Existence of Multiple Solutions for a 𝑝-Kirchhoff Problem with Nonlinear Boundary Conditions

Abstract

The paper considers the existence of multiple solutions of the singular nonlocal elliptic problem −𝑀(∫_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝)div(|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2∇𝑢)=𝜆ℎ(𝑥)|𝑢|^𝑟−2𝑢, 𝑥∈Ω, 𝑀(∫_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝)|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2 (𝜕𝑢/𝜕𝜈) = 𝑔(𝑥)|𝑢|^𝑞−2𝑢, on 𝜕Ω, where 1<(𝑁+1)/2<𝑝<𝑁, 𝑎<(𝑁−𝑝)/𝑝. By the variational method on the Nehari manifold, we prove that the problem has at least two positive solutions when some conditions are satisfied.

1. Introduction and Main Result

In this paper, we consider the existence of multiple solutions for the singular elliptic problem: −𝑀_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥div|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2(𝑥𝑢|∇𝑢=𝜆ℎ)|^𝑟−2𝑀𝑢,𝑥∈Ω,_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2𝜕𝑢(𝑥𝑢|𝜕𝜈=𝑔)|^𝑞−2𝑢,on𝜕Ω, (1) where 1<(𝑁+1)/2<𝑝<𝑁, 𝑎<(𝑁−𝑝)/𝑝, 𝜆>0, Ω is an exterior domain of ℝ^𝑁: that is, and Ω=ℝ^𝑁⧵𝐷, where 𝐷 is a bounded domain in ℝ^𝑁 with the smooth boundary 𝜕𝐷(=𝜕Ω), and 0∈Ω. 𝑔(𝑥) and ℎ(𝑥) are continuous functions, 𝑀(𝑠)=𝛼𝑠+𝛽 with the parameters 𝛼,𝛽>0.

Problem like (1) is usually called nonlocal problem because of the presence of the integral over the entire domain, and this implies that (1) is no longer a pointwise identity. In fact, such kind of problem can be traced back to the work of Kirchhoff. In [1], Kirchhoff investigated the model of the form 𝜌𝜕²𝑢𝜕𝑡²−𝑃₀ℎ+𝐸2𝐿^𝐿₀𝜕𝑢𝜕𝑥²𝜕𝑑𝑥²𝑢𝜕𝑥²=0, (2) where 𝜌, 𝑃₀, ℎ, 𝐸, and 𝐿 are all positive constants. This equation extends the classical d’Alembert’s wave equation by considering the effects of changes in the length of the strings during the vibrations. Problem (1) is related to the stationary analogue of problem (2). After Kirchhoff’s work, various models of Kirchhoff-type have been studied by many authors: we refer the readers to [2–9]. In [4], by the variational methods, Bensedik and Bouchekif considered the problem −𝑀_Ω|∇𝑢|²𝑑𝑥Δ𝑢=𝑓(𝑥,𝑢),𝑥∈Ω,𝑢=0,on𝜕Ω, (3) where Ω is a bounded domain in ℝ^𝑁. One of the assumptions made on 𝑓(𝑥,𝑡) in (3) is that (𝐟_𝟏)𝑓(𝑥,𝑡) is continuous function on Ω×ℝ such that 𝑓(𝑥,𝑡)≥0,∀𝑡≥0,𝑥∈𝑓()Ω,𝑥,𝑡=0,∀𝑡≤0,𝑥∈Ω. (4) The authors proved that problem (3) has a positive solution or has no solution when some other assumptions are fulfilled. In our paper, however, the weight functions ℎ(𝑥) and 𝑔(𝑥) are permitted to change sign. Thus, the methods in [4] cannot be directly applied on (1).

In recent years, some other authors considered the Kirchhoff-type equations with 𝑝-Laplacian [10–13]. In fact, motivated by [4, 5] and our previous work [14], we consider the existence of multiple solutions for problem (1) on the Nehari manifold by variational methods. We prove that problem (1) has at least two positive solutions. Since Ω⊂ℝ^𝑁 is an unbounded domain and the problem is singular, the loss of compactness of the Sobolev embedding renders variational technique more delicate.

In order to state our result, we introduce a weighted Sobolev space 𝐸=𝑊_𝑎^1,𝑝(Ω), which is the completion of the space 𝐶^∞₀(Ω) with the norm of ‖𝑢‖_𝐸=_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥^1/𝑝. (5) For 𝜌≥1 and 𝑓=𝑓(𝑥)>0 in Ω, we define the space 𝐿^𝜌(Ω,𝑓) as being the set of Lebesgue measurable functions 𝑢∶Ω→ℝ¹, which satisfies ‖𝑢‖_𝐿^𝜌_(Ω,𝑓)=_Ω𝑓(𝑥)|𝑢|^𝜌𝑑𝑥^1/𝜌<∞. (6) The following weighted Sobolev-Hardy inequality is due to Caffarelli et al. [15], which is called the Caffarelli-Kohn-Nirenberg inequality. There is a constant 𝑆₁>0 such that _ℝ𝑁|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^1/𝑝∗≤𝑆_1ℝ𝑁|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥^1/𝑝,∀𝑢∈𝐶^∞₀ℝ^𝑁, (7) where −∞<𝑎<(𝑁−𝑝)/𝑝, 𝑎≤𝑏<𝑎+1, 𝑑=𝑎+1−𝑏, and 𝑝^∗=𝑝𝑁/(𝑁−𝑝𝑑). Throughout this paper, we make the following assumptions:  (𝐴₁)ℎ(𝑥)|𝑥|^𝑏𝑟∈𝐿^𝜂(Ω)∩𝐿^∞(Ω) with 𝜂=𝑝^∗/(𝑝^∗−𝑟), ℎ^±=max{±ℎ(𝑥),0}≢0, (𝐴₂)𝑔(𝑥)∈𝐶(𝜕Ω)∩𝐿^∞(𝜕Ω), (𝐴₃)1<𝑟<𝑝<2𝑝≤𝑞<𝑝_∗=𝑝(𝑁−1)/(𝑁−𝑝).

Now, we give the definition of weak solution for problem (1).

