Multiple Solutions to Elliptic Equations on ℝ^𝑁 with Combined Nonlinearities

Abstract

In this paper, we are concerned with the multiplicity of nontrivial radial solutions for the following elliptic equations (𝑃)_𝜆: −Δ𝑢+𝑉(𝑥)𝑢=𝜆𝑄(𝑥)|𝑢|^𝑞−2𝑢+𝑄(𝑥)𝑓(𝑢),𝑥∈ℝ^𝑁; 𝑢(𝑥)→0, as |𝑥|→+∞, where 1<𝑞<2,0<𝜆∈ℝ,𝑁≥3, 𝑉, and 𝑄 are radial positive functions, which can be vanishing or coercive at infinity, and 𝑓 is asymptotically linear at infinity.

1. Introduction and Main Results

In this paper, we deal with the multiplicity of nontrivial radial solutions for the following elliptic equations: −Δ𝑢+𝑉(𝑥)𝑢=𝜆𝑄(𝑥)|𝑢|^𝑞−2𝑢+𝑄(𝑥)𝑓(𝑢)𝑥∈ℝ^𝑁,𝑢(𝑥)⟶0,as|𝑥|⟶+∞, (P)_𝜆 where 1<𝑞<2, 0<𝜆∈ℝ,  𝑁≥3, 𝑉, and 𝑄 are radial positive functions, which can be vanishing or coercive at infinity.

When Ω is a smooth bounded domain in ℝ^𝑁, the problem −Δ𝑢=±𝜆|𝑢|^𝑞−2()𝑢(𝑥)𝑢+𝑓𝑥,𝑢𝑥∈Ω,=0,𝑥∈Ω, P^′_±𝜆 where 1<𝑞<2, 0<𝜆∈ℝ, and 𝑁≥3, has been widely studied in the literature and plays a central role in modern mathematical sciences, in the theory of heat conduction in electrically conduction materials and in the study of non-Newtonian fluids. However, it is not possible to give here a complete bibliography. Here we just list some representative results. In the case where 𝑓 is superlinear near infinity, problem (P^′₊)_𝜆 is the famous concave-convex problem; after the celebrated work [1, 2], this kind of problem has drawn much attention. In the case where 𝑓 is linear in 𝑢, the authors in [3] have proved that there exist at least two nonnegative solutions for a more general question: −Δ𝑢=ℎ(𝑥)𝑢^𝑞+𝑓(𝑥,𝑢),0≤𝑢∈𝐻¹₀(Ω),0<𝑞<1, (1) where ℎ(𝑥)∈𝐿^∞(Ω) satisfies some additional conditions. For problem (P^′₋)_𝜆, in the special case 𝑓(𝑢)=𝑎𝑢+|𝑢|^𝑝, where 2<𝑝<2^∗, one nonnegative solution for any 𝑎∈ℝ and 𝜆>0 was found in [4] via Mountain Pass Theorem. In the last years, several papers have also been devoted to the study of nonlinearities with indefinite sign, for example, [5, 6] and the references therein.

When Ω=ℝ^𝑁, there are a large number of papers devoted to the following equation: −Δ𝑢+𝑉(𝑥)𝑢=𝑓(𝑥,𝑢)with𝑢∈𝑊^1,2ℝ^𝑁. (R) So far, in almost all the results concerning (R), the nonlinear function 𝑓 is assumed to be globally superlinear, that is, lim_|𝑢|→0(𝑓(𝑥,𝑢)/𝑢)=0 and there exists 𝜃>2 such that 0<𝜃𝐹(𝑥,𝑢)≤𝑢𝑓(𝑥,𝑢) for all (𝑥,𝑢)∈ℝ^𝑁×(ℝ⧵{0}), where 𝐹(𝑥,𝑢)=∫^𝑢₀𝑓(𝑥,𝑡)𝑑𝑡. The case in which 𝑉(𝑥)→+∞, |𝑥|→∞, and 𝑓 is globally superlinear was first studied by Rabinowitz in [7]. The assumptions in [7] ensure that the associated functional of the equation satisfies the Palais-Smale condition; this fact was observed in [8, 9] where the results in [7] were generalized. For a radially symmetric Schrödinger equation with an asymptotically linear term, one radial solution has been obtained in [10, 11] by Stuart and Zhou and their results were generalized to more general situations in [12–15].

