>𝑏₁𝛾+𝑏₂𝛾^𝑞−2,(40) where 𝑏₁=𝑛2^𝑛−1/2𝑎₁𝑐₁𝐶^∗𝜔_𝑛−1𝑚_𝜆𝑅^𝑛,𝑏₂=𝑛2^{𝑛+(𝑞−2)/2}𝑎₂𝑐^𝑞_𝑞𝐶^∗𝑞𝜔_𝑛−1𝑚^𝑞_𝜆𝑅^𝑛,(41) and two positive constants 𝐶^∗ and 𝐶_∗ are as in (28) and (36), respectively. Then, for every 𝜇 belonging to Λ∶=𝑛2^𝑛−1𝐶^∗𝜔_𝑛−1𝑅^𝑛𝛿²inf_𝑥∈Ω𝐹(),𝑥,𝛿𝑛2^𝑛−1𝐶^∗𝜔_𝑛−1𝑅^𝑛1𝑏₁/𝛾+𝑏₂𝛾^𝑞−2,(42) problem (2) possesses at least three weak solutions in 𝑋₀.

Proof. Let us apply Theorem 3 with 𝑋₀ and 𝒥_𝜆,𝜇(𝑢)∶=Φ_𝜆(𝑢)−𝜇Ψ(𝑢),𝑢∈𝑋₀,(43) where Φ_𝜆(𝑢)1∶=2_ℝ^𝑛_×ℝ^𝑛𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑛+2𝑠𝜆𝑑𝑥𝑑𝑦−2_Ω|𝑢(𝑥)|²Ψ(𝑢)𝑑𝑥,∶=_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥.(44) For each 𝑢,𝑣∈𝑋₀, one has Φ^′_𝜆(𝑢)(𝑣)=_{ℝ𝑛×ℝ𝑛}𝑢(𝑥)−𝑢𝑦𝑣(𝑥)−𝑣𝑦𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦−𝜆_Ω𝑢(𝑥)𝑣(𝑥)Ψ𝑑𝑥,_′(𝑢)(𝑣)=_Ω𝑓(𝑥,𝑢(𝑥))𝑣(𝑥)𝑑𝑥.(45)
From the proof of Theorem 1 in [16], we obtain that Φ_𝜆 is coercive, continuously Gâteaux differentiable, and sequentially weakly lower semicontinuous functional. Moreover, similar to the proof of proposition in [17], we get by (12) that Φ^′_𝜆(𝑢)−Φ^′_𝜆(𝑣)()=𝑢−𝑣_{ℝ𝑛×ℝ𝑛}(𝑢−𝑣)(𝑥)−(𝑢−𝑣)𝑦²𝑥−𝑦^𝑛+2𝑠𝑑𝑥𝑑𝑦−𝜆_Ω(𝑢(𝑥)−𝑣(𝑥))²=‖‖𝑑𝑥𝑢−𝑣²_𝑋0,𝜆≥𝑚²_𝜆‖𝑢−𝑣‖²_𝑋0,(46) for every 𝑢 and 𝑣 belonging to 𝑋₀. This actually means that Φ^′_𝜆 is a uniformly monotone operator in 𝑋₀. In addition, standard arguments ensure that Φ^′_𝜆 also turns out to be coercive and hemicontinuos in 𝑋₀. Therefore, Φ^′_𝜆 admits that a continuous inverse in 𝑋^∗₀ follows immediately by applying Theorem 26. A. of [18]. Furthermore, the functional Ψ is well defined, continuously Gâteaux differentiable with compact derivative and Φ_𝜆(0)=Ψ(0)=0.
By [16] we know that being 𝑢 a weak solution of problem (2) is equivalent to being a critical point of the functional ℐ_𝜆,𝜇. Since 0<𝜆<𝜆₁, from Lemma 2, one has Φ_𝜆𝑢_𝛿=12𝑢_𝛿²_𝑋0,𝜆≥12𝐶_∗𝑚²_𝜆𝛿².(47) Bearing in mind that 𝛿>2/𝐶_∗(𝛾/𝑚_𝜆) (H4), it follows that Φ_𝜆(𝑢_𝛿)>𝛾². By (H2), we obtain Ψ𝑢_𝛿=_Ω𝐹𝑥,𝑢_𝛿(𝑥)𝑑𝑥≥_{𝐵(𝑥0,𝑅/2)}𝐹(𝑥,𝛿)𝑑𝑥≥inf_𝑥∈Ω𝐹()⋅𝜔𝑥,𝛿^𝑛−1𝑛𝑅2^𝑛.(48) By Lemma 2, we have Φ_𝜆𝑢_𝛿=12𝑢_𝛿²_𝑋0,𝜆≤12𝐶^∗𝛿².(49) So, by (48) and (49), one has Ψ𝑢_𝛿Φ_𝜆𝑢_𝛿≥𝜔_𝑛−1𝑅^𝑛𝑛2^𝑛−1𝐶^∗inf_𝑥∈Ω𝐹(𝑥,𝛿)𝛿².(50) Thanks to (H1), one has 𝐹(𝑥,𝑢)≤𝑎₁|𝑢|+𝑎₂𝑞|𝑢|^𝑞,(𝑥,𝑢)∈Ω×ℝ.(51) Thus, by (15) and (51), for every 𝑢∈𝑋₀∶Φ_𝜆(𝑢)≤𝑟, we obtain Ψ(𝑢)=_Ω𝐹(𝑥,𝑢(𝑥))𝑑𝑥≤𝑎₁‖𝑢‖_𝐿¹_(Ω)+𝑎₂𝑞‖𝑢‖^𝑞_𝐿𝑞(Ω)≤√2𝑎₁𝑐₁𝑚_𝜆√2𝑟+^𝑞/2𝑐^𝑞_𝑞𝑎₂𝑞𝑚^𝑞_𝜆𝑟^𝑞/2.(52) Therefore sup_{𝑢∈Φ𝜆−1(]−∞,𝑟])}Ψ(𝑢)≤√2𝑎₁𝑐₁𝑚_𝜆√2𝑟+^𝑞/2𝑐^𝑞_𝑞𝑎₂𝑞𝑚^𝑞_𝜆𝑟^𝑞/2.(53)
Denote the function 𝜒(𝑟)∶=sup_{𝑢∈Φ𝜆}⁻¹_{(]−∞,𝑟])}Ψ(𝑢)𝑟,𝑟>0.(54) By (53), we have 𝜒(𝑟)≤√2𝑎₁𝑐₁𝑚_𝜆𝑟^−1/2+2^𝑞/2𝑐^𝑞_𝑞𝑎₂𝑞𝑚^𝑞_𝜆𝑟^𝑞/2−1,𝑟>0.(55)
Owing to (50), (55), and (H4), we have 𝜒𝛾²≤√2𝑎₁𝑐₁𝑚_𝜆𝛾+2^𝑞/2𝑐^𝑞_𝑞𝑎₂𝑞𝑚^𝑞_𝜆𝛾^𝑞−2=𝜔_𝑛−1𝑅^𝑛𝑛2^𝑛−1𝐶^∗𝑏₁𝛾+𝑏₂𝛾^𝑞−2<𝜔_𝑛−1𝑅^𝑛𝑛2^𝑛−1𝐶^∗inf_𝑥∈Ω𝐹(𝑥,𝛿)𝛿²≤Ψ𝑢_𝛿Φ_𝜆𝑢_𝛿.(56) Hence, the assumption (i) of Theorem 3 is satisfied.
Furthermore, if 𝑝<2, for each 𝑢∈𝑋₀, |𝑢|^𝑝∈𝐿^2/𝑝(Ω), Hölder's inequality and (15) give _Ω|𝑢(𝑥)|^𝑝𝑑𝑥≤‖𝑢‖^𝑝_𝐿2(Ω)meas(Ω)^(2−𝑝)/2≤𝑐^𝑝₂𝑚^𝑝_𝜆‖𝑢‖^𝑝_𝑋0,𝜆meas(Ω)^(2−𝑝)/2,∀𝑢∈𝑋₀.(57) Due to (H3) and (57), we deduce that 𝒥_𝜆,𝜇(𝑢)≥12‖𝑢‖²_𝑋0,𝜆−𝜇𝑏𝑐^𝑝₂𝑚^𝑝_𝜆meas(Ω)^(2−𝑝)/2‖𝑢‖^𝑝_𝑋0,𝜆−𝜇𝑏⋅meas(Ω),∀𝑢∈𝑋₀.(58) Hence, 𝒥_𝜆,𝜇(𝑢) is a coercive functional for every positive parameter 𝜇, in particular, for each 𝜇∈Λ⊂]Φ_𝜆(𝑢_𝛿)/Ψ(𝑢_𝛿),𝛾²/sup_{Φ𝜆(𝑢)≤𝛾2}Ψ(𝑢)[. So also condition (ii) holds. So all the assumptions of Theorem 3 are satisfied. Thus, for each 0<𝜆<𝜆₁ there exists 𝜇>0, depending on 𝜆, such that, for any 𝜇∈Λ, the functional 𝒥_𝜆,𝜇(𝑢) has at least three distinct critical points that are weak solutions to problem (2).

