Existence of Nontrivial Solutions and High Energy Solutions for a Class of Quasilinear Schrödinger Equations via the Dual-Perturbation Method

Abstract

We study the quasilinear Schrödinger equation of the form −Δ𝑢+𝑉(𝑥)𝑢−Δ(𝑢²)𝑢=ℎ(𝑥,𝑢), 𝑥∈𝑅^𝑁. Under appropriate assumptions on 𝑉(𝑥) and ℎ(𝑥,𝑢), existence results of nontrivial solutions and high energy solutions are obtained by the dual-perturbation method.

1. Introduction and Preliminaries

In this paper we consider the quasilinear Schrödinger equation of the form −Δ𝑢+𝑉(𝑥)𝑢−Δ𝑢²𝑢=ℎ(𝑥,𝑢),𝑥∈𝑅^𝑁, (1) where ℎ∈𝐶(𝑅^𝑁×𝑅,𝑅) and 𝑉∈𝐶(𝑅^𝑁,𝑅). Solutions of (1) are standing waves of the following quasilinear Schrödinger equation: 𝑖𝜓_𝑡+Δ𝜓−𝑉(𝑥)𝜓+𝑘Δ𝛼𝜓²𝛼^′𝜓²𝜓+𝑔𝑥,𝜓=0,𝑥∈𝑅^𝑁, (2) where 𝑉(𝑥) is a given potential, 𝑘 is a real constant, and 𝛼 and 𝑔 are real functions. The quasilinear Schrödinger equations (2) are derived as models of several physical phenomena; for example, see [1–5]. Several methods can be used to solve (1). For instance, the existence of a positive ground state solution has been proved in [6, 7] by using a constrained minimization argument; the problem is transformed to a semilinear one in [8–11] by a change of variables (dual approach); Nehari method is used to get the existence results of ground state solutions in [12, 13].

Recently, some new methods have been applied to these equations. In [14], the authors prove that the critical points are 𝐿^∞ functions by the Moser’s iteration; then the existence of multibump type solutions is constructed for this class of quasilinear Schrödinger equations. In [15], by analysing the behavior of the solutions for subcritical case, the authors pass to the limit as the exponent approaches to the critical exponent in order to establish the existence of both one-sign and nodal ground state solutions. Another new method which works for these equations is perturbations. In [16] 4-Laplacian perturbations are led into these equations; then high energy solutions are obtained on bounded smooth domain.

In this paper, the perturbation, combined with dual approach, is applied to search the existence of nontrivial solution and sequence of high energy solutions of (1) on the whole space 𝑅^𝑁. For simplicity we call this method the dual-perturbation method.

We need the following several notations. Let 𝐶^∞_𝑐(𝑅^𝑁) be the collection of smooth functions with compact support. Let 𝐻¹(𝑅)∶=𝑢∈𝐿²𝑅^𝑁∶|∇𝑢|∈𝐿²𝑅^𝑁, (3) with the inner product ⟨𝑢,𝑣⟩_𝐻¹=_𝑅𝑁[∇𝑢⋅∇𝑣+𝑢𝑣]𝑑𝑥 (4) and the norm ‖𝑢‖_𝐻¹=⟨𝑢,𝑢⟩_𝐻^1/2₁. (5) Let the following assumption (𝑉) hold:(𝑉)𝑉∈𝐶(𝑅^𝑁,𝑅) satisfies inf_𝑥∈𝑅^𝑁𝑉(𝑥)≥𝑎₀>0 and lim_|𝑥|→∞𝑉(𝑥)=+∞.

Set 𝐸∶=𝑢∈𝐻¹𝑅^𝑁∶_ℝ𝑁𝑉(𝑥)𝑢²𝑑𝑥<+∞ (6) with the inner product ⟨𝑢,𝑣⟩_𝐸=_𝑅𝑁[∇𝑢⋅∇𝑣+𝑉(𝑥)𝑢𝑣]𝑑𝑥 (7) and the norm ‖𝑢‖_𝐸=⟨𝑢,𝑢⟩_𝐸^1/2. (8) Then both 𝐻¹(𝑅^𝑁) and 𝐸 are Hilbert spaces.

By the continuity of the embedding𝐸↪𝐿^𝑠(ℝ^𝑁) for 𝑠∈[2,2^∗] we know that, for each 𝑠∈[2,2^∗], there exists constant 𝑎_𝑠>0 such that ‖𝑢‖_𝑠≤𝑎_𝑠‖𝑢‖_𝐸,∀𝑢∈𝐸, (9) where ‖⋅‖_𝑠 denotes the 𝐿^𝑠-norm. In the following, we use 𝐶 or 𝐶_𝑖 to denote various positive constants. Moreover, we need the following assumptions:(ℎ₁) there exist4<𝑝<2(2^∗) if 𝑁≥3 and 4<𝑝<∞ if 𝑁=2 such that |ℎ(𝑥,𝑠)|≤𝐶1+|𝑠|^𝑝−1,∀𝑠∈𝑅, (10)(ℎ₂)lim_𝑠→0ℎ(𝑥,𝑠)/𝑠=0 uniformly in 𝑥∈𝑅^𝑁,(ℎ₃) there exist 𝜇>4 and 𝑟>0 such that 𝑐₀∶=inf_{𝑥∈𝑅𝑁,|𝑠|=𝑟}𝐻()()()𝑠𝑥,𝑠>0,𝜇𝐻𝑥,𝑠≤ℎ𝑥,𝑠 (11)for all 𝑥∈𝑅^𝑁 and |𝑠|≥𝑟, where 𝐻(𝑥,𝑠)=∫^𝑠₀ℎ(𝑥,𝑡)𝑑𝑡.

By Lemma 3.4 in [17] we know that, under the assumption (𝑉), the embedding 𝐸↪𝐿^𝑠(𝑅^𝑁) is compact for each 2≤𝑠<2^∗.

Equation (1) is the Euler-Lagrange equation of the energy functional 𝐽(𝑢)=12_𝑅^𝑁1+2𝑢²|∇𝑢|²+𝑉(𝑥)𝑢²−𝑑𝑥_𝑅𝑁𝐻(𝑥,𝑢)𝑑𝑥, (12) where 𝐻(𝑥,𝑢)=∫^𝑢₀ℎ(𝑥,𝑡)𝑑𝑡. Due to the presence of the term ∫_𝑅𝑁𝑢²|∇𝑢|²𝑑𝑥, 𝐽(𝑢) is not well defined in 𝐸. To overcome this difficulty, a dual approach is used in [9, 10]. Following the idea from these papers, let 𝑓 be defined by 𝑓^′(𝑡)=11+2𝑓²(𝑡) (13) on [0,+∞), 𝑓(0)=0 and 𝑓(−𝑡)=−𝑓(𝑡) on (−∞,0]. Then 𝑓 has the following properties:(𝑓₁)𝑓 is uniquely defined 𝐶^∞ function and invertible;(𝑓₂)0<𝑓^′(𝑡)≤1 for all 𝑡∈𝑅;(𝑓₃)|𝑓(𝑡)|≤|𝑡| for all 𝑡∈𝑅;(𝑓₄)lim_𝑡→0𝑓(𝑡)/𝑡=1;(𝑓₅)lim_𝑡→+∞𝑓(𝑡)/√𝑡=2^1/4, lim_{𝑡→−∞}𝑓(𝑡)/|𝑡|=−2^1/4;(𝑓₆)(1/2)𝑓(𝑡)≤𝑡𝑓^′(𝑡)≤𝑓(𝑡) for all 𝑡≥0 and 𝑓(𝑡)≤𝑡𝑓^′(𝑡)≤(1/2)𝑓(𝑡) for all 𝑡≤0;(𝑓₇)|𝑓(𝑡)|≤2^1/4|𝑡| for all 𝑡∈𝑅;(𝑓₈) the function 𝑓²(𝑡) is strictly convex;(𝑓₉) there exists a positive 𝐶 such that 𝑓(𝑡≥𝐶|𝑡|,|𝑡|𝐶|𝑡|)≤1,_1/2,|𝑡|≥1; (14)(𝑓₁₀) there exist positive constants 𝐶₁ and 𝐶₂ such that |𝑡|≤𝐶₁𝑓(𝑡)+𝐶₂𝑓(𝑡)² (15) for all 𝑡∈𝑅;(𝑓₁₁)|𝑓(𝑡)𝑓^′(𝑡)|≤1/√2 for all 𝑡∈𝑅;(𝑓₁₂) for each 𝜉>0, there exists 𝐶(𝜉)>0 such that 𝑓²(𝜉𝑡)≤𝐶(𝜉)𝑓²(𝑡).

The properties (𝑓₁)--(𝑓₁₁) have been proved in [8–11]. It suffices to prove (𝑓₁₂).

