Existence Results for Quasilinear Elliptic Equations with Indefinite Weight

Abstract

We provide the existence of a solution for quasilinear elliptic equation −div(𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢+𝑎(𝑥,|∇𝑢|)∇𝑢)=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢+𝑓(𝑥,𝑢)+ℎ(𝑥) in Ω under the Neumann boundary condition. Here, we consider the condition that 𝑎(𝑥,𝑡)=𝑜(𝑡^𝑝−2) as 𝑡→+∞ and 𝑓(𝑥,𝑢)=𝑜(|𝑢|^𝑝−1) as |𝑢|→∞. As a special case, our result implies that the following 𝑝-Laplace equation has at least one solution: −Δ_𝑝𝑢=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢+𝜇|𝑢|^𝑟−2𝑢+ℎ(𝑥) in Ω,𝜕𝑢/𝜕𝜈=0 on 𝜕Ω for every 1<𝑟<𝑝<∞, 𝜆∈ℝ, 𝜇≠0 and 𝑚,ℎ∈𝐿^∞(Ω) with ∫_Ω𝑚𝑑𝑥≠0. Moreover, in the nonresonant case, that is, 𝜆 is not an eigenvalue of the 𝑝-Laplacian with weight 𝑚, we present the existence of a solution of the above 𝑝-Laplace equation for every 1<𝑟<𝑝<∞, 𝜇∈ℝ and 𝑚,ℎ∈𝐿^∞(Ω).

1. Introduction

In this paper, we consider the existence of a solution for the following quasilinear elliptic equation: −div𝐴(𝑥,∇𝑢)=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢+𝑓(𝑥,𝑢)+ℎ(𝑥)inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω, (𝑃;𝜆,𝑚,ℎ)

where Ω⊂ℝ^𝑁 is a bounded domain with 𝐶² boundary 𝜕Ω, 𝜈 denotes the outward unit normal vector on 𝜕Ω, 𝜆∈ℝ, 1<𝑝<∞ and 𝑚,ℎ∈𝐿^∞(Ω). We assume that 𝑓 is a Carathéodory function on Ω×ℝ satisfying lim_|𝑡|→∞𝑓(𝑥,𝑡)|𝑡|^𝑝−2𝑡=0uniformlyin𝑥∈Ω, (1.1) and that 𝑓(𝑥,𝑡) is bounded on a bounded set (admitting 𝑓≡0 in the nonresonant case). Here, 𝐴∶Ω×ℝ^𝑁→ℝ^𝑁 is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (𝐴)). The equation (𝑃;𝜆,𝑚,ℎ) contains the corresponding 𝑝-Laplacian problem as a special case. Although the operator 𝐴 is nonhomogeneous in the second variable in general, we assume that 𝐴(𝑥,𝑦) is asymptotically (𝑝−1)-homogeneous at infinity in the following sense (𝐴𝐻).

Throughout this paper, we assume that the map 𝐴 satisfies the following assumptions (𝐴𝐻) and (𝐴):(𝐴𝐻) there exist a positive function 𝑎_∞∈𝐶¹(Ω,ℝ) and a continuous function 𝑎(𝑥,𝑡) on Ω×ℝ such that 𝐴𝑥,𝑦=𝑎_∞(𝑥)𝑦^𝑝−2𝑦+𝑎𝑥,𝑦𝑦forevery𝑥∈Ω,𝑦∈ℝ^𝑁,lim_𝑡→+∞𝑎(𝑥,𝑡)𝑡^𝑝−2=0uniformlyin𝑥∈Ω. (1.2)(𝐴)𝐴(𝑥,𝑦)=𝑎(𝑥,|𝑦|)𝑦, where 𝑎(𝑥,𝑡)>0 for all (𝑥,𝑡)∈Ω×(0,+∞) and (i)𝐴∈𝐶⁰(Ω×ℝ^𝑁,ℝ^𝑁)∩𝐶¹(Ω×(ℝ^𝑁⧵{0}),ℝ^𝑁); (ii) there exists 𝐶₁>0 such that 𝐷_𝑦𝐴𝑥,𝑦≤𝐶₁𝑦^𝑝−2forevery𝑥∈Ω,𝑦∈ℝ^𝑁⧵{0}; (1.3)(iii) there exists 𝐶₀>0 such that 𝐷_𝑦𝐴𝑥,𝑦𝜉⋅𝜉≥𝐶₀𝑦^𝑝−2𝜉²forevery𝑥∈Ω,𝑦∈ℝ^𝑁⧵{0},𝜉∈ℝ^𝑁; (1.4)(iv) there exists 𝐶₂>0 such that 𝐷_𝑥𝐴𝑥,𝑦≤𝐶₂1+𝑦^𝑝−1forevery𝑥∈Ω,𝑦∈ℝ^𝑁⧵{0}. (1.5)

A similar hypothesis to (𝐴) is considered in the study of quasilinear elliptic problems (cf. [1, Example  2.2], [2–6]). It is easily seen that many examples as in the above references satisfy the condition (𝐴𝐻). Also, the following example satisfies our hypotheses: div|∇𝑢|^𝑝−2+|∇𝑢|^𝑞−21+|∇𝑢|^{𝑞(𝑝−𝑞)/𝑞}∇𝑢for1<𝑝≤𝑞<∞. (1.6) In particular, for 𝐴(𝑥,𝑦)=|𝑦|^𝑝−2𝑦, that is, div𝐴(𝑥,∇𝑢) stands for the usual 𝑝-Laplacian Δ_𝑝𝑢, we can take 𝐶₀=𝐶₁=𝑝−1 in (𝐴). Conversely, in the case where 𝐶₀=𝐶₁=𝑝−1 holds in (𝐴), by the inequalities in Remark 1.4 (ii) and (iii), we see 𝑎(𝑥,𝑡)=|𝑡|^𝑝−2 whence 𝐴(𝑥,𝑦)=|𝑦|^𝑝−2𝑦.

Concerning the weight 𝑚, throughout this paper, we assume that |{𝑚>0}|∶=|{𝑥∈Ω;𝑚(𝑥)>0}|>0 (1.7) holds, where |𝑋| denotes the Lebesgue measure of a measurable set 𝑋.

Because 𝐴(𝑥,𝑦) is asymptotically (𝑝−1)-homogeneous at infinity, the solvability of our equation is related to the following homogeneous equation (see Theorem 1.1): −div𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω, (𝐸𝑉;𝑚) where 𝑎_∞ is the positive function as in (𝐴𝐻). We say that 𝜆∈ℝ is an eigenvalue of (𝐸𝑉;𝑚) if the equation (𝐸𝑉;𝑚) has a nontrivial solution.

There are few existence results of a solution to our equation (and also the 𝑝-Laplace equation). For example, if 𝜆<0 and 𝑚≡1 hold, then the standard argument guarantees the existence of a solution. For the 𝑝-Laplacian as a special case of our problem, it is shown in [7] that the equation −Δ_𝑝𝑢=𝜆𝑚|𝑢|^𝑝−2𝑢+ℎinΩ𝜕𝑢𝜕𝜈=0on𝜕Ω (1.8) has a unique positive solution provided 0<𝜆<𝜆^∗(𝑚), ∫_Ω𝑚𝑑𝑥<0 and 0≢ℎ∈𝐿^∞(Ω)₊, where 𝜆^∗(𝑚) is the principal eigenvalue defined in Section 2.1 with 𝑎_∞≡1. In [8], although the resonant case where 𝜆=𝜆₁(𝑚) or 𝜆=𝜆₂(𝑚) is considered under the assumptions to 𝑓(𝑥,𝑢)=𝑓(𝑢), its result does not cover the case of 𝑓(𝑢)=|𝑢|^𝑟−2𝑢 with 1<𝑟<𝑝, where 𝜆_𝑖(𝑚) (𝑖=1, 2) is 𝑖th eigenvalue of the 𝑝-Laplacian with weight 𝑚. For the Laplace problem under the Neumann boundary condition, we can refer to [9, 10]. Under the Dirichlet boundary condition, the existence results for the Laplace problem are well known when 𝑚≡1 and 𝜆 is not an eigenvalue of the Laplacian (cf. [11]). Moreover, under the Dirichlet (or blow-up) boundary condition, many authors study various equations involving the 𝑝-Laplace (Laplace) operator with (indefinite) weight. For example, we refer to [12] for boundary blow-up problems with Laplacian, [13] for periodic reaction-diffusion problems and [14, 15] for singular quasilinear elliptic problems.

Recently, the present author shows the existence of a solution for our problem in the case where 𝜆 is between the principal eigenvalue and the second eigenvalue in [6] (for 𝑓≡0). In addition, a similar situation is treated in [5]. However, existence results are not seen in the case when 𝜆 is greater than the second eigenvalue for our problem. Therefore, the first purpose of this paper is to present an existence result of a solution in the nonresonant case where 𝜆 is not an eigenvalue of (𝐸𝑉;𝑚). Then, it studied the existence of at least one solution in the resonant case under assumptions that cover the case 𝑓(𝑢)=𝜇|𝑢|^𝑟−2𝑢 with 1<𝑟<𝑝 and 𝜇≠0.

For the proof of our result, it is necessary to study the weighted eigenvalue problem (𝐸𝑉;𝑚). Thus, in Section 2, we introduce two sequences {𝜆_𝑛(𝑚)}_𝑛 and {𝜇_𝑛(𝑚)}_𝑛 of an eigenvalue of (𝐸𝑉;𝑚) defined by Ljusternik-Schnirelman theory or Drábek-Robinson's method (cf. [16]), respectively. Then, we show several properties of above eigenvalues. In Section 3, we give the proof in the nonresonant case by using {𝜇_𝑛(𝑚)}_𝑛. In Sections 4 and 5, we handle the resonant case.

1.1. Statements of Our Existence Results

First, we state the existence result of a solution in the nonresonant case.

Theorem 1.1. Assume that 𝜆∈ℝ is not an eigenvalue of (𝐸𝑉;𝑚). Then, (𝑃;𝜆,𝑚,ℎ) has at least one solution.

To state our existence result in the resonant case, we introduce some conditions. Set 𝐹(𝑥,𝑢)∶=^𝑢₀𝑓(𝑥,𝑠)𝑑𝑠,𝐺𝑥,𝑦∶=₀^|𝑦|𝑎(𝑥,𝑡)𝑡𝑑𝑡, (1.9) where 𝑎 is the function as in (𝐴𝐻). (𝐻+) there exist 0≤𝑞≤𝑝−1 and 𝐻₀>0 such that lim_𝑦→∞𝑝𝐺𝑥,𝑦−𝑎𝑥,𝑦𝑦²𝑦^1+𝑞𝑓()()=+∞uniformlyina.e.𝑥∈Ω,𝑥,𝑡𝑡−𝑝𝐹𝑥,𝑡≥−𝐻₀1+|𝑡|^1+𝑞fora.e.𝑥∈Ω,every𝑡∈ℝ; (1.10)(𝐻−) there exist 0≤𝑞≤𝑝−1 and 𝐻₀>0 such that lim_𝑦→∞𝑝𝐺𝑥,𝑦−𝑎𝑥,𝑦𝑦²𝑦^1+𝑞𝑓()()=−∞uniformlyina.e.𝑥∈Ω,𝑥,𝑡𝑡−𝑝𝐹𝑥,𝑡≤𝐻₀|𝑡|^1+𝑞+1fora.e.𝑥∈Ω,every𝑡∈ℝ; (1.11)(𝐻𝐹+) there exist 0≤𝑞≤𝑝−1 and 𝐻₀>0 such that 𝑝𝐺𝑥,𝑦−𝑎𝑥,𝑦𝑦²≥−𝐻₀1+𝑦^1+𝑞forevery𝑥∈Ω,𝑦∈ℝ^𝑁,lim_|𝑡|→∞𝑓(𝑥,𝑡)𝑡−𝑝𝐹(𝑥,𝑡)|𝑡|^1+𝑞=+∞uniformlyina.e.𝑥∈Ω; (1.12)(𝐻𝐹−) there exist 0≤𝑞≤𝑝−1 and 𝐻₀>0 such that 𝑝𝐺𝑥,𝑦−𝑎𝑥,𝑦𝑦²≤𝐻₀1+𝑦^1+𝑞forevery𝑥∈Ω,𝑦∈ℝ^𝑁,lim_|𝑡|→∞𝑓(𝑥,𝑡)𝑡−𝑝𝐹(𝑥,𝑡)|𝑡|^1+𝑞=−∞uniformlyina.e.𝑥∈Ω. (1.13)

Theorem 1.2. Assume one of the following conditions: (i)𝜆=0 and (𝐻𝐹+) or (𝐻𝐹−) hold; (ii)𝜆≠0, ∫_Ω𝑚𝑑𝑥≠0 and one of (𝐻+), (𝐻−), (𝐻𝐹+) and (𝐻𝐹−) hold; (iii)𝜆≠0, ∫_Ω𝑚𝑑𝑥=0 and (𝐻+) or (𝐻𝐹+) hold; Then, (𝑃;𝜆,𝑚,ℎ) has at least one solution.

In the special case where 𝑎(𝑥,𝑡)≡0 and 𝑓(𝑥,𝑢)=𝜇|𝑢|^𝑟−2𝑢 for 1<𝑟<𝑝, we easily see that (𝐻𝐹+) or (𝐻𝐹−) holds with 0≤𝑞<𝑟−1 provided 𝜇<0 or 𝜇>0, respectively. Therefore, the following result is proved according to Theorem 1.2.

Corollary 1.3. Let 1<𝑟<𝑝<∞, 𝜇≠0 and ∫_Ω𝑚𝑑𝑥≠0. Then, the following equation has at least one solution: −div𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢+𝜇|𝑢|^𝑟−2𝑢+ℎ(𝑥)inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω. (1.14)

1.2. Properties of the Map 𝐴

In what follows, the norm on 𝑊^1,𝑝(Ω) is given by ‖𝑢‖^𝑝∶=‖∇𝑢‖^𝑝_𝑝+‖𝑢‖^𝑝_𝑝, where ‖𝑢‖_𝑞 denotes the norm of 𝐿^𝑞(Ω) for 𝑢∈𝐿^𝑞(Ω) (1≤𝑞≤∞). Setting 𝐺(𝑥,𝑦)∶=∫₀^|𝑦|𝑎(𝑥,𝑡)𝑡𝑑𝑡, then we can easily see that ∇_𝑦𝐺𝑥,𝑦=𝐴𝑥,𝑦,𝐺(𝑥,0)=0 (1.15) for every 𝑥∈Ω.

