Existence of Solutions for a Fractional Laplacian Equation with Critical Nonlinearity

Abstract

We study the fractional Laplacian equation (−Δ)^𝑠𝑢+𝜆𝐴(𝑥)𝑢=𝜇𝑢+|𝑢|^{2∗(𝑠)−2}𝑢, 𝑥∈ℝ^𝑁, here 𝑁>2𝑠, 𝑠∈(0,1), 2^∗(𝑠)=2𝑁/(𝑁−2𝑠) is the critical exponent, and 𝐴(𝑥)≥0 is a real potential function. Employing the variational method we prove the existence of nontrivial solutions for 𝜇 small and 𝜆 large.

1. Introduction

We consider the nonlinear Schrödinger equation: 𝑖ℏ𝜕𝜓𝜕𝑡=−ℏ²Δ𝜓+𝐴(𝑥)𝜓−𝜓^𝑝−2𝜓,𝑥∈ℝ^𝑁, (1) whereℏis the Planck constant. When looking for stationary waves of the form𝜓(𝑡,𝑥)=𝑒^{−𝑖𝜇(ℏ𝑡)}𝜑(𝑥)with𝜇∈ℝ^𝑁, one is led to considering the elliptic equation inℝ^𝑁; namely, replacingℏby𝜀, one sees that𝜑must satisfy −𝜀²Δ𝜑+𝐴(𝑥)𝜑=𝜀²𝜇𝜑+𝜑^𝑝−2𝜑. (2) Setting𝑢(𝑥)∶=𝜀^{−2/(𝑝−2)}𝜑(𝑥)and𝜆=𝜀⁻², this equation is transformed into −Δ𝑢+𝜆𝐴(𝑥)𝑢=𝜇𝑢+|𝑢|^2∗−2𝑢,𝑥∈ℝ^𝑁. (3) Problem (3) has been widely studied in the literature (see, for instance, [1, 2] and references therein), where2^∗=2𝑁/(𝑁−2)is the critical exponent𝑁≥4,and𝐴(𝑥)≥0is a potential well.

The study of existence and concentration of the semiclassical states of Schrödinger equation goes back to the pioneer work [3] by Floer and Weinstein. Ever since then, equations of (3) type with subcritical nonlinearities (𝑝<2^∗=2𝑁/(𝑁−2)for𝑁≥3) have been studied by many authors. For critical nonlinearity (𝑝=2^∗for𝑁≥4), Clapp and Ding [1, 2] established the existence and multiplicity of positive solutions and minimal nodal solutions which localize near the potential well for𝜇small and𝜆large.

The fractional Schrödinger equation is a fundamental equation of fractional quantum mechanics. It was discovered by Nick Laskin as a result of extending the Feynman path integral, from the Brownian-like to Lévy-like quantum mechanical paths. The term fractional Schrödinger equation was coined by Nick Laskin.

Recently, a great attention has been devoted to the fractional and nonlocal operators of elliptic type, both for their interesting theoretical structure and in view of concrete applications in many fields such as combustion and dislocations in mechanical systems. This type of operator seems to have a prevalent role in physical situations and has been studied by many authors [4–9] and references therein. In [5], Di Nezza et al. deal with the fractional Sobolev space𝑊^𝑠,𝑝and analyze their role in the trace theory. They prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. In [8], Felmer et al. proved the existence of positive solutions of nonlinear Schrödinger equation involving the fractional Laplacian inℝ^𝑁. They further analyzed regularity, decay, and symmetry properties of these solutions. Servadei and Valdinoci [9] studied the existence of nontrivial solutions for equations driven by a nonlocal integrodifferential operator𝐿_𝐾with homogeneous Dirichlet boundary conditions. They give more general and more precise results about the eigenvalues of a linear operator.

The aim of this paper is to study the fractional Laplacian equation: (−Δ)^𝑠𝑢+𝜆𝐴(𝑥)𝑢=𝜇𝑢+|𝑢|^{2∗(𝑠)−2}𝑢inℝ^𝑁, (4) where𝑁>2𝑠,𝜆>0,𝜇∈ℝ,  𝑠∈(0,1),and𝐻^𝑠(ℝ^𝑁)is the usual fractional Sobolev space, and2^∗(𝑠)=2𝑁/(𝑁−2𝑠)is the corresponding critical exponent. Suppose𝐴(𝑥)satisfies the following assumptions. (𝐴1)𝐴∈𝐶(ℝ^𝑁,ℝ),𝐴≥0,Ω∶=int𝐴⁻¹(0) is a nonempty bounded set with smooth boundary, andΩ=𝐴⁻¹(0). (𝐴2)There exists𝑀₀>0such that 𝐿𝑥∈ℝ^𝑁∶𝐴(𝑥)≤𝑀₀<∞, (5) where𝐿denotes the Lebesgue measure inℝ^𝑁.

The fractional Laplace operator(−Δ)^𝑠in (4) can be defined as −(−Δ)^𝑠𝑢(𝑥)=12_ℝ^𝑁𝑢𝑥+𝑦+𝑢𝑥−𝑦−2𝑢(𝑥)𝑦^𝑁+2𝑠𝑑𝑦,𝑥∈ℝ^𝑁. (6)

We say that a function𝑢∈𝐻^𝑠(ℝ^𝑁)solves (4) in the weak sense if _ℝ2𝑁𝑢(𝑥)−𝑢𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢(𝑥)𝜑(𝑥)𝑑𝑥=𝜇_ℝ𝑁𝑢(𝑥)𝜑(𝑥)+𝑑𝑥_ℝ𝑁|𝑢(𝑥)|^{2∗(𝑠)−2}𝑢(𝑥)𝜑(𝑥)𝑑𝑥,∀𝜑∈𝐻^𝑠ℝ^𝑁. (7)