Definition 1. A function 𝑢∈𝐸 is said to be a weak solution of problem (1) if for any 𝜑∈𝐶^∞₀(Ω)𝛼‖𝑢‖^𝑝_𝐸+𝛽_Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2−∇𝑢⋅∇𝜑𝑑𝑥_𝜕Ω𝑔(𝑥)|𝑢|^𝑞−2𝑢𝜑𝑑𝜎=𝜆_Ωℎ(𝑥)|𝑢|^𝑟−2𝑢𝜑𝑑𝑥. (8) The assumptions (𝐴₁)--(𝐴₃) mean that all the integrals in (8) are well defined and convergent.
In view of (𝐴₃), it follows from the compact trace embedding 𝑊_𝑎^1,𝑝(Ω)↪𝐿^𝑞(𝜕Ω) [16] that _𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎≤𝑆₂𝑔_∞‖𝑢‖^𝑞_𝐸, (9) for some constant 𝑆₂>0, and ‖𝑔‖_∞≜‖𝑔‖_{𝐿∞(𝜕Ω)}.

Our main result is in the following.

Theorem 2. Assume (𝐴₁)--(𝐴₃); there exists 𝜆^∗>0 such that problem (1) has at least two positive solutions for all 𝜆>0 satisfying 0<𝜆<𝜆^∗.

This paper is organized as follows. In Section 2, we give some properties of the Nehari manifold and set up the variational framework for problem (1). In Section 3, we consider the multiplicity results and prove Theorem 2.

2. Preliminary Results

It is clear that problem (1) has a variational structure. Let 𝐽_𝜆(𝑢)∶𝐸→ℝ¹ be the corresponding Euler functional of problem (1), which is defined by 𝐽_𝜆(𝑢)=1𝑝𝑀‖𝑢‖^𝑝_𝐸−1𝑞_𝜕Ω𝑔(𝑥)|𝑢|^𝑞−1𝑑𝜎𝑟_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥, (10) where 𝑀(𝑡)=∫^𝑡₀𝑀(𝑠)𝑑𝑠. Then, we see that the functional 𝐽_𝜆(𝑢)∈𝐶¹(𝐸,ℝ¹), and for ∀𝜑∈𝐸, there holds 𝐽^′_𝜆(𝑢),𝜑=𝑀‖𝑢‖^𝑝_𝐸Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−2−∇𝑢⋅∇𝜑𝑑𝑥_𝜕Ω𝑔|𝑢|^𝑞−2𝑢𝜑𝑑𝜎−𝜆_Ωℎ|𝑢|^𝑟−2𝑢𝜑𝑑𝑥. (11) Particularly, we have 𝐽^′_𝜆(𝑢),𝑢=𝑀‖𝑢‖^𝑝_𝐸Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝−𝑑𝑥_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎−𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥. (12) It is well known that the weak solution of problem (1) is the critical point of 𝐽_𝜆(𝑢). Thus, to prove the existence of weak solutions for problem (1), it is sufficient to show that 𝐽_𝜆(𝑢) admits a sequence of critical points. Since 𝐽_𝜆(𝑢) is not bounded below on 𝐸, it is useful to consider the functional 𝐽_𝜆(𝑢) on the Nehari manifold [17, 18]: 𝑁_𝜆=𝑢∈𝐸⧵{0}∣𝐽^′_𝜆(𝑢),𝑢=0, (13) where ⟨,⟩ denotes the usual duality. Then, it follows from (12) that 𝑢∈𝑁_𝜆 if and only if 𝑀‖𝑢‖^𝑝_𝐸‖𝑢‖^𝑝_𝐸−_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎−𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥=0. (14) Then, we get from (10)–(14) that 𝐽_𝜆(𝑢)=1−12𝑝𝑟𝛼‖𝑢‖_𝐸^2𝑝−1𝑟−1𝑝𝛽‖𝑢‖^𝑝_𝐸+1𝑟−1𝑞_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎=1−12𝑝𝑞𝛼‖𝑢‖_𝐸^2𝑝+1𝑝−1𝑞𝛽‖𝑢‖^𝑝_𝐸−1𝑟−1𝑞𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥. (15)(16) We define Φ_𝜆(𝑢)=𝐽^′_𝜆(𝑢),𝑢. (17) Then, (14) implies that Φ^′_𝜆(𝑢),𝑢=𝑝𝑀^′‖𝑢‖^𝑝_𝐸‖𝑢‖_𝐸^2𝑝+𝑝𝑀‖𝑢‖^𝑝_𝐸‖𝑢‖^𝑝_𝐸−𝑞_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎−𝑟𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥=2𝑝−𝑞𝛼‖𝑢‖_𝐸^2𝑝+𝑝−𝑞𝛽‖𝑢‖^𝑝_𝐸+𝑞−𝑟𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥=2𝑝−𝑟𝛼‖𝑢‖_𝐸^2𝑝+𝑝−𝑟𝛽‖𝑢‖^𝑝_𝐸−𝑞−𝑟_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎. (18)(19)(20) It is natural to split 𝑁_𝜆 into three parts: 𝑁⁺_𝜆=𝑢∈𝑁_𝜆∣Φ^′_𝜆(𝑢),𝑁,𝑢>0_−𝜆=𝑢∈𝑁_𝜆∣Φ^′_𝜆(𝑢),𝑁,𝑢<0_0𝜆=𝑢∈𝑁_𝜆∣Φ^′_𝜆(𝑢),𝑢=0. (21) Now, we give some important properties of 𝑁⁺_𝜆, 𝑁⁻_𝜆, and 𝑁⁰_𝜆.

Lemma 3. Let 1<𝑟<𝑝<𝑞 and 𝑞≥2𝑝. Then, 𝐽_𝜆(𝑢) is coercive and bounded below on 𝑁_𝜆.