Remark 1.4. It is easily seen that the following assertions hold under condition (𝐴): (i)for all 𝑥∈Ω, 𝐴(𝑥,𝑦) is maximal monotone and strictly monotone in 𝑦; (ii)|𝐴(𝑥,𝑦)|≤(𝐶₁/(𝑝−1))|𝑦|^𝑝−1 for every (𝑥,𝑦)∈Ω×ℝ^𝑁; (iii)𝐴(𝑥,𝑦)𝑦≥(𝐶₀/(𝑝−1))|𝑦|^𝑝 for every (𝑥,𝑦)∈Ω×ℝ^𝑁; (iv)𝐺(𝑥,𝑦) is convex in 𝑦 for all 𝑥 and satisfies the following inequalities: 𝐴≥𝐶𝑥,𝑦𝑦≥𝐺𝑥,𝑦⁰𝑝𝑦𝑝−1^𝑝≤𝐶,𝐺𝑥,𝑦¹𝑝𝑦𝑝−1^𝑝, (1.16) for every (𝑥,𝑦)∈Ω×ℝ^𝑁, where 𝐶₀ and 𝐶₁ are the positive constants in (𝐴).

The following result is proved in [3]. It plays an important role for our poof.

Proposition 1.5 (see [3, Proposition 1]). Let 𝐴∶𝑊^1,𝑝(Ω)→𝑊^1,𝑝(Ω)^∗ be the map defined by ⟨𝐴(𝑢),𝑣⟩=_Ω𝐴(𝑥,∇𝑢)∇𝑣𝑑𝑥, (1.17) for 𝑢, 𝑣∈𝑊^1,𝑝(Ω). Then, 𝐴 has the (𝑆)₊ property, that is, any sequence {𝑢_𝑛} weakly convergent to 𝑢 with limsup_𝑛→∞⟨𝐴(𝑢_𝑛),𝑢_𝑛−𝑢⟩≤0 strongly converges to 𝑢.

2. The Weighted Eigenvalue Problems

2.1. Preliminaries

The following lemmas can be easily shown by way of contradiction because ∫_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥 is equivalent to ‖∇𝑢‖^𝑝_𝑝 (note that 𝑎_∞ is positive). Here, we omit the proofs (refer to [7]).

Lemma 2.1. Assume ∫_Ω𝑚𝑑𝑥<0. Then, there exists a constant 𝑐>0 such that ∫_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥≥𝑐‖𝑢‖^𝑝_𝑝 for every 𝑢∈𝑊^1,𝑝(Ω) with ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0.

Lemma 2.2. Assume that ∫_Ω𝑚𝑑𝑥≠0 and 𝜉>0. Then, there exists a constant 𝑏(𝑚,𝜉)>0 such that _Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜉_Ω𝑚|𝑢|^𝑝𝑑𝑥≥𝑏(𝑚,𝜉)_Ω|𝑢|^𝑝𝑑𝑥 (2.1) for every 𝑢∈𝐵(𝑚)∶={𝑢∈𝑊^1,𝑝(Ω);∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0}.

Lemma 2.3. Assume that 𝑚≥0 in Ω. Then, for every 𝜉>0 there existed 𝑑(𝑚,𝜉)>0 such that _Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜉_Ω𝑚|𝑢|^𝑝𝑑𝑥≥𝑑(𝑚,𝜉)_Ω|𝑢|^𝑝𝑑𝑥 (2.2) for every 𝑢∈𝑊^1,𝑝(Ω).

First, we recall the following principle eigenvalue 𝜆^∗(𝑚): 𝜆^∗(𝑚)∶=inf_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥;𝑢∈𝑊^1,𝑝(Ω),_Ω𝑚|𝑢|^𝑝𝑑𝑥=1. (2.3) Because of ∞>sup_𝑥∈Ω𝑎_∞(𝑥)≥inf_𝑥∈Ω𝑎_∞(𝑥)>0, we have the following result as the same argument as in the case of the 𝑝-Laplacian.

Proposition 2.4 (see [7, Proposition  2.2]). The following assertions hold: (i)If ∫_Ω𝑚𝑑𝑥≥0 holds, then 𝜆^∗(𝑚)=0; (ii)If ∫_Ω𝑚𝑑𝑥<0 holds, then 𝜆^∗(𝑚)>0 is a simple eigenvalue and it admits a positive eigenfunction. In addition, the open interval (0,𝜆^∗(𝑚)) contains no eigenvalues of (𝐸𝑉;𝑚).

Lemma 2.5. Assume ∫_Ω𝑚𝑑𝑥<0. Then, one has 𝜆^∗(𝑚+𝜀)<𝜆^∗(𝑚)<𝜆^∗(𝑚−𝜀^′) for every 𝜀>0 and 𝜀^′>0 with |{𝑚−𝜀^′>0}|>0.

Proof. We choose a minimizer 𝑢 for 𝜆^∗(𝑚) because Proposition 2.4 guarantees the existence of it. Then, for every 𝜀>0, we have 𝜆^∗()≤∫𝑚+𝜀^Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥∫_Ω(𝑚+𝜀)|𝑢|^𝑝<∫𝑑𝑥^Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥∫_Ω𝑚|𝑢|^𝑝=𝑑𝑥^Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥=𝜆^∗(𝑚) (2.4) by the definition of 𝜆^∗(𝑚+𝜀). By applying the same argument to a minimizer for 𝜆^∗(𝑚−𝜀), we obtain 𝜆^∗(𝑚)<𝜆^∗(𝑚−𝜀^′) for 𝜀^′>0 with |{𝑚−𝜀^′>0}|>0.

2.2. Other Eigenvalues

Here, we introduce two unbounded sequences {𝜆_𝑛(𝑚)}_𝑛 and {𝜇_𝑛(𝑚)}_𝑛 as follows: 𝐽(𝑢)∶=_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥for𝑢∈𝑊^1,𝑝(Ω),𝐽̃∶=𝐽∣_𝑆(𝑚),𝑆(𝑚)∶=𝑢∈𝑊^1,𝑝(Ω);_Ω𝑚|𝑢|^𝑝,𝒮𝑑𝑥=1_𝑛(𝑚)(𝑚)(𝑋),ℱ∶=𝑋⊂𝑆;compact,symmetricand𝛾≥𝑛_𝑛(𝑚)∶=𝑔∈𝐶𝑆^𝑛−1(𝑚),𝜆,𝑆;𝑔isodd_𝑛(𝑚)∶=inf_{𝑋∈𝒮𝑛(𝑚)}max_𝑢∈𝑋𝐽̃(𝑢),𝜇_𝑛(𝑚)∶=inf_{𝑔∈ℱ𝑛(𝑚)}max_{𝑧∈𝑆𝑛−1}𝐽̃𝑔(𝑧), (2.5) where 𝛾(𝑋) denotes the Krasnoselskii genus of 𝑋 (see [17, Definition 5.1] for the definition) and 𝑆^𝑛−1 denotes the usual unit sphere in ℝ^𝑛. We see that 𝜆_𝑛(𝑚) is defined by Ljusternik-Schnirelman theory and it is known that the definition of 𝜇_𝑛(𝑚) is introduced by Drábek and Robinson ([16]) under the 𝑝-Laplace Dirichlet problem with 𝑚≡1.

Remark 2.6. The following assertions can be shown easily: (i)𝜆₁(𝑚)=𝜇₁(𝑚)=𝜆^∗(𝑚); (ii)𝒮_𝑛(𝑚)≠∅ and ℱ_𝑛(𝑚)≠∅ for every 𝑛∈ℕ; (iii)𝑔(𝑆^𝑛−1)⊂𝒮_𝑛(𝑚) for every 𝑔∈ℱ_𝑛(𝑚); (iv)𝜇_𝑛(𝑚)≥𝜆_𝑛(𝑚) for every 𝑛∈ℕ; (v)𝜆_𝑛+1(𝑚)≥𝜆_𝑛(𝑚) and 𝜇_𝑛+1(𝑚)≥𝜇_𝑛(𝑚) for every 𝑛∈ℕ, see [18] for the proof of (ii).

Define a 𝐶¹ functional Φ_𝑚 on 𝑊^1,𝑝(Ω) by Φ_𝑚(𝑢)∶=∫_Ω𝑚|𝑢|^𝑝𝑑𝑥 for 𝑢∈𝑊^1,𝑝(Ω). Because 1∈ℝ is a regular value of Φ_𝑚, it is well known that the norm of the derivative at 𝑢∈𝑆(𝑚) of the restriction of 𝐽 to 𝑆(𝑚) is defined as follows: 𝐽^′(𝑢)_∗∶=min𝐽^′(𝑢)−𝑡Φ^′_𝑚(𝑢)_{𝑊1,𝑝(Ω)∗}𝐽;𝑡∈ℝ=sup^′(𝑢),𝑣;𝑣∈𝑇_𝑢(𝑆(𝑚)),‖𝑣‖=1, (2.6) where 𝑇_𝑢(𝑆(𝑚)) denotes the tangent space of 𝑆(𝑚) at 𝑢, that is, 𝑇_𝑢(𝑆(𝑚))={𝑣∈𝑊^1,𝑝(Ω);∫_Ω𝑚|𝑢|^𝑝−2𝑢𝑣𝑑𝑥=0}. Here, we recall the definition of the Palais-Smale condition for 𝐽̃.

Definition 2.7. 𝐽̃ is said to satisfy the bounded Palais-Smale condition if any bounded sequence 𝑢_𝑛∈𝑆(𝑚) such that ‖𝐽^′(𝑢_𝑛)‖_∗→0 has a convergent subsequence. Moreover, we say that 𝐽̃ satisfies the Palais-Smale condition at level 𝑐∈ℝ if any sequence 𝑢_𝑛∈𝑆(𝑚) such that 𝐽(̃𝑢_𝑛)→𝑐 and ‖𝐽^′(𝑢_𝑛)‖_∗→0 as 𝑛→∞ has a convergent subsequence. In addition, we say that 𝐽̃ satisfies the Palais-Smale condition if 𝐽̃ satisfies the Palais-Smale condition for every 𝑐∈ℝ.

The following result can be proved by the same argument as in [19, Proposition 3.3] (which treats the case of the 𝑝-Laplacian, i.e., 𝑎_∞≡1) because of ∞>sup_𝑥∈Ω𝑎_∞(𝑥)≥inf_𝑥∈Ω𝑎_∞(𝑥)>0. Here, we omit the proof.

Lemma 2.8. The following assertions hold: (i)𝐽̃ satisfies the bounded Palais-Smale condition; (ii)𝐽̃ satisfies the Palais-Smale condition provided ∫_Ω𝑚𝑑𝑥≠0.

Proposition 2.9. 𝜆_𝑛(𝑚) and 𝜇_𝑛(𝑚) are eigenvalues of (𝐸𝑉;𝑚) such that lim_𝑛→∞𝜆_𝑛(𝑚)=lim_𝑛→∞𝜇_𝑛(𝑚)=+∞. (2.7)

Proof. In the case of ∫_Ω𝑚𝑑𝑥≠0, since 𝐽̃ satisfies the Palais-Smale condition, we can apply the first deformation lemma on 𝐶¹ manifold (refer to [20]). Thus, by the standard argument, we can prove that 𝜆_𝑛(𝑚) and 𝜇_𝑛(𝑚) are critical values of 𝐽̃. This means that 𝜆_𝑛(𝑚) and 𝜇_𝑛(𝑚) are eigenvalues of (𝐸𝑉;𝑚) by the Lagrange multiplier rule. In addition, we can easily show lim_𝑛→∞𝜆_𝑛(𝑚)=+∞ by the standard argument via the first deformation lemma on 𝐶¹ manifold (refer to [21, Proposition  3.14.7], [22] or [17] in the case of a Banach space). Hence, lim_𝑛→∞𝜇_𝑛(𝑚)=+∞ holds because of 𝜇_𝑛(𝑚)≥𝜆_𝑛(𝑚) for every 𝑛∈ℕ.
In the case of ∫_Ω𝑚𝑑𝑥=0, by the same argument as in [18], our conclusion can be proved. For readers' convenience, we give a sketch of the proof. For 𝜀>0, we define 𝐽_𝜀(𝑢)∶=𝐽(𝑢)+𝜀‖𝑢‖^𝑝_𝑝 and 𝐽_𝜀∶=𝐽_𝜀∣_𝑆(𝑚). Moreover, we set minimax values 𝜆^𝜀_𝑛(𝑚) and 𝜇^𝜀_𝑛(𝑚) of 𝐽_𝜀 by 𝜆^𝜀_𝑛(𝑚)∶=inf_{𝑋∈𝒮𝑛(𝑚)}max_𝑢∈𝑋𝐽_𝜀(𝑢),𝜇^𝜀_𝑛(𝑚)∶=inf_{𝑔∈ℱ𝑛(𝑚)}max_{𝑧∈𝑆𝑛−1}𝐽_𝜀𝑔(𝑧). (2.8) Because any Palais-Smale sequence of 𝐽_𝜀 is bounded, it is easily shown that 𝐽_𝜀 satisfies the Palais-Smale condition (refer to [19, Proposition  3.3]) Hence, it can be proved that 𝜆^𝜀_𝑛(𝑚) and 𝜇^𝜀_𝑛(𝑚) are critical values of 𝐽_𝜀. Furthermore, it follows from the argument as in [18, Lemma  3.5] that 𝜆^𝜀_𝑛(𝑚)→𝜆_𝑛(𝑚) and 𝜇^𝜀_𝑛(𝑚)→𝜇_𝑛(𝑚) as 𝜀→0+. Therefore, by noting that 𝐽_𝜀 is 𝑝-homogeneous, we can obtain a solution 𝑢_𝜀 with ‖𝑢_𝜀‖=1 for −div(𝑎_∞|∇𝑢|^𝑝−2∇𝑢)=𝑐_𝜀𝑚|𝑢|^𝑝−2𝑢 in Ω, 𝜕𝑢/𝜕𝜈=0 on 𝜕Ω, where 𝑐_𝜀=𝜆^𝜀_𝑛(𝑚) or 𝜇^𝜀_𝑛(𝑚). Because of ‖𝑢_𝜀‖=1, it follows from the standard argument that 𝑢_𝜀 has a subsequence strongly convergent to a solution 𝑢 for −div𝑎_∞|∇𝑢|^𝑝−2∇𝑢=𝑐𝑚|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω, (2.9) where 𝑐=lim_𝜀→0+𝑐_𝜀. Thus, 𝜆_𝑛(𝑚) and 𝜇_𝑛(𝑚) are eigenvalues of (𝐸𝑉;𝑚). To prove lim_𝑛→∞𝜆_𝑛(𝑚)=+∞, by considering a function 𝑚_𝛿(𝑥)∶=max{𝑚(𝑥),𝛿} for 𝛿>0, we have 𝜆_𝑛(𝑚_𝛿)≤𝜆_𝑛(𝑚) (refer to Proposition 2.10). Because we can apply our fist assertion to 𝑚_𝛿 (note ∫_Ω𝑚_𝛿𝑑𝑥>0), we obtain lim_𝑛→∞𝜇_𝑛(𝑚)≥lim_𝑛→∞𝜆_𝑛(𝑚)≥lim_𝑛→∞𝜆_𝑛(𝑚_𝛿)=+∞.