Since the class Sobolev embedding is 𝑊^1,2(ℝ^𝑁)↪𝐿^𝑝(ℝ^𝑁),𝑝∈(2,2𝑁/(𝑁−2)), we cannot study the sublinear problems in 𝑊^1,2(ℝ^𝑁) via variation method. In order to overcome this obstacle, a regular way is to add some restrictions on potentials 𝑉 and 𝑄. For example, in [16], the authors obtained the existence of infinitely many nodal solutions for problem (R), where 𝑉∈𝐶(ℝ^𝑁,ℝ), 𝑉(𝑥)≥1,∫_ℝ𝑁(1/𝑉(𝑥))𝑑𝑥<+∞, and the nonlinearity 𝑓 is symmetric in the sense of being odd in 𝑢 and may involve a combination of concave and convex terms. There are also some other results about concave and convex problem on ℝ^𝑁, such as [17–19] and the references therein. However, as we have known, there are few results about problem (P)_𝜆 with both sublinear terms and asymptotically linear terms.

Recently, in [20], the authors established a weighted Sobolev type embedding of radially symmetric functions which provides a basic tool to study quasilinear elliptic equations with sublinear nonlinearities. Motivated by the works of [20], we consider (P)_𝜆 with more general potentials and combined nonlinearities. In our paper, we assume the following.(𝑉)𝑉(𝑥)∈𝐶(ℝ^𝑁,(0,+∞)) is radially symmetric and there exists 𝑎₁∈ℝ such that liminf_|𝑥|→∞𝑉(𝑥)|𝑥|^𝑎1>0. (2)(𝑄)𝑄(𝑥)∈𝐶(ℝ^𝑁,(0,+∞)) is radially symmetric and there exists 𝑎₂∈ℝ such that limsup_|𝑥|→∞𝑄(𝑥)|𝑥|^𝑎2<∞. (3)It is clear that the indexes 𝑎₁ and 𝑎₂ describe the behavior of 𝑉 and 𝑄 near infinity. On 𝑎₁,𝑎₂, we assume the following:(𝐴₁)𝑎₂≥((2(𝑁−1)+𝑎₁)/2)−𝑁,  ((𝑁−2)/2)−𝑁≤𝑎₂≤−2;(𝐴₂)𝑎₂<((2(𝑁−1)+𝑎₁)/4)−𝑁,  ((𝑁−2)/2)−𝑁≤𝑎₂≤−2;(𝐴₃)𝑎₁≤−2,  ((𝑁−2)/2)−𝑁<𝑎₂<((2(𝑁−1)+𝑎₁)/2)−𝑁;(𝐴₄)𝑎₂≤((𝑁−2)/2)−𝑁,  ((2(𝑁−1)+𝑎₁)/4)−𝑁≤𝑎₂<((2(𝑁−1)+𝑎₁)/2)−𝑁;(𝐴₅)𝑎₁≥−2,  ((2(𝑁−1)+𝑎₁)/4)−𝑁≤𝑎₂<((2(𝑁−1)+𝑎₁)/2)−𝑁. According to the indexes 𝑎₁,𝑎₂, we define the bottom index 2_∗: 2_∗=2𝑎₂+𝑁,𝑁−2if𝑎₁,𝑎₂∈𝐴_𝑖4𝑎,𝑖=1,2,3;₂+𝑁2(𝑁−1)+𝑎₁,if𝑎₁,𝑎₂∈𝐴_𝑖,𝑖=4,5. (4)