Proof. For 1<𝑟<𝑝<𝑞, we obtain from (𝐴₁), the Hölder and Caffarelli-Kohn-Nirenberg inequalities that _Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥≤_Ω|ℎ(𝑥)||𝑥|^𝑏𝑟𝜂𝑑𝑥^1/𝜂×_Ω|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥^𝑟/𝑝∗≤𝑆^𝑟₁ℎ_𝜂Ω|𝑥|^−𝑎𝑝|∇𝑢|^𝑝𝑑𝑥^𝑟/𝑝≤𝑆^𝑟₁ℎ_𝜂‖𝑢‖^𝑟_𝐸, (22) where ℎ_𝜂=(∫_Ω(|𝑔(𝑥)||𝑥|^𝑏𝑟)^𝜂𝑑𝑥)^1/𝜂, 𝜂=𝑝^∗/(𝑝^∗−𝑟). Thus, we get from (16) that 𝐽_𝜆(𝑢)=1−12𝑝𝑞𝛼‖𝑢‖_𝐸^2𝑝+1𝑝−1𝑞𝛽‖𝑢‖^𝑝_𝐸−1𝑟−1𝑞_Ωℎ(𝑥)|𝑢|^𝑟≥‖𝑢‖𝑑𝑥_𝑝𝐸1−12𝑝𝑞𝛼‖𝑢‖^𝑝_𝐸+1𝑝−1𝑞𝛽−1𝑟−1𝑞𝜆𝑆^𝑟₁ℎ_𝜂‖𝑢‖^𝑟_𝐸. (23) Then, one can obtain by the Young inequality and 𝑞≥2𝑝 that 𝐽_𝜆(𝑢) is coercive and bounded below on 𝑁_𝜆.
Let 𝜆₀=2𝑞−2𝑝𝑞−𝑝𝑞−𝑟𝑆^𝑟₂ℎ_𝜂×2𝛼𝛽2𝑝−𝑟𝑝−𝑟𝑞−𝑟𝑔_∞^{(3𝑝−2𝑟)/(2𝑞−3𝑝)},𝑞>2𝑝,𝑝𝛽2𝑝−𝑟𝑆^𝑟₁ℎ_𝜂×2𝛼𝛽2𝑝−𝑟𝑝−𝑟2𝑝−𝑟𝑔_∞^{2(𝑝−𝑟)/𝑝},𝑞=2𝑝. (24) Then, we have the following result.

Lemma 4. Let 𝑟<𝑝<𝑞,𝑞≥2𝑝. Then, 𝑁⁰_𝜆=∅ for 0<𝜆<𝜆₀.

Proof. Suppose that there exists 𝑢∈𝑁⁰_𝜆. If 𝑞>2𝑝, then it follows from (19) and (21) that 2𝛽𝑞−2𝑝𝑞−𝑝‖𝑢‖_𝐸^3𝑝/2≤𝑞−2𝑝𝛼‖𝑢‖_𝐸^2𝑝+𝛽𝑞−𝑝‖𝑢‖^𝑝_𝐸=𝑞−𝑟𝜆_Ωℎ(𝑥)|𝑢|^𝑟≤𝑑𝑥𝑞−𝑟𝜆𝑆^𝑟₁ℎ_𝜂‖𝑢‖^𝑟_𝐸, (25) which implies that ‖𝑢‖_𝐸≤𝑞−𝑟𝜆𝑆^𝑟₁ℎ_𝜂2𝑞−2𝑝𝑞−𝑝^{2/(3𝑝−2𝑟)}. (26) On the other hand, we can similarly get from (20) and (21) that 2𝛼𝛽2𝑝−𝑟𝑝−𝑟‖𝑢‖_𝐸^3𝑝/2≤2𝑝−𝑟𝛼‖𝑢‖_𝐸^2𝑝+𝑝−𝑟‖𝑢‖^𝑝_𝐸=𝑞−𝑟_𝜕Ω𝑔(𝑥)|𝑢|^𝑞≤𝑔𝑑𝜎𝑞−𝑟_∞‖𝑢‖^𝑞_𝐸, (27) which yields that ‖𝑢‖_𝐸≥2𝛼𝛽2𝑝−𝑟𝑝−𝑟𝑞−𝑟𝑔_∞^{2/(2𝑞−3𝑝)}. (28) Thus, inequalities (26) and (28) show that 𝜆≥𝜆₀, which contradicts the hypothesis of 𝜆.
For 𝑞=2𝑝, we can similarly obtain from (19) and (21) that ‖𝑢‖_𝐸≤𝑞−𝑟𝜆𝑆^𝑟₁ℎ_𝜂𝑝𝛽^{1/(𝑝−𝑟)}. (29) Thus, (28) and (29) imply that 𝜆>𝜆₀, which is also a contradiction. Therefore, we complete the proof.

Lemma 5. If 𝑢₀ is a local minimizer of 𝐽_𝜆(𝑢) on 𝑁_𝜆 and 𝑢₀∉𝑁⁰_𝜆, then 𝑢₀ is a critical point of 𝐽_𝜆(𝑢).

Proof. Let 𝐹(𝑢)=𝑀‖𝑢‖^𝑝_𝐸‖𝑢‖^𝑝_𝐸−_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎−𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥,∀𝑢∈𝐸. (30) Consider the following minimizing problem: min_{𝑢∈𝑁𝜆}𝐽_𝜆(𝑢),subjectto𝐹(𝑢)=0. (31) By the Lagrange multiplier principle, there exists 𝛾∈ℝ¹ such that 𝐽^′_𝜆𝑢₀=𝛾𝐹^′𝑢₀,𝐽^′_𝜆𝑢₀,𝑢₀=𝛾𝐹^′𝑢₀,𝑢₀=𝛾Φ^′_𝜆𝑢₀,𝑢₀. (32)(33) Since 𝑢₀∈𝑁_𝜆 and 𝑢₀∉𝑁⁰_𝜆, it follows from (33) that 𝛾=0; furthermore, 𝐽^′_𝜆(𝑢₀)=0.

Now, we write 𝑁_𝜆=𝑁⁺_𝜆∪𝑁⁻_𝜆 for 0<𝜆<𝜆₀ and define 𝛿⁺_𝜆=inf_{𝑢∈𝑁+𝜆}𝐽_𝜆(𝑢),𝛿⁻_𝜆=inf_{𝑢∈𝑁−𝜆}𝐽_𝜆(𝑢). (34) Denote 𝜆₁=𝑟𝑞−𝑝𝛽𝑝𝑞−𝑟𝑆^𝑟₁ℎ_𝜂𝛽𝑝−𝑟𝑆₂𝑞−𝑟𝑔_∞^{(𝑝−𝑟)/(𝑞−𝑝)}. (35) Then, the following results on 𝛿⁺_𝜆 and 𝛿⁻_𝜆 are established.