Proposition 2.10. Let 1<𝑟<∞ if 𝑁≤𝑝 and 𝑝^∗/(𝑝^∗−𝑝)≤𝑟<∞ if 𝑁>𝑝. Then, the following assertions hold: (i)if 𝑚^′≥𝑚 in Ω, then 𝜇_𝑘(𝑚^′)≤𝜇_𝑘(𝑚); (ii)if lim_𝑛→∞𝑚_𝑛=𝑚 in 𝐿^𝑟(Ω), then limsup_𝑛→∞𝜇_𝑘(𝑚_𝑛)≤𝜇_𝑘(𝑚); (iii)if ∫_Ω𝑚𝑑𝑥≠0 and lim_𝑛→∞𝑚_𝑛=𝑚 in 𝐿^𝑟(Ω), then lim_𝑛→∞𝜇_𝑘(𝑚_𝑛)=𝜇_𝑘(𝑚). Moreover, the same conclusion holds for 𝜆_𝑘(𝑚).

Proof. We only treat 𝜇_𝑘(𝑚) because we can give the proof for 𝜆_𝑘(𝑚) similarly.(i) Let 𝑚^′≥𝑚 in Ω. Fix an arbitrary 𝜀>0. Then, by the definition of 𝜇_𝑘(𝑚), there exists a 𝑔∈ℱ_𝑘(𝑚) such that max_𝑧∈𝑆^𝑘−1𝐽(𝑔(𝑧))<𝜇_𝑘(𝑚)+𝜀. Set 𝑔(𝑧)∶=𝑔(𝑧)/(∫_Ω𝑚^′|𝑔(𝑧)|^𝑝𝑑𝑥)^1/𝑝 for 𝑧∈𝑆^𝑘−1 (note ∫_Ω𝑚^′|𝑔(𝑧)|^𝑝𝑑𝑥≥∫_Ω𝑚|𝑔(𝑧)|^𝑝𝑑𝑥=1), then 𝑔∈ℱ_𝑘(𝑚^′) holds. Therefore, by the definition of 𝜇_𝑘(𝑚^′), we have 𝜇_𝑘𝑚^′≤max_{𝑧∈𝑆𝑘−1}𝐽𝑔(𝑧)=max_{𝑧∈𝑆𝑘−1}𝐽𝑔(𝑧)∫_Ω𝑚^′𝑔(𝑧)^𝑝𝑑𝑥≤max_{𝑧∈𝑆𝑘−1}𝐽𝑔(𝑧)<𝜇_𝑘(𝑚)+𝜀. (2.10) because of ∫_Ω𝑚^′|𝑔(𝑧)|^𝑝𝑑𝑥≥∫_Ω𝑚|𝑔(𝑧)|^𝑝𝑑𝑥=1 for every 𝑧∈𝑆^𝑘−1. Since 𝜀>0 is arbitrary, we obtain 𝜇_𝑘(𝑚^′)≤𝜇_𝑘(𝑚).(ii) Let lim_𝑛→∞𝑚_𝑛=𝑚 in 𝐿^𝑟(Ω) and fix an arbitrary 𝜀>0. By the definition of 𝜇_𝑘(𝑚), there exists a 𝑔∈ℱ_𝑘(𝑚) such that max_𝑧∈𝑆^𝑘−1𝐽(𝑔(𝑧))<𝜇_𝑘(𝑚)+𝜀/2. Since 𝑔(𝑆^𝑘−1) is compact and 𝑝𝑟^′∶=𝑝𝑟/(𝑟−1)≤𝑝^∗, we set 𝑀∶=max_{𝑢∈𝑔(𝑆}^𝑘−1₎‖𝑢‖_𝑝𝑟′. Then, due to Hölder's inequality and 𝑚_𝑛→𝑚 in 𝐿^𝑟(Ω), there exists an 𝑛₀∈ℕ such that _Ω𝑚_𝑛|𝑢|^𝑝𝑑𝑥=1+_Ω𝑚_𝑛−𝑚|𝑢|^𝑝𝑑𝑥≥1−𝑚_𝑛−𝑚_𝑟𝑀^𝑝>0 (2.11) for every 𝑢∈𝑔(𝑆^𝑘−1) and 𝑛≥𝑛₀. Therefore, by a similar argument to (i), we obtain 𝜇_𝑘𝑚_𝑛≤max_{𝑧∈𝑆𝑘−1}𝐽𝑔(𝑧)∫_Ω𝑚_𝑛𝑔(𝑧)^𝑝≤𝜇𝑑𝑥^𝑘(𝑚)+𝜀/21−𝑚_𝑛−𝑚_𝑟𝑀^𝑝<𝜇_𝑘(𝑚)+𝜀 (2.12) for sufficiently large 𝑛. Hence, limsup_𝑛→∞𝜇_𝑘(𝑚_𝑛)≤𝜇_𝑘(𝑚)+𝜀 follows. Since 𝜀>0 is arbitrary, our conclusion is proved.(iii) Let lim_𝑛→∞𝑚_𝑛=𝑚 in 𝐿^𝑟(Ω) and ∫_Ω𝑚𝑑𝑥≠0. We fix an arbitrary 𝜀>0. Due to our assertion (ii), there exists an 𝑛₁∈ℕ such that 𝜇_𝑘(𝑚_𝑛)≤𝜇_𝑘(𝑚)+𝜀/2. For every 𝑛≥𝑛₁, by the definition of 𝜇_𝑘(𝑚_𝑛), we can take 𝑔_𝑛∈ℱ_𝑘(𝑚_𝑛) satisfying max_𝑧∈𝑆^𝑘−1𝐽(𝑔_𝑛(𝑧))<𝜇_𝑘(𝑚_𝑛)+𝜀/2.
Here, we will prove sup_𝑛≥𝑛1max‖𝑢‖_𝑝;𝑢∈𝑔_𝑛𝑆^𝑘−1<∞. (2.13)
If 𝑢∈𝑔_𝑛(𝑆^𝑘−1) satisfies ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0, then we obtain 𝑏(𝑚,1)‖𝑢‖^𝑝_𝑝≤𝐽(𝑢)−_Ω𝑚|𝑢|^𝑝𝑑𝑥=𝐽(𝑢)−_Ω𝑚_𝑛|𝑢|^𝑝𝑑𝑥+_Ω𝑚_𝑛−𝑚|𝑢|^𝑝𝑑𝑥≤𝜇_𝑘𝑚_𝑛+𝜀2−1+𝑚_𝑛−𝑚_𝑟‖𝑢‖^𝑝_𝑝𝑟′≤𝜇_𝑘(𝑚)+𝜀+𝐶𝑚_𝑛−𝑚_𝑟‖𝑢‖^𝑝_𝑝+𝐶𝐽(𝑢)𝑚_𝑛−𝑚_𝑟inf_Ω𝑎_∞≤𝐶𝑚1+^𝑛−𝑚_𝑟inf_Ω𝑎_∞𝜇_𝑘(𝑚)+𝜀+𝐶𝑚_𝑛−𝑚_𝑟‖𝑢‖^𝑝_𝑝 (2.14) by Lemma 2.2 and Hölder's inequality (note ‖∇𝑢‖^𝑝_𝑝≤𝐽(𝑢)/inf_Ω𝑎_∞ and 𝜇_𝑘(𝑚_𝑛)≤𝜇_𝑘(𝑚)+𝜀/2), where 𝐶>0 is a constant (independent of 𝑛 and 𝑢) obtained by the continuity of 𝑊^1,𝑝(Ω) into 𝐿^𝑝𝑟′(Ω). Therefore, if we take an 𝑛₂≥𝑛₁ satisfying 𝐶‖𝑚_𝑛−𝑚‖_𝑟≤𝑏(𝑚,1)/2 for every 𝑛≥𝑛₂, then we obtain ‖𝑢‖^𝑝_𝑝≤2𝑏()𝑏()𝑚,11+𝑚,12inf_Ω𝑎_∞𝜇_𝑘(𝑚)+𝜀 (2.15) for every 𝑢∈𝑔_𝑛(𝑆^𝑘−1) provided ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0 and 𝑛≥𝑛₂. Similarly, in the case where 𝑚 changes sign, for every 𝑢∈𝑔_𝑛(𝑆^𝑘−1) satisfying ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0, we have 𝑏(−𝑚,1)‖𝑢‖^𝑝_𝑝≤𝐽(𝑢)−_Ω(−𝑚)|𝑢|^𝑝≤𝐶𝑚𝑑𝑥1+^𝑛−𝑚_𝑟inf_Ω𝑎_∞𝜇_𝑘(𝑚)+𝜀+1+𝐶𝑚_𝑛−𝑚_𝑟‖𝑢‖^𝑝_𝑝. (2.16) Hence, by taking a sufficiently large 𝑛₃≥𝑛₂, we get the inequality ‖𝑢‖^𝑝_𝑝≤2𝑏()𝑏()−𝑚,11+−𝑚,12inf_Ω𝑎_∞𝜇_𝑘(𝑚)+𝜀+1, (2.17) for every 𝑢∈𝑔_𝑛(𝑆^𝑘−1) with ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0 and 𝑛≥𝑛₃. In the case of 𝑚≥0 in Ω, by using Lemma 2.3 instead of Lemma 2.2, we have a similar inequality ‖𝑢‖^𝑝_𝑝≤2𝑑()𝑑()𝑚,11+𝑚,12inf_Ω𝑎_∞𝜇_𝑘(𝑚)+𝜀+1, (2.18) for every 𝑢∈𝑔_𝑛(𝑆^𝑘−1) provided 𝑛≥𝑛₄ (some sufficiently large 𝑛₄≥𝑛₃). Consequently, our claim follows from (2.15), (2.17), and (2.18).
Let us return to the proof of (iii). Because sup‖𝑢‖_𝑝𝑟′;𝑢∈𝑔_𝑛𝑆^𝑘−1,𝑛≥𝑛₁=∶𝑅<+∞ (2.19) holds by (2.13), 𝐽(𝑢)≤𝜇_𝑘(𝑚)+𝜀/2 and the continuity of 𝑊^1,𝑝(Ω) into 𝐿^𝑝𝑟′(Ω), we see the inequality _Ω𝑚|𝑢|^𝑝𝑑𝑥=1−_Ω𝑚_𝑛−𝑚|𝑢|^𝑝𝑑𝑥>1−𝑚_𝑛−𝑚_𝑟𝑅^𝑝>0, (2.20) for every 𝑢∈𝑔_𝑛(𝑆^𝑘−1) and 𝑛≥𝑛₅ (some sufficiently large 𝑛₅≥𝑛₄). By considering 𝑔_𝑛(⋅)∶=𝑔_𝑛(⋅)/(∫_Ω𝑚|𝑔_𝑛(⋅)|^𝑝𝑑𝑥)^1/𝑝∈ℱ_𝑘(𝑚), we obtain 𝜇_𝑘(𝑚)≤max_{𝑧∈𝑆𝑘−1}𝐽𝑔_𝑛(𝑧)≤max_𝑧∈𝑆^𝑘−1𝐽𝑔_𝑛(𝑧)1−𝑚_𝑛−𝑚_𝑟𝑅^𝑝≤𝜇_𝑘𝑚_𝑛+𝜀/21−𝑚_𝑛−𝑚_𝑟𝑅^𝑝. (2.21) Because of ‖𝑚_𝑛−𝑚‖_𝑟→0, we get 𝜇_𝑘(𝑚_𝑛)≥𝜇_𝑘(𝑚)−𝜀 for sufficiently large 𝑛, and hence our conclusion holds.

Finally, we recall the second eigenvalue of (𝐸𝑉;𝑚) obtained by the mountain pass theorem. Σ(𝑚)](𝑚(0)(1)∈(,𝑐(𝑚)∶=𝜂∈𝐶([0,1,𝑆));𝜂∈𝑃,𝜂−𝑃)∶=inf_{(𝑚)𝜂∈Σ}max_[]𝑡∈0,1𝐽̃𝜂(𝑡), (2.22) where 𝑃∶={𝑢∈𝑊^1,𝑝(Ω);𝑢(𝑥)≥0fora.e.𝑥∈Ω}.

Since ∞>sup_𝑥∈Ω𝑎_∞(𝑥)≥inf_𝑥∈Ω𝑎_∞(𝑥)>0 holds, the following result can be shown by the same argument as in [19] (although they handle the asymmetry case, it is sufficient to consider the case of 𝑚≡𝑛 in this paper). See [19, Theorem 3.2] for the proof.

Theorem 2.11. 𝑐(𝑚) is an eigenvalue of (𝐸𝑉;𝑚) which satisfies 𝜆^∗(𝑚)<𝑐(𝑚). Moreover, there is no eigenvalues of (𝐸𝑉;𝑚) between 𝜆^∗(𝑚) and 𝑐(𝑚).

Now, we have the following result.

Proposition 2.12. 𝜆₂(𝑚)=𝜇₂(𝑚)=𝑐(𝑚) (2.23) holds, where 𝑐(𝑚) is a minimax value defined by (2.22).