Indeed, by (𝑓₁), (𝑓₄), and (𝑓₅), there exist 𝛿>0 and 𝑀>0 such that, for |𝑡|≤𝛿, 12𝑡²≤𝑓²(𝑡)≤32𝑡², (16) and for |𝑡|≥𝑀, √22|𝑡|≤𝑓²(𝑡)≤3√22|𝑡|. (17) Since there exists a 𝐶₀>0 such that 𝑓²(2𝑡)≤𝐶₀𝑓²(𝑡) (see [10]), we can assume that 0<𝜉<1. For |𝑡|≤𝛿, we have |𝜉𝑡|≤𝛿, and hence 𝑓²()≤3𝜉𝑡2𝜉²𝑡²≤3𝜉²𝑓²(𝑡); (18) for |𝑡|≥𝑀/𝜉>𝑀, one has |𝜉𝑡|≥𝑀, and hence 𝑓²()≤3√𝜉𝑡22𝜉|𝑡|≤3𝜉𝑓²(𝑡); (19) and for 𝛿≤|𝑡|≤𝑀/𝜉, there exist 𝑚(𝜉)>0 and 𝑀(𝜉)>0 such that 𝑓²(𝜉𝑡)≤𝑀(𝜉) and 𝑓²(𝑡)≥𝑚(𝜉). Then we have 𝑓²()(𝜉)≤𝑀(𝜉)𝜉𝑡≤𝑀𝑚(𝜉)𝑓²(𝑡). (20) Hence 𝑓²(𝜉𝑡)≤𝐶(𝜉)𝑓²(𝑡), where 𝐶(𝜉)=max{3𝜉²,𝑀(𝜉)/𝑚(𝜉)}.

After the change of variable, 𝐽(𝑢) can be reduced to 𝐼(𝑣)𝑓(𝑣=1∶=𝐽)2_𝑅^𝑁|∇𝑣|²+𝑉(𝑥)𝑓²(𝑣−)𝑑𝑥_𝑅𝑁𝐻𝑥,𝑓(𝑣)𝑑𝑥. (21) From [8, 9, 11] we know that if 𝑣∈𝐸 is a critical point of 𝐼, that is, 𝐼^′(𝑣),𝜑=_𝑅𝑁∇𝑣∇𝜑𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)−𝜑𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜑𝑑𝑥=0 (22) for all 𝜑∈𝐸, then 𝑢∶=𝑓(𝑣) is a weak solution of (1). Particularly, if 𝑣∈𝐻¹(𝑅^𝑁)∩𝐶²(𝑅^𝑁) is a critical point of 𝐼, then 𝑢∶=𝑓(𝑣) is a classical solution of (1).

A sequence {𝑢_𝑛}⊂𝐸 is called a Cerami sequence of 𝐽 if {𝐽(𝑢_𝑛)} is bounded and (1+‖𝑢_𝑛‖)𝐽^′(𝑢_𝑛)→0 in 𝐸^∗. We say that 𝐽 satisfies the Cerami condition if every Cerami sequence possesses a convergent subsequence.

2. Some Lemmas

Consider the following perturbation functional 𝐼_𝜃 defined by 𝐼_𝜃(𝑣)(𝑣)+𝜃=𝐼2_𝑅^𝑁𝑉(𝑥)𝑣²𝑑𝑥, (23) where 𝜃∈(0,1]. We have the following lemmas.

Lemma 1. If assumptions (𝑉), (ℎ₁), and (ℎ₂) hold, then the functional 𝐼_𝜃 is well defined on 𝐸 and 𝐼_𝜃∈𝐶¹(𝐸,𝑅).

Proof. By conditions (ℎ₁) and (ℎ₂), the properties (𝑓₂), (𝑓₃), (𝑓₇), and (𝑓₁₁) imply that there exists 𝛿>0 such that ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)≤𝑓(𝑣)𝑓^′(𝑣)≤|𝑣||𝑣|ℎ(𝑣)𝑓for<𝛿,𝑥,𝑓^′(𝑣)≤𝐶𝑓(𝑣)^𝑝−1𝑓^′(𝑣)≤𝐶𝑓(𝑣)^𝑝−2≤𝐶|𝑣|^(𝑝−2)/2for|𝑣|≥𝛿. (24) Hence ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)≤𝐶|𝑣|+|𝑣|^(𝑝/2)−1𝐻(𝑣)|𝑣|,𝑥,𝑓≤𝐶²+|𝑣|^𝑝/2 (25)(26) for all 𝑣∈𝑅. By (26) and the continuity of the embedding 𝐸↪𝐿^𝑠(𝑅^𝑁) (𝑠∈[2,2^∗]), _𝑅𝑁𝐻𝑥,𝑓(𝑣)𝑑𝑥<+∞,∀𝑣∈𝐸. (27) Hence 𝐼_𝜃 is well defined in 𝐸.
Now, we prove that 𝐼_𝜃∈𝐶¹(𝐸,𝑅). It suffices to prove that Ψ₁(𝑣)∶=_𝑅𝑁𝐻𝑥,𝑓(𝑣)𝑑𝑥∈𝐶¹(),Ψ𝐸,𝑅₂(𝑣)∶=_𝑅𝑁𝑉(𝑥)𝑓²(𝑣)𝑑𝑥∈𝐶¹(𝐸,𝑅). (28)
For any 𝑣,𝜙∈𝐸 and 0<|𝑡|<1, by the mean value theorem, (25) and (𝑓₂)-(𝑓₃), we have 𝐻𝑥,𝑓𝑣+𝑡𝜙−𝐻𝑥,𝑓(𝑣)|𝑡|≤¹₀ℎ𝑓𝑥,𝑣+𝑠𝑡𝜙𝑓^′𝜙𝑣𝜙+𝜙𝑣+𝑠𝑡𝜙𝑑𝑠≤𝐶||²+|𝑣|^(𝑝−2)/2𝜙+𝜙^𝑝/2,𝑉(𝑥)𝑓²𝑣+𝑡𝜙−𝑉(𝑥)𝑓²(𝑣)|𝑡|≤2¹₀𝑉(𝑥)𝑓𝑣+𝑠𝑡𝜙𝑓^′𝜙(𝑥)𝑣+𝑠𝑡𝜙𝑑𝑠≤2𝑉¹₀𝜙(𝑥𝑣𝜙+𝜙𝑣+𝑠𝑡𝜙𝑑𝑠≤2𝑉)||². (29) The Hölder inequality implies that 𝐶|𝑣|𝜙+𝜙²+|𝑣|^(𝑝−2)/2𝜙+𝜙^𝑝/2∈𝐿¹𝑅^𝑁,2𝑉(𝑥)|𝑣|𝜙+𝜙²∈𝐿¹𝑅^𝑁. (30) Hence, by the Lebesgue theorem, we have Ψ^′₁(𝑣),𝜙=_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)Ψ𝜙𝑑𝑥,_′2(𝑣),𝜙=2_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)𝜙𝑑𝑥 (31) for all 𝜙∈𝐸. Now, we show that Ψ^′_𝑖(⋅)∶𝐸→𝐸^∗, 𝑖=1,2, are continuous. Indeed, if 𝑣_𝑛→𝑣 in 𝐸, then 𝑣_𝑛→𝑣 in 𝐿^𝑠(𝑅^𝑁) for all 𝑠∈[2,2^∗].
On the space 𝐿^𝑝1(𝑅^𝑁)∩𝐿^𝑝2(𝑅^𝑁), we define the norm ‖𝑣‖_{𝑝1∧𝑝2}=‖𝑣‖_𝑝1+‖𝑣‖_𝑝2. (32) Then 𝑣_𝑛⟶𝑣in𝐿²𝑅^𝑁∩𝐿^(𝑝/2)−1𝑅^𝑁. (33) Moreover, on the space 𝐿^𝑝1(𝑅^𝑁)+𝐿^𝑝2(𝑅^𝑁), we define the norm ‖𝑣‖_{𝑝1∨𝑝2}=inf‖𝑢‖_𝑝1+‖𝑤‖_𝑝2∶𝑣=𝑢+𝑤,𝑢∈𝐿^𝑝1𝑅^𝑁,𝑤∈𝐿^𝑝2𝑅^𝑁. (34) By (25), we have ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)≤𝐶|𝑣|+|𝑣|^(𝑝/2)−1≤𝐶|𝑣|^2/2+|𝑣|^𝑞/𝑟, (35) where 𝑞=𝑝/2 and 𝑟=𝑝/(𝑝−2). Then Theorem A.4 in [18] implies ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛−ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)⟶0in𝐿²𝑅^𝑁+𝐿^𝑟𝑅^𝑁 (36) as 𝑛→+∞. If ℎ(𝑥,𝑓(𝑣_𝑛))𝑓^′(𝑣_𝑛)−ℎ(𝑥,𝑓(𝑣))𝑓^′(𝑣)=𝑦_𝑛+𝑧_𝑛 with 𝑦_𝑛∈𝐿²(𝑅^𝑁) and 𝑧_𝑛∈𝐿^𝑟(𝑅^𝑁), one has _𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛−ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣≤)𝜙𝑑𝑥_𝑅𝑁𝑦_𝑛𝜙+𝑧_𝑛𝜙𝑦𝑑𝑥≤𝐶_𝑛2+𝑧_𝑛𝑟𝜙_𝐸. (37) Hence _𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛−ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣ℎ𝑣)𝜙𝑑𝑥≤𝐶𝑥,𝑓_𝑛𝑓^′𝑣_𝑛−ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)_2∨𝑟𝜙_𝐸, (38) and hence Ψ^′₁𝑣_𝑛−Ψ^′₁(𝑣)⟶0 (39) as 𝑛→∞. Therefore, Ψ₁∈𝐶¹(𝐸,𝑅).
Define 𝐿^𝑠_𝑉𝑅^𝑁=𝑢∶𝑅^𝑁⟶𝑅∶𝑢ismeasurableand_𝑅𝑁𝑉(𝑥)𝑢^𝑠𝑑𝑥<∞ (40) with the norm ‖𝑢‖_𝐿^𝑠_𝑉=(∫_𝑅𝑁𝑉(𝑥)𝑢^𝑠𝑑𝑥)^1/𝑠. On the space 𝐿^𝑝1_𝑉(𝑅^𝑁)∩𝐿^𝑝2_𝑉(𝑅^𝑁), we define the norm ‖𝑣‖_{𝑝1∧𝑝2}=‖𝑣‖_𝐿^𝑝_1𝑉+‖𝑣‖_𝐿^𝑝_2𝑉. (41) On the space 𝐿^𝑝1_𝑉(𝑅^𝑁)+𝐿^𝑝2_𝑉(𝑅^𝑁), we define the norm ‖𝑣‖_{𝑝1∨𝑝2}=inf‖𝑣‖_{𝑝1∨𝑝2}=‖𝑢‖_𝐿^𝑝_1𝑉+‖𝑤‖_𝐿^𝑝_2𝑉∶𝑣=𝑢+𝑤,𝑢∈𝐿^𝑝1_𝑉𝑅^𝑁,𝑤∈𝐿^𝑝2_𝑉𝑅^𝑁. (42) From 𝑣_𝑛→𝑣 in 𝐸, one has 𝑣_𝑛,𝑣∈𝐿²_𝑉(𝑅^𝑁) and 𝑣_𝑛⟶𝑣in𝐿²_𝑉𝑅^𝑁∩𝐿²_𝑉𝑅^𝑁 (43) as 𝑛→∞. Since |𝑓(𝑣)𝑓^′(𝑣)|≤|𝑣|, by the following Lemma 2, we have 𝑓𝑣_𝑛𝑓^′𝑣_𝑛⟶𝑓(𝑣)𝑓^′(𝑣)in𝐿²_𝑉𝑅^𝑁+𝐿²_𝑉𝑅^𝑁. (44) If 𝑓(𝑣_𝑛)𝑓^′(𝑣_𝑛)−𝑓(𝑣)𝑓^′(𝑣)=𝑦_𝑛+𝑧_𝑛 with 𝑦_𝑛∈𝐿²_𝑉(𝑅^𝑁) and 𝑧_𝑛∈𝐿²_𝑉(𝑅^𝑁), one has _𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛−𝑓(𝑣)𝑓^′(𝑣≤)𝜙𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑦_𝑛𝜙+𝑉(𝑥)𝑧_𝑛𝜙≤𝑦𝑑𝑥_{𝑛𝐿2𝑉}+𝑧_{𝑛𝐿2𝑉}𝜙_𝐸. (45) Hence _𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛−𝑓(𝑣)𝑓^′(𝑣≤)𝜙𝑑𝑥𝑓(𝑣_𝑛)𝑓^′(𝑣_𝑛)−𝑓(𝑣)𝑓^′(𝑣)_2∨2𝜙_𝐸, (46) and hence Ψ^′₂𝑣_𝑛−Ψ^′₂(𝑣)⟶0 (47) as 𝑛→∞. Therefore, Ψ₂∈𝐶¹(𝐸,𝑅). This completes the proof.