Remark 5. Similar to Example 3.1 in [15], we can give a concrete example of function satisfying hypotheses (H1)–(H4). Set 𝑞∈(2,2^∗_𝑠), 𝑠∈(0,1), and 0<𝑝<2 and let 1ℎ∶=max1,𝑚_𝜆2𝐶_∗,𝐴₁+𝐴₂^1/(𝑞−2)𝑞^1/(𝑞−2),(59) where 𝐴₁=𝑛2^𝑛−1/2𝑐₁𝐶^∗𝜔_𝑛−1𝑚_𝜆𝑅^𝑛,𝐴₂=𝑛2^{𝑛+(𝑞−2)/2}𝑐^𝑞_𝑞𝐶^∗𝑞𝜔_𝑛−1𝑚^𝑞_𝜆𝑅^𝑛.(60) From (H4) we know that 𝑏₁=𝑎₁𝐴₁ and 𝑏₂=𝑎₂𝐴₂. Let 𝑟 be a positive constant such that 𝑟>ℎ and consider the following continuous and positive function 𝑓∶Ω×ℝ→ℝ: 𝑓()|𝑣|𝑥,𝑣∶=1+^𝑞−1if𝑣≤𝑟,1+𝑟^𝑞−𝑝𝑣^𝑝−1if𝑣>𝑟.(61) Obviously, 𝑓(𝑥,𝑣)≤1+|𝑣|^𝑞−1 for each (𝑥,𝑣)∈Ω×ℝ, and (H1) holds. Furthermore, for every 𝜂∈ℝ, we have 𝐹≤𝑟𝑥,𝜂𝑟+^𝑞𝑝1+𝜂^max{1,𝑝}.(62) Thus the conditions (H2) and (H3) are satisfied. Moreover, 𝑟>ℎ≥(1/𝑚_𝜆)2/𝐶_∗ and ∫^𝑟₀𝑓(𝑥,𝑡)𝑑𝑡𝑟²=𝑟^𝑞−2𝑞+1𝑟>𝐴₁+𝐴₂,(63) which implies that (H4) holds.

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 Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212).