Let 𝐶^∞₀(ℝ^𝑁) denote the collection of smooth functions with compact support and 𝐶^∞_0,𝑟ℝ^𝑁=𝑢∈𝐶^∞₀ℝ^𝑁∣𝑢isradial. (5) Denote by 𝐷_𝑟^1,2(ℝ^𝑁) the completion of 𝐶^∞_0,𝑟(ℝ^𝑁) under the norm ‖𝑢‖_𝐷^1,2=_ℝ𝑁|∇𝑢|²𝑑𝑥^1/2. (6) Define 𝑊_𝑟^1,2ℝ^𝑁;𝑉∶=𝑢∈𝐷_𝑟^1,2ℝ^𝑁∣_ℝ𝑁𝑉(𝑥)𝑢²𝑑𝑥<∞, (7) which is a Hilbert space [21, 22] equipped with the norm ‖𝑢‖=_ℝ𝑁|∇𝑢|²+𝑉(𝑥)𝑢²𝑑𝑥^1/2. (8) Let 𝐿^𝑝ℝ^𝑁;𝑄∶=𝑢∶ℝ^𝑁⟼ℝ∣𝑢beLebesguemeasurabe,_ℝ𝑁𝑄(𝑥)|𝑢|^𝑝𝑑𝑥<∞, (9) which is a Banach space equipped with the norm ‖𝑢‖_𝐿^𝑝_(ℝ^𝑁_;𝑄)=_ℝ𝑁𝑄(𝑥)|𝑢|^𝑝𝑑𝑥^1/𝑝. (10)

Following Theorem  1.2 in [20], under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, it holds that the embedding 𝑊_𝑟^1,2(ℝ^𝑁;𝑉)↪𝐿^𝑝(ℝ^𝑁;𝑄) is compact for 𝑝∈(2_∗,2𝑁/(𝑁−2)). We remark that the index 2_∗<2 by (𝐴_𝑖), 𝑖=1,…,5, so it is possible to study (P)_𝜆 with sublinear nonlinearities. We make the following assumptions on 𝑓:(𝑓₁)𝑓(𝑢)∈𝐶(ℝ,ℝ);(𝑓₂)lim_|𝑢|→∞(2𝐹(𝑢)/|𝑢|²)=𝑏.

Since under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, it holds that the embedding 𝑊_𝑟^1,2(ℝ^𝑁;𝑉)↪𝐿²(ℝ^𝑁;𝑄) is compact, the eigenvalue problem −Δ𝑢+𝑉(𝑥)𝑢=𝜇𝑄(𝑥)𝑢𝑥∈ℝ^𝑁,𝑢(𝑥)⟶0,as|𝑥|⟶+∞ (P)_𝜇 has the eigenvalue sequence 0<𝜇₁<𝜇₂≤𝜇₃≤⋯⟶+∞. (11) Similar to the eigenvalue problem on bounded domain, 𝜇₁>0 is simple and isolated and has an associated eigenfunction 𝜙₁ which is positive in ℝ^𝑁.

Our main results are the following.

Theorem 1. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies (𝑓₁), (𝑓₂) with 𝜇₁<𝑏<+∞, moreover(𝑓₃)𝐹(𝑢)=∫^𝑢₀𝑓(𝑠)𝑑𝑠≥0, 𝑢∈ℝ;(𝑓₄)there exist 𝐶^′∈(0,𝜇₁) and 𝑟₀>0 small, such that |𝑓(𝑢)|≤𝐶^′|𝑢|, |𝑢|≤𝑟₀,then there exists 𝜆₁>0 such that, for any 𝜆∈(0,𝜆₁), (P)_𝜆 has at least four nontrivial solutions.

Theorem 2. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies (𝑓₁), (𝑓₂) with 𝜇_𝑘+1<𝑏<+∞ for some 𝑘∈𝑁, moreover, (𝑓₅)𝑏 is not an eigenvalue, 𝐹(𝑢)≥(𝜇_𝑚/2)𝑢², 𝑢∈ℝ, limsup_𝑢→0(2𝐹(𝑢)/𝑢²)<𝜇_𝑚+1, for some 𝑚∈𝑁,𝑚≤𝑘, then there exists 𝜆₂>0 such that, for 0<𝜆<𝜆₂, (P)_𝜆 has at least one nontrivial solution.

Remark 3. In Theorem 1, 𝑓 may be assumed as superlinear near zero; we can get four nontrivial solutions by Mountain Pass Theorem and Ekeland’s variational principle and truncation technique. In Theorem 2, under the assumptions on 𝑓 near zero, the functional associated to problem (P)_𝜆 enjoys linking structure, and (P)_𝜆 has a linking solution.

Remark 4. In Theorem 1, 𝑏 may be an eigenvalue of problem (P)_𝜇; then problem (P)_𝜆 may be resonant near infinity.