Lemma 6. Let 𝑟<𝑝<𝑞,𝑞≥2𝑝, and 0<𝜆<𝜆₁; then(i)𝛿⁺_𝜆<0,(ii)there exists constant 𝑘₀>0 such that 𝛿⁻_𝜆>𝑘₀.

Proof. (i) For ∀𝑢∈𝑁⁺_𝜆, we have from (19) and (21) that 𝑞−2𝑝𝛼‖𝑢‖2𝑝𝑞^𝐸2𝑝+𝑞−𝑝𝛽‖𝑢‖𝑝𝑞^𝑝𝐸≤𝑞−𝑟𝜆𝑝𝑞_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝜆𝑑𝜎,_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝛽𝑑𝜎≥𝑞−𝑝‖𝑢‖𝑞−𝑟^𝑝𝐸. (36) Thus, (36) and (16) give that 𝐽_𝜆(𝑢)<𝑞−𝑟−𝑝𝑞𝑞−𝑟𝜆𝑞𝑟_Ωℎ(𝑥)|𝑢|^𝑟𝛽𝑑𝑥≤−𝑝−𝑟𝑞−𝑝‖𝑢‖𝑞𝑝𝑟^𝑝𝐸<0, (37) which implies that 𝛿⁺_𝜆=inf_{𝑢∈𝑁+𝜆}𝐽_𝜆(𝑢)<0. (38)
(ii) Let 𝑢∈𝑁⁺_𝜆; one can deduce from (20) and (21) that ‖𝑢‖_𝐸≥(𝑝−𝑟)𝛽𝑆₂(𝑞−𝑟)𝑔_∞^{1/(𝑞−𝑝)}. (39) On the other hand, we obtain from (16), (26), and (39) that 𝐽_𝜆(𝑢)≥1−12𝑝𝑞𝛼‖𝑢‖_𝐸^2𝑝+1𝑝−1𝑞𝛽‖𝑢‖^𝑝_𝐸−𝑞−𝑟𝜆𝑆𝑟𝑞^𝑟1ℎ_𝜂‖𝑢‖^𝑟_𝐸≥‖𝑢‖^𝑟_𝐸𝑞−𝑝𝛽‖𝑢‖𝑝𝑞^{𝐸𝑝−𝑟}−𝑞−𝑟𝜆𝑆^𝑟₁ℎ_𝜂≥‖𝑢‖𝑞𝑟_𝑟𝐸𝑞−𝑝𝛽𝛽𝑝𝑞𝑝−𝑟𝑆₂𝑞−𝑟𝑔_∞^{(𝑝−𝑟)/(𝑞−𝑝)}−𝑞−𝑟𝜆𝑆^𝑟₁ℎ_𝜂.𝑟𝑞 (40) Therefore, if 0<𝜆<𝜆₁, there exists 𝑘₀>0 such that 𝛿⁻_𝜆>𝑘₀.

For each 𝑢∈𝐸 with ∫_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎>0, we define 𝑚_𝛼(𝑡)=𝛼𝑡^2𝑝−𝑟‖𝑢‖_𝐸^2𝑝+𝛽𝑡^𝑝−𝑟‖𝑢‖^𝑝_𝐸−𝑡^𝑞−𝑟_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎. (41) Then, 𝑚_𝛼(0)=0, 𝑚_𝛼(𝑡)→−∞ as 𝑡→+∞, and 𝑚_𝛼(𝑡) gets its unique maximum at the critical point 𝑡_𝛼,max. Particularly, 𝑚₀(𝑡) gets its unique maximum at 𝑡_0,max=𝛽𝑝−𝑟‖𝑢‖^𝑝_𝐸𝑞−𝑟∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎^{1/(𝑞−𝑝)}, (42) and we have the following results.

Lemma 7. Let 𝑟<𝑝<𝑞<𝑝_∗, 𝑞≥2𝑝, and 0<𝜆<(𝑝/𝑟)𝜆₁. For each 𝑢∈𝐸 with ∫_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎>0, one has the following:(i)if ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥≤0, then there exists 𝑡⁻>𝑡_𝛼,max such that 𝑡⁻𝑢∈𝑁⁻_𝜆 and 𝐽_𝜆(𝑡⁻𝑢)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢); (43)(ii)if ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥>0, then there exists 0<𝑡⁺<𝑡_𝛼,max<𝑡⁻ such that 𝑡⁺𝑢∈𝑁⁺_𝜆,𝑡⁻𝑢∈𝑁⁻_𝜆, and 𝐽_𝜆𝑡⁺𝑢=inf_{0<𝑡≤𝑡𝛼,max}𝐽_𝜆(),𝐽𝑡𝑢_𝜆(𝑡⁻𝑢)=sup_{𝑡≥𝑡𝛼,max}𝐽_𝜆(𝑡𝑢). (44)