Proof. First, we prove the inequality 𝑐(𝑚)≥𝜇₂(𝑚). Because 𝑐(𝑚) is an eigenvalue (note that the following equation is homogeneous), we can choose a solution 𝑢∈𝑊^1,𝑝(Ω) with ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥=1 for −div𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢=𝑐(𝑚)𝑚(𝑥)|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω. (2.24) Note that 𝑢 is a sign-changing function because any eigenfunction associated with any eigenvalue greater than the principal eigenvalue changes sign (refer to [18, Proposition  4.3]). Thus, we have 0<_Ω𝑎_∞∇𝑢_±^𝑝𝑑𝑥=𝑐(𝑚)_Ω𝑚𝑢^𝑝_±𝑑𝑥 (2.25) by taking ±𝑢_± as test function (recall that 𝑢_±∶=max{±𝑢,0}). Hence, we may assume that ∫_Ω𝑚𝑢^𝑝_±𝑑𝑥=1 by the normalization. Set 𝑋∶={𝑠𝑢₊−𝑡𝑢₋;|𝑠|^𝑝+|𝑡|^𝑝=1}⊂𝑆(𝑚). Then, because 𝑋 is homeomorphic to 𝑆¹, there exists 𝑔∈ℱ₂(𝑚) such that 𝑔(𝑆¹)=𝑋. Since the value of 𝐽 is equal to 𝑐(𝑚) on 𝑋, we obtain 𝜇₂(𝑚)≤max_𝑧∈𝑆1𝐽̃𝑔(𝑧)=𝑐(𝑚) (2.26) by the definition of 𝜇₂(𝑚) and 𝑋.
Next, we will prove the inequality 𝑐(𝑚)≤𝜆₂(𝑚) by dividing into two cases: ∫_Ω𝑚𝑑𝑥≠0 and ∫_Ω𝑚𝑑𝑥=0.
Case of ∫_Ω𝑚𝑑𝑥≠0: by way of contradiction, we assume that 𝜆₂(𝑚)<𝑐(𝑚). Then, 𝜆^∗(𝑚)=𝜆₁(𝑚)=𝜆₂(𝑚) follows from Theorem 2.11. Note that 𝐽̃ satisfies the Palais-Smale condition in this case (see Lemma 2.8), and hence we can apply the first deformation lemma to 𝐽̃. Therefore, by the standard argument (cf. [22], [17, Lemma 5.6]), we see that 𝛾(𝐾)≥2, where 𝐾∶={𝑢∈𝑆(𝑚);𝐽^′(𝑢)=0,𝐽(̃𝑢)=𝜆^∗(𝑚)}. This means that 𝐾 is an infinite set, that is, the following equation has infinite many solutions: −div𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢=𝜆^∗(𝑚)𝑚(𝑥)|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω (2.27) due to the Lagrange multiplier's rule. This contradicts to the fact described as in Proposition 2.4 that 𝜆^∗(𝑚) is simple. As a result, we have shown that 𝑐(𝑚)=𝜆₂(𝑚)=𝜇₂(𝑚) holds in the case of ∫_Ω𝑚𝑑𝑥≠0 (note 𝜆_𝑛(𝑚)≤𝜇_𝑛(𝑚)).
Case of ∫_Ω𝑚𝑑𝑥=0: According to Proposition 2.10 (i) for 𝜆₂(𝑚), we have 𝜆₂(𝑚)≥𝜆₂(𝑚+𝜀)=𝑐(𝑚+𝜀) for every 𝜀>0 since we can apply the first result to 𝑚+𝜀. Because we prove lim_𝜀→0+𝑐(𝑚+𝜀)=𝑐(𝑚) by the same argument as in [6, Lemma 2.9] (for the case 𝑎_∞≡1), our conclusion is proved by taking 𝜀↓0 in the inequality 𝜆₂(𝑚)≥𝑐(𝑚+𝜀).

3. Proof of Theorem 1.1

We define a functional 𝐼_𝜆,𝑚 on 𝑊^1,𝑝(Ω) as follows: 𝐼_𝜆,𝑚(𝑢)=_Ω𝐺()𝜆𝑥,∇𝑢𝑑𝑥−𝑝_Ω𝑚|𝑢|^𝑝𝑑𝑥−_Ω𝐹(𝑥,𝑢)𝑑𝑥−_Ω=1ℎ𝑢𝑑𝑥𝑝_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥+_Ω𝐺()𝜆𝑥,∇𝑢𝑑𝑥−𝑝_Ω𝑚|𝑢|^𝑝−𝑑𝑥_Ω𝐹(𝑥,𝑢)𝑑𝑥−_Ωℎ𝑢𝑑𝑥 (3.1) for 𝑢∈𝑊^1,𝑝(Ω) ((1.15) or (1.9) for the definition of 𝐺, 𝐺, and 𝐹). It is easily seen that 𝐼_𝜆,𝑚 is well defined and class of 𝐶¹ on 𝑊^1,𝑝(Ω) by (1.1), (1.16) and the continuity of 𝑊^1,𝑝(Ω)↪𝐿^𝑝(Ω).

Remark 3.1. Let 𝑢∈𝑊^1,𝑝(Ω) be a critical point of 𝐼_𝜆,𝑚, namely, 𝑢 satisfies the equality _Ω𝐴(𝑥,∇𝑢)∇𝜑𝑑𝑥=𝜆_Ω𝑚|𝑢|^𝑝−2𝑢𝜑𝑑𝑥+_Ω𝑓(𝑥,𝑢)𝜑𝑑𝑥+_Ωℎ𝜑𝑑𝑥 (3.2) for every 𝜑∈𝑊^1,𝑝(Ω). Then, 𝑢∈𝐿^∞(Ω) by the Moser iteration process (refer to Theorem C in [4]). Therefore, 𝑢∈𝐶^1,𝛼(Ω) (0<𝛼<1) follows from the regularity result in [23]. Furthermore, due to [24, Theorem 3], 𝑢 satisfies (𝑃;𝜆,𝑚,ℎ) in the distribution sense and the boundary condition 0=𝜕𝑢𝜕𝜈_𝐴=𝐴(⋅,∇𝑢)𝜈=𝑎(⋅,|∇𝑢|)𝜕𝑢𝜕𝜈in𝑊^{−1/𝑞,𝑞}(𝜕Ω) (3.3) for every 1<𝑞<∞ (see [24] for the definition of 𝑊^{−1/𝑞,𝑞}(𝜕Ω)). Since 𝑢∈𝐶^1,𝛼(Ω) and 𝑎(𝑥,𝑡)>0 for every 𝑡≠0, 𝑢 satisfies the Neumann boundary condition, that is, (𝜕𝑢/𝜕𝜈)(𝑥)=0 for every 𝑥∈𝜕Ω.

3.1. The Palais-Smale Condition in the Nonresonant Case

First, we recall the definition of the Palais-Smale condition.

Definition 3.2. A 𝐶¹ functional Ψ on a Banach space 𝑋 is said to satisfy the Palais-Smale condition at 𝑐∈ℝ if a Palais-Smale sequence {𝑢_𝑛}⊂𝑋 at level 𝑐, namely, Ψ𝑢_𝑛⟶𝑐,Ψ^′𝑢_𝑛𝑋∗⟶0as𝑛⟶∞ (3.4) has a convergent subsequence. We say that Ψ satisfies the Palais-Smale condition if Ψ satisfies the Palais-Smale condition at any 𝑐∈ℝ. Moreover, we say that Ψ satisfies the bounded Palais-Smale condition if any bounded sequence {𝑢_𝑛} such that {Ψ(𝑢_𝑛)} is bounded and ‖Ψ^′(𝑢_𝑛)‖_𝑋∗→0 as 𝑛→∞ has a convergent subsequence.

Concerning the Palais-Smale condition, we state the following result developed from [6, Proposition 7].

Proposition 3.3. If 𝜆 is not an eigenvalue of (𝐸𝑉;𝑚), then 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition.

Proof. Let {𝑢_𝑛} be a Palais-Smale sequence of 𝐼_𝜆,𝑚, namely, 𝐼_𝜆,𝑚𝑢_𝑛⟶𝑐,𝐼^′_𝜆,𝑚𝑢_{𝑛𝑊1,𝑝(Ω)∗}⟶0as𝑛⟶∞ (3.5) for some 𝑐∈ℝ. It is sufficient to prove only the boundedness of ‖𝑢_𝑛‖ because the operator 𝐴∶𝑊^1,𝑝(Ω)→𝑊^1,𝑝(Ω)^∗ described in Proposition 1.5 has the (𝑆)₊ property.
To prove the boundedness of ‖𝑢_𝑛‖, it suffices to show that ‖𝑢_𝑛‖_𝑝 is bounded because of the inequality |𝑓(𝑥,𝑢)|≤𝐶(|𝑢|^𝑝−1+1) (obtained by (1.1)) and the following inequality: 𝐼^′_𝜆,𝑚𝑢_𝑛,𝑢_𝑛+𝜆_Ω𝑚𝑢_𝑛^𝑝𝑑𝑥+_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛𝑑𝑥+_Ωℎ𝑢_𝑛=𝑑𝑥,_Ω𝐴𝑥,∇𝑢_𝑛∇𝑢_𝑛𝐶𝑑𝑥≥⁰𝑝−1∇𝑢_𝑛^𝑝_𝑝, (3.6) where we use Remark 1.4 (iii) in the last inequality. By way of contradiction, we may assume that ‖𝑢_𝑛‖_𝑝→∞ as 𝑛→∞ by choosing a subsequence if necessary. Set 𝑣_𝑛∶=𝑢_𝑛/‖𝑢_𝑛‖_𝑝. Then, since the inequality (3.6) guarantees that {𝑣_𝑛} is bounded in 𝑊^1,𝑝(Ω), we may suppose, by choosing a subsequence, that 𝑣_𝑛⇀𝑣₀ in 𝑊^1,𝑝(Ω) and 𝑣_𝑛→𝑣₀ in 𝐿^𝑝(Ω) for some 𝑣₀.
Here, we will prove that lim_𝑛→∞𝑓⋅,𝑢_𝑛𝑝′𝑢_𝑛𝑝^𝑝−1=0, (3.7) where 𝑝^′=𝑝/(𝑝−1). Fix an arbitrary 𝜀>0. It follows from (1.1) that there exists a 𝐶_𝜀>0 such that 𝑓(𝑥,𝑢)≤𝜀|𝑢|^𝑝−1+𝐶_𝜀forevery𝑢∈ℝ,a.𝑒.𝑥∈Ω. (3.8) Then, we obtain _Ω𝑓𝑥,𝑢_𝑛^𝑝′𝑑𝑥≤2^𝑝′_Ω𝜀^𝑝′𝑢_𝑛^𝑝+𝐶^𝑝′_𝜀𝑑𝑥≤2^𝑝′𝜀^𝑝′𝑢_𝑛^𝑝_𝑝+2^𝑝′𝐶^𝑝′_𝜀|Ω|. (3.9) Since we are assuming that ‖𝑢_𝑛‖_𝑝→∞ as 𝑛→∞, there exists 𝑛₀∈ℕ such that for every 𝑛≥𝑛₀𝑓⋅,𝑢_𝑛𝑝′𝑢_𝑛𝑝^𝑝−1≤4𝜀 (3.10) holds. This shows that lim_𝑛→∞‖𝑓(⋅,𝑢_𝑛)‖_𝑝′/‖𝑢_𝑛‖_𝑝^𝑝−1=0 because 𝜀>0 is arbitrary.
Here, we recall the following result proved in [6]: lim_𝑛→∞Ω𝑎𝑥,∇𝑢_𝑛∇𝑢_𝑛𝑢_𝑛𝑝^𝑝−1∇𝑣_𝑛−𝑣₀𝑑𝑥=lim_𝑛→∞Ω𝑎𝑥,∇𝑢_𝑛∇𝑢_𝑛𝑢_𝑛𝑝^𝑝−1∇𝜑𝑑𝑥=0, (3.11) for every 𝜑∈𝑊^1,𝑝(Ω). Thus, by considering 𝑜(1)=𝐼^′_𝜆,𝑚𝑢_𝑛,𝑣_𝑛−𝑣₀𝑢_𝑛𝑝^𝑝−1=_Ω𝑎_∞∇𝑣_𝑛^𝑝−2∇𝑣_𝑛∇𝑣_𝑛−𝑣₀𝑑𝑥+𝑜(1), (3.12) we see that 𝑣_𝑛 strongly converges to 𝑣₀ in 𝑊^1,𝑝(Ω) (note that 𝑝-Laplacian has the (𝑆)₊ property). Therefore, by taking a limit in 𝑜(1)=⟨𝐼^′_𝜆,𝑚(𝑢_𝑛),𝜑⟩/‖𝑢_𝑛‖_𝑝^𝑝−1 for any 𝜑∈𝑊^1,𝑝(Ω) and by noting (3.7) and (3.11), we know that 𝑣₀ is a nontrivial solution (note ‖𝑣₀‖_𝑝=1) of −div𝑎_∞|∇𝑢|^𝑝−2∇𝑢=𝜆𝑚|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω. (3.13) This means that 𝜆 is an eigenvalue of (𝐸𝑉;𝑚). This is a contradiction. Hence, ‖𝑢_𝑛‖_𝑝 is bounded.

3.2. Key Lemmas

To show the linking lemma, we define 𝑌𝜇,𝑚∶=𝑢∈𝑊^1,𝑝(Ω);_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥≥𝜇_Ω𝑚|𝑢|^𝑝𝑑𝑥 (3.14)for𝜇∈ℝ.

Lemma 3.4. Let 𝑔₀∈𝐶(𝑆^𝑘−1,𝑊^1,𝑝(Ω)⧵{0}) be odd and 0<𝜇≤𝜇_𝑘+1(𝑚). Then, 𝑔(𝑆^𝑘₊)∩𝑌(𝜇,𝑚)≠∅ for every 𝑔∈𝐶(𝑆^𝑘₊,𝑊^1,𝑝(Ω)) with 𝑔∣_𝑆^𝑘−1=𝑔₀, where 𝑌(𝜇,𝑚) is the set introduced in (3.14) and 𝑆^𝑘₊ is the upper hemisphere in ℝ^𝑘+1 with boundary 𝑆^𝑘−1.

Proof. Fix any 𝑔∈𝐶(𝑆^𝑘₊,𝑊^1,𝑝(Ω)) such that 𝑔∣_𝑆^𝑘−1=𝑔₀. If 𝑢∈𝑔(𝑆^𝑘₊) satisfies ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0, then 𝑢∈𝑌(𝜇,𝑚) holds. So, we may assume that ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0 for every 𝑢∈𝑔(𝑆^𝑘₊). Define 𝑔∈ℱ_𝑘+1(𝑚) as follows: 𝑔(𝑧)𝑔(𝑧)∶=∫_Ω𝑚𝑔(𝑧)^𝑝𝑑𝑥^1/𝑝if𝑧∈𝑆^𝑘₊,−𝑔(−𝑧)∫_Ω𝑚𝑔(−𝑧)^𝑝𝑑𝑥^1/𝑝if𝑧∈𝑆^𝑘₋. (3.15) By the definition of 𝜇_𝑘+1(𝑚), there exists 𝑧₀∈𝑆^𝑘 such that 𝐽(̃𝑔(𝑧₀))≥𝜇_𝑘+1(𝑚). Since 𝑔 is odd and 𝐽 is even, we may suppose 𝑧₀∈𝑆^𝑘₊. So, this yields the inequality 𝐽(𝑔(𝑧₀))≥𝜇_𝑘+1(𝑚)∫_Ω𝑚|𝑔(𝑧₀)|^𝑝𝑑𝑥≥𝜇∫_Ω𝑚|𝑔(𝑧₀)|^𝑝𝑑𝑥, whence 𝑔(𝑧₀)∈𝑌(𝜇,𝑚) holds.

Lemma 3.5. Let 𝜇_𝑘(𝑚)<𝜆. Then, there exists 𝑔₀∈ℱ_𝑘(𝑚) such that max_{𝑧∈𝑆𝑘−1}𝐽𝑔₀(𝑧)<𝜆,max_{𝑧∈𝑆𝑘−1}𝐼_𝜆,𝑚𝑇𝑔₀(𝑧)⟶−∞as|𝑇|⟶∞, (3.16) where 𝜇_𝑘(𝑚) is defined by (2.5).