Lemma 2. Assume that 1≤𝑝,𝑞,𝑟,𝑠<+∞, 𝑔∈𝐶(𝑅^𝑁×𝑅) and 𝑔(𝑥,𝑣)≤𝐶|𝑣|^𝑝/𝑟+|𝑣|^𝑞/𝑠. (48) Then, for every 𝑣∈𝐿^𝑝_𝑉(𝑅^𝑁)∩𝐿^𝑞_𝑉(𝑅^𝑁), 𝑔(⋅,𝑣)∈𝐿^𝑟_𝑉(𝑅^𝑁)+𝐿^𝑠_𝑉(𝑅^𝑁), and the operator 𝐴∶𝐿^𝑝_𝑉𝑅^𝑁∩𝐿^𝑞_𝑉𝑅^𝑁⟶𝐿^𝑟_𝑉𝑅^𝑁+𝐿^𝑠_𝑉𝑅^𝑁∶𝑣⟼𝑔(𝑥,𝑣) (49) is continuous.

Proof. Let 𝜂(𝑠) be a smooth cut-off function such that 𝜂(𝑠)=1 for |𝑠|≤1 and 𝜂(𝑠)=0 for |𝑠|≥2. Define 𝑔₁()(𝑣)𝑔(),𝑔𝑥,𝑣∶=𝜂𝑥,𝑣₂(𝑥,𝑣)∶=1−𝜂(𝑣)𝑔(𝑥,𝑣). (50) We can assume that 𝑝/𝑟≤𝑞/𝑠. Hence 𝑔₁(𝑥,𝑣)≤𝐶|𝑣|^𝑝/𝑟,𝑔₂(𝑥,𝑣)≤𝐶|𝑣|^𝑞/𝑠 (51) for all (𝑥,𝑣)∈𝑅^𝑁×𝑅. Assume 𝑣_𝑛→𝑣 in 𝐿^𝑝_𝑉(𝑅^𝑁)∩𝐿^𝑞_𝑉(𝑅^𝑁). Then 𝑣_𝑛→𝑣 in 𝐿^𝑝_𝑉(𝑅^𝑁) and 𝑔(⋅,𝑣_𝑛)→𝑔(⋅,𝑣) in 𝐿^𝑟_𝑉(𝑅^𝑁). As in the proof of Lemma A.1 in [18], there exists a subsequence {𝑤_𝑛} of {𝑣_𝑛} and 𝛼∈𝐿^𝑝_𝑉(𝑅^𝑁) such that 𝑤_𝑛(𝑥)→𝑣(𝑥) and |𝑣(𝑥)|,|𝑤_𝑛(𝑥)|≤𝛼(𝑥) for a.e. 𝑥∈𝑅^𝑁. Hence, from (51), one has 𝑔₁𝑥,𝑤_𝑛−𝑔₁(𝑥,𝑣)^𝑟≤2^𝑟𝐶|𝛼(𝑥)|^𝑝 (52) a.e. on 𝑅^𝑁. It follows from the Lebesgue theorem that 𝑔₁(⋅,𝑤_𝑛)→𝑔₁(⋅,𝑣) in 𝐿^𝑟_𝑉(𝑅^𝑁). Consequently, 𝑔₁(⋅,𝑣_𝑛)→𝑔₁(⋅,𝑣) in 𝐿^𝑟_𝑉(𝑅^𝑁). Similarly, we can prove 𝑔₂(⋅,𝑣_𝑛)→𝑔₂(⋅,𝑣) in 𝐿^𝑠_𝑉. Since 𝑔⋅,𝑣_𝑛−𝑔(⋅,𝑣)_𝑟∨𝑠≤𝑔₁⋅,𝑣_𝑛−𝑔₁(⋅,𝑣)_𝐿𝑟𝑉+𝑔₂⋅,𝑣_𝑛−𝑔₂(⋅,𝑣)_𝐿𝑠𝑉, (53) it follows that 𝑔(⋅,𝑣_𝑛)→𝑔(⋅,𝑣) in 𝐿^𝑟_𝑉+𝐿^𝑠_𝑉. This completes the proof.

Lemma 3. Let (𝑉), (ℎ₁), and (ℎ₂) hold. Then every bounded sequence {𝑣_𝑛}⊂𝐸 with 𝐼^′_𝜃(𝑣_𝑛)→0 possesses a convergent subsequence.