Define the energy functional by 𝐼_𝜆(𝑢)=12_ℝ^2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠+1𝑑𝑥𝑑𝑦2𝜆_ℝ𝑁𝐴(𝑥)|𝑢(𝑥)|²−1𝑑𝑥2𝜇_ℝ𝑁|𝑢(𝑥)|²1𝑑𝑥−2^∗(𝑠)_ℝ^𝑁|𝑢(𝑥)|^2∗(𝑠)𝑑𝑥. (8) Then we know the critical points of𝐼_𝜆are exactly the weak solutions of (7). In this sense we will prove the existence of the critical points of the functional𝐼_𝜆. Fréchet derivative of𝐼_𝜆is 𝐼^′_𝜆(𝑢),𝜑=_ℝ2𝑁𝑢(𝑥)−𝑢𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢(𝑥)𝜑(𝑥)𝑑𝑥−𝜇_ℝ𝑁𝑢(𝑥)𝜑(𝑥)−𝑑𝑥_ℝ𝑁|𝑢(𝑥)|^{2∗(𝑠)−2}𝑢(𝑥)𝜑(𝑥)𝑑𝑥∀𝜑∈𝐻^𝑠ℝ^𝑁. (9)

Concerning the Schrödinger equation: −Δ𝑢+𝜆𝐴(𝑥)𝑢=𝜇𝑢+|𝑢|^2∗−2𝑢inℝ^𝑁,𝑢>0inℝ^𝑁,𝑢∈𝐻¹ℝ^𝑁. (10) Clapp and Ding [1] proved the following.(a)Assume(𝐴1)and(𝐴2)hold and𝑁≥4. Then, for every0<𝜇<𝜇¹₁(Ω), there exists𝜆(𝜇)>0such that (4) has a least solution𝑢_𝜆for each𝜆≥𝜆(𝜇), where𝜇¹₁(Ω)is the first eigenvalue of−ΔonΩwith boundary condition𝑢=0.(b)Assume(𝐴1)and(𝐴2)hold and𝑁≥4. Then, there exist0<𝜇^∗<𝜇¹₁(Ω)and for each0<𝜇≤𝜇^∗there exist two numbersΛ(𝜇)>0and0<𝑐(𝜇)<(1/𝑁)𝑆^𝑁/2such that if𝜆≥Λ(𝜇), then (4) has at least𝑐𝑎𝑡(Ω)(the number of solutions is bounded from below by a topological invariant) solutions with energy𝐼_𝜆,𝜇≤𝑐(𝜇).(c)Every sequence of solutions(𝑢_𝑛)of (10) such that0<𝜇<𝜇¹₁(Ω),𝜆_𝑛→∞and𝐼_{𝜆𝑛,𝜇}(𝑢_𝑛)→𝑐<(1/𝑁)𝑆^𝑁/2as𝑛→∞concentrates at a solution of −Δ𝑢=𝜇𝑢+|𝑢|^2∗−2𝑢inΩ,𝑢>0inΩ,𝑢=0on𝜕Ω, (11) where𝑆is the best Sobolev constant.

Our aim is to show that (a) and (c) can be extended to problem (4). In this paper, we have the following results.

Theorem 1. Assume(𝐴1)and(𝐴2)hold𝑁>2𝑠and𝑠∈(0,1). Then, for every0<𝜇<𝜇₁(Ω), there exists𝜆(𝜇)>0such that (4) has at least a solution𝑢for each𝜆≥𝜆(𝜇), where𝜇₁(Ω)is the first eigenvalue of(−Δ)^𝑠onΩwith boundary condition𝑢=0. There is a great deal of work on𝜇₁(Ω); see for example [9]. We have 𝜇₁(Ω)=inf_{𝑢∈𝐻𝑠0(Ω)⧵{0}}∫_ℝ2𝑁𝑢(𝑥)−𝑢𝑦²/𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦∫_Ω|𝑢(𝑥)|².𝑑𝑥 (12)

Theorem 2. Every sequence of solutions(𝑢_𝑛)of (4) such that0<𝜇<𝜇₁(Ω),𝜆_𝑛→∞,and𝐼_𝜆(𝑢_𝑛)→𝑐<(𝑠/𝑁)𝑆_𝑠^𝑁/2𝑠as𝑛→∞concentrates at a solution of (−Δ)^𝑠𝑢=𝜇𝑢+|𝑢|^{2∗(𝑠)−2}𝑢𝑖𝑛Ω,𝑢=0𝑜𝑛ℝ^𝑁⧵Ω, (13) whereΩis defined as in(𝐴1).

Here𝑆_𝑠is defined as 𝑆_𝑠∶=inf_{{0}𝑢∈𝐸⧵}∫_ℝ2𝑁𝑢(𝑥)−𝑢𝑦²/𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦|𝑢|²₂^∗_(𝑠), (14) where𝐸is an𝐿²(ℝ^𝑁)space with potential and will be defined in Section 2.