Proof. (i) For ∀𝑢∈𝐸, we define 𝜙₀(𝑡)=𝐽^′_𝜆(𝑡𝑢),𝑡𝑢=𝑡^2𝑝𝛼‖𝑢‖_𝐸^2𝑝+𝑡^𝑝𝛽‖𝑢‖^𝑝_𝐸−𝑡^𝑞×_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎−𝑡^𝑟𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥,𝜙₁(𝑡)=Φ^′_𝜆(𝑡𝑢),𝑡𝑢=2𝑝𝛼𝑡^2𝑝‖𝑢‖_𝐸^2𝑝+𝑡^𝑝𝛽‖𝑢‖^𝑝_𝐸−𝑡^𝑞_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎−𝑡^𝑟𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥,𝜙₂(𝑡)=𝐽_𝜆()=𝛼𝑡𝑢𝑡2𝑝^2𝑝‖𝑢‖_𝐸^2𝑝+𝛽𝑝𝑡^𝑝‖𝑢‖^𝑝_𝐸−1𝑞𝑡^𝑞_𝜕Ω𝑔|𝑢|^𝑞𝜆𝑑𝜎−𝑟𝑡^𝑟_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥. (45)(46)(47) Since ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥≤0, there exists unique 𝑡⁻>𝑡_𝛼,max such that 𝑚^′_𝛼(𝑡⁻)<0,𝑚_𝛼(𝑡⁻)=_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥. (48) Thus, 𝜙₀(𝑡⁻)=𝑡^𝑟(𝑚_𝛼(𝑡⁻)−∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥)=0, which implies that 𝑡⁻𝑢∈𝑁_𝜆. It follows from (20) that 𝜙₁(𝑡⁻)=2𝑝−𝑟𝛼(𝑡⁻)^2𝑝‖𝑢‖_𝐸^2𝑝+𝑝−𝑟𝛽(𝑡⁻)^𝑝‖𝑢‖^𝑝_𝐸−𝑞−𝑟(𝑡⁻)^𝑞_𝜕Ω𝑔|𝑢|^𝑞=(𝑡𝑑𝜎₋)^𝑟+1𝑚^′_𝛼(𝑡⁻)<0; (49) that is, 𝑡⁻𝑢∈𝑁⁻_𝜆. By (47), we obtain that 𝜙^′₂(𝑡)=𝛼𝑡^2𝑝−1‖𝑢‖_𝐸^2𝑝+𝛽𝑡^𝑝−1‖𝑢‖^𝑝_𝐸−𝑡^𝑞−1_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎−𝜆𝑡^𝑟−1_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥=𝑡^𝑟−1𝑚_𝛼(𝑡)−𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥, (50) which shows that 𝜙₂(𝑡) increases for 𝑡∈[0,𝑡⁻] and decreases for 𝑡∈[𝑡⁻,+∞). Therefore, 𝐽_𝜆(𝑡⁻𝑢)=𝜙₂(𝑡⁻)=sup_𝑡≥0𝐽_𝜆(𝑡𝑢). (51)
(ii) We firstly want to prove that 0<∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥≤𝑚_𝛼(𝑡_𝛼,max). In fact, 𝑚₀𝑡_0,max𝛽‖𝑢‖=𝛽𝑝−𝑟^𝑝_𝐸𝑞−𝑟∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎^{(𝑝−𝑟)/(𝑞−𝑝)}‖𝑢‖^𝑝_𝐸−𝛽𝑝−𝑟‖𝑢‖^𝑝_𝐸𝑞−𝑟∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎^{(𝑞−𝑟)/(𝑞−𝑝)}×_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎=𝛽^{(𝑞−𝑟)/(𝑞−𝑝)}‖𝑢‖_𝐸^{𝑝(𝑞−𝑟)/(𝑞−𝑝)}𝑝−𝑟𝑞−𝑟^{(𝑝−𝑟)/(𝑞−𝑝)}×𝑞−𝑝1𝑞−𝑟𝑔_∞𝑆₂‖𝑢‖^𝑞_𝐸^{(𝑝−𝑟)/(𝑞−𝑝)}≥𝛽^{(𝑞−𝑟)/(𝑞−𝑝)}‖𝑢‖^𝑟_𝐸𝑞−𝑝×𝑞−𝑟𝑝−𝑟𝑞−𝑟𝑔_∞^{(𝑝−𝑟)/(𝑞−𝑝)}. (52) Then, if 0<𝜆<(𝑝/𝑟)𝜆₁, we have that 𝑚_𝛼(0)=0<𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥≤𝛽^{(𝑞−𝑟)/(𝑞−𝑝)}‖𝑢‖^𝑟_𝐸𝑞−𝑝𝑞−𝑟𝑝−𝑟𝑞−𝑟𝑔_∞^{(𝑝−𝑟)/(𝑞−𝑝)}≤𝑚₀𝑡_0,max<𝑚_𝛼𝑡_𝛼,max. (53) Since ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥>0, there exists 0<𝑡⁺<𝑡_𝛼,max<𝑡⁻ such that 𝑚_𝛼𝑡⁺=𝑚_𝛼(𝑡⁻)=𝜆_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥, (54) and 𝑚^′_𝛼(𝑡⁺)>0, 𝑚^′_𝛼(𝑡⁻)<0. Similar to the proof of (i), we get that 𝑡⁺𝑢∈𝑁⁺_𝜆, 𝑡⁻𝑢∈𝑁⁻_𝜆. We can deduce from (50) that 𝜙₂(𝑡) decreases for 𝑡∈[0,𝑡⁺] and increases for 𝑡∈[𝑡⁺,𝑡_𝛼,max). Therefore, 𝐽_𝜆𝑡⁺𝑢=𝜙₂𝑡⁺=inf_{0≤𝑡≤𝑡𝛼,max}𝐽_𝜆(𝑡𝑢). (55) Similarly, 𝐽_𝜆(𝑡⁻𝑢)=𝜙₂(𝑡⁻)=sup_{𝑡≥𝑡𝛼,max}𝐽_𝜆(𝑡𝑢). (56) Then, we complete the proof.

For each 𝑢∈𝐸 with ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥>0, we define 𝑚_𝛼(𝑡)=𝛼𝑡^2𝑝−𝑞‖𝑢‖_𝐸^2𝑝+𝛽𝑡^𝑝−𝑞‖𝑢‖^𝑝_𝐸−𝜆𝑡^𝑟−𝑞_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥. (57) Then, 𝑚_𝛼(𝑡)→−∞ as 𝑡→0⁺ and 𝑚_𝛼(𝑡)→0 as 𝑡→∞. Furthermore, 𝑚_𝛼(𝑡) gets its unique maximum at some certain point 𝑡_𝛼,max. Particularly, 𝑚₀(𝑡) gets its unique maximum at the critical point 𝑡₀=𝑡_0,max(𝑢)=𝜆𝑞−𝑟∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥𝛽𝑞−𝑝‖𝑢‖^𝑝_𝐸^{1/(𝑝−𝑟)}, (58) then we have the following results.