Remark 5. As we have known, there are few results about problem on ℝ^𝑛 with both sublinear and asymptotically linear nonlinearities at the same time.

The paper is organized as follows. In Section 2, we give some preliminary results. The proof of our main results will be given in Section 3.

2. Preliminary

In this section we give some preliminaries that will be used to prove the main results of the paper. We begin with a special case of results on Sobolev type embedding which is due to [20].

Lemma 6 (see [20]). Let (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, be satisfied; the space 𝑊_𝑟^1,2(ℝ^𝑁;𝑉) is compactly embedded in 𝐿^𝑝(ℝ^𝑁;𝑄), for any 𝑝 such that 2_∗<𝑝<2𝑁/(𝑁−2).

For 𝑊_𝑟^1,2(ℝ^𝑁;𝑉), we denote 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄(𝑥)|𝑢|^𝑞𝑑𝑥−_ℝ𝑁𝑄(𝑥)𝐹(𝑢)𝐼𝑑𝑥,_±𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄(𝑥)𝑢^±𝑞−𝑑𝑥_ℝ𝑁𝑄(𝑥)𝐹𝑢^±𝑑𝑥, (12) where 𝑢⁺=max{𝑢,0}, 𝑢⁻=min{𝑢,0}; then, under conditions (𝑓₁) and (𝑓₂), 𝐼_𝜆 and 𝐼^±_𝜆∈𝐶¹(𝑊_𝑟^1,2(ℝ^𝑁;𝑉),ℝ).

Recall that a sequence {𝑢_𝑛} is a (PS)_𝑐 sequence for the functional 𝐼, if 𝐼𝑢_𝑛⟶𝑐,𝐼^′𝑢_𝑛⟶0,𝑛⟶∞. (13) A sequence {𝑢_𝑛} is a (C)_𝑐 sequence for the functional 𝐼, if 𝐼𝑢_𝑛⟶𝑐,1+𝑢_𝑛𝐼^′𝑢_𝑛⟶0,𝑛⟶∞. (14)

Definition 7. Assume 𝑋 is a Banach space, 𝐼∈𝐶¹(𝑋,ℝ); one says that 𝐼 satisfies the (PS)_𝑐 condition, if every (PS)_𝑐 sequence {𝑢_𝑛} has a convergent subsequence. 𝐼 satisfies (PS) condition if 𝐼 satisfies (PS)_𝑐 at any 𝑐∈ℝ.

Definition 8. Assume 𝑋 is a Banach space, 𝐼∈𝐶¹(𝑋,ℝ); one says that 𝐼 satisfies the (C)_𝑐 condition, if every (C)_𝑐 sequence {𝑢_𝑛} has a convergent subsequence. 𝐼 satisfies (C) condition if 𝐼 satisfies (C)_𝑐 at any 𝑐∈ℝ.

Lemma 9 (Ekeland’s variational principle, [23]). Let 𝑉 be a complete metric space and let 𝐼∶𝑉→ℝ∪{+∞} be lower semicontinuous, bounded from below. For any 𝜀>0, there is some point 𝑣∈𝑉 with 𝐼(𝑣)≤inf_𝑉𝐼+𝜀,𝐼(𝑤)≥𝐼(𝑣)−𝜀𝑑(𝑣,𝑤)∀𝑤∈𝑉. (15)

Lemma 10 (Mountain Pass Theorem, Ambrosetti-Rabinowitz, 1973, [24]). Let 𝑋 be a Banach space, 𝐼∈𝐶¹(𝑋,ℝ). Let 𝑒∈𝑋 and 𝑟>0 be such that ‖𝑒‖>𝑟 and 𝑏∶=inf_{‖𝑢‖=𝑟}𝐼(𝑢)>𝐼(0)≥𝐼(𝑒). (16) If 𝐼 satisfies the (𝑃𝑆)_𝑐 condition with 𝑐∶=inf_𝛾∈Γmax_𝑡∈[0,1]𝐼𝛾(𝑡),Γ∶=𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒, (17) then 𝑐 is a critical value of 𝐼.