There is a great deal of work on (13); see, for example, [4, 6, 7] and the references therein. Among them Servadei and Valdinoci [4, 6, 7] studied the problem 𝐿_𝐾𝑢+𝜆𝑢+|𝑢|^{2∗(𝑠)−2}𝑢+𝑓(𝑥,𝑢)=0inΩ,𝑢=0inℝ^𝑁⧵Ω, (15) whereΩis an open bounded set with Lipschitz boundary inℝ^𝑁,  𝑁>2𝑠,𝑠∈(0,1),𝜆>0is a real parameter.𝐿_𝐾is defined as follows: 𝐿_𝐾𝑢(𝑥)=12_ℝ^𝑁𝑢𝑥+𝑦+𝑢𝑥−𝑦−𝑢(𝑥)𝐾𝑦𝑑𝑥𝑑𝑦,𝑥∈ℝ^𝑁. (16) Here𝐾∶ℝ^𝑁⧵{0}→(0,+∞)is a function such that 𝑚(𝑥)⋅𝐾∈𝐿¹ℝ^𝑁,where𝑚(𝑥)=min|𝑥|²,1; (17) there exists𝜃>0such that𝐾(𝑥)≥𝜃|𝑥|^{−(𝑁+2𝑠)}and𝐾(𝑥)=𝐾(−𝑥)for any𝑥∈ℝ^𝑁⧵{0}. They proved that problem (15) admits a nontrivial solution for any𝜆>0. They also studied the case𝑓(𝑥,𝑢)≡0and𝐾(𝑥)=|𝑥|^{−(𝑁+2𝑠)}, respectively.

Clapp and Ding [1] proved the existence of minimizing sequence for energy function of (10) on Nehari manifold and assumed that it is a Palais Smale sequence by Ekeland’s variational principle. Since Palais Smale conditions hold, this finished the proof of (a). For (c), they analyzed the problem directly. We will show that their method can be extended to the case0<𝑠<1.

This paper is organized as follows. In Section 2, we give some preliminary results. In Section 3, we finish the proof of Theorem 1. In Section 4, we finish the proof of Theorem 2.

2. Preliminary Results

Throughout this paper we write|⋅|_𝑞for the𝐿^𝑞norm for𝑞∈[1,∞]. We always assume that(𝐴1)-(𝐴2)hold,𝑁>2𝑠,𝜆>0,𝜇∈ℝ,and𝑠∈(0,1).𝜇₁(Ω)is the first eigenvalue of(−Δ)^𝑠onΩ.Ωis a nonempty bounded set with smooth boundary.

We consider the fractional Sobolev space: 𝐻^𝑠ℝ^𝑁=𝑢∈𝐿²ℝ^𝑁∣_ℝ2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠+𝑑𝑥𝑑𝑦_ℝ𝑁|𝑢(𝑥)|²𝑑𝑥<+∞ (18) with norm ‖𝑢‖_𝐻^𝑠=_ℝ2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+_ℝ𝑁|𝑢(𝑥)|²𝑑𝑥^1/2. (19) And let 𝐸=𝑢∈𝐻^𝑠ℝ^𝑁∣_ℝ𝑁𝐴(𝑥)𝑢²𝑑𝑥<+∞ (20) be the Hilbert space equipped with norm ‖𝑢‖_𝐸=‖𝑢‖²_𝐻^𝑠+_ℝ𝑁𝐴(𝑥)𝑢²𝑑𝑥^1/2. (21) If𝜆>0, then it is equivalent to the norms ‖𝑢‖_𝜆=‖𝑢‖²_𝐻^𝑠+𝜆_ℝ𝑁𝐴(𝑥)𝑢²𝑑𝑥^1/2. (22) Thus𝐸is continuously embedded in𝐻^𝑠(ℝ^𝑁).

Remark 3. We know the embedding𝐻^𝑠(ℝ^𝑁)↪𝐿^𝜈(ℝ^𝑁)is continuous; see [5] or [8]. So the embedding𝐸↪𝐿^𝜈(ℝ^𝑁)is also continuous for any𝜈∈[2,2^∗(𝑠)].

Thanks to Remark 3, we can define the constant𝑆_𝑠as in formula (14) and get that𝑆_𝑠>0.

Lemma 4. Let𝑢_𝑛∈𝐸be such that𝜆_𝑛→∞and‖𝑢_𝑛‖²_𝜆𝑛<𝐶. Then, there is a𝑢∈𝐻^𝑠₀(Ω)such that, up to a subsequence,𝑢_𝑛→𝑢in𝐿²(ℝ^𝑁).

Proof. If𝑢_𝑛→𝑢strongly in𝐿²(ℝ^𝑁), we prove𝑢∈𝐻^𝑠₀(Ω). Set𝐹_𝑚={𝑥∶|𝑥|≤𝑚,𝐴(𝑥)≥1/𝑚},and𝑚∈ℕ. For𝑛large enough that𝜆_𝑛≥1, thanks to𝜆_𝑛→∞. So‖𝑢_𝑛‖²_𝐸≤‖𝑢_𝑛‖²_𝜆𝑛<𝐶, we get _𝐹𝑚𝑢_𝑛²𝑑𝑥≤𝑚_𝐹𝑚𝐴(𝑥)𝑢_𝑛²𝑑𝑥≤𝑚𝐶𝜆_𝑛⟶0as𝑛⟶∞ (23) for every𝑚. This implies that𝑢(𝑥)=0for a.e.𝑥∈ℝ^𝑁⧵Ω. Hence, since𝜕Ωis smooth,𝑢∈𝐻^𝑠₀(Ω).
We will show that𝑢_𝑛→𝑢strongly in𝐿²(ℝ^𝑁). Let𝐹={𝑥∈ℝ^𝑁∶𝐴(𝑥)≤𝑀₀}with𝑀₀as in(𝐴2), and let𝐹^𝑐=ℝ^𝑁⧵𝐹. Then _𝐹𝑐𝑢²_𝑛1𝑑𝑥≤𝜆_𝑛𝑀_0𝐹^𝑐𝜆_𝑛𝐴(𝑥)𝑢²_𝑛𝐶𝑑𝑥≤𝜆_𝑛𝑀₀⟶0 (24) as𝑛→∞. Setting𝐵^𝑐_𝑅=ℝ^𝑁⧵𝐵_𝑅, where𝐵_𝑅={𝑥∈ℝ^𝑁∶|𝑥|≤𝑅}, and choosing𝑟∈(1,𝑁/(𝑁−2𝑠)),and   𝑟^′=𝑟/(𝑟−1), we have _{𝐵𝑐𝑅∩𝐹}𝑢_𝑛−𝑢²𝑑𝑥≤𝑢_𝑛−𝑢²_2𝑟𝐿𝐵^𝑐_𝑅∩𝐹^1/𝑟′≤𝐶₁𝑢_𝑛−𝑢²_𝐸𝐿𝐵^𝑐_𝑅∩𝐹^1/𝑟′⟶0 (25) as𝑅→∞, thanks to(𝐴2). Since𝑢_𝑛→𝑢in𝐿²_loc(ℝ^𝑁), _𝐵𝑅𝑢_𝑛−𝑢²𝑑𝑥⟶0 (26) as𝑛→∞. By𝑢∈𝐻^𝑠₀(Ω), _ℝ𝑁𝑢_𝑛−𝑢²𝑑𝑥=_𝐹𝑐𝑢²_𝑛𝑑𝑥+_𝐹𝑢_𝑛−𝑢²≤𝑑𝑥_𝐹𝑐𝑢²_𝑛𝑑𝑥+_{𝐵𝑐𝑅∩𝐹}𝑢_𝑛−𝑢²+𝑑𝑥_𝐵𝑅𝑢_𝑛−𝑢²𝑑𝑥⟶0 (27) as𝑛→∞. Thus𝑢_𝑛→𝑢strongly in𝐿²(ℝ^𝑁).