Proof. Choose 𝜀₀>0 such that 𝜇_𝑘(𝑚)+𝜀₀<𝜆. By the definition of 𝜇_𝑘(𝑚), there exists 𝑔₀∈ℱ_𝑘(𝑚) such that max_{𝑧∈𝑆𝑘−1}𝐽𝑔₀(𝑧)<𝜇_𝑘(𝑚)+𝜀₀. (3.17) Due to the compactness of 𝑔₀(𝑆^𝑘−1), we put 𝑀∶=max_𝑧∈𝑆^𝑘−1‖𝑔₀(𝑧)‖_𝑝. By the property of the function 𝑎 as in (𝐴𝐻) and Young's inequality, for every 𝜀>0 there exist constants 𝐶_𝜀>0 and 𝐶^′_𝜀>0 such that 𝐺≤𝜀𝑥,𝑦2𝑦^𝑝+𝐶_𝜀𝑦≤𝜀𝑦^𝑝+𝐶^′_𝜀≤𝜀inf_Ω𝑎_∞𝑎_∞(𝑥)𝑦^𝑝+𝐶^′_𝜀 (3.18) for every 𝑥∈Ω and 𝑦∈ℝ^𝑁. Moreover, the hypothesis (1.1) ensures that for every 𝜀^′>0 there exist constants 𝐷_𝜀′>0 satisfying |𝐹(≤𝜀𝑥,𝑢)|^′2|𝑢|^𝑝+𝐷_𝜀′|𝑢|≤𝜀^′|𝑢|^𝑝+𝐷^′_𝜀′ (3.19) for every 𝑢∈ℝ and a.e. 𝑥∈Ω. Hence, we have 𝐼_𝜆,𝑚()≤𝑇𝑇𝑢^𝑝𝑝1+𝑝𝜀𝑎_Ω𝑎_∞|∇𝑢|^𝑝𝑇𝑑𝑥−^𝑝𝜆−𝑝𝜀^′𝑀^𝑝𝑝+𝑇‖ℎ‖_∞‖𝑢‖₁≤𝑇+𝐶_𝑝𝑝1+𝑝𝜀𝑎𝜇_𝑘(𝑚)+𝜀₀−𝜆+𝑝𝑀^𝑝𝜀^′+𝑇𝑀‖ℎ‖_∞|Ω|^{(𝑝−1)/𝑝}+𝐶 (3.20) for every 𝑇>0, 𝑢∈𝑔₀(𝑆^𝑘−1), 𝜀>0 and 𝜀^′>0 since 𝑔₀(𝑆^𝑘−1)⊂𝑆(𝑚), (3.17), (3.18) and (3.19), where 𝐶=(𝐶^′_𝜀+𝐷^′_𝜀′)|Ω| and 𝑎=inf_𝑥∈Ω𝑎_∞(𝑥)>0. By taking 𝜀>0 and 𝜀^′>0 satisfying (1+𝑝𝜀/𝑎)(𝜇_𝑘(𝑚)+𝜀₀)−𝜆+𝑝𝑀^𝑝𝜀^′<0, we show that max_𝑧∈𝑆^𝑘−1𝐼_𝜆,𝑚(𝑇𝑔₀(𝑧))→−∞ as 𝑇→+∞. Thus, our conclusion follows because 𝑔₀(𝑆^𝑘−1) is symmetric.

3.3. The Case ∫_Ω𝑚𝑑𝑥≠0

Lemma 3.6. Let ∫_Ω𝑚𝑑𝑥<0 and 0<𝜆<𝜆^∗(𝑚). Then, 𝐼_𝜆,𝑚 is bounded from below, coercive and weakly lower semicontinuous (w.l.s.c.) on 𝑊^1,𝑝(Ω).

Proof. Φ(𝑢)∶=∫_Ω𝐺(𝑥,∇𝑢)𝑑𝑥 is w.l.s.c. on 𝑊^1,𝑝(Ω) because Φ is convex and continuous on 𝑊^1,𝑝(Ω) (cf. [25, Theorem 1.2]). Thus, 𝐼_𝜆,𝑚 is also w.l.s.c. on 𝑊^1,𝑝(Ω) since the inclusion from 𝑊^1,𝑝(Ω) to 𝐿^𝑝(Ω) is compact.
Choose 𝜀>0 such that 𝑝𝜀<𝑎(1−𝜆/𝜆^∗(𝑚)), where 𝑎∶=inf_Ω𝑎_∞. By an easy estimation, (3.18) and (3.19) as in Lemma 3.5, we have 𝐼_𝜆,𝑚(𝑢)≥𝑎−𝜀𝑝𝑝𝑎_Ω𝑎_∞|∇𝑢|^𝑝𝜆𝑑𝑥−𝑝_Ω𝑚|𝑢|^𝑝𝑑𝑥−𝜀^′‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖_𝑝|Ω|^{(𝑝−1)/𝑝}−𝐶^′_𝜀+𝐷^′_𝜀′|Ω| (3.21) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝜀^′>0.
Let 𝑢∈𝑊^1,𝑝(Ω) satisfy ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0. Then, the following inequality follows from Lemma 2.2: 𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜆_Ω𝑚|𝑢|^𝑝𝐷𝑑𝑥≥⁰2_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥+𝑏(𝑚,𝜉)‖𝑢‖^𝑝_𝑝, (3.22) where 𝑏(𝑚,𝜉) is a positive constant independent of 𝑢 with 𝜉=2𝜆/𝐷₀ and 𝐷₀=(𝑎−𝜀𝑝)/𝑎.
For every 𝑢∈𝑊^1,𝑝(Ω) such that ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0, we obtain 𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜆_Ω𝑚|𝑢|^𝑝𝑑𝑥≥𝐷₀−𝜆𝜆^∗(𝑚)_Ω𝑎_∞|∇𝑢|^𝑝≥1𝑑𝑥2𝐷₀−𝜆𝜆^∗(𝑚)_Ω𝑎_∞|∇𝑢|^𝑝𝑐𝑑𝑥+2𝐷₀−𝜆𝜆^∗(𝑚)‖𝑢‖^𝑝_𝑝 (3.23) by the definition of 𝜆^∗(𝑚), Lemma 2.1 and 𝐷₀−𝜆/𝜆^∗(𝑚)>0, where 𝑐>0 is a constant obtained by Lemma 2.1.
Consequently, if we choose a 𝜀^′>0 satisfying 𝜀^′<min{𝑏(𝑚,𝜉)/𝑝,𝑐(𝐷₀−𝜆/𝜆^∗(𝑚))/(2𝑝)}, then we obtain positive constants 𝑑₁ and 𝑑₂ (independent of 𝑢) such that 𝐼_𝜆,𝑚(𝑢)≥𝑑_1Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥+𝑑₂‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖_𝑝|Ω|^{(𝑝−1)/𝑝}−𝐶^′_𝜀+𝐷^′_𝜀′|Ω|≥min𝑎𝑑₁,𝑑₂‖𝑢‖^𝑝−‖ℎ‖_∞‖𝑢‖|Ω|^{(𝑝−1)/𝑝}−𝐶^′_𝜀+𝐷^′_𝜀′|Ω| (3.24) for every 𝑢∈𝑊^1,𝑝(Ω) by (3.21), (3.22), and (3.23). Because of 𝑝>1, our conclusion is shown.

Lemma 3.7. Let 𝑚≥0 in Ω and 𝑚≢0. If 𝜆<0 holds, then 𝐼_𝜆,𝑚 is bounded from below, coercive and w.l.s.c. on 𝑊^1,𝑝(Ω).

Proof. First, as the same reason in Lemma 3.6, it follows that 𝐼_𝜆,𝑚 is w.l.s.c. on 𝑊^1,𝑝(Ω). By a similar argument to Lemma 3.6, for every 𝜀^′>0 and 0<𝜀<𝑎/𝑝 where 𝑎=inf_Ω𝑎_∞, we obtain 𝐼_𝜆,𝑚(𝑢)≥𝑎−𝜀𝑝𝑝𝑎_Ω𝑎_∞|∇𝑢|^𝑝|𝜆|𝑑𝑥+𝑝_Ω𝑚|𝑢|^𝑝𝑑𝑥−𝜀^′‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖_𝑝|Ω|^{(𝑝−1)/𝑝}−𝐶^′_𝜀+𝐷^′_𝜀′|Ω| (3.25) for every 𝑢∈𝑊^1,𝑝(Ω) (note 𝜆<0). Here, from Lemma 2.3, 𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥+|𝜆|_Ω𝑚|𝑢|^𝑝𝐷𝑑𝑥≥⁰2_Ω𝑎_∞|∇𝑢|^𝑝𝐷𝑑𝑥+⁰2𝑏(𝜉,𝑚)‖𝑢‖^𝑝_𝑝 (3.26) for every 𝑢∈𝑊^1,𝑝(Ω) follows, where 𝐷₀∶=(𝑎−𝜀𝑝)/𝑎, 𝜉∶=2|𝜆|/𝐷₀ and 𝑏(𝜉,𝑚) is a constant obtained in Lemma 2.3. Therefore, by choosing a 𝜀^′ such that 0<𝜀^′<𝐷₀𝑏(𝜉,𝑚)/2, we can prove our conclusion.

Lemma 3.8. Let ∫_Ω𝑚𝑑𝑥≠0 and 0<𝜆<𝜇. Then, 𝐼_𝜆,𝑚 is bounded from below on 𝑌(𝜇,𝑚), where 𝑌(𝜇,𝑚) is the set introduced in (3.14).

Proof. Due to the same inequalities concerning 𝐺 and 𝐹 as in Lemma 3.5, for every 𝜀>0 and 𝜀^′>0, there exists 𝐶=𝐶(𝜀,𝜀^′)>0 such that 𝐼_𝜆,𝑚(𝑢)≥𝑎−𝑝𝜀𝑝𝑎_Ω𝑎_∞|∇𝑢|^𝑝𝜆𝑑𝑥−𝑝_Ω𝑚|𝑢|^𝑝𝑑𝑥−𝜀^′‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖₁−𝐶|Ω| (3.27) for every 𝑢∈𝑊^1,𝑝(Ω), where 𝑎∶=inf_𝑥∈Ω𝑎_∞(𝑥). Choose positive constants 𝜀 and 𝛿 such that 𝐷₀∶=1−𝑝𝜀/𝑎>𝛿>𝜆/𝜇 (note 𝜆/𝜇<1).
First, we consider the case of 𝑚≥0 in Ω. For every 𝑢∈𝑌(𝜇,𝑚), we obtain 𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜆_Ω𝑚|𝑢|^𝑝≥𝐷𝑑𝑥₀−𝛿_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥+𝛿𝜇−𝜆_Ω𝑚|𝑢|^𝑝𝑑𝑥≥𝑑𝑚,𝜉₁𝐷₀−𝛿‖𝑢‖^𝑝_𝑝 (3.28) by Lemma 2.3 with 𝜉₁=(𝛿𝜇−𝜆)/(𝐷₀−𝛿) (note 𝛿𝜇−𝜆>0 and 𝐷₀−𝛿>0).
Next, we handle with the case where 𝑚 changes sign. Let 𝑢∈𝑊^1,𝑝(Ω) satisfy ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥≤0. Then, we have for such 𝑢𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜆_Ω𝑚|𝑢|^𝑝𝑑𝑥≥𝑏𝑚,𝜉₂𝐷₀‖𝑢‖^𝑝_𝑝 (3.29) by Lemma 2.2, where 𝐷₀=1−𝑝𝜀/𝑎 and 𝜉₂∶=𝜆/𝐷₀.
On the other hand, for 𝑢∈𝑌(𝜇,𝑚) with ∫_Ω𝑚|𝑢|^𝑝𝑑𝑥>0, the following inequality follows from Lemma 2.2: 𝐷_0Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝜆_Ω𝑚|𝑢|^𝑝≥𝐷𝑑𝑥₀−𝛿_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥−𝛿𝜇−𝜆_Ω(−𝑚)|𝑢|^𝑝𝑑𝑥≥𝑏−𝑚,𝜉₁𝐷₀−𝛿‖𝑢‖^𝑝_𝑝. (3.30) Consequently, by (3.27), (3.29), (3.28), and (3.30), there exists 𝑑>0 independent of 𝑢 such that 𝐼_𝜆,𝑚(𝑢)≥𝑑−𝜀^′‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖_𝑝|Ω|^{(𝑝−1)/𝑝}−𝐶|Ω| (3.31) for every 𝑢∈𝑌(𝜇,𝑚). Hence, our conclusion is shown by taking 𝜀^′>0 satisfying 𝜀^′<𝑑.

Proof of Theorem 1.1 in the Case ∫_Ω𝑚𝑑𝑥≠0. First, if either 𝑚≥0 on Ω and 𝜆<0 or 0<𝜆<𝜆^∗(𝑚)=𝜇₁(𝑚) (i.e., ∫_Ω𝑚𝑑𝑥<0) holds, then Lemma 3.7 or Lemma 3.6 guarantees the existence of a global minimizer of 𝐼_𝜆,𝑚, respectively (cf. [25, Theorem 1.1]). Hence, (𝑃;𝜆,𝑚,ℎ) has a solution.
Since 𝜆 is an eigenvalue of (𝐸𝑉;𝑚) if and only if −𝜆 is one of (𝐸𝑉;−𝑚), it suffices to consider the case of 𝜆>𝜆^∗(𝑚)≥0. Furthermore, by Proposition 2.9, Remark 2.6 (i), and our hypothesis that 𝜆 is not an eigenvalue of (𝐸𝑉;𝑚), we may assume that there exists a 𝑘∈ℕ such that 𝜇_𝑘(𝑚)<𝜆<𝜇_𝑘+1(𝑚). By Lemmas 3.5 and 3.8, we can choose 𝑇>0 and 𝑔₀∈ℱ_𝑘(𝑚) satisfying max_{𝑧∈𝑆𝑘−1}𝐼_𝜆,m𝑇𝑔₀(𝑧)<inf𝐼_𝜆,𝑚(𝑢);𝑢∈𝑌𝜇_𝑘+1(𝑚),𝑚=∶𝛼. (3.32) Put Σ∶=𝑔∈𝐶𝑆^𝑘₊,𝑊^1,𝑝(Ω);𝑔∣_𝑆^𝑘−1=𝑇𝑔₀,𝑐∶=inf_𝑔∈Σmax_{𝑧∈𝑆𝑘+}𝐼_𝜆,𝑚𝑔(𝑧). (3.33) Then, it follows from Lemma 3.4 and (3.32) that 𝑐≥𝛼>max_𝑧∈𝑆^𝑘−1𝐼_𝜆,𝑚(𝑇𝑔₀(𝑧)) holds. Since 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition by Proposition 3.3, the minimax theorem guarantees (cf. [25, Theorem 4.6]) that 𝑐 is a critical value of 𝐼_𝜆,𝑚. Hence, (𝑃;𝜆,𝑚,ℎ) has at least one solution.