Lemma 11 (Linking Theorem, Rabinowitz, 1978, [24]). Let 𝑋=𝑌⨁𝑍 be a Banach space with dim𝑌<∞. Let 𝑅>𝑟>0 and 𝑧∈𝑍 be such that ‖𝑧‖=𝑟. Define 𝑀∶={𝑢=𝑦+𝑡𝑧∣‖𝑢‖≤𝑅,𝑡≥0,𝑦∈𝑌},  𝑀₀∶={𝑢=𝑦+𝑡𝑧∣𝑦∈𝑌,‖𝑢‖=𝑅𝑎𝑛𝑑𝑡≥0𝑜𝑟‖𝑢‖≤𝑅𝑎𝑛𝑑𝑡=0},  𝑁∶={𝑢∈𝑍∣‖𝑢‖=𝑟}. Let 𝐼∈𝐶¹(𝑋,ℝ) be such that 𝑑∶=inf_𝑁𝐼>𝑎∶=max_𝑀0𝐼. (18) If 𝐼 satisfies the (𝑃𝑆)_𝑐 condition with 𝑐=∶inf_𝛾∈Γmax_𝑢∈𝑀𝐼𝛾(𝑢),Γ∶=𝛾∈𝐶(𝑀,𝑋)∶𝛾_𝑀0=𝑖𝑑, (19) then 𝑐 is a critical value of 𝐼.

It is well known that the above two minimax theorems are still valid under (C)_𝑐 condition. In our paper, we denote 𝑋∶=𝑊_𝑟^1,2(ℝ^𝑁;𝑉); 𝐶 is denoted to be various positive constants.

Lemma 12. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies (𝑓₁), (𝑓₄), and (𝑓₂) with 𝜇₁<𝑏<+∞, then there exists 𝜆^∗>0 such that, for 0<𝜆<𝜆^∗, one has the following.(i)There exist 𝜌^±_𝜆,𝛽^±_𝜆>0, such that 𝐼^±_𝜆(𝑢)≥𝛽^±_𝜆>0∀𝑢∈𝑋𝑤𝑖𝑡ℎ‖𝑢‖=𝜌^±_𝜆. (20)(ii)There exists 𝑒^±_𝜆∈𝑋 with ‖𝑒^±_𝜆‖>𝜌^±_𝜆 such that 𝐼^±_𝜆(𝑒^±_𝜆)<0.