We denote𝐴_𝜆∶=(−Δ)^𝑠+𝜆𝐴(𝑥)and by⟨⋅,⋅⟩the𝐿²-inner product and write 𝐴_𝜆𝑢,𝑣=_ℝ2𝑁𝑢(𝑥)−𝑢𝑦𝑣(𝑥)−𝑣𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢𝑣𝑑𝑥 (28) for𝑢,𝑣∈𝐸. Set𝑎_𝜆∶=inf𝜎_𝑝(𝐴_𝜆), the infimum of the point spectrum of𝐴_𝜆. Observe that 0≤𝑎_𝜆=inf𝐴_𝜆𝑢,𝑢∶𝑢∈𝐸,|𝑢|₂=1 (29) and that𝑎_𝜆is nondecreasing in𝜆.

Lemma 5. For each0<𝜇<𝜇₁(Ω), there exists𝜆(𝜇)>0such that𝑎_𝜆≥(𝜇+𝜇₁(Ω))/2for𝜆≥𝜆(𝜇). Consequently, 𝑐_𝜇‖𝑢‖²_𝜆≤𝐴_𝜆−𝜇𝑢,𝑢 (30) for all𝑢∈𝐸,  𝜆≥𝜆(𝜇), where𝑐_𝜇>0is a constant.

Proof. Assume, by contradiction, that there exists a sequence𝜆_𝑛→∞such that𝑎_𝜆𝑛<(𝜇+𝜇₁(Ω))/2for all𝑛and𝑎_𝜆𝑛→𝑐_𝜆≤(𝜇+𝜇₁(Ω))/2. Let𝑢_𝑛∈𝐸be such that|𝑢_𝑛|₂=1and⟨(𝐴_𝜆𝑛−𝑎_𝜆𝑛)𝑢_𝑛,𝑢_𝑛⟩→0. Then 𝑢_𝑛²_𝜆𝑛=_ℝ2𝑁𝑢_𝑛(𝑥)−𝑢_𝑛𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+_ℝ𝑁𝑢_𝑛(𝑥)²𝑑𝑥+𝜆_𝑛ℝ𝑁𝐴(𝑥)𝑢_𝑛(𝑥)²=𝐴𝑑𝑥_𝜆𝑛−𝑎_𝜆𝑛𝑢_𝑛,𝑢_𝑛+1+𝑎_𝜆𝑛𝑢_𝑛²₂≤21+𝜇₁(Ω) (31) for all𝑛large. By Lemma 4 there is a𝑢∈𝐻^𝑠₀(Ω)such that, up to a subsequence,𝑢_𝑛→𝑢in𝐿²(ℝ^𝑁), and thus|𝑢|₂=1. Using Fatou’s theorem, we know _ℝ2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦−𝑐_𝜆Ω|𝑢(𝑥)|²𝑑𝑥≤lim_𝑛→∞inf_ℝ2𝑁𝑢_𝑛(𝑥)−𝑢_𝑛𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦−𝑎_{𝜆𝑛ℝ𝑁}𝑢_𝑛(𝑥)²𝑑𝑥≤lim_𝑛→∞inf𝐴_𝜆𝑛−𝑎_𝜆𝑛𝑢_𝑛,𝑢_𝑛=0. (32) Consequently, _ℝ2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦≤𝑐_𝜆<𝜇+𝜇₁(Ω)2<𝜇₁(Ω). (33) Since𝜇₁(Ω)is the first eigenvalue of(−Δ)^𝑠onΩwith boundary condition𝑢=0, we have𝜇₁(Ω)≤∫_ℝ2𝑁(|𝑢(𝑥)−𝑢(𝑦)|²/|𝑥−𝑦|^𝑁+2𝑠)𝑑𝑥𝑑𝑦. This is a contradiction.

In the following, enlarging𝜆(𝜇)if necessary, we assume𝜆(𝜇)≥𝜇/𝑀₀; thus 𝜆𝑀₀−𝜇≥0∀𝜆≥𝜆𝜇. (34)

3. The Proof of Theorem 1

In this section we will finish the proof of Theorem 1.