3.4. The Case ∫_Ω𝑚𝑑𝑥=0

First, we introduce an approximate functional 𝐼⁺_{𝜆,𝑚,𝑛} as follows: 𝐼⁺_{𝜆,𝑚,𝑛}(𝑢)∶=𝐼_𝜆,𝑚(𝑢)+1‖𝑢‖𝑝𝑛^𝑝𝑝=𝐼_{𝜆,𝑚−1/(𝜆𝑛)}(𝑢)for𝑢∈𝑊^1,𝑝(Ω). (3.34)

Lemma 3.9. Let 0<𝜆<𝜇. Then, there exists an 𝑛₀∈ℕ such that for each 𝑛≥𝑛₀, 𝐼⁺_{𝜆,𝑚,𝑛} is bounded from below on 𝑌(𝜇,𝑚−1/𝜆𝑛), where 𝑌(𝜇,𝑚−1/𝜆𝑛) is the set introduced in (3.14).

Proof. Choose 𝑛₀∈ℕ such that 1/𝑛₀<𝜆esssup_x∈Ω𝑚(𝑥)/2. Then, for every 𝑛≥𝑛₀, Lemma 3.8 guarantees that 𝐼⁺_{𝜆,𝑚,𝑛}=𝐼_{𝜆,𝑚−1/(𝜆𝑛)} bounded from below on 𝑌(𝜇,𝑚−1/(𝜆𝑛)) because of ∫_Ω(𝑚−1/(𝜆𝑛))𝑑𝑥<0 and |{𝑚−1/(𝜆𝑛)>0}|>0.

Proof of Theorem 1.1 in the Case ∫_Ω𝑚𝑑𝑥=0. By noting that 𝜆𝑚=(−𝜆)(−𝑚) and 𝜇₁(𝑚)=𝜆^∗(𝑚)=0, we may assume that 𝜇_𝑘(𝑚)<𝜆<𝜇_𝑘+1(𝑚) for some 𝑘∈ℕ. Let 𝑛₀ be a natural number obtained by Lemma 3.9. Due to Proposition 2.10 (i) and (ii), there exists an 𝑛₁≥𝑛₀ such that 𝜇_𝑘(𝑚)≤𝜇_𝑘1𝑚−𝑛𝜆≤𝜇_𝑘1𝑚−𝑛₁𝜆<𝜆<𝜇_𝑘+1(𝑚)≤𝜇_𝑘+11𝑚−𝑛𝜆 (3.35) for every 𝑛≥𝑛₁. Thus, for every 𝑛≥𝑛₁, we can take 𝑇_𝑛>0 and 𝑔_𝑛∈ℱ_𝑘(𝑚−1/(𝑛𝜆)) satisfying max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛}𝑇_𝑛𝑔_𝑛(𝑧)<inf𝐼_{𝜆,𝑚,𝑛}(𝑢);𝑢∈Y𝜇_𝑘+11𝑚−()1𝑛𝜆,𝑚−(𝑛𝜆) (3.36) by applying Lemmas 3.5 and 3.9 to 𝐼⁺_{𝜆,𝑚,𝑛}=𝐼_{𝜆,𝑚−1/(𝑛𝜆)} (note (3.35)). Set Σ_𝑛∶=𝑔∈𝐶𝑆^𝑘₊,𝑊^1,𝑝(Ω);𝑔∣_𝑆^𝑘−1=𝑇_𝑛𝑔_𝑛,𝑐_𝑛∶=inf_{𝑔∈Σ𝑛}max_{𝑧∈𝑆𝑘+}𝐼⁺_{𝜆,𝑚,𝑛}𝑔(𝑧) (3.37) for each 𝑛≥𝑛₁. Then, for each 𝑛≥𝑛₁, we can obtain 𝑢_𝑛 satisfying 𝐼⁺_{𝜆,𝑚,𝑛}𝑢_𝑛−𝑐_𝑛<1𝑛,𝐼⁺_{𝜆,𝑚,𝑛}^′𝑢_{𝑛𝑊1,𝑝(Ω)}<1𝑛 (3.38) by applying Ekeland's variational principle to each 𝐼⁺_{𝜆,𝑚,𝑛} (refer to [25, Theorem 4.3]). In addition, we can see that {𝑢_𝑛} is bounded in 𝑊^1,𝑝(Ω). Indeed, if there exists a subsequence {𝑢_𝑛𝑙}_𝑙 satisfying ‖𝑢_𝑛𝑙‖_𝑝→∞ as 𝑙→∞, then we can show that 𝜆 is an eigenvalue of (𝐸𝑉;𝑚) by the same argument as in Proposition 3.3. This contradicts to our assumption that 𝜆 is not an eigenvalue of (𝐸𝑉;𝑚). Moreover, the boundedness of ‖∇𝑢_𝑛‖_𝑝 follows from a similar inequality to (3.6) as in Proposition 3.3 under the boundedness of ‖𝑢_𝑛‖_𝑝.
Therefore, we may assume, by choosing a subsequence that {𝑢_𝑛} is a Palais-Smale sequence of 𝐼_𝜆,𝑚 since 𝐼_𝜆,𝑚 is bounded on a bounded set and according to the following inequality: 𝐼^′_𝜆,𝑚(𝑢_𝑛)_{(𝑊1,𝑝(Ω))∗}≤𝐼^′_𝜆,𝑚𝑢_𝑛−𝐼⁺_{𝜆,𝑚,𝑛}^′𝑢_{𝑛(𝑊1,𝑝(Ω))∗}+1𝑛≤1𝑛𝑢_𝑛𝑝^𝑝−1+1𝑛. (3.39) Therefore, because 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition by Proposition 3.3, 𝐼_𝜆,𝑚 has a critical point, whence (𝑃;𝜆,𝑚,ℎ) has at least one solution.

4. Proof of Theorem 1.2

First, we will prove the following result concerning the Palais-Smale condition under the additional hypothesis (𝐻±) or (𝐻𝐹±).

Proposition 4.1. Assume that one of the following conditions hold: (i)𝜆=0 and (𝐻𝐹+) or (𝐻𝐹−); (ii)𝜆≠0 and one of (𝐻+), (𝐻−), (𝐻𝐹+) and (𝐻𝐹−). Then, 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition.

Proof. As the same reason in Proposition 3.3, it suffices to prove the boundedness of a Palais-Smale sequence {𝑢_𝑛} such that 𝐼_𝜆,𝑚(𝑢_𝑛)→𝑐 (for some 𝑐∈ℝ) and ‖𝐼^′_𝜆,𝑚(𝑢_𝑛)‖_𝑊∗→0 as 𝑛→∞. By way of contradiction, we may assume that ‖𝑢_𝑛‖_𝑝→∞ as 𝑛→∞ by choosing a subsequence. Set 𝑣_𝑛∶=𝑢_𝑛/‖𝑢_𝑛‖_𝑝. Then, by the same argument as in Proposition 3.3, {𝑣_𝑛} has a subsequence strongly convergent to 𝑣₀ being a nontrivial solution of −div𝑎_∞(𝑥)|∇𝑢|^𝑝−2∇𝑢=𝜆𝑚(𝑥)|𝑢|^𝑝−2𝑢inΩ,𝜕𝑢𝜕𝜈=0on𝜕Ω. (4.1) To simplify the notation, we denote the above subsequence strongly convergent to 𝑣₀ by {𝑣_𝑛}, again. Thus, |𝑢_𝑛(𝑥)|→∞ as 𝑛→∞ for a.e. 𝑥∈Ω₀∶={𝑥^′∈Ω;𝑣₀(𝑥^′)≠0} (note ‖𝑣₀‖_𝑝=1).
Assume (𝐻𝐹+) or (𝐻𝐹−). Then, we can obtain (𝐼)∶=_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝑝𝐹𝑥,𝑢_𝑛𝑢_𝑛𝑝^1+𝑞𝑑𝑥⟶±∞if(𝐻𝐹±),respectively. (4.2) Indeed, it follows from (𝐻𝐹+) that there exist 𝑅>0 and 𝐶>0 independent of 𝑛 such that 𝑓(𝑥,𝑡)𝑡−𝑝𝐹(𝑥,𝑡)≥0 if |𝑡|≥𝑅 and a.e. 𝑥∈Ω, and |𝑓(𝑥,𝑡)𝑡−𝑝𝐹(𝑥,𝑡)|≤𝐶 for every |𝑡|≤𝑅 and a.e. 𝑥∈Ω. Therefore, since |𝑢_𝑛(𝑥)|→∞ a.e. 𝑥∈Ω₀ and |Ω₀|>0 (note ‖𝑣₀‖_𝑝=1), we have (4.2) if (𝐻𝐹+) holds, by applying Fatou's lemma to the following inequality: (𝐼)≥_Ω0𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝑝𝐹𝑥,𝑢_𝑛𝑢_𝑛^1+𝑞𝑣_𝑛^1+𝑞𝐶𝑑𝑥−Ω⧵Ω₀𝑢_𝑛𝑝^1+𝑞. (4.3) In the case of (𝐻𝐹−), by considering −𝑓 instead of 𝑓 as in the above argument, we can show our claim (4.2).
Furthermore, by Hölder's inequality, we have (𝐼𝐼)∶=_Ω𝑝𝐺𝑥,∇𝑢_𝑛−𝑎𝑥,∇𝑢_𝑛∇𝑢_𝑛²𝑢_𝑛𝑝^1+𝑞𝑑𝑥≤𝐻_0Ω∇𝑣_𝑛^1+𝑞+1𝑢_𝑛𝑝^1+𝑞𝑑𝑥≤𝐻₀∇𝑣_𝑛𝑝^1+𝑞|Ω|^{(𝑝−1−𝑞)/𝑝}+𝑜(1)≤𝐻₀∇𝑣_0𝑝^1+𝑞|Ω|^{(𝑝−1−𝑞)/𝑝}+𝑜(1) (4.4) in the case of (𝐻𝐹−) because 𝑣_𝑛→𝑣₀ in 𝑊^1,𝑝(Ω), where 𝑞∈[0,𝑝−1] and 𝐻₀>0 are constants as in (𝐻𝐹−). Similarly, we obtain (𝐼𝐼)≥−𝐻₀∇𝑣_0𝑝^1+𝑞|Ω|^{(𝑝−1−𝑞)/𝑝}+𝑜(1) (4.5) in the case of (𝐻𝐹+).
Hence, we have a contradiction because of (4.2), (4.4), or (4.5) by taking a limit inferior or superior in the following equality: 𝑜(1)=𝑝𝐼_𝜆,𝑚𝑢_𝑛−𝐼^′_𝜆,𝑚𝑢_𝑛,𝑢_𝑛𝑢_𝑛𝑝^1+𝑞=(𝐼𝐼)+(𝐼)+1−𝑝_Ωℎ𝑣_𝑛𝑢_𝑛^𝑞_𝑝𝑑𝑥, (4.6) where we use the fact that ‖𝑢_𝑛‖/‖𝑢_𝑛‖_𝑝^1+𝑞=‖𝑣_𝑛‖/‖𝑢_𝑛‖^𝑞_𝑝 is bounded because of 𝑞≥0.
Assume 𝜆≠0 and (𝐻+) or (𝐻−): because 𝑣₀ is a nontrivial solution of (4.1) with 𝜆≠0, 𝑣₀ is not a constant function, that is, ‖∇𝑣₀‖_𝑝>0. Therefore, we have |∇𝑢_𝑛(𝑥)|→∞ as 𝑛→∞ for a.e. 𝑥∈Ω₀∶={𝑥^′∈Ω;|∇𝑣₀(𝑥^′)|≠0}. Because of |Ω₀|>0, we can show _Ω𝑝𝐺𝑥,∇𝑢_𝑛−𝑎𝑥,∇𝑢_𝑛∇𝑢_𝑛²𝑢_𝑛𝑝^1+𝑞𝑑𝑥⟶±∞if(𝐻±),respectively, (4.7) by a similar argument to one for 𝑓 in the above. In addition, we can easily obtain the following inequality: ±_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝑝𝐹𝑥,𝑢_𝑛𝑢_𝑛𝑝^1+𝑞𝑑𝑥≥−𝐻₀𝑣_𝑛^1+𝑞_1+𝑞+o(1)=−𝐻₀𝑣₀^1+𝑞_1+𝑞+𝑜(1) (4.8) in the case of (𝐻±), respectively. Hence, we have a contradiction by considering 𝑜(1)=(𝑝𝐼_𝜆,𝑚(𝑢_𝑛)−⟨𝐼^′_𝜆,𝑚(𝑢_𝑛),𝑢_𝑛⟩)/‖𝑢_𝑛‖_𝑝^1+𝑞.

By a similar way to the case ∫_Ω𝑚𝑑𝑥=0, we introduce the following approximate functionals on 𝑊^1,𝑝(Ω): 𝐼^±_{𝜆,𝑚,𝑛}(𝑢)∶=𝐼_𝜆,𝑚(𝑢)±1‖𝑢‖𝑝𝑛^𝑝𝑝for𝑢∈𝑊^1,𝑝(Ω). (4.9) Note 𝐼^±_{𝜆,𝑚,𝑛}(𝑢)=𝐼_{𝜆,𝑚∓1/(𝜆𝑛)}(𝑢) on 𝑊^1,𝑝(Ω) provided 𝜆≠0.

Proposition 4.2. If either 𝜆≠0 and (𝐻+) or (𝐻𝐹+) (resp., either 𝜆≠0 and (𝐻−) or (𝐻𝐹−)) and {𝑢_𝑛} satisfies sup_𝑛∈ℕ𝐼⁺_{𝜆,𝑚,𝑛}𝑢_𝑛<+∞,lim_𝑛→∞𝐼⁺_{𝜆,𝑚,𝑛}^′𝑢_{𝑛𝑊1,𝑝(Ω)∗}=0,resp.inf_𝑛∈ℕ𝐼⁻_{𝜆,𝑚,𝑛}𝑢_𝑛>−∞,lim_𝑛→∞I⁻_{𝜆,𝑚,𝑛}^′𝑢_{𝑛𝑊1,𝑝(Ω)∗}=0, (4.10)(4.11) then {𝑢_𝑛} is bounded in 𝑊^1,𝑝(Ω).