Proof . We only prove the above results for 𝐼⁺_𝜆.
(i) By (𝑓₁), (𝑓₂), and (𝑓₄), there exists 𝐶>0 and 𝑝∈(2,2𝑁/(𝑁−2)), such that |𝐹(𝑢⁺)|≤(𝐶^′/2)|𝑢⁺|²+(𝐶/2)|𝑢⁺|^𝑝. Then 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄(𝑥)𝑢^+𝑞𝑑𝑥−_ℝ𝑁𝑄(𝑥)𝐹𝑢⁺≥1𝑑𝑥2‖𝑢‖²−𝐶^′2_ℝ^𝑁𝑄(𝑥)|𝑢|²𝐶𝑑𝑥−2_ℝ^𝑁𝑄(𝑥)|𝑢|^𝑝−𝜆𝑑𝑥𝑞_ℝ^𝑁𝑄(𝑥)|𝑢|^𝑞≥𝜇𝑑𝑥₁−𝐶^′2𝜇₁−𝐶₁2‖𝑢‖^𝑝−2−𝐶₂𝜆𝑞‖𝑢‖^𝑞−2‖𝑢‖². (21) Set 𝑔(𝑡)=𝐶₁2𝑡^𝑝−2+𝐶₂𝜆𝑞𝑡^𝑞−2for𝑡>0, (22) where 𝑞∈(2_∗,2) and 𝑝∈(2,2𝑁/(𝑁−2)). By 𝑔^′(𝑡₀)=0, we have 𝑡₀=2𝐶₁2−𝑞𝑞𝐶₂𝜆𝑝−2^{1/(𝑝−𝑞)}. (23) Then there exists 𝐶₀>0 such that 𝑔(𝑡₀)=𝐶₀𝜆^{(𝑝−2)/(𝑝−𝑞)}. Thus, there exists 𝜆_∗>0 such that, for 𝜆∈(0,𝜆_∗), (𝜇₁−𝐶^′)/2𝜇₁>𝐶₀𝜆^{(𝑝−2)/(𝑝−𝑞)}. Furthermore, set 𝜌⁺_𝜆=𝑡₀; we have 𝐼⁺_𝜆(𝑢)≥𝜇₁−𝐶^′2𝜇₁−𝐶₀𝜆^{(𝑝−2)/(𝑝−𝑞)}2𝐶₁2−𝑞𝑞𝐶₂𝜆𝑝−2^{2/(𝑝−𝑞)}=𝛽⁺_𝜆>0∀𝑢∈𝑋with‖𝑢‖=𝜌⁺_𝜆. (24)
(ii) Let 𝜙₁>0 be a 𝜇₁-eigenfunction; for 𝑡>0 we have 𝐼⁺_𝜆𝑡𝜙₁=𝑡²2𝜙₁²−𝜆𝑡^𝑞𝑞_ℝ^𝑁𝑄(𝑥)𝜙₁^𝑞−𝑑𝑥_ℝ𝑁𝑄(𝑥)𝐹𝑡𝜙₁=𝑡𝑑𝑥₂𝜙₁²2×1−2𝜆𝑡^𝑞−2𝑞𝜙₁^2ℝ𝑁𝑄(𝑥)𝜙₁^𝑞−2𝑑𝑥𝑡²𝜙₁^2ℝ𝑁𝑄(𝑥)𝐹𝑡𝜙₁𝑑𝑥. (25) By (𝑓₂), 𝜇₁<𝑏<+∞ and 𝑞<2; then there exists 𝑇_0,𝜆>0 large enough such that 1−2𝜆𝑇^𝑞−2_0,𝜆𝑞𝜙₁^2ℝ𝑁𝑄(𝑥)𝜙₁^𝑞−2𝑑𝑥𝑇²_0,𝜆𝜙₁^2ℝ𝑁𝑄(𝑥)𝐹𝑇_0,𝜆𝜙₁𝑑𝑥<0. (26) So, we can choose 𝑒_𝜆=𝑇_0,𝜆𝜙₁; then (ii) is proved.

Lemma 13. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies (𝑓₁)–(𝑓₃) with 𝜇_𝑘+1<𝑏<+∞ and (𝑓₅), then(i)for any given 𝜆 and 𝑢∈𝑋_𝑚∶=⨁^𝑚_𝑗=1ker(−Δ+𝑉−𝜇_𝑗𝑄), we have 𝐼_𝜆(𝑢)≤0; (27)(ii)there exists 𝜆^∗∗ satisfying the fact that for 𝜆∈(0,𝜆^∗∗) there exist two positive constants 𝑑(𝜆) and 𝑟(𝜆) such that for all 𝑢∈𝑁∶={𝑢∈𝑋^⟂_𝑚,‖𝑢‖=𝑟(𝜆)}, one has 𝐼_𝜆(𝑢)≥𝑑(𝜆)>0; (28)(iii)there exists 𝑅>0 such that, for any given 𝜆 and 𝑢∈𝑋_𝑚+1, and ‖𝑢‖≥𝑅, we have 𝐼_𝜆(𝑢)≤0.