The critical points of𝐼_𝜆lie on the Nehari manifold 𝑀=𝑢∈𝐸⧵{0}∶𝐼^′_𝜆(𝑢),𝑢=0. (35) Since0<𝜇<𝜇₁(Ω)and2<2^∗(𝑠), the function𝑡∈ℝ₊→𝐼_𝜆(𝑡𝑢)has a unique maximum point𝑡(𝑢)>0and𝑡(𝑢)𝑢∈𝑀. Define 𝑐₁∶=inf_𝑀𝐼_𝜆, (36) and we observe that 𝑐₁=inf_{𝑢∈𝐸,𝑢≠0}max_𝑡≥0𝐼_𝜆(𝑡𝑢). (37) From Lemma 5, the constant𝑐₁is positive. On the other hand, we define 𝑐=inf_𝛾∈Γmax_[]𝑡∈0,1𝐼_𝜆𝛾(𝑡), (38) where Γ∶=𝛾∈𝐶([0,1],𝐸)∶𝛾(0)=0,𝐼_𝜆𝛾(1)<0. (39)

Proposition 6. Consider𝑐=𝑐₁.

Proof. Proposition is proved, for instance, in [8, see Section2].

𝑀is radially diffeomorphic𝑉={𝑣∈𝐸∶|𝑣|₂^∗_(𝑠)=1}. For𝑢∈𝑀, the functional𝐼_𝜆is 𝐼_𝜆(𝑢)=𝑠𝑁𝐴_𝜆=𝑠−𝜇𝑢,𝑢𝑁|𝑢|^2∗₂^(𝑠)_∗(𝑠). (40) So, 𝑐₁∶=inf_𝑢∈𝑀𝐼_𝜆(𝑢)=inf_𝑣∈𝑉𝑠𝑁𝐴_𝜆−𝜇𝑣,𝑣^𝑁/2𝑠. (41)

We consider the functional 𝐼_Ω(𝑢)=12_ℝ^2𝑁𝑢(𝑥)−𝑢𝑦²𝑥−𝑦^𝑁+2𝑠1𝑑𝑥𝑑𝑦−2𝜇_Ω|𝑢(𝑥)|²−1𝑑𝑥2^∗(𝑠)_Ω|𝑢(𝑥)|^2∗(𝑠)=1𝑑𝑥2𝐴₀−1−𝜇𝑢,𝑢2^∗(𝑠)|𝑢|^2∗₂^(𝑠)_∗(𝑠) (42) on𝐻^𝑠₀(Ω). Its Nehari manifold 𝑀_Ω=𝑢∈𝐻^𝑠₀(Ω)⧵{0}∶𝐼^′_Ω(𝑢),𝑢=0 (43) is radially diffeomorphic𝑉_Ω={𝑣∈𝐻^𝑠₀(Ω)∶|𝑣|₂^∗_(𝑠)=1}. Set 𝑐(Ω)∶=inf_{𝑢∈𝑀Ω}𝐼_Ω(𝑢)=inf_{𝑣∈𝑉Ω}𝑠𝑁𝐴₀−𝜇𝑣,𝑣^𝑁/2𝑠. (44)

Proposition 7. If0<𝜇<𝜇₁(Ω)and𝜆≥𝜆(𝜇),then 𝑠𝑁𝑐_𝜇𝑆_𝑠^𝑁/2𝑠(Ω)<𝑠≤𝑐<𝑐𝑁𝑆_𝑠^𝑁/2𝑠, (45) where𝑆_𝑠is defined in formula (14) and𝑐_𝜇is given in Lemma 5.

Proof. By Lemma 5,𝑐_𝜇‖𝑣‖²_𝐸≤𝑐_𝜇‖𝑣‖²_𝜆≤⟨(𝐴_𝜆−𝜇)𝑣,𝑣⟩for all𝑣∈𝐸. Taking infima over𝑣∈𝑉gives the first inequality. Since𝑉_Ω⊂𝑉and⟨𝐴_𝜆𝑣,𝑣⟩=⟨𝐴₀𝑣,𝑣⟩for𝑣∈𝑉_Ω, it follows that𝑐≤𝑐(Ω). By [6, see Section7] and [10, see Section8], we know𝑐(Ω)<(𝑠/𝑁)𝑆_𝑠^𝑁/2𝑠and𝑐(Ω)is achieved at some𝑢₀. Thus𝑐<𝑐(Ω),because other𝑐would be also achieved at𝑢₀which vanishes outsideΩ, contradicting the maximum principle.
Hence, Proposition 7 is proved.

By definition of𝑐₁and Proposition 6, there exists a minimizing sequence for𝐼_𝜆on𝑀,and we note{𝑢_𝑗}. By Ekeland’s variational principle, we may assume that it is a Palais Smale sequence. So we have 𝐼_𝜆𝑢_𝑗𝐼⟶𝑐,sup^′_𝜆𝑢_𝑗,𝜑∶𝜑∈𝐸,𝜑_𝜆=1⟶0 (46)(47) as𝑗→+∞.

Proposition 8. 𝐼_𝜆has at least one critical point with critical value𝑐for each0<𝜇<𝜇₁(Ω)and𝜆≥𝜆(𝜇).