Proof. Since {𝑣_𝑛}⊂𝐸 is bounded, then, by the compactness of the embedding 𝐸↪𝐿^𝑠(𝑅^𝑁) (2≤𝑠<2^∗), passing to a subsequence, one has 𝑣_𝑛⇀𝑣 in 𝐸, 𝑣_𝑛→𝑣 in 𝐿^𝑠(𝑅^𝑁) for all 2≤𝑠<2^∗, and 𝑣_𝑛(𝑥)→𝑣(𝑥) for a.e. 𝑥∈𝑅^𝑁. By (25) _𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝑣−𝑣_𝑛≤𝑑𝑥_𝑅𝑁𝐶𝑣_𝑛+𝑣_𝑛^(𝑝/2)−1𝑣_𝑛𝑣−𝑣𝑑𝑥≤𝐶_𝑛2𝑣_𝑛−𝑣₂+𝑣_𝑛^(𝑝/2)−1_𝑝/2𝑣_𝑛−𝑣_𝑝/2≤𝐶𝑣_𝑛𝐸𝑣_𝑛−𝑣₂+𝑣_𝑛𝐸^(𝑝/2)−1𝑣_𝑛−𝑣_𝑝/2⟶0 (54) as 𝑛→∞. Similarly, ∫_𝑅𝑁ℎ(𝑥,𝑓(𝑣))𝑓^′(𝑣)(𝑣−𝑣_𝑛)𝑑𝑥→0 as 𝑛→∞. Hence, by the property of (𝑓₈), we have ⟨𝐼^′_𝜃𝑣_𝑛−𝐼^′_𝜃(𝑣),𝑣_𝑛=−𝑣⟩_𝑅𝑁∇𝑣_𝑛−𝑣²𝑑𝑥+𝜃_𝑅𝑁𝑉(𝑥)𝑣_𝑛−𝑣²+𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛−𝑓(𝑣)𝑓^′(𝑣×𝑣_𝑛)−−𝑣𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛−ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝑣_𝑛𝑣−𝑣𝑑𝑥≥𝜃_𝑛−𝑣²_𝐸−𝑜_𝑛(1), (55) where 𝑜_𝑛(1)→0 as 𝑛→∞. This shows that ‖𝑣_𝑛−𝑣‖²_𝐸→0 as 𝑛→∞. This completes the proof.

The following Lemma 4 has been proved in [10] (see Proposition 2.1(3) in [10]).

Lemma 4. If 𝑣_𝑛(𝑥)→𝑣(𝑥) a.e. in 𝑅^𝑁 and lim_𝑛→∞∫_𝑅𝑁𝑉(𝑥)𝑓²(𝑣_𝑛)𝑑𝑥=∫_𝑅𝑁𝑉(𝑥)𝑓²(𝑣)𝑑𝑥, then ∫_𝑅𝑁𝑉(𝑥)𝑓²(𝑣_𝑛−𝑣)𝑑𝑥→0 as 𝑛→∞.

3. Main Results

Theorem 5. Assume conditions (𝑉), (ℎ₁)--(ℎ₃) hold. Let {𝜃_𝑛}⊂(0,1] be such that 𝜃_𝑛→0. Let 𝑣_𝑛∈𝐸 be a critical point of 𝐼_𝜃𝑛 with 𝐼_𝜃𝑛(𝑣_𝑛)≤𝑐 for some constant 𝑐 independent of 𝑛. Then, up to subsequence, one has 𝑣_𝑛→𝑣 in 𝐸, 𝐼_𝜃𝑛(𝑣_𝑛)→𝐼(𝑣) and 𝑣 is a critical point of 𝐼.