Lemma 8. Let 1<𝑟<𝑝<𝑞, 𝑞≥2𝑝, and 0<𝜆<𝑝𝜆₁/𝑟. For each 𝑢∈𝐸 with ∫_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥>0 one has the following:(i)if ∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎<0, then there exists unique 0<𝑡⁺<𝑡_𝛼,max such that 𝑡⁺𝑢∈𝑁⁺_𝜆 and 𝐽_𝜆𝑡⁺𝑢=inf_𝑡≥0𝐽_𝜆(𝑡𝑢); (59)(ii)if ∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎≥0, then there exists 0<𝑡⁺<𝑡_𝛼,max<𝑡⁻ such that 𝑡⁺𝑢∈𝑁⁺_𝜆, 𝑡⁻𝑢∈𝑁⁻_𝜆, and 𝐽_𝜆𝑡⁺𝑢=inf_{0<𝑡≤𝑡𝛼,max}𝐽_𝜆(𝑡𝑢),𝐽_𝜆(𝑡⁻𝑢)=sup_{𝑡≥𝑡𝛼,max}𝐽_𝜆(𝑡𝑢). (60)

Proof. (i) Since ∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎<0, there exists unique 0<𝑡⁺<𝑡_𝛼,max such that 𝑚^′_𝛼𝑡⁺>0,𝑚_𝛼𝑡⁺=_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎. (61) Note that (45) can be rewritten as 𝜙₀(𝑡)=𝑡^𝑞𝑚_𝛼(𝑡)−_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎. (62) Thus, (61) shows that 𝑡⁺𝑢∈𝑁_𝜆. By virtue of (14) and (46), we get that 𝜙₁𝑡⁺=Φ^′_𝜆𝑡⁺𝑢,𝑡⁺𝑢=𝑡^+𝑞+12𝑝−𝑞𝛼𝑡^{+2𝑝−𝑞−1}+𝑝−𝑞𝛽𝑡^{+𝑝−𝑞−1}‖𝑢‖^𝑝_𝐸−𝜆𝑟−𝑞𝑡^{+𝑟−𝑞−1}_Ωℎ(𝑥)|𝑢|^𝑟=𝑡𝑑𝑥_+𝑞+1𝑚^′_𝛼𝑡⁺>0, (63) which implies that 𝑡⁺𝑢∈𝑁⁺_𝜆. We have from (47) that 𝜙^′₂(𝑡)=𝑡^𝑞−1𝑚_𝛼(𝑡)−_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎. (64) Then, 𝜙₂(𝑡) decreases for 𝑡∈[0,𝑡⁺] and increases for 𝑡∈[𝑡⁺,+∞); that is, 𝐽_𝜆𝑡⁺𝑢=inf_𝑡≥0𝐽_𝜆(𝑡𝑢). (65)
(ii) We need also to prove that 0<∫_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎<𝑚_𝛼(𝑡). In fact, we have that 𝑚_𝛼𝑡_𝛼,max≥𝑚₀𝑡₀𝜆∫=𝛽𝑞−𝑟_Ωℎ|𝑢|^𝑟𝑑𝑥𝛽‖𝑢‖𝑞−𝑝^{𝑝𝐸(𝑝−𝑞)/(𝑝−𝑟)}‖𝑢‖_𝐸^2𝑝𝜆∫−𝜆𝑞−𝑟_Ωℎ|𝑢|^𝑟𝑑𝑥𝛽‖𝑢‖𝑞−𝑝^{𝑝𝐸(𝑟−𝑞)/(𝑝−𝑟)}=𝜆^{(𝑝−𝑞)/(𝑝−𝑟)}𝛽^{(𝑞−𝑟)/(𝑝−𝑟)}‖𝑢‖_𝐸^{𝑝(𝑞−𝑟)/(𝑝−𝑟)}×𝑞−𝑝𝑞−𝑟^{(𝑞−𝑝)/(𝑝−𝑟)}×𝑝−𝑟𝑞−𝑟_Ωℎ|𝑢|^𝑟𝑑𝑥^{(𝑝−𝑞)/(𝑝−𝑟)}≥𝜆^{(𝑝−𝑞)/(𝑝−𝑟)}𝛽^{(𝑞−𝑟)/(𝑝−𝑟)}𝑞−𝑟×𝑝−𝑟𝑞−𝑝𝑞−𝑟𝑆^𝑟₁ℎ_𝜂^{(𝑞−𝑝)/(𝑝−𝑟)}‖𝑢‖^𝑞_𝐸≥_𝜕Ω𝑔|𝑢|^𝑞𝑑𝜎≥0 (66) for 0<𝜆≤(𝑝/𝑟)𝜆₁.
The rest of the proof is similar to that of (ii) in Lemma 7, and here we omit the proof.

Lemma 9. Assume (𝐴₁)--(𝐴₃). If 𝑢_𝑛⇀𝑢 in 𝐸, then there exists a subsequence of {𝑢_𝑛}, still denoted by {𝑢_𝑛}, such that _Ωℎ(𝑥)𝑢_𝑛^𝑟𝑑𝑥⟶_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥,_𝜕Ω𝑔(𝑥)𝑢_𝑛^𝑞𝑑𝜎⟶_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎. (67)

Proof. By the assumption (𝐴₂), we have ℎ(𝑥)|𝑥|^𝑏𝑟∈𝐿^𝜂(Ω)∩𝐿^∞(Ω), and then for any 𝜀>0, there exists 𝑅_𝜀>0 large enough such that _{𝐵𝑐𝑅𝜀}|ℎ(𝑥)||𝑥|^𝑏𝑟𝜂𝑑𝑥≤𝜀^𝜂, (68) where 𝐵_𝑅={𝑥∈ℝ^𝑁∣|𝑥|≤𝑅}, 𝐵^𝑐_𝑅={𝑥∈ℝ^𝑁∣|𝑥|>𝑅} for 𝑅>0.
The compact embedding theorem 𝑊_𝑎^1,𝑝(Ω)↪𝐿^𝑟(Ω,|𝑥|^−𝛼) (Theorem 2.1 in [19]) implies that 𝑢_𝑛⟶𝑢a.e.inΩ. (69) Inequality (7) shows that {𝑢_𝑛} is bounded in 𝐿^𝑝∗(ℝ^𝑁,|𝑥|^𝑏𝑟), which implies that 𝑢_𝑛⇀𝑢 in 𝐿^𝑝∗(ℝ^𝑁,|𝑥|^𝑏𝑟). Thus, we can obtain that _{𝐵𝑅𝜀⧵𝐷}|𝑥|^{−𝑏𝑝∗}𝑢_𝑛−𝑢^𝑝∗𝑑𝑥<𝜀^𝑝∗,_Ω|𝑥|^{−𝑏𝑝∗}𝑢_𝑛^𝑝∗𝑑𝑥≤𝑀^𝑝∗,_Ω|𝑥|^{−𝑏𝑝∗}|𝑢|^𝑝∗𝑑𝑥≤𝑀^𝑝∗ (70) for some 𝑀>0. On the other hand, we get from the Höder inequality and (68)–(70) that _{𝐵𝑐𝑅𝜀}ℎ𝑢_𝑛−𝑢^𝑟≤𝑑𝑥_{𝐵𝑐𝑅𝜀}|ℎ||𝑥|^𝑏𝑟𝜂𝑑𝑥^1/𝜂×_{𝐵𝑐𝑅𝜀}|𝑥|^{−𝑏𝑝∗}𝑢_𝑛−𝑢^𝑝∗𝑑𝑥^𝑟/𝑝∗≤2^𝑟𝑀^𝑟𝜀,_{𝐵𝑅𝜀⧵𝐷}ℎ𝑢_𝑛−𝑢^𝑟≤𝑑𝑥_{𝐵𝑅𝜀⧵𝐷}|ℎ||𝑥|^𝑏𝑟𝜂𝑑𝑥^1/𝜂×_{𝐵𝑅𝜀⧵𝐷}|𝑥|^{−𝑏𝑝∗}𝑢_𝑛−𝑢^𝑝∗𝑑𝑥^𝑟/𝑝∗≤𝑐₀𝜀^𝑟 (71) for some constant 𝑐₀>0 and large 𝑛.
Thus, (71) implies that _Ωℎ(𝑥)𝑢_𝑛−𝑢^𝑟𝑑𝑥⟶0as𝑛⟶∞; (72) that is, _Ωℎ(𝑥)𝑢_𝑛^𝑟𝑑𝑥⟶_Ωℎ(𝑥)|𝑢|^𝑟𝑑𝑥as𝑛⟶∞. (73) Since 𝜕Ω is compact and 𝑔∈𝐶(𝜕Ω)∩𝐿^∞(𝜕Ω), we obtain by the trace embedding theorem in [20] that 𝜆_𝜕Ω𝑔(𝑥)𝑢_𝑛^𝑞𝑑𝜎⟶𝜆_𝜕Ω𝑔(𝑥)|𝑢|^𝑞𝑑𝜎as𝑛⟶∞. (74) This concludes the proof.

3. Existence of Solutions

In this part, we will give the proof of the existence of nonnegative and nontrivial solutions. Before this, we need to prove the following two important lemmas.

Lemma 10. Assume (𝐴₃) and 0<𝜆<(𝑝/𝑟)𝜆₁. Then, the functional 𝐽_𝜆(𝑢) has a minimizer 𝑢⁺₀∈𝑁⁺_𝜆 and(i)𝐽_𝜆(𝑢⁺₀)=𝛿⁺_𝜆, (ii)𝑢⁺₀ is a positive weak solution of problem (1).

Proof. Since 𝐽_𝜆(𝑢) is bounded in 𝑁⁺_𝜆, there exists a minimizing sequence of {𝑢_𝑘}∈𝑁⁺_𝜆 such that lim_𝑘→∞𝐽_𝜆𝑢_𝑘=inf_{𝑢∈𝑁+𝜆}𝐽_𝜆(𝑢). (75) We can get that {𝑢_𝑘} is bounded in 𝐸 and 𝑢_𝑘⇀𝑢⁺₀ weakly in 𝐸 since 𝐽_𝜆(𝑢) is coercive. It follows from Lemma 6 that 𝐽_𝜆𝑢_𝑘⟶𝛿⁺_𝜆<0as𝑘⟶+∞. (76) Furthermore, equality (16) and Lemma 9 imply that ∫_Ωℎ(𝑥)|𝑢⁺₀|^𝑟𝑑𝑥>0. In the following, we prove that 𝑢_𝑘→𝑢⁺₀ in 𝐸. Suppose otherwise that 𝑢⁺_0𝐸<lim_𝑘→+∞inf𝑢_𝑘𝐸. (77) Let 𝑚_𝛼,𝑢⁺₀(𝑡)=𝛼𝑡^2𝑝−𝑞𝑢⁺_0𝐸^2𝑝+𝛽𝑡^𝑝−𝑞𝑢⁺₀^𝑝_𝐸−𝜆𝑡^𝑟−𝑞_Ωℎ(𝑥)𝑢⁺₀^𝑟𝜑𝑑𝑥,₀(𝑡)=𝛼𝑡^2𝑝−𝑞𝑢⁺_0𝐸^2𝑝+𝛽𝑡^𝑝−𝑞𝑢⁺₀^𝑝_𝐸−𝜆𝑡^𝑟−𝑞_Ωℎ(𝑥)𝑢⁺₀^𝑟𝑑𝑥−_𝜕Ω𝑔(𝑥)𝑢⁺₀^𝑞𝑑𝜎. (78) It is obvious that 𝑚_𝛼,𝑢⁺₀(𝑡) has the unique critical point 𝑡_𝛼,max(𝑢⁺₀) and 𝑚_𝛼,𝑢⁺₀(𝑡) increases for 𝑡∈[0,𝑡_𝛼,max(𝑢⁺₀)] and decreases for 𝑡∈[𝑡_𝛼,max(𝑢⁺₀),∞). Particularly, 𝑚_0,𝑢⁺₀(𝑡) has the unique critical point at 𝑡₀(𝑢⁺₀)=𝑡_0,max(𝑢⁺₀). Since ∫_Ωℎ(𝑥)|𝑢⁺₀|^𝑟𝑑𝑥>0, it follows from Lemma 8 that there exists 𝑡⁺₀<𝑡_𝛼,max(𝑢⁺₀) such that 𝑡⁺₀𝑢⁺₀∈𝑁⁺_𝜆 and 𝐽_𝜆𝑡⁺₀𝑢⁺₀=inf_{0≤𝑡≤𝑡𝛼,max𝑢+0}𝐽_𝜆𝑡𝑢⁺₀. (79) By virtue of 𝑡⁺₀𝑢⁺₀∈𝑁⁺_𝜆, we obtain that 𝜑₀𝑡⁺₀=𝑡⁺₀^−𝑞𝐽_𝜆𝑡⁺₀𝑢⁺₀,𝑡⁺₀𝑢⁺₀=0. (80) Similarly, we set 𝜑_𝑘(𝑡)=𝛼𝑡^2𝑝−𝑞𝑢_𝑘𝐸^2𝑝+𝛽𝑡^𝑝−𝑞𝑢_𝑘^𝑝_𝐸−𝜆𝑡^𝑟−𝑞_Ωℎ(𝑥)𝑢_𝑘^𝑟−𝑑𝑥_𝜕Ω𝑔(𝑥)𝑢_𝑘^𝑞𝑑𝜎. (81) In view of (77), we get that 𝜑_𝑘(𝑡⁺₀)>𝜑₀(𝑡⁺₀)=0 for large 𝑘. Since 𝜑_𝑘(1)=0 and 𝜑^′_𝑘(𝑡)≥𝑚^′_𝛼,𝑢⁺₀(𝑡)>0 for 𝑡∈(0,𝑡_𝛼,max(𝑢⁺₀)], we obtain that 1<𝑡⁺₀<𝑡_𝛼,max(𝑢⁺₀) and 𝜑_𝑘(𝑡)<0 for 𝑡∈[0,1]. Then, by (77), (79), and Lemma 9, we get that 𝐽_𝜆𝑡⁺₀𝑢⁺₀≤𝐽_𝜆𝑢⁺₀<lim_𝑘→+∞𝐽_𝜆𝑢_𝑘=𝛿⁺_𝜆, (82) which is a contradiction. Hence, 𝑢_𝑘→𝑢⁺₀ strongly in 𝐸, and 𝐽_𝜆𝑢_𝑘⟶𝐽_𝜆𝑢⁺₀=𝛿⁺_𝜆as𝑘⟶+∞. (83) Thus, 𝑢⁺₀ is a minimum of 𝐽_𝜆(𝑢) on 𝑁⁺_𝜆. Since 𝐽_𝜆(𝑢⁺₀)=𝐽_𝜆(|𝑢⁺₀|) and |𝑢⁺₀|∈𝑁⁺_𝜆, we may assume by Lemma 5 that 𝑢⁺₀ is a positive solution of problem (1).