Proof. We proceed by steps.
Step 1. The sequence{𝑢_𝑗}is bounded in𝐸.
Proof. For any𝑗∈ℕby (46) and (47) it easily follows that there exists𝐶₁>0such that 𝐼_𝜆𝑢_𝑗≤𝐶₁,𝐼^′_𝜆𝑢_𝑗,𝑢_𝑗𝑢_𝑗𝜆≤𝐶₁. (48) As a consequence of (48) we have 𝐼_𝜆𝑢_𝑗−12^∗(𝑠)𝐼^′_𝜆𝑢_𝑗,𝑢_𝑗=𝑠𝑁𝐴_𝜆−𝜇𝑢_𝑗,𝑢_𝑗≤𝐶₁1+𝑢_𝑗𝜆. (49) By (49) and the definition of𝐼_𝜆we have 𝑢_𝑗²_𝜆≤𝐶₂1+𝑢_𝑗𝜆. (50)
Thus{𝑢_𝑗}is bounded in𝐸.
Step 2. Problem (7) admits a solution𝑢_∞∈𝐸.
Proof. By Step1and𝐸is a reflexive space, up to a subsequence, still denoted by𝑢_𝑗, there exists𝑢_∞∈𝐸such that𝑢_𝑗→𝑢_∞weakly in𝐸; that is, _ℝ2𝑁𝑢_𝑗(𝑥)−𝑢_𝑗𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_𝑗(𝑥)𝜑(𝑥)𝑑𝑥⟶_ℝ2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_∞(𝑥)𝜑(𝑥)𝑑𝑥 (51) as𝑗→+∞. Since Step1and Remark 3, we have that𝑢_𝑗is bounded in𝐿^2∗(𝑠)(ℝ^𝑁). Since𝐿^2∗(𝑠)(ℝ^𝑁)is a reflexive space, up to a subsequence 𝑢_𝑗⟶𝑢_∞weaklyin𝐿^2∗(𝑠)ℝ^𝑁 (52) as𝑗→+∞. While by Lemma 4, up to a subsequence, 𝑢_𝑗(𝑥)⟶𝑢_∞(𝑥)in𝐿²ℝ^𝑁𝑢_𝑗,⟶𝑢_∞a.e.inℝ^𝑁 (53)(54) as𝑗→+∞. By (52) and the fact that|𝑢_𝑗|^{2∗(𝑠)−2}𝑢_𝑗is bounded in𝐿^{2∗(𝑠)/(2∗(𝑠)−1)}(ℝ^𝑁),we have 𝑢_𝑗^{2∗(𝑠)−2}𝑢_𝑗⟶𝑢_∞^{2∗(𝑠)−2}𝑢_∞weaklyin𝐿^{2∗(𝑠)/(2∗(𝑠)−1)}ℝ^𝑁 (55) as𝑗→+∞.
Since (47) holds true, for any𝜑∈𝐸0⟵𝐼^′_𝜆𝑢_𝑗=,𝜑_ℝ2𝑁𝑢_𝑗(𝑥)−𝑢_𝑗𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_𝑗(𝑥)𝜑(𝑥)𝑑𝑥−𝜇_ℝ𝑁𝑢_𝑗(𝑥)𝜑(𝑥)−𝑑𝑥_ℝ𝑁𝑢_𝑗(𝑥)^{2∗(𝑠)−2}𝑢_𝑗(𝑥)𝜑(𝑥)𝑑𝑥. (56)
Passing to the limit in this expression as𝑗→+∞and taking into account (51), (53), and (55), we get _ℝ2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_∞(𝑥)𝜑(𝑥)𝑑𝑥−𝜇_ℝ𝑁𝑢_∞(𝑥)𝜑(𝑥)−𝑑𝑥_ℝ𝑁𝑢_∞(𝑥)^{2∗(𝑠)−2}𝑢_∞(𝑥)𝜑(𝑥)𝑑𝑥=0 (57) for any𝜑∈𝐸; that is,𝑢_∞is a solution of problem (7).
Step 3. The following equality holds true: 𝐼_𝜆𝑢_∞=𝑠𝑁_ℝ^𝑁𝑢_∞(𝑥)^2∗(𝑠)𝑑𝑥≥0. (58)
Proof. By Step2, taking𝜑=𝑢_∞∈𝐸as a test function in (7), we have _ℝ2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_∞(𝑥)²𝑑𝑥=𝜇_ℝ𝑁𝑢_∞(𝑥)²𝑑𝑥+_ℝ𝑁𝑢_∞(𝑥)^2∗(𝑠)𝑑𝑥. (59) So we get 𝐼_𝜆𝑢_∞=𝑠𝑁_ℝ^𝑁𝑢_∞(𝑥)^2∗(𝑠)𝑑𝑥≥0. (60) Hence, Step3is proved.
Now, we conclude the proof of Proposition 8.