Proof. By (ℎ₂), for 0<𝜀₀<(1/4)(1/2−1/𝜇)𝑎₀, there exists 𝛿₀>0 such that 1𝜇𝑠ℎ(𝑥,𝑠)−𝐻(𝑥,𝑠)≤𝜀₀𝑠²,∀𝑠∈−𝛿₀,𝛿₀. (56) By (ℎ₁), for 𝛿₀≤|𝑠|≤𝑟 (𝑟 is the constant appearing in condition (ℎ₃)), we have 1𝜇()(1𝑠ℎ𝑥,𝑠−𝐻𝑥,𝑠)≤2𝐶𝛿²₀+𝑟^𝑝−2𝑠², (57) where 𝐶 is the constant appearing in condition (ℎ₁). Hence 1𝜇𝑠ℎ(𝑥,𝑠)−𝐻(𝑥,𝑠)≤𝜀₀𝑠²1+2𝐶𝛿²₀+𝑟^𝑝−2𝑠²,∀𝑠∈[−𝑟,𝑟]. (58) Since lim_|𝑥|→∞𝑉(𝑥)=+∞, there exists 𝜌₀>0 such that 1412−1𝜇𝑉(𝑥)1>2𝐶𝛿²₀+𝑟^𝑝−2 (59) for all |𝑥|≥𝜌₀. Hence 12−1𝜇_𝑅^𝑁𝑉(𝑥)𝑓²(𝑣)+𝑑𝑥_{{𝑥∶|𝑓(𝑣)|≤𝑟}}1𝜇𝑓(𝑣)ℎ(𝑣)(𝑣)≥1𝑥,𝑓−𝐻𝑥,𝑓𝑑𝑥212−1𝜇_𝑅^𝑁𝑉(𝑥)𝑓²(𝑣)1𝑑𝑥−2𝐶𝛿²₀+𝑟^𝑝−2𝑟²𝐵_𝜌0. (60) Since 𝑣_𝑛 is a critical point of 𝐼_𝜃𝑛, 𝐼^′_𝜃𝑛𝑣_𝑛,𝜙=_𝑅𝑁∇𝑣_𝑛+∇𝜙𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜙𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛−𝜙𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜙𝑑𝑥=0 (61) for all 𝜙∈𝐸. Consequently, taking 𝜙=𝑓(𝑣_𝑛)/𝑓^′(𝑣_𝑛)∈𝐸, by (ℎ₃) and (𝑓₆) we have 𝑐≥𝐼_𝜃𝑛𝑣_𝑛=𝐼_𝜃𝑛𝑣_𝑛−1𝜇𝐼^′_𝜃𝑛𝑣_𝑛,𝑓𝑣_𝑛𝑓^′𝑣_𝑛=_𝑅𝑁12−1𝜇1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛∇𝑣_𝑛²+1𝑑𝑥2−1𝜇_𝑅^𝑁𝑉(𝑥)𝑓²𝑣_𝑛+𝜃𝑑𝑥_𝑛2_𝑅^𝑁𝑉(𝑥)𝑣²_𝑛−𝜃𝑑𝑥_𝑛𝜇_𝑅^𝑁𝑉(𝑥)𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛+𝑑𝑥_𝑅𝑁1𝜇ℎ𝑥,𝑓𝑣_𝑛𝑓𝑣_𝑛−𝐻𝑥,𝑓𝑣_𝑛≥1𝑑𝑥2−2𝜇_𝑅^𝑁∇𝑣_𝑛²1𝑑𝑥+2−1𝜇_𝑅^𝑁𝑉(𝑥)𝑓²𝑣_𝑛+1𝑑𝑥2−2𝜇𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣²_𝑛+𝑑𝑥_{{𝑥∶|𝑓(𝑣𝑛)|≤𝑟}}1𝜇ℎ𝑥,𝑓𝑣_𝑛𝑓𝑣_𝑛−𝐻𝑥,𝑓𝑣_𝑛≥1𝑑𝑥2−2𝜇_𝑅^𝑁∇𝑣_𝑛²+1𝑑𝑥212−1𝜇_𝑅^𝑁𝑉(𝑥)𝑓²𝑣_𝑛+1𝑑𝑥2−2𝜇𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣²_𝑛1𝑑𝑥−2𝐶𝛿²₀+𝑟^𝑝−2𝑟²𝐵_𝜌0≥12−2𝜇_𝑅^𝑁∇𝑣_𝑛²+1𝑑𝑥4−12𝜇_𝑅^𝑁𝑉(𝑥)𝑓²𝑣_𝑛+1𝑑𝑥2−2𝜇𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣²_𝑛𝑑𝑥−𝐶₁, (62) and hence _𝑅𝑁∇𝑣_𝑛²𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣²_𝑛𝑑𝑥≤𝐶 (63) for some constant 𝐶 independent of 𝑛. By the boundedness of ∫_𝑅𝑁|∇𝑣_𝑛|²𝑑𝑥, there exists 𝐶₂>0 such that 2_ℝ𝑁𝑓²𝑣_𝑛∇𝑓𝑣_𝑛²≤𝑑𝑥_ℝ𝑁1+2𝑓²𝑣_𝑛∇𝑓𝑣_𝑛²=𝑑𝑥_ℝ𝑁∇𝑣_𝑛²𝑑𝑥≤𝐶₂ (64) for all 𝑛. Hence, by the Sobolev embedding theorem, one has 𝑓(𝑣_𝑛)⁴_2(2∗)=𝑓²(𝑣_𝑛)²_2∗≤𝐶₃∇𝑓²𝑣_𝑛²₂≤𝐶. (65)
Next, we prove that 𝑓(𝑣_𝑛)∈𝐿^∞(𝑅^𝑁) and ‖𝑓(𝑣_𝑛)‖_𝐿∞≤𝐶, where the positive constant 𝐶 is independent of 𝑛. Setting 𝑇>2, 𝑟>0, define 𝑣^𝑇_𝑛̃=𝑏(𝑣_𝑛), where 𝑏∶𝑅→𝑅 is a smooth function satisfying 𝑏(𝑠)=𝑠 for |𝑠|≤𝑇−1, 𝑏(−𝑠)=−𝑏(𝑠); 𝑏^′(𝑠)=0 for 𝑠≥𝑇, and 𝑏^′(𝑠) is decreasing in [𝑇−1,𝑇].
This means that 𝑣^𝑇_𝑛̃=𝑣_𝑛, for |𝑣_𝑛|≤𝑇−1; |𝑣^𝑇_𝑛̃|=|𝑏(𝑣_𝑛)|≤|𝑣_𝑛|, for 𝑇−1≤|𝑣_𝑛|≤𝑇; |𝑣^𝑇_𝑛̃|=𝐶_𝑇>0, for |𝑣_𝑛|≥𝑇, where 𝑇−1≤𝐶_𝑇≤𝑇.
Let 𝜙=(𝑓(𝑣_𝑛)/𝑓^′(𝑣_𝑛))|𝑓(𝑣^𝑇_𝑛̃)|^2𝑟; then 𝜙∈𝐸. By (61) ⟨𝐼^′(𝑣_𝑛),𝜙⟩=0. Hence 𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅=_ℝ𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟𝑑𝑥, (66) where 𝐼₅∶=_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟≥𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟𝐼𝑑𝑥,₁∶=_{{𝑥∶|𝑣𝑛|≥𝑇}}1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟∇𝑣_𝑛²≥𝑑𝑥_{{𝑥∶|𝑣𝑛|≥𝑇}}𝑓𝑣̃^𝑇_𝑛^2𝑟∇𝑣_𝑛²𝑑𝑥=_{{𝑥∶|𝑣𝑛|≥𝑇}}1+2𝑓²𝑣_𝑛×∇𝑓𝑣_𝑛²𝑓𝑣̃^𝑇_𝑛^2𝑟𝑑𝑥≥2_{{𝑥∶|𝑣𝑛|≥𝑇}}𝑓²𝑣_𝑛∇𝑓𝑣_𝑛²𝑓𝑣̃^𝑇_𝑛^2𝑟=1𝑑𝑥2_{{𝑥∶|𝑣𝑛|≥𝑇}}∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²𝐼𝑑𝑥,₂∶=_{{𝑥∶|𝑣𝑛|≤𝑇−1}}2𝑟+1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛×𝑓𝑣̃^𝑇_𝑛^2𝑟∇𝑣_𝑛²≥𝑑𝑥_{{𝑥∶|𝑣𝑛|≤𝑇−1}}2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^2𝑟∇𝑣_𝑛²≥𝑑𝑥_{{𝑥∶|𝑣𝑛|≤𝑇−1}}𝑓𝑣_𝑛^2𝑟+2∇𝑓𝑣_𝑛²=1𝑑𝑥(𝑟+2)^{2{𝑥∶|𝑣}_𝑛^{|≤𝑇−1}}∇𝑓^𝑟+2𝑣_𝑛²=1𝑑𝑥(𝑟+2)^{2{𝑥∶|𝑣}_𝑛^{|≤𝑇−1}}∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²𝐼𝑑𝑥,₃∶=_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛𝑓^2𝑟𝑣^𝑇_𝑛̃∇𝑣_𝑛²≥𝑑𝑥_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛𝑓^2𝑟𝑣^𝑇_𝑛̃∇𝑣_𝑛²=1𝑑𝑥2_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^𝑟𝑣^𝑇_𝑛̃∇𝑓²𝑣_𝑛²𝐼𝑑𝑥,₄∶=2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^2𝑟−1𝑣^𝑇_𝑛̃𝑓^′𝑣^𝑇_𝑛̃×𝑏^′𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛∇𝑣_𝑛²𝑑𝑥. (67)(68) For 𝑇−1≤|𝑣_𝑛|≤𝑇, |𝑣^𝑇_𝑛̃|=|𝑏(𝑣_𝑛)|≤|𝑣_𝑛|. By the properties of 𝑓 and 𝑏, the mean value theorem implies 𝑓𝑏𝑣_𝑛≥𝑓^′𝑏𝑣_𝑛𝑏^′𝑣_𝑛𝑣_𝑛≥1√2𝑓^′𝑏𝑣_𝑛𝑏^′𝑣_𝑛𝑓²𝑣_𝑛. (69) Hence 𝐼₄=2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^2𝑟−1𝑣^𝑇_𝑛̃𝑓^′𝑣^𝑇_𝑛̃×𝑏^′𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛∇𝑣_𝑛²𝑑𝑥=2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^2𝑟−1𝑏𝑣_𝑛𝑓^′𝑏𝑣_𝑛×𝑏^′𝑣_𝑛𝑓𝑣_𝑛×1+2𝑓²𝑣_𝑛∇𝑣_𝑛²𝑑𝑥≥2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^𝑟−1𝑏𝑣_𝑛𝑓^′𝑏𝑣_𝑛𝑏^′𝑣_𝑛²×𝑓𝑣_𝑛𝑣_𝑛2𝑓²𝑣_𝑛∇𝑣_𝑛²𝑑𝑥≥2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^𝑟−1𝑏𝑣_𝑛𝑓^′𝑏𝑣_𝑛𝑏^′𝑣_𝑛²×𝑓⁴𝑣_𝑛∇𝑣_𝑛²𝑑𝑥=2𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓⁴𝑣_𝑛×𝑓^𝑟−1𝑏𝑣_𝑛∇𝑓𝑏𝑣_𝑛²=2𝑑𝑥𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓²𝑣_𝑛∇𝑓^𝑟𝑣̃^𝑇_𝑛²𝑑𝑥. (70) Consequently, 𝐼₃+𝐼₄=12_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓^𝑟𝑣^𝑇_𝑛̃∇𝑓²𝑣_𝑛²+2𝑑𝑥𝑟_{{𝑥∶𝑇−1≤|𝑣𝑛|≤𝑇}}𝑓²𝑣_𝑛∇𝑓^𝑟𝑣̃^𝑇_𝑛²≥1𝑑𝑥(𝑟+2)^{2{𝑥∶𝑇−1≤|𝑣}_𝑛^|≤𝑇}2𝑓^𝑟𝑣^𝑇_𝑛̃∇𝑓²𝑣_𝑛²+2𝑓²𝑣_𝑛∇𝑓^𝑟𝑣̃^𝑇_𝑛²≥1𝑑𝑥(𝑟+2)^{2{𝑥∶𝑇−1≤|𝑣}_𝑛^|≤𝑇}∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²𝑑𝑥. (71) Combining (67) and (68), we have 𝐼₁+𝐼₂+𝐼₃+𝐼₄≥1(𝑟+2)^2ℝ𝑁∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²𝑑𝑥. (72) For any 𝜀>0, by (ℎ₁) and (ℎ₂), there exists 𝐶(𝜀)>0 such that |ℎ(𝑥,𝑠)|≤𝜀|𝑠|+𝐶(𝜀)|𝑠|^𝑝−1. (73) Combining (66), (72), and (73), one has 1(𝑟+2)^2𝑅𝑁∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²(𝜀)𝑑𝑥≤𝐶_𝑅𝑁𝑓𝑣_𝑛^𝑝𝑓𝑣̃^𝑇_𝑛^2𝑟𝑑𝑥. (74) By the Hölder inequality and (65), _𝑅𝑁𝑓𝑣_𝑛^𝑝𝑓𝑣̃^𝑇_𝑛^2𝑟=𝑑𝑥_𝑅𝑁𝑓𝑣_𝑛^𝑝−4𝑓𝑣̃^𝑇_𝑛^2𝑟𝑓⁴𝑣_𝑛≤𝑑𝑥_𝑅𝑁𝑓𝑣_𝑛^{(𝑝−4)⋅(4𝑁/(𝑝−4)(𝑁−2))}𝑑𝑥^{(𝑝−4)(𝑁−2)/4𝑁}⋅_𝑅𝑁𝑓𝑣̃^𝑇_𝑛^2𝑟×𝑓⁴𝑣_𝑛^{4𝑁/(4𝑁−(𝑝−4)(𝑁−2))}𝑑𝑥^{(4𝑁−(𝑝−4)(𝑁−2))/4𝑁}=_𝑅𝑁𝑓𝑣_𝑛^22∗𝑑𝑥^{((𝑝−4)(𝑁−2))/4𝑁}⋅_𝑅𝑁𝑓𝑣̃^𝑇_𝑛^𝑟×𝑓²𝑣_𝑛^{8𝑁/(4𝑁−(𝑝−4)(𝑁−2))}𝑑𝑥^{(4𝑁−(𝑝−4)(𝑁−2))/4𝑁}≤𝐶_𝑅𝑁𝑓𝑣̃^𝑇_𝑛^𝑟×𝑓²𝑣_𝑛^{8𝑁/(4𝑁−(𝑝−4)(𝑁−2))}𝑑𝑥^{(4𝑁−(𝑝−4)(𝑁−2))/4𝑁}. (75) Moreover, 1(𝑟+2)^2𝑅𝑁∇𝑓²𝑣_𝑛𝑓^𝑟𝑣̃^𝑇_𝑛²≥𝐶𝑑𝑥(𝑟+2)^2𝑅𝑁𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^𝑟2∗𝑑𝑥^2/2∗. (76) Hence_𝑅𝑁𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^𝑟2∗𝑑𝑥^2/2∗≤𝐶(𝑟+2)²×_𝑅𝑁𝑓𝑣̃^𝑇_𝑛^𝑟×𝑓²𝑣_𝑛^{8𝑁/(4𝑁−(𝑝−4)(𝑁−2))}𝑑𝑥^{(4𝑁−(𝑝−4)(𝑁−2))/4𝑁}. (77)Since 