Lemma 11. Assume (𝐴₃) and 0<𝜆<(𝑝/𝑟)𝜆₁. Then, the functional 𝐽_𝜆(𝑢) has a minimizer 𝑢⁻₀∈𝑁⁻_𝜆 and (i)𝐽_𝜆(𝑢⁻₀)=𝛿⁻_𝜆, (ii)𝑢⁻₀ is a positive weak solution of problem (1).

Proof. Since 𝐽_𝜆(𝑢) is bounded on 𝑁⁻_𝜆, there exists a minimizing sequence {𝑢_𝑘}∈𝑁⁻_𝜆 such that lim_𝑘→∞𝐽_𝜆𝑢_𝑘=inf_{𝑢∈𝑁−𝜆}𝐽_𝜆(𝑢). (84) Similar to the proof of Lemma 10, we may assume that 𝑢_𝑘⇀𝑢⁻₀ weakly in 𝐸. For 𝑢_𝑘∈𝑁⁻_𝜆, we deduce by Lemma 6 and (15) that ∫_𝜕Ω𝑔(𝑥)|𝑢_𝑘|^𝑞𝑑𝜎>0; furthermore, ∫_𝜕Ω𝑔(𝑥)|𝑢⁻₀|^𝑞𝑑𝜎>0. We also want to prove that 𝑢_𝑘→𝑢⁻₀ strongly in 𝐸. In fact, if not, we have 𝑢⁺_0𝐸<lim_𝑘→+∞inf𝑢_𝑘𝐸. (85) By virtue of Lemma 7, there exists 𝑡⁻₀>0 such that 𝐽_𝜆𝑡⁻₀𝑢=sup_𝑡≥0𝐽_𝜆𝑡𝑢_𝑘, (86) and a simple transformation shows that 𝐽_𝜆𝑢_𝑘≥𝐽_𝜆𝑡𝑢_𝑘,∀𝑢_𝑘∈𝑁⁻_𝜆. (87) Therefore, Lemma 9 together with (84)–(87) gives that 𝐽_𝜆𝑡⁻₀𝑢⁻₀<lim_𝑘→+∞𝐽_𝜆𝑡⁻₀𝑢_𝑘≤lim_𝑘→+∞𝐽_𝜆𝑢_𝑘=𝛿⁻_𝜆, (88) which is a contradiction, and we complete the proof.

Proof of Theorem 2. We set 𝜆^∗=min{𝜆₁,𝜆₀}. When 0<𝜆<𝜆^∗, by Lemmas 10 and 11, we obtain that problem (1) has two nontrivial nonnegative solutions 𝑢⁺₀∈𝑁⁺_𝜆, 𝑢⁻₀∈𝑁⁻_𝜆. Lemma 4 and the assumptions of Theorem 2 imply that 𝑁⁺_𝜆∩𝑁⁻_𝜆=∅; then 𝑢⁺₀ and 𝑢⁻₀ are distinct. Furthermore, since 𝐽_𝜆(𝑢^±₀)=𝐽_𝜆(|𝑢^±₀|) and |𝑢^±₀|∈𝑁^±_𝜆, we can assume that the solutions 𝑢⁺₀ and 𝑢⁻₀ are positive. This completes the proof.

4. Conclusions

The object of this paper is to prove the existence of multiple solutions for the nonlinear Kirchhoff-type problem (1). By the variational methods, we discuss the problem on the Nehari manifold and give the sufficient conditions for the existence of solutions. We overcome the difficulty due to the loss of compactness on the unbounded domain. In the future work, we are interested to consider similar problems, but the term on the right will be replaced by abstract functions.

Acknowledgment

The authors would like to express their sincere gratitude to the anonymous reviewers for the valuable comments and suggestions.