We write𝑣_𝑗∶=𝑢_𝑗−𝑢_∞, and then𝑣_𝑗→0weakly in𝐸. Moreover, since (54) holds true, by the Brézis-Lieb Lemma, we get _ℝ2𝑁𝑢_𝑗(𝑥)−𝑢_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_𝑗(𝑥)²=𝑑𝑥_ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²+𝑑𝑥_ℝ2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_∞(𝑥)²𝑑𝑥+∘_ℝ𝑁(1),𝑢_𝑗(𝑥)²𝑑𝑥=_ℝ𝑁𝑣_𝑗(𝑥)²+𝑑𝑥_ℝ𝑁𝑢_∞(𝑥)²𝑑𝑥+∘_ℝ𝑁(1),𝑢_𝑗(𝑥)^2∗(𝑠)𝑑𝑥=_ℝ𝑁𝑣_𝑗(𝑥)^2∗(𝑠)+𝑑𝑥_ℝ𝑁𝑢_∞(𝑥)^2∗(𝑠)𝑑𝑥+∘(1),as𝑗⟶+∞. (61) Then, 𝑐⟵𝐼_𝜆𝑢_𝑗=12_ℝ^2𝑁𝑢_𝑗(𝑥)−𝑢_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠+1𝑑𝑥𝑑𝑦2𝜆_ℝ𝑁𝐴(𝑥)𝑢_𝑗(𝑥)²−1𝑑𝑥2𝜇_ℝ𝑁𝑢_𝑗(𝑥)²−1𝑑𝑥2^∗(𝑠)_ℝ^𝑁𝑢_𝑗(𝑥)^2∗(𝑠)=1𝑑𝑥2_ℝ^2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠+1𝑑𝑥𝑑𝑦2𝜆_ℝ𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²+1𝑑𝑥2_ℝ^2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦²𝑥−𝑦^𝑁+2𝑠+1𝑑𝑥𝑑𝑦2_ℝ^𝑁𝐴(𝑥)𝑢_∞(𝑥)²−1𝑑𝑥2𝜇_ℝ𝑁𝑢_∞(𝑥)²−1𝑑𝑥2𝜇_ℝ𝑁𝑣_𝑗(𝑥)²−1𝑑𝑥2^∗(𝑠)_ℝ^𝑁𝑣_𝑗(𝑥)^2∗(𝑠)−1𝑑𝑥2^∗(𝑠)_ℝ^𝑁𝑢_∞(𝑥)^2∗(𝑠)𝑑𝑥+∘(1)=𝐼_𝜆𝑢_∞+12_ℝ^2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠+1𝑑𝑥𝑑𝑦2_ℝ^𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²−1𝑑𝑥2𝜇_ℝ𝑁𝑣_𝑗(𝑥)²−1𝑑𝑥2^∗(𝑠)_ℝ^𝑁𝑣_𝑗(𝑥)^2∗(𝑠)(1),𝐼𝑑𝑥+∘_′𝜆𝑢_𝑗,𝑢_𝑗=_ℝ2𝑁𝑢_𝑗(𝑥)−𝑢_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_𝑗(𝑥)²𝑑𝑥−𝜇_ℝ𝑁𝑢_𝑗(𝑥)²𝑑𝑥−_ℝ𝑁𝑢_𝑗(𝑥)^2∗(𝑠)=𝑑𝑥_ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²𝑑𝑥+_ℝ2𝑁𝑢_∞(𝑥)−𝑢_∞𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑢_∞(𝑥)²𝑑𝑥−𝜇_ℝ𝑁𝑢_∞(𝑥)²𝑑𝑥−𝜇_ℝ𝑁𝑣_𝑗(𝑥)²−𝑑𝑥_ℝ𝑁𝑣_𝑗(𝑥)^2∗(𝑠)−𝑑𝑥_ℝ𝑁𝑢_∞(𝑥)^2∗(𝑠)(1)=𝐼𝑑𝑥+∘_′𝜆𝑢_∞,𝑢_∞+_ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²𝑑𝑥−𝜇_ℝ𝑁𝑣_𝑗(𝑥)²−𝑑𝑥_ℝ𝑁𝑣_𝑗(𝑥)^2∗(𝑠)𝑑𝑥+∘(1). (62)(63) By⟨𝐼^′_𝜆(𝑢_∞),𝑢_∞⟩=0and⟨𝐼^′_𝜆(𝑣_𝑗),𝑣_𝑗⟩→0, we get _ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_ℝ𝑁𝐴(𝑥)𝑣_𝑗(𝑥)²𝑑𝑥−𝜇_ℝ𝑁𝑣_𝑗(𝑥)²𝑑𝑥⟶𝑏,_ℝ𝑁𝑣_𝑗(𝑥)^2∗(𝑠)𝑑𝑥⟶𝑏. (64) As in the proof of Lemma 4 one shows that _𝐹𝑣_𝑗(𝑥)²𝑑𝑥⟶0 (65) as𝑗→∞, where𝐹={𝑥∈ℝ^𝑁∶𝐴(𝑥)≤𝑀₀}. Let𝐹^𝑐=ℝ^𝑁⧵𝐹. Then, by (34), 𝑆_𝑠𝑣_𝑗²_{𝐿2∗(𝑠)ℝ𝑁}≤_ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠≤𝑑𝑥𝑑𝑦_ℝ2𝑁𝑣_𝑗(𝑥)−𝑣_𝑗𝑦²𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+_𝐹𝑐𝜆𝐴(𝑥)−𝜇𝑣_𝑗(𝑥)²≤𝐴𝑑𝑥_𝜆−𝜇𝑣_𝑗,𝑣_𝑗+𝜇_𝐹𝑣_𝑗(𝑥)²=𝐴𝑑𝑥_𝜆−𝜇𝑣_𝑗,𝑣_𝑗+∘(1). (66) Passing to the limit yields𝑏≥𝑆_𝑠𝑏^2/2∗(𝑠). Either𝑏=0or𝑏≥𝑆_𝑠^𝑁/2𝑠. If𝑏=0, the proof is complete. Assuming𝑏≥𝑆_𝑠^𝑁/2𝑠, we obtain from Step3, (45), and (62) that 𝑠𝑁𝑆_𝑠^𝑁/2𝑠≤12−12^∗(𝑠)𝑠𝑏=𝑐<𝑁𝑆_𝑠^𝑁/2𝑠, (67) which is a contradiction. Thus𝑏=0, and 𝑢_𝑗−𝑢_∞𝜆⟶0 (68) as𝑗→+∞. This ends the proof of Proposition 8.