4<𝑝<2(2^∗), 𝑑=2^∗/(8𝑁/(4𝑁−(𝑝−4)(𝑁−2)))=2^∗/2−𝑝/4+1>1. Set 𝑞=8𝑁/(4𝑁−(𝑝−4)(𝑁−2)). Then _𝑅𝑁𝑓²𝑣_𝑛𝑓𝑣̃^𝑇_𝑛^𝑟𝑞𝑑𝑑𝑥^{1/𝑞𝑑(𝑟+2)}≤𝐶(𝑟+2)^21/2(𝑟+2)×_𝑅𝑁𝑓𝑣̃^𝑇_𝑛^𝑟𝑓²𝑣_𝑛^𝑞𝑑𝑥^{1/𝑞(𝑟+2)}. (78) Take 𝑟=𝑟₀ such that (2+𝑟₀)𝑞=2(2^∗). Since |𝑣^𝑇_𝑛̃|=|𝑏(𝑣_𝑛)|≤|𝑣_𝑛|, |𝑓(𝑣^𝑇_𝑛̃)|≤|𝑓(𝑣_𝑛)|. Hence, from (65), we have _𝑅𝑁𝑓𝑣̃^𝑇_𝑛^𝑟0𝑓²𝑣_𝑛^𝑞𝑑𝑥≤_𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟0)𝑞}𝑑𝑥<𝐶. (79) Since 𝑓(𝑣^𝑇_𝑛̃)→𝑓(𝑣_𝑛) as 𝑇→+∞, taking 𝑇→+∞ in (78) with 𝑟=𝑟₀, we have _𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟0)𝑞𝑑}𝑑𝑥^{1/𝑞𝑑(𝑟0+2)}≤𝐶𝑟₀+2^{21/2(𝑟0+2)}×_𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟0)𝑞}𝑑𝑥^{1/𝑞(𝑟0+2)}. (80) Set 2+𝑟₁=(2+𝑟₀)𝑑. Then _𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟1)𝑞}𝑑𝑥^{1/𝑞(𝑟1+2)}≤𝐶𝑟₀+2^{21/2(𝑟0+2)}×_𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟0)𝑞}𝑑𝑥^{1/𝑞(𝑟0+2)}. (81) Inductively, we have _𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟𝑘+1)𝑞}𝑑𝑥^{1/𝑞(𝑟𝑘+1+2)}≤𝐶𝑟_𝑘+2^{21/2(𝑟𝑘+2)}×_𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟𝑘)𝑞}𝑑𝑥^{1/𝑞(𝑟𝑘+2)}≤^𝑘_𝑖=0𝐶𝑟_𝑖+2^{21/2(𝑟𝑖+2)}×_𝑅𝑁𝑓𝑣_𝑛^{(2+𝑟0)𝑞}𝑑𝑥^{1/𝑞(𝑟0+2)}, (82) where (2+𝑟_𝑖)=𝑑^𝑖(2+𝑟₀)  (𝑖=0,1,…,𝑘), and ^𝑘_𝑖=0𝐶𝑟_𝑖+2²12(𝑟_𝑖+2)=exp^𝑘_𝑖=0ln√𝐶𝑑^𝑖𝑟₀+2𝑑^𝑖𝑟₀+2=exp^𝑘_𝑖=0ln√𝐶𝑟₀+2𝑑^𝑖𝑟₀++2𝑖ln𝑑𝑑^𝑖𝑟₀+2 (83) is convergent as 𝑘→∞. Let 𝐶_𝑘=∏^𝑘_𝑖=0[𝐶(𝑟_𝑖+2)²]^{1/2(𝑟𝑖+2)}. Then 𝐶_𝑘→𝐶_∞>0 as 𝑘→∞. Hence 𝑓(𝑣_𝑛)_{𝐿0(2+𝑟)𝑞𝑑}^𝑘+1≤𝐶_𝑘𝑓𝑣_𝑛𝐿^∗₎₂₍₂. (84) Let 𝑘→∞; by (65), we have 𝑓(𝑣_𝑛)_𝐿∞≤𝐶_∞𝑓(𝑣_𝑛)_𝐿^∗₎₂₍₂≤𝐶,𝑓(𝑣)_𝐿∞≤𝐶. (85) Hence, by (𝑓₉) and (85), we have _𝑅𝑁𝑉(𝑥)𝑣²_𝑛=𝑑𝑥_{{𝑥∶|𝑣𝑛(𝑥)|≤1}}𝑉(𝑥)𝑣²_𝑛+𝑑𝑥_{{𝑥∶|𝑣𝑛(𝑥)|>1}}𝑉(𝑥)𝑣²_𝑛≤1𝑑𝑥𝐶_{{𝑥∶|𝑣𝑛(𝑥)|≤1}}𝑉(𝑥)𝑓²𝑣_𝑛+1𝑑𝑥𝐶_{{𝑥∶|𝑣𝑛(𝑥)|>1}}𝑉(𝑥)𝑓⁴𝑣_𝑛≤1𝑑𝑥𝐶_{{𝑥∶|𝑣𝑛(𝑥)|≤1}}𝑉(𝑥)𝑓²𝑣_𝑛𝑑𝑥+𝐶_{{𝑥∶|𝑣𝑛(𝑥)|≥1}}𝑉(𝑥)𝑓²𝑣_𝑛𝑑𝑥≤𝐶_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛𝑑𝑥. (86) By (63) we know that ∫_𝑅𝑁𝑉(𝑥)𝑣²_𝑛𝑑𝑥 is bounded, and hence {𝑣_𝑛} is bounded in 𝐸. Up to subsequence, one has 𝑣_𝑛⇀𝑣 in 𝐸, 𝑣_𝑛→𝑣 in 𝐿^𝑠(𝑅^𝑁) for 𝑠∈[2,2^∗), and 𝑣_𝑛(𝑥)→𝑣(𝑥) a.e. 𝑥∈𝑅^𝑁.
Now, we show that 𝑣 is a critical point of 𝐼. For any 𝜓∈𝐶^∞₀(𝑅^𝑁) with 𝜓≥0, by (85), we know that 𝜙=𝜓exp(−𝑓(𝑣_𝑛))∈𝐸. Take 𝜙=𝜓exp(−𝑓(𝑣_𝑛)) as the test function in (61); we have 0=_𝑅𝑁exp−𝑓𝑣_𝑛∇𝑣_𝑛−⋅∇𝜓𝑑𝑥_𝑅𝑁∇𝑣_𝑛²𝜓exp−𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛𝜓exp−𝑓𝑣_𝑛+𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜓exp−𝑓𝑣_𝑛−𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜓exp−𝑓𝑣_𝑛𝑑𝑥. (87) By |∇(𝑣_𝑛−𝑣)|²𝜓exp(−𝑓(𝑣_𝑛))𝑓^′(𝑣_𝑛)≥0, one has limsup_{𝑛→∞𝑅𝑁}−∇𝑣_𝑛²𝜓exp−𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝑑𝑥≤−_𝑅𝑁|∇𝑣|²𝜓exp−𝑓(𝑣)𝑓^′(𝑣)𝑑𝑥. (88) Since 𝜃_𝑛→0, by (63) 𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛𝜓exp−𝑓𝑣_𝑛𝑑𝑥⟶0 (89) as 𝑛→∞. Moreover, notice that 𝑣_𝑛⇀𝑣 in 𝐸, 𝑣_𝑛→𝑣 in 𝐿^𝑠(𝑅^𝑁) for 𝑠∈[2,2^∗), and 𝑣_𝑛(𝑥)→𝑣(𝑥) a.e. 𝑥∈𝑅^𝑁; by Hölder inequality and Lebesgue theorem, we have exp−𝑓𝑣_𝑅𝑁𝑛∇𝑣_𝑛⟶⋅∇𝜓𝑑𝑥_𝑅𝑁exp−𝑓(𝑣)∇𝑣⋅∇𝜓𝑑𝑥,_𝑅𝑁𝑉(𝑥)𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜓exp−𝑓𝑣_𝑛⟶𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)𝜓exp−𝑓(𝑣)𝑑𝑥,_𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝜓exp−𝑓𝑣_𝑛𝑑𝑥⟶_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜓exp−𝑓(𝑣)𝑑𝑥. (90) Hence, from (87), we have 0≤_𝑅𝑁(𝑣)−exp−𝑓∇𝑣⋅∇𝜓𝑑𝑥_𝑅𝑁|∇𝑣|²𝜓exp−𝑓(𝑣)𝑓^′(𝑣)𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)(𝑣)+𝑣𝜓exp−𝑓𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)(𝑣)−𝜓exp−𝑓𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜓exp−𝑓(𝑣)𝑑𝑥=_𝑅𝑁∇𝑣⋅∇𝜓exp−𝑓(𝑣)𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)𝜓exp−𝑓(𝑣)𝑑𝑥−_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜓exp−𝑓(𝑣)𝑑𝑥. (91) For any 𝜑∈𝐸 with 𝜑≥0, by (85) we know that 𝜁∶=𝜑exp(𝑓(𝑣))∈𝐸. By Theorem 2.8 in [19], there exists a sequence {𝜓_𝑛}⊂𝐶^∞₀(𝑅^𝑁) such that 𝜓_𝑛≥0 and 𝜓_𝑛→𝜁 and 𝜓_𝑛(𝑥)→𝜁(𝑥) for a.e. 𝑥∈𝑅^𝑁. Take 𝜓=𝜓_𝑛 in (91), and let 𝑛→∞; we have 0≤_𝑅𝑁∇𝑣⋅∇𝜑𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)−𝜑𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜑𝑑𝑥. (92) The opposite inequality can be obtained by taking 𝜙=𝜓exp(𝑓(𝑣_𝑛)) and 𝜁=𝜑exp(−𝑓(𝑣)). Consequently, _𝑅𝑁∇𝑣⋅∇𝜑𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓(𝑣)𝑓^′(𝑣)−𝜑𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓^′(𝑣)𝜑𝑑𝑥=0,∀𝜑∈𝐸. (93) This shows that 𝑣∈𝐸 is a critical point of 𝐼, and by taking 𝜑=𝑓(𝑣)/𝑓^′(𝑣)∈𝐸, one has _𝑅𝑁1+2𝑓²(𝑣)1+2𝑓²(𝑣)|∇𝑣|²𝑑𝑥+_𝑅𝑁𝑉(𝑥)𝑓²(𝑣)𝑑𝑥−_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓(𝑣)𝑑𝑥=0. (94)
Finally, taking 𝜙=𝑓(𝑣_𝑛)/𝑓^′(𝑣_𝑛) as the test function in (61), we have _𝑅𝑁1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛∇𝑣_𝑛²𝑑𝑥+𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛+𝑑𝑥_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛−𝑑𝑥_𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓𝑣_𝑛𝑑𝑥=0. (95) Since _𝑅𝑁ℎ𝑥,𝑓𝑣_𝑛𝑓𝑣_𝑛𝑑𝑥⟶_𝑅𝑁ℎ𝑥,𝑓(𝑣)𝑓(𝑣)d𝑥,_𝑅𝑁1+2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛∇𝑣_𝑛−𝑣²𝑑𝑥≥0, (96) by Fatou’s Lemma, (63), (94), (95), up to subsequence, one has 𝜃_𝑛𝑅𝑁𝑉(𝑥)𝑣_𝑛𝑓𝑣_𝑛𝑓^′𝑣_𝑛𝑑𝑥⟶0,_𝑅𝑁∇𝑣_𝑛²𝑑𝑥⟶_𝑅𝑁|∇𝑣|²𝑑𝑥,_𝑅𝑁2𝑓²𝑣_𝑛1+2𝑓²𝑣_𝑛∇𝑣_𝑛²𝑑𝑥⟶_𝑅𝑁2𝑓²(𝑣)1+2𝑓²(𝑣)|∇𝑣|²𝑑𝑥,_𝑅𝑁𝑉(𝑥)𝑓²𝑣_𝑛𝑑𝑥⟶_𝑅𝑁𝑉(𝑥)𝑓²(𝑣)𝑑𝑥. (97)(98) Hence 𝐼_𝜃𝑛(𝑣_𝑛)→𝐼(𝑣) as 𝑛→∞. Set 𝑤_𝑛∶=𝑣_𝑛−𝑣∈𝐸. By (𝑓₈), (𝑓₁₂), and (85), one has 𝑓²𝑤_𝑛=𝑓²𝑤2⋅^𝑛21≤𝐶2𝑓²𝑣_𝑛+12𝑓²(𝑣𝑓)≤𝐶²𝑣_𝑛+𝑓²(𝑣)≤𝐶. (99) Consequently, by (𝑓₉), (98), and Lemma 4, one has _𝑅𝑁𝑉(𝑥)𝑤_𝑛²=𝑑𝑥_{{𝑥∶|𝑤𝑛|≤1}}𝑉(𝑥)𝑤_𝑛²+𝑑𝑥_{{𝑥∶|𝑤𝑛|≥1}}𝑉(𝑥)𝑤_𝑛²𝑑𝑥≤𝐶_{{𝑥∶|𝑤𝑛|≤1}}𝑉(𝑥)𝑓²𝑤_𝑛𝑑𝑥+𝐶_{{𝑥∶|𝑤𝑛|≥1}}𝑉(𝑥)𝑓⁴𝑤_𝑛𝑑𝑥≤𝐶_{{𝑥∶|𝑤𝑛|≤1}}𝑉(𝑥)𝑓²𝑤_𝑛𝑑𝑥+𝐶_{{𝑥∶|𝑤𝑛|≥1}}𝑉(𝑥)𝑓²𝑤_𝑛𝑑𝑥=𝐶_𝑅𝑁𝑉(𝑥)𝑓²𝑤_𝑛𝑑𝑥⟶0 (100) as 𝑛→∞. Therefore, 𝑣_𝑛→𝑣 in 𝐸. This completes the proof.