Proof. First, we note that the boundedness of ‖𝑢_𝑛‖_𝑝 guarantees that ‖𝑢_𝑛‖ is bounded by lim_𝑛→∞‖(𝐼^±_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)‖_{𝑊1,𝑝(Ω)∗}=0 (refer to (3.6) as in the proof of Proposition 3.3). Moreover, because of the following equality: 𝑝𝐼^±_{𝜆,𝑚,𝑛}𝑢_𝑛−𝐼^±_{𝜆,𝑚,𝑛}^′𝑢_𝑛,𝑢_𝑛𝑢_𝑛𝑝^1+𝑞=1−𝑝_Ωℎ𝑣_𝑛𝑢_𝑛^𝑞_𝑝+𝑑𝑥,_Ω𝑝𝐺𝑥,∇𝑢_𝑛−𝑎𝑥,∇𝑢_𝑛∇𝑢_𝑛²𝑢_𝑛𝑝^1+𝑞𝑑𝑥+_Ω𝑓𝑥,𝑢_𝑛𝑢_𝑛−𝑝𝐹𝑥,𝑢_𝑛𝑢_𝑛𝑝^1+𝑞𝑑𝑥, (4.12) we can prove the boundedness of ‖𝑢_𝑛‖_𝑝 by the same argument as in Proposition 4.1.

Proof of Theorem 1.2. Because of 𝜆𝑚=(−𝜆)(−𝑚), we may assume 𝜆≥0. In the case where ∫_Ω𝑚𝑑𝑥≠0 and 𝜇_𝑘(𝑚)<𝜆<𝜇_𝑘+1(𝑚) for some 𝑘∈ℕ, the proof of Theorem 1.1 implies the existence of a critical point of 𝐼_𝜆,𝑚 because 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition by Proposition 4.1. Concerning other cases, in the next section, we will prove the existence of a bounded sequence {𝑢_𝑛} satisfying (𝐼⁺_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)→0 or (𝐼⁻_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)→0 in 𝑊^1,𝑝(Ω)^∗ as 𝑛→∞. Because 𝐼_𝜆,𝑚 is bounded on a bounded set, we may assume that 𝐼_𝜆,𝑚(𝑢_𝑛) converges to some 𝑐∈ℝ by choosing a subsequence. In addition, by noting the inequality ‖𝐼^′_𝜆,𝑚(𝑢_𝑛)‖_{𝑊1,𝑝(Ω)∗}≤‖(𝐼^±_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)‖_{𝑊1,𝑝(Ω)∗}+‖𝑢_𝑛‖_𝑝^𝑝−1/𝑛, we easily see that {𝑢_𝑛} is a bounded Palais-Smale sequence of 𝐼_𝜆,𝑚. Therefore, 𝐼_𝜆,𝑚 has a critical point since 𝐼_𝜆,𝑚 satisfies the Palais-Smale condition by Proposition 4.1.

5. Construction of a Bounded Palais-Smale Sequence

In this section, due to the reason stated in the proof of Theorem 1.2, we will construct a bounded sequence {𝑢_𝑛} satisfying (𝐼⁺_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)→0 or (𝐼⁻_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)→0 in 𝑊^1,𝑝(Ω)^∗ as 𝑛→∞. It implies the existence of a bounded Palais-Smale sequence of 𝐼_𝜆,𝑚.

5.1. The Case 𝜆=0

Assume (𝐻𝐹+)
In this c ase, we can show that for each 𝑛∈ℕ, 𝐼⁺_{𝜆,𝑚,𝑛} has a global minimizer 𝑢_𝑛. Indeed, for 0<𝜀<1/(𝑝𝑛), there exists 𝐶_𝜀>0 such that 𝐼⁺_{𝜆,𝑚,𝑛}(𝑢)≥𝐶₀‖∇𝑢‖^𝑝_𝑝/(𝑝(𝑝−1))+(1/(𝑝𝑛)−𝜀)‖𝑢‖^𝑝_𝑝−‖ℎ‖_∞‖𝑢‖₁−𝐶_𝜀 for every 𝑢∈𝑊^1,𝑝(Ω) by (1.1), (1.16) and 𝜆=0 (refer to the inequality as in the proof of Lemma 3.5). This means that 𝐼⁺_{𝜆,𝑚,𝑛} is coercive and bounded from below on 𝑊^1,𝑝(Ω). Therefore, 𝐼⁺_{𝜆,𝑚,𝑛} has a global minimizer 𝑢_𝑛 since 𝐼⁺_𝜆,𝑚,n is w.l.s.c. on 𝑊^1,𝑝(Ω) as the same reason in Lemma 3.6.
Furthermore, because of (𝐼⁺_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)=0 in 𝑊^1,𝑝(Ω)^∗ and 𝐼⁺_{𝜆,𝑚,𝑛}(𝑢_𝑛)=min_𝑊^1,𝑝_(Ω)𝐼⁺_{𝜆,𝑚,𝑛}≤𝐼⁺_{𝜆,𝑚,𝑛}(0)=0, it follows from Proposition 4.2 that {𝑢_𝑛} is bounded.

Assume (𝐻𝐹−)
Choose 𝑛₀∈ℕ such that 1/𝑛₀<𝑐(1)=𝜇₂(1), where 𝑐(1) is the second eigenvalue of (𝐸𝑉;1) (so the weight function 𝑚≡1 and see (2.22) for the definition). Then, by noting that 𝐼⁻_0,𝑚,𝑛0=𝐼_1/𝑛0,1, we have 𝛼∶=inf𝐼⁻_0,𝑚,𝑛0(𝑢);𝑢∈𝑌(𝑐(1),1)>−∞ (5.1) by Lemma 3.8, where 𝑌(𝑐(1),1) is a subset defined by (3.14) with the weight 𝑚≡1, that is, 𝑌(𝑐(1),1)∶=𝑢∈𝑊^1,𝑝(Ω);_Ω𝑎_∞|∇𝑢|^𝑝𝑑𝑥≥𝑐(1)‖𝑢‖^𝑝_𝑝. (5.2) Moreover, inf{𝐼⁻_0,𝑚,𝑛(𝑢);𝑢∈𝑌(𝑐(1),1)}≥𝛼 for every 𝑛≥𝑛₀ holds because 𝐼⁻_0,𝑚,𝑛(𝑢)≥𝐼⁻_0,𝑚,𝑛0(𝑢) for every 𝑢∈𝑊^1,𝑝(Ω). Since ∫_Ω𝐹(𝑥,𝑢)𝑑𝑥=𝑜(1)‖𝑢‖^𝑝_𝑝 as ‖𝑢‖_𝑝→∞ by (1.1), there exists 𝑇_𝑛>0 such that 𝐼⁻_0,𝑚,𝑛(±𝑇_𝑛)=−𝑇^𝑝_𝑛(|Ω|/(𝑛𝑝)−𝑜(1))<𝛼−2.
Define Σ_𝑛∶=𝑔∈𝐶[0,1],𝑊^1,𝑝(Ω);𝑔(0)=𝑇_𝑛,𝑔(1)=−𝑇_𝑛,𝑐_𝑛∶=inf_{𝑔∈Σ𝑛}max_[]𝑡∈0,1𝐼⁻_0,𝑚,𝑛𝑔(𝑡) (5.3) for 𝑛≥𝑛₀. By the definition of 𝑐(1), we easily see that 𝑔([0,1])∩𝑌(𝑐(1),1)≠∅ for every 𝑔∈Σ_𝑛 (refer to [6] or Lemma 3.4). Hence, 𝑐_𝑛≥inf𝐼⁻_0,𝑚,𝑛(𝑢);𝑢∈𝑌(𝑐(1),1)≥𝛼>𝐼_0,𝑚,𝑛±𝑇_𝑛 (5.4) holds, whence 𝑐_𝑛 is bounded from below. Moreover, by applying Ekeland's variational principle to each 𝐼⁻_0,𝑚,𝑛, we can obtain a sequence {𝑢_𝑛} satisfying |𝐼⁻_0,𝑚,𝑛(𝑢_𝑛)−𝑐_𝑛|<1/𝑛 and ‖(𝐼⁻_0,𝑚,𝑛)^′(𝑢_𝑛)‖_{𝑊1,𝑝(Ω)∗}<1/𝑛. Since 𝑐_𝑛 is bounded from below, it follows from Proposition 4.2 that {𝑢_𝑛} is bounded. As a result, we can construct a bounded sequence {𝑢_𝑛} satisfying (𝐼⁻_0,𝑚,𝑛)^′(𝑢_𝑛)→0 as 𝑛→∞ in 𝑊^1,𝑝(Ω)^∗.

5.2. The Case 𝜆=𝜆^∗(𝑚)=𝜇₁(𝑚) with ∫_Ω𝑚𝑑𝑥<0

Assume (𝐻+) or (𝐻𝐹+)
Since we see that 𝐼⁺_{𝜆,𝑚,𝑛}=𝐼_{𝜆,𝑚−1/(𝑛𝜆)} and 𝜆^∗(𝑚−1/(𝑛𝜆))>𝜆^∗(𝑚)=𝜆>0 (according to Lemma 2.5), 𝐼⁺_{𝜆,𝑚,𝑛} is coercive, bounded from below and w.l.s.c. on 𝑊^1,𝑝(Ω) by Lemma 3.6. Thus, we obtain a global minimizer 𝑢_𝑛 of 𝐼⁺_{𝜆,𝑚,𝑛} for sufficiently large 𝑛 such that |{𝑚−1/(𝑛𝜆)>0}|>0. Because of 𝐼⁺_{𝜆,𝑚,𝑛}(𝑢_𝑛)≤𝐼⁺_{𝜆,𝑚,𝑛}(0)=0 for every 𝑛, Proposition 4.2 guarantees that {𝑢_𝑛} is bounded.

Assume (𝐻−) or (𝐻𝐹−)
First, we note that 𝐼⁻_{𝜆,𝑚,𝑛}=𝐼_{𝜆,𝑚+1/(𝑛𝜆)} and 0<𝜆^∗(𝑚+1/(𝑛𝜆))<𝜆^∗(𝑚)=𝜆 by Lemma 2.5 for sufficiently large 𝑛 such that ∫_Ω(𝑚+1/(𝑛𝜆))𝑑𝑥<0. Moreover, it follows from Proposition 2.10 and 𝜇₁(𝑚)<𝜇₂(𝑚) that there exists an 𝑛₀∈ℕ satisfying ∫_Ω𝑚+1/(𝑛₀𝜆)𝑑𝑥<0 and 𝜆^∗1𝑚+𝑛𝜆<𝜆=𝜇₁(𝑚)<𝜇₂1𝑚+𝑛₀𝜆≤𝜇₂1𝑚+𝑛𝜆≤𝜇₂(𝑚) (5.5) for every 𝑛≥𝑛₀. By applying Theorem 1.1 to each case of a weight 𝑚+1/(𝑛𝜆) (note that 𝜆 is not an eigenvalue of (𝐸𝑉;𝑚+1/(𝑛𝜆)) by (5.5), there exists 𝑢_𝑛 satisfying (𝐼⁻_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)=0 (note 𝐼⁻_{𝜆,𝑚,𝑛}=𝐼_{𝜆,𝑚+1/(𝑛𝜆)}) and 𝐼⁻_{𝜆,𝑚,𝑛}𝑢_𝑛=𝑐_𝑛≥inf𝐼⁻_{𝜆,𝑚,𝑛}(𝑢);𝑢∈𝑌𝜇₂𝑚_𝑛0,𝑚_𝑛0, (5.6) where the last inequality follows from Lemma 3.4 with 𝑚_𝑛0∶=𝑚+1/(𝑛₀𝜆). On the other hand, because 𝐼⁻_{𝜆,𝑚,𝑛}(𝑢)≥𝐼⁻_{𝜆,𝑚,𝑛0}(𝑢)=𝐼_{𝜆,𝑚𝑛0}(𝑢) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝑛≥𝑛₀, we have 𝑐_𝑛≥inf𝐼_{𝜆,𝑚𝑛0}(𝑢);𝑢∈𝑌𝜇₂𝑚_𝑛0,𝑚_𝑛0>−∞ (5.7) for every 𝑛≥𝑛₀, where the last inequality follows from Lemma 3.8. Thus, 𝑐_𝑛 is bounded from below. Hence, Proposition 4.2 guarantees the boundedness of {𝑢_𝑛}.

5.3. The Case 𝜆=𝜇_𝑘+1(𝑚) with ∫_Ω𝑚𝑑𝑥≠0

Assume (𝐻+) or (𝐻𝐹+)
We may assume 𝜇_𝑘(𝑚)<𝜇_𝑘+1(𝑚)=𝜆 by taking 𝑘 anew if necessary (note that we have already proved the case of 𝜇_𝑘(𝑚)<𝜆<𝜇_𝑘+1(𝑚) in Section 4). Here, we can choose an 𝑛₀∈ℕ such that ∫_Ω(𝑚−1/(𝑛𝜆))𝑑𝑥≠0, |{𝑚−1/(𝑛𝜆)>0}|>0 and 𝜇_𝑘1𝑚−𝑛𝜆≤𝜇_𝑘1𝑚−𝑛₀𝜆1<𝜆−𝑛‖𝑚‖_∞<𝜆=𝜇_𝑘+1(𝑚)≤𝜇_𝑘+11𝑚−𝑛𝜆 (5.8) for every 𝑛≥𝑛₀ by ∫_Ω𝑚𝑑𝑥≠0 and Proposition 2.10 (i), (iii). Note the following inequality: 𝐼⁺_{𝜆,𝑚,𝑛0}(𝑢)≥𝐼⁺_{𝜆,𝑚,𝑛}(𝑢)≥𝐼_{𝜆−1/(𝑛‖𝑚‖∞),𝑚}(𝑢) (5.9) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝑛≥𝑛₀, where the last inequality is obtained by ‖𝑢‖^𝑝_𝑝≥∫_Ω𝑚|𝑢|^𝑝𝑑𝑥/‖𝑚‖_∞. Let 𝑛≥𝑛₀. It follows from Lemma 3.8 and (5.8) that 𝐼_{𝜆−1/(𝑛‖𝑚‖∞),𝑚} is bounded from below on 𝑌(𝜆,𝑚). Hence, (5.9) yields that 𝐼⁺_{𝜆,𝑚,𝑛} is also bounded from below on 𝑌(𝜆,𝑚), namely, 𝛼_𝑛∶=inf𝐼⁺_{𝜆,𝑚,𝑛}(𝑢);𝑢∈𝑌(𝜆,𝑚)>−∞. (5.10) On the other hand, because of 𝜇_𝑘(𝑚−1/(𝑛₀𝜆))<𝜆 (see (5.8)), Lemma 3.5 guarantees the existence of 𝑔₀∈ℱ_𝑘(𝑚−1/(𝑛₀𝜆)) satisfying max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑇𝑔₀(𝑧)=max_{𝑧∈𝑆𝑘−1}𝐼_{𝜆,𝑚−1/(𝑛0𝜆)}𝑇𝑔₀(𝑧)⟶−∞as|𝑇|⟶∞. (5.11) Thus, for each 𝑛≥𝑛₀, we can take 𝑇_𝑛>0 such that max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛}𝑇_𝑛𝑔₀(𝑧)≤max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑇_𝑛𝑔₀(𝑧)≤𝛼_𝑛−1, (5.12) (note (5.9) for the first inequality). Set Σ_𝑛∶=𝑔∈𝐶𝑆^𝑘₊,𝑊^1,𝑝(Ω);𝑔∣_𝑆^𝑘−1=𝑇_𝑛𝑔₀,𝑐⁺_𝑛∶=inf_{𝑔∈Σ𝑛}max_{𝑧∈𝑆𝑘+}𝐼⁺_{𝜆,𝑚,𝑛}𝑔(𝑧) (5.13) for 𝑛≥𝑛₀. Since 𝑔(𝑆^𝑘₊)∩𝑌(𝜆,𝑚)≠∅ for every 𝑔∈Σ_𝑛 by Lemma 3.4 and 𝜆=𝜇_𝑘+1(𝑚), we have 𝑐⁺_𝑛≥𝛼_𝑛>max_𝑧∈𝑆^𝑘−1𝐼⁺_{𝜆,𝑚,𝑛}(𝑇_𝑛𝑔₀(𝑧)). Therefore, Ekeland's variational principle (refer to [25, Theorem  4.3]) guarantees the existence of 𝑢_𝑛 satisfying |𝐼⁺_{𝜆,𝑚,𝑛}(𝑢_𝑛)−𝑐_𝑛|<1/𝑛 and ‖(𝐼⁺_{𝜆,𝑚,𝑛})^′(𝑢_𝑛)‖_{𝑊1,𝑝(Ω)∗}<1/𝑛.