We have finished the proof of Theorem 1 by Proposition 8.

4. The proof of Theorem 2

Proof of Theorem 2. Let(𝑢_𝑛)be a sequence of solutions of (4) such that0<𝜇<𝜇₁(Ω),𝜆_𝑛→∞,and𝑁𝐼_𝜆𝑛(𝑢_𝑛)=⟨(𝐴_𝜆𝑛−𝜇)𝑢_𝑛,𝑢_𝑛⟩→𝑁𝑐<𝑠𝑆_𝑠^𝑁/2𝑠. Then, by Lemma 4, there is a𝑢∈𝐻^𝑠₀(Ω)such that, up to a subsequence,𝑢_𝑛→𝑢in𝐸. By𝑢_𝑛that is a solution of (4), we have _ℝ2𝑁𝑢_𝑛(𝑥)−𝑢_𝑛𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦+𝜆_𝑛ℝ𝑁𝐴(𝑥)𝑢_𝑛(𝑥)𝜑(𝑥)𝑑𝑥−𝜇_ℝ𝑁𝑢_𝑛(𝑥)𝜑(𝑥)=𝑑𝑥_ℝ𝑁𝑢_𝑛(𝑥)^{2∗(𝑠)−2}𝑢_𝑛(𝑥)𝜑(𝑥)𝑑𝑥 (69) for any𝜑∈𝐸. If𝜑∈𝐻^𝑠₀(Ω),then𝜆_𝑛∫_ℝ𝑁𝐴(𝑥)𝑢_𝑛(𝑥)𝜑(𝑥)𝑑𝑥=0for all𝑛, so letting𝑛→∞we obtain _ℝ2𝑁𝑢(𝑥)−𝑢𝑦𝜑(𝑥)−𝜑𝑦𝑥−𝑦^𝑁+2𝑠𝑑𝑥𝑑𝑦−𝜇_ℝ𝑁𝑢(𝑥)𝜑(𝑥)𝑑𝑥=_ℝ𝑁|𝑢(𝑥)|^{2∗(𝑠)−2}𝑢(𝑥)𝜑(𝑥)𝑑𝑥 (70) for any𝜑∈𝐻^𝑠₀(Ω). So,𝑢is a solution of (13). We write𝑣_𝑛∶=𝑢_𝑛−𝑢. Then,𝑣_𝑛→0in𝐿²(ℝ^𝑁).
Since𝐴(𝑥)=0for𝑥∈Ω, we get 𝐴_𝜆𝑛−𝜇𝑢_𝑛,𝑢_𝑛=𝐴₀−𝜇𝑢,𝑢+𝐴_𝜆𝑛−𝜇𝑣_𝑛,𝑣_𝑛. (71) By𝑣_𝑛→0in𝐸and the Brézis-Lieb Lemma, we have _ℝ𝑁𝑢_𝑛(𝑥)^2∗(𝑠)=𝑑𝑥_ℝ𝑁|𝑢(𝑥)|^2∗(𝑠)𝑑𝑥+_ℝ𝑁𝑣_𝑛(𝑥)^2∗(𝑠)𝑑𝑥+∘(1). (72) So, we can get 𝐴_𝜆𝑛−𝜇𝑣_𝑛,𝑣_𝑛−_ℝ𝑁𝑣_𝑛(𝑥)^2∗(𝑠)𝑑𝑥=∘(1). (73) We claim that∫_ℝ𝑁|𝑣_𝑛(𝑥)|^2∗(𝑠)𝑑𝑥→0. Assume∫_ℝ𝑁|𝑣_𝑛(𝑥)|^2∗(𝑠)𝑑𝑥→𝑏>0. Then, 𝑆_𝑠ℝ𝑁𝑣_𝑛(𝑥)^2∗(𝑠)𝑑𝑥^2/2∗(𝑠)≤_ℝ2𝑁𝑣_𝑛(𝑥)−𝑣_𝑛𝑦²𝑥−𝑦^𝑁+2𝑠≤𝐴𝑑𝑥𝑑𝑦_𝜆𝑛−𝜇𝑣_𝑛,𝑣_𝑛=_ℝ𝑁𝑣_𝑛(𝑥)^2∗(𝑠)𝑑𝑥+∘(1), (74) thanks to (73). It follows that 𝑆_𝑠≤_ℝ𝑁𝑣_𝑛(𝑥)^2∗(𝑠)𝑑𝑥^{(2∗(𝑠)−2)/2∗(𝑠)}(1)≤+∘_ℝ𝑁𝑢_𝑛(𝑥)^2∗(𝑠)𝑑𝑥^{(2∗(𝑠)−2)/2∗(𝑠)}(1),𝑆+∘_{𝑠𝑁/2𝑠}≤lim_{𝑛→∞ℝ𝑁}𝑢_𝑛(𝑥)^2∗(𝑠)𝑑𝑥=lim_𝑛→∞𝐴_𝜆𝑛−𝜇𝑢_𝑛,𝑢_𝑛=𝑁𝑐<𝑠𝑆_𝑠^𝑁/2𝑠. (75) This is a contradiction. Thus∫_ℝ𝑁|𝑣_𝑛(𝑥)|^2∗(𝑠)𝑑𝑥→0and⟨(𝐴_𝜆𝑛−𝜇)𝑣_𝑛,𝑣_𝑛⟩→0, by (73). Hence, by (71) lim_𝑛→∞𝐴_𝜆𝑛−𝜇𝑢_𝑛,𝑢_𝑛=𝐴₀−𝜇𝑢,𝑢. (76) Since𝑢_𝑛=𝑣_𝑛inℝ^𝑁⧵Ωand𝐴(𝑥)=0for𝑥∈Ω, _ℝ𝑁𝐴(𝑥)𝑢_𝑛(𝑥)²𝑑𝑥≤_ℝ𝑁𝜆_𝑛𝐴(𝑥)𝑢_𝑛(𝑥)²=𝑑𝑥_ℝ𝑁𝜆_𝑛𝐴(𝑥)𝑣_𝑛(𝑥)²≤𝐴𝑑𝑥_𝜆𝑛−𝜇𝑣_𝑛,𝑣_𝑛. (77) Therefore,∫_ℝ𝑁𝐴(𝑥)|𝑢_𝑛(𝑥)|²𝑑𝑥→0and (76) implies that𝑢_𝑛→𝑢in𝐸.

Acknowledgment

This paper is supported by the National Nature Science Foundation of China (11271331).