Theorem 6. Assume conditions (𝑉), (ℎ₁)--(ℎ₃) hold; then (1) has a weak solution.

Proof. First, we prove that, for each 𝜃∈(0,1], 𝐼_𝜃 satisfy the Cerami condition. Indeed, let {𝑣_𝑛}⊂𝐸 be an arbitrary Cerami sequence of 𝐼_𝜃. Set 𝜙=𝑓(𝑣_𝑛)/𝑓^′(𝑣_𝑛). Then ‖𝜙‖_𝐸≤𝐶‖𝑣_𝑛‖_𝐸. Similar to the proof of (63), we can prove that {𝑣_𝑛} is bounded in 𝐸. Hence, by Lemma 3, the sequence {𝑣_𝑛} possesses a convergent subsequence in 𝐸. This shows that 𝐼_𝜃 satisfy the Cerami condition.
Next, for any 𝜀>0, by (ℎ₁), (ℎ₂), (𝑓₃), and (𝑓₇), there exists 𝐶(𝜀)>0 such that 𝐻𝑥,𝑓(𝑡)≤𝜀𝑡²+𝐶(𝜀)|𝑡|^𝑝/2 (101) for all (𝑥,𝑡)∈𝑅^𝑁×𝑅. For small 0<𝜌≪1, set 𝑆_𝜌=𝑣∈𝐸∶‖𝑣‖_𝐸=𝜌. (102) Then, from (101), for 𝑣∈𝑆_𝜌, 𝐼_𝜃(𝑣)=12_𝑅^𝑁|∇𝑣|²+𝑉(𝑥)𝑓²(𝑣+𝜃)𝑑𝑥2_𝑅^𝑁𝑉(𝑥)𝑣²𝑑𝑥−_𝑅𝑁𝐻(𝑣≥𝜃𝑥,𝑓)𝑑𝑥2_𝑅^𝑁|∇𝑣|²+𝑉(𝑥)𝑣²𝑑𝑥−𝜀_𝑅𝑁𝑣²𝑑𝑥−𝐶(𝜀)_𝑅𝑁|𝑣|^𝑝/2≥𝜃𝑑𝑥2‖𝑣‖²_𝐸−𝜀𝑎²₂‖𝑣‖²_𝐸−𝐶(𝜀)𝑎^𝑝/2_𝑝/2‖𝑣‖_𝐸^𝑝/2≥𝜌²𝜃4−𝐶𝜌^(𝑝−4)/2≥𝛿>0 (103) for small 𝜀>0 and 𝜌>0. Moreover, by (ℎ₃), for any (𝑥,𝑧)∈𝑅^𝑁×𝑅 with |𝑧|≥𝑟, one has 𝐻(𝑥,𝑧)≥𝑐₀|𝑧|^𝜇. (104) Since 𝜇>4, there exists a constant 4<𝛼<min{𝜇,2(2^∗)}. Hence, by (𝑓₅), we have lim_|𝑡|→∞𝐻𝑥,𝑓(𝑡)|𝑡|^𝛼/2=lim_|𝑡|→∞𝐻𝑥,𝑓(𝑡)𝑓(𝑡)^𝛼⋅𝑓(𝑡)^𝛼|𝑡|^𝛼/2=+∞ (105) uniformly in 𝑥∈𝑅^𝑁. Consequently, there exist constants 𝜏>1 such that 𝐻𝑥,𝑓(𝑡)≥|𝑡|^𝛼/2,∀|𝑡|≥𝜏, (106) for all 𝑥∈𝑅^𝑁. For any finite-dimensional subspace 𝐸⊂𝐸, by the equivalency of all norms in the finite-dimensional space, there is a constant 𝑎>0 such that ‖𝑣‖_𝛼/2≥𝑎‖𝑣‖_𝐸,∀𝑣∈𝐸. (107) By (ℎ₁), (ℎ₂), and (106), there exists a positive constant 𝐶>0 such that 𝐻𝑥,𝑓(𝑡)≥|𝑡|^𝛼/2−𝐶𝑡²,∀(𝑥,𝑡)∈𝑅^𝑁×𝑅. (108) Since 4<𝛼<2(2^∗), by (𝑓₃), (107), and (108), we have 𝐼_𝜃(𝑣)=12_𝑅^𝑁|∇𝑣|²+𝑉(𝑥)𝑓²(𝑣+𝜃)𝑑𝑥2_𝑅^𝑁𝑉(𝑥)𝑣²𝑑𝑥−_𝑅𝑁𝐻(𝑣≤‖𝑣‖𝑥,𝑓)𝑑𝑥_2𝐸−‖𝑣‖^𝛼/2_𝛼/2+𝐶‖𝑣‖²₂≤𝐶‖𝑣‖²_𝐸−𝑎^𝛼/2‖𝑣‖_𝐸^𝛼/2 (109) for all 𝑣∈𝐸. Hence there exists a large 𝑅>0 such that 𝐼_𝜃<0 on 𝐸⧵𝐵_𝑅. Set a fixed 𝑒∈𝐸 with ‖𝑒‖_𝐸=1. For any fixed 𝑇>𝜌, define the path ℎ_𝑇∶[0,1]↦𝐸⊂𝐸 by ℎ_𝑇(𝑡)=𝑡𝑇𝑒. Then for large 𝑇>0, by (109), one has 𝐼_𝜃ℎ_𝑇(1)≤𝐶𝑇²−𝑎^𝛼/2𝑇^𝛼/2ℎ<0,_𝑇(1)_𝐸=𝑇>𝜌,sup_𝑡∈[0,1]𝐼_𝜃ℎ_𝑇(𝑡)≤𝐶𝑇²<+∞. (110) Hence by Theorem 2.2 with the Cerami condition in [20], 𝐼_𝜃 possesses a critical value 𝑐_𝜃∶=inf_𝛾∈Γmax_[]𝑡∈0,1𝐼_𝜃𝛾(𝑡)𝑐≥𝛿>0,_𝜃≤sup_𝑡∈[0,1]𝐼_𝜃ℎ_𝑇(𝑡)≤𝐶𝑇², (111) where Γ=𝛾∈𝐶([0,1],𝐸)∶𝛾(0)=0,𝛾(1)=ℎ_𝑇(1). (112) Consequently, by Theorem 5, we know that (1) has a weak solution. This completes the proof of Theorem 6.