Proof. (i) Let 𝑢∈𝑋_𝑚; by (𝑓₅), 𝐹(𝑢)≥(1/2)𝜇_𝑚𝑢², 𝑢∈ℝ, then 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄|𝑢|^𝑞𝑑𝑥−_ℝ𝑁(𝑢)≤1𝑄𝐹𝑑𝑥2‖𝑢‖²−𝜇_𝑚2_ℝ^𝑁𝑄|𝑢|²𝑑𝑥≤0. (29)
(ii) Let 𝑢∈𝑋^⟂_𝑚; by (𝑓₁) and (𝑓₂) with 𝜇_𝑘+1<𝑏<+∞, and limsup_𝑢→0(𝐹(𝑢)/𝑢²)<(1/2)𝜇_𝑚+1, we have that there exist 𝜀₀>0, 𝐶>0, 𝑝>2, such that 𝐹(𝑢)≤(1/2)(𝜇_𝑚+1−𝜀₀)𝑢²+𝐶|𝑢|^𝑝. Then 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄|𝑢|^𝑞𝑑𝑥−_ℝ𝑁(𝑢)≥1𝑄𝐹𝑑𝑥2‖𝑢‖²−12𝜇_𝑚+1−𝜀_0ℝ𝑁𝑄|𝑢|²𝑑𝑥−𝐶_ℝ𝑁𝑄|𝑢|^𝑞𝜆𝑑𝑥−𝑞_ℝ^𝑁𝑄|𝑢|^𝑞≥1𝑑𝑥2𝜇1−^𝑚+1−𝜀₀𝜇_𝑚+1‖𝑢‖²−𝐶‖𝑢‖^𝑝−𝐶𝜆‖𝑢‖^𝑞. (30) The rest of the proof is similar to the proof of (i) of Lemma 12.
(iii) For any 𝑢∈𝑋_𝑚+1, set 𝑓(𝑢)=𝑏𝑢+𝑔(𝑢); by (𝑓₂), we have 𝐺(𝑢)/𝑢²→0, as |𝑢|→∞, where 𝐺(𝑢)=∫^𝑢₀𝑔(𝑠)𝑑𝑠. Then 𝐼_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄|𝑢|^𝑞𝑏𝑑𝑥−2_ℝ^𝑁𝑄|𝑢|²−𝑑𝑥_ℝ𝑁𝑄𝐺(𝑢)𝑑𝑥. (31) Since 𝑏>𝜇_𝑚+1, for every 𝑧∈span{𝜙_𝑚+1}, 𝑡∈ℝ, 𝑤∈𝑋_𝑚, 𝑡²‖𝑧‖²+‖𝑤‖²−𝑏_ℝ𝑁𝑄(𝑡𝑧+𝑤)²𝑑𝑥<0,for𝑡𝑧+𝑤≠0. (32) Arguing by contradiction, we find a sequence {𝑢_𝑛}, satisfying ‖𝑢_𝑛‖→∞, 𝑢_𝑛=𝑡_𝑛𝑧₀+𝑤_𝑛, where 𝑧₀∈span{𝜙_𝑚+1}, 𝑡_𝑛∈ℝ, 𝑤_𝑛∈𝑋_𝑚, such that 𝐼_𝜆𝑢_𝑛=12𝑡²_𝑛𝑧₀²+12𝑤_𝑛²−𝜆𝑞_ℝ^𝑁𝑄𝑢_𝑛^𝑞−𝑑𝑥_ℝ𝑁𝑄𝐹𝑢_𝑛𝑑𝑥≥0. (33) Dividing ‖𝑢_𝑛‖² in both sides of the above equality, there holds 𝐼_𝜆𝑢_𝑛𝑢_𝑛²=12𝜏²_𝑛𝑧₀²+12𝑣_𝑛²−𝜆𝑞𝑢_𝑛^2ℝ𝑁𝑄𝑢_𝑛^𝑞−𝑑𝑥_ℝ𝑁𝑄𝐹𝑢_𝑛𝑢_𝑛²𝑑𝑥≥0, (34) where 𝜏_𝑛∶=𝑡_𝑛/‖𝑢_𝑛‖, 𝑣_𝑛∶=𝑤_𝑛/‖𝑢_𝑛‖. Since 𝜏²_𝑛‖𝑧₀‖²+‖𝑣_𝑛‖²=1, after passing to a subsequence 𝜏_𝑛→𝜏, in ℝ, 𝑣_𝑛→𝑣 in 𝑋_𝑚. Let 𝑢^′=𝜏𝑧₀+𝑣; by (32), there exists a bounded domain Ω⊂ℝ^𝑁, such that 𝜏²𝑧₀²+‖𝑣‖²−𝑏_Ω𝑄𝜏𝑧₀+𝑣²𝑑𝑥<0. (35) As 𝐹(𝑢)=(1/2)𝑏𝑢²+𝐺(𝑢), it follows from (34) that 10≤2𝜏²_𝑛𝑧₀²+12𝑣_𝑛²−_Ω𝑄𝐹𝑢_𝑛𝑢_𝑛²−𝜆𝑑𝑥𝑞𝑢_𝑛^2ℝ𝑁𝑄𝑢_𝑛^𝑞=1𝑑𝑥2𝜏²_𝑛𝑧₀²+