Finally, to show the boundedness of {𝑢_𝑛} due to Proposition 4.2, we will prove that 𝑐⁺_𝑛 is bounded from above. For each 𝑛≥𝑛₀, we define a continuous map 𝑔_𝑛 from 𝑆^𝑘₊ to 𝑊^1,𝑝(Ω) by 𝑔_𝑛(𝑧)∶=1−𝑧_𝑘+1𝑇_𝑛𝑔₀𝑧^′1−𝑧²_𝑘+1for𝑧=𝑧^′,𝑧_𝑘+1∈𝑆^𝑘₊with0≤𝑧_𝑘+1𝑧<1,0for𝑧=^′,𝑧_𝑘+1∈𝑆^𝑘₊with𝑧_𝑘+1=1. (5.14) Then, 𝑔_𝑛∈Σ_𝑛 holds. This leads to 𝑐⁺_𝑛≤sup_{𝑡≥0,𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛}𝑡𝑔₀(𝑧)≤sup_{𝑡≥0,𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑡𝑔₀(𝑧)<+∞ (5.15) because of (5.9), (5.11), and the compactness of 𝑔₀(𝑆^𝑘−1).

Assume (𝐻−) or (𝐻𝐹−)
Because the case of 𝜇₁(𝑚)=𝜆^∗(𝑚) is already shown (see Sections 5.1 and 5.2), We may assume (0<)𝜇_𝑘(𝑚)=𝜆<𝜇_𝑘+1(𝑚) for some 𝑘≥2 by taking 𝑘 anew if necessary. Here, we can choose an 𝑛₀∈ℕ such that ∫_Ω(𝑚+1/(𝑛𝜆))𝑑𝑥≠0 and 𝜇_𝑘1𝑚+𝑛𝜆≤𝜇_𝑘(𝑚)=𝜆<𝜇_𝑘+11𝑚+𝑛₀𝜆≤𝜇_𝑘+11𝑚+𝑛𝜆≤𝜇_𝑘+1(𝑚) (5.16) for every 𝑛≥𝑛₀ by ∫_Ω𝑚𝑑𝑥≠0 and Proposition 2.10 (i), (iii). Moreover, we note the following inequality: 𝐼⁻_{𝜆,𝑚,𝑛0}(𝑢)≤𝐼⁻_{𝜆,𝑚,𝑛}(𝑢)=𝐼_{𝜆,𝑚+1/(𝑛𝜆)}(𝑢)≤𝐼_{𝜆+1/(𝑛‖𝑚‖∞),𝑚}(𝑢) (5.17) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝑛≥𝑛₀. It follows from Lemma 3.8 and (5.16) (note (5.17) also) that 𝐼⁻_{𝜆,𝑚,𝑛0}=𝐼_𝜆,𝑚0 is bounded from below on 𝑌(𝜇_𝑘+1(𝑚₀),𝑚₀) with 𝑚₀∶=𝑚+1/(𝑛₀𝜆). Hence, (5.17) implies inf𝐼⁻_{𝜆,𝑚,𝑛}(𝑢);𝑢∈𝑌𝜇_𝑘+1𝑚₀,𝑚₀≥inf𝐼⁻_{𝜆,𝑚,𝑛0}(𝑢);𝑢∈𝑌𝜇_𝑘+1𝑚₀,𝑚₀=∶𝛼₀>−∞ (5.18) for every 𝑛≥𝑛₀. Because of 𝜆+1/(𝑛‖𝑚‖_∞)>𝜆=𝜇_𝑘(𝑚), there exist 𝑔_𝑛∈ℱ_𝑘(𝑚) and 𝑇_𝑛>0 such that max_{𝑧∈𝑆𝑘−1}𝐼⁻_{𝜆,𝑚,𝑛}𝑇_𝑛𝑔_𝑛(𝑧)≤max_{𝑧∈𝑆𝑘−1}𝐼_{𝜆+1/(𝑛‖𝑚‖∞),𝑚}𝑇_𝑛𝑔_𝑛(𝑧)<𝛼₀−1 (5.19) by Lemma 3.5. Define Σ_𝑛∶=𝑔∈𝐶𝑆^𝑘₊,𝑊^1,𝑝(Ω);𝑔∣_𝑆^𝑘−1=𝑇_𝑛𝑔_𝑛,𝑐⁻_𝑛∶=inf_{𝑔∈Σ𝑛}max_{𝑧∈𝑆𝑘+}𝐼⁻_{𝜆,𝑚,𝑛}𝑔(𝑧) (5.20) for 𝑛≥𝑛₀. Then, 𝑐⁻_𝑛≥𝛼₀ occurs (see (5.18)) since 𝑔(𝑆^𝑘₊)∩𝑌(𝜇_𝑘+1(𝑚₀),𝑚₀)≠∅ for every 𝑔∈Σ_𝑛 by Lemma 3.4. This means that 𝑐⁻_𝑛 is bounded from below. Consequently, we can obtain a desired bounded sequence by the same argument as in Sections 5.1 and 5.2.

5.4. The Case (iii) as in Theorem 1.2

First, note that we are assuming the hypothesis (𝐻+) or (𝐻𝐹+) in this case (iii). In addition, as the reason in the proof of Theorem 1.2, it suffices to handle with 𝜆>0.

Let 𝑘∈ℕ satisfy 𝜇_𝑘(𝑚)<𝜆≤𝜇_𝑘+1(𝑚). According to Proposition 2.10 (i) and (ii), we can take an 𝑛₀∈ℕ such that |{𝑚−1/(𝑛𝜆)>0}|>0 and 𝜇_𝑘1𝑚−2𝑛𝜆≤𝜇_𝑘1𝑚−𝑛₀𝜆1<𝜆−2𝑛‖𝑚‖_∞<𝜆≤𝜇_𝑘+1(𝑚)≤𝜇_𝑘+11𝑚−2𝑛𝜆 (5.21) for every 𝑛≥𝑛₀. The following inequality follows from the easy estimates: 𝐼⁺_{𝜆,𝑚,𝑛0}(𝑢)≥𝐼⁺_{𝜆,𝑚,𝑛}(𝑢)=𝐼_{𝜆,𝑚−1/(𝑛𝜆)}(𝑢)≥𝐼_{𝜆−1/(2𝑛‖𝑚‖∞),𝑚−1/(2𝑛𝜆)}(𝑢) (5.22) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝑛≥𝑛₀. Let 𝑛≥𝑛₀ and set 𝑚_𝑛∶=𝑚−1/(2𝑛𝜆). Because of (5.21), Lemma 3.8 implies that 𝐼_{𝜆−1/(2𝑛‖𝑚‖∞),𝑚𝑛} is bounded from below on 𝑌(𝜇_𝑘+1(𝑚_𝑛),𝑚_𝑛) with (note ∫_Ω𝑚_𝑛𝑑𝑥≠0). Hence, (5.22) yields that 𝛼_𝑛∶=inf𝐼⁺_{𝜆,𝑚,𝑛}(𝑢);𝑢∈𝑌𝜇_𝑘+1𝑚_𝑛,𝑚_𝑛>−∞ (5.23) for each 𝑛≥𝑛₀. On the other hand, because of 𝜇_𝑘(𝑚−1/(𝑛₀𝜆))<𝜆 (see (5.21)), Lemma 3.5 guarantees the existence of 𝑔₀∈ℱ_𝑘(𝑚−1/(𝑛₀𝜆)) satisfying max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑇𝑔₀(𝑧)=max_{𝑧∈𝑆𝑘−1}𝐼_{𝜆,𝑚−1/(𝑛0𝜆)}𝑇𝑔₀(𝑧)⟶−∞as𝑇⟶∞. (5.24) Therefore, for each 𝑛≥𝑛₀, we can choose 𝑇_𝑛>0 such that max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛}𝑇_𝑛𝑔₀(𝑧)≤max_{𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑇_𝑛𝑔₀(𝑧)≤𝛼_𝑛−1, (5.25) (note (5.22) for the first inequality). Set Σ_𝑛∶=𝑔∈𝐶𝑆^𝑘₊,𝑊^1,𝑝(Ω);𝑔∣_𝑆^𝑘−1=𝑇_𝑛𝑔₀,𝑐⁺_𝑛∶=inf_{𝑔∈Σ𝑛}max_{𝑧∈𝑆𝑘+}𝐼⁺_{𝜆,𝑚,𝑛}𝑔(𝑧) (5.26) for 𝑛≥𝑛₀. Since 𝑔(𝑆^𝑘₊)∩𝑌(𝜇_𝑘+1(𝑚_𝑛),𝑚_𝑛)≠∅ for every 𝑔∈Σ_𝑛 by Lemma 3.4, we have 𝑐⁺_𝑛≥𝛼_𝑛>max_𝑧∈𝑆^𝑘−1𝐼⁺_{𝜆,𝑚,𝑛}(𝑇_𝑛𝑔₀(𝑧)). Moreover, by the same argument as in Section 5.3 (note (5.24)), we have 𝑐⁺_𝑛≤sup_{𝑡≥0,𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛}𝑡𝑔₀(𝑧)≤sup_{𝑡≥0,𝑧∈𝑆𝑘−1}𝐼⁺_{𝜆,𝑚,𝑛0}𝑡𝑔₀(𝑧)<+∞, (5.27) and hence our conclusion is shown.

Remark 5.1. If ∫_Ω𝑚𝑑𝑥=0 holds, then we can not show the continuity of 𝜇_𝑘(𝑚) with respect to 𝑚 (refer to Proposition 2.10). Hence, we are not able to construct a bounded Palais-Smale sequence under (𝐻−) or (𝐻𝐹−). However, if we have the additional information about the existence of a suitable 𝑚^′∈𝐿^∞(Ω) such that 𝑚^′≥𝑚 in Ω, ∫_Ω𝑚^′𝑑𝑥≠0 and 𝜇_𝑘(𝑚)≤𝜆<𝜇_𝑘+1(𝑚^′) when 𝜇_𝑘(𝑚)≤𝜆<𝜇_𝑘+1(𝑚) occurs, then we can still easily prove that equation (𝑃;𝜆,𝑚,ℎ) has a solution in the case also where 𝜆≠0, ∫_Ω𝑚𝑑𝑥=0 and (𝐻−) or (𝐻𝐹−). In fact, let 0<𝜇_𝑘(𝑚)≤𝜆<𝜇_𝑘+1(𝑚^′) for some 𝑘≥2. Note the following inequality: 𝐼_{𝜆+1/(𝑛‖𝑚‖∞),𝑚}(𝑢)≥𝐼⁻_{𝜆,𝑚,𝑛}(𝑢)≥𝐼_𝜆,𝑚′(𝑢)−1‖𝑢‖𝑛𝑝^𝑝𝑝=𝐼_{𝜆,𝑚′−1/(𝑛𝜆)}(𝑢) (5.28) for every 𝑢∈𝑊^1,𝑝(Ω) and 𝑛. Fix 𝑛₀∈ℕ such that ∫_Ω𝑚^′−1/(𝑛₀𝜆)𝑑𝑥>0 and |{𝑚^′−1/(𝑛₀𝜆)>0}|>0. Set 𝑚^′₀∶=𝑚^′−1/(𝑛₀𝜆). Because of 𝜆<𝜇_𝑘+1(𝑚^′)≤𝜇_𝑘+1(𝑚^′₀) (the last inequality follows from Proposition 2.10 (i)), Lemma 3.8 implies that 𝐼_{𝜆,𝑚′0} is bounded from below on 𝑌(𝜇_𝑘+1(𝑚^′₀),𝑚^′₀) (note ∫_Ω𝑚^′₀𝑑𝑥>0). By combining this fact and (5.28), we have inf_𝑛≥𝑛0inf𝐼⁻_{𝜆,𝑚,𝑛}(𝑢);𝑢∈𝑌𝜇_𝑘+1𝑚^′₀,𝑚^′₀≥inf𝐼_{𝜆,𝑚′0}(𝑢);𝑢∈𝑌𝜇_𝑘+1𝑚^′₀,𝑚^′₀>−∞. (5.29) Because of 𝜆+1/(𝑛‖𝑚‖_∞)>𝜆≥𝜇_𝑘(𝑚), for each 𝑛≥𝑛₀, we can take a 𝑔_𝑛∈ℱ_𝑘(𝑚) satisfying max_{𝑧∈𝑆𝑘−1}𝐼⁻_{𝜆,𝑚,𝑛}𝑇𝑔_𝑛(𝑧)≤max_{𝑧∈𝑆𝑘−1}𝐼_{𝜆+1/(𝑛‖𝑚‖∞),𝑚}𝑇𝑔_𝑛(𝑧)⟶−∞ (5.30) as 𝑇→∞ by Lemma 3.5.
Since any extension 𝑔∈𝐶(𝑆^𝑘₊,𝑊^1,𝑝(Ω)) of 𝑇𝑔_𝑛 (𝑇>0) links 𝑌(𝜇_𝑘+1(𝑚^′₀),𝑚^′₀) by Lemma 3.4, we can construct a desired sequence by the same argument as in Section 5.3 under (𝐻−) or (𝐻𝐹−).

Acknowledgments

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks the referees for his helpful comments and suggestions.