Remark 7. Let 𝑣⁺=max{𝑣,0} and 𝑣⁻=max{−𝑣,0}. Set 𝐼^±(𝑢)=12_𝑅^𝑁|∇𝑣|²+𝑉(𝑥)𝑓²(𝑣−)𝑑𝑥_𝑅𝑁𝐻𝑥,𝑓𝑣^±𝐼𝑑𝑥,_±𝜃(𝑣)=12𝜃_𝑅𝑁𝑉(𝑥)𝑣²𝑑𝑥+𝐼^±(𝑣) (113) instead of 𝐼(𝑢) and 𝐼_𝜃(𝑢), respectively. Then, under the conditions of Theorem 6, we can obtain the existence of a positive solution and a negative solution for (1).

Theorem 8. Assume conditions (𝑉), (ℎ₁)--(ℎ₃) hold. If ℎ(𝑥,𝑠) is odd in 𝑠, then (1) has a sequence {𝑣_𝑚} of solutions such that 𝐼(𝑣_𝑚)→+∞.

Proof. Consider the eigenvalue of the operate 𝐿=−Δ+𝑉. By assumption (𝑉) and the compactness of the embedding 𝐸↪𝐿²(𝑅^𝑁), we know that the spectrum 𝜎(𝐿)={𝜆₁,𝜆₂,…,𝜆_𝑛⋯} of 𝐿 with 0<𝜆₁<𝜆₂<⋯<𝜆_𝑛<⋯ (114) and 𝜆_𝑛→+∞ as 𝑛→+∞ (see page 3820 in [21]). Let 𝜑_𝑛 be the eigenfunction corresponding to 𝜆_𝑛. By regularity argument we know that 𝜑_𝑛∈𝐸. Set 𝐸_𝑛=span{𝜑₁,𝜑₂,…,𝜑_𝑛}. Then we can decompose the space 𝐸 as 𝐸=𝐸_𝑛⊕𝑊_𝑛 for 𝑛=1,2,…, where 𝑊_𝑛 is orthogonal to 𝐸_𝑛 in 𝐸. For 𝜌>0, set 𝑄_𝜌=𝑣∈𝐸∶_𝑅𝑁|∇𝑣|²+𝑉(𝑥)𝑓²(𝑣)𝑑𝑥≤𝜌². (115) By (109) there exists 𝑟_𝑛>0 independent of 𝜃 such that 𝐼_𝜃(𝑣)<0,∀𝑣∈𝐸_𝑛⧵𝑄_𝑟𝑛. (116) Set 𝐷_𝑛=𝐸_𝑛∩𝑄_𝑟𝑛,𝐺_𝑛=𝜑∈𝐶𝐷_𝑛,𝐸∶𝜑isoddand𝜑|_{𝜕𝑄𝑟𝑛∩𝐸𝑛},Γ=𝑖𝑑_𝑗=𝜑𝐷_𝑛⧵𝑄_𝑟𝑛∶𝜑∈𝐺_𝑛,𝑛≥𝑗,𝐴=−𝐴⊂𝐸_𝑛∩𝑄_𝑟𝑛isclosedand𝛾(𝐴)≤𝑛−𝑗, (117) where 𝛾(⋅) is the genus. Let 𝑐_𝑗(𝜃)=inf_{𝐵∈Γ𝑗}sup_𝑣∈𝐵𝐼_𝜃,𝑗=1,2…. (118) We have the following three facts (we refer the reader to [16] for their proofs).  Fact (1). For each 𝐵∈Γ_𝑗, if 0<𝜌<𝑟_𝑛 for all 𝑛≥𝑗, then 𝐵∩𝜕𝑄_𝜌∩𝑊_𝑗−1≠∅.  Fact (2). There exist constants 𝛼_𝑗≤𝛽_𝑗 such that 𝑐_𝑗(𝜃)∈[𝛼_𝑗,𝛽_𝑗] and 𝛼_𝑗→+∞ as 𝑗→+∞.  Fact (3).  𝑐_𝑗(𝜃), 𝑗=1,2,… are critical values of 𝐼_𝜃.Consequently, Theorem 8 follows from Theorem 5 and the above Facts (2)-(3). This completes the proof.

Corollary 9. If the following conditions (ℎ₄) and (ℎ₅) are used in place of (ℎ₃); then the conclusions of Theorem 5, Theorem 6, and Theorem 8 hold:(ℎ₄)lim_{|𝑠|→+∞}inf𝐻(𝑥,𝑠)>0 uniformly in 𝑥∈ℝ^𝑁,(ℎ₅) there exist 𝜇>4 and 𝜏>0 such that 𝜇𝐻(𝑥,𝑠)≤ℎ(𝑥,𝑠)𝑠 (119)for all 𝑥∈ℝ^𝑁 and |𝑠|≥𝜏.

Proof. By (ℎ₄), there are constants 𝜆>0 and 𝑟₁>0 such that whenever |𝑠|≥𝑟₁, one has 𝐻(𝑥,𝑠)>𝜆,∀𝑥∈ℝ^𝑁. (120) Set 𝑟=max{𝜏,𝑟₁}. Then, by (ℎ₅), 𝑐₀∶=inf_{𝑥∈ℝ𝑁,|𝑠|=𝑟}𝐻()()()𝑠𝑥,𝑠≥𝜆>0,𝜇𝐻𝑥,𝑠≤ℎ𝑥,𝑠 (121) for all 𝑥∈ℝ^𝑁 and |𝑠|≥𝑟. Therefore, condition (ℎ₃) holds. This completes the proof.

Acknowledgment

This work was supported partially by the National Natural Science Foundation of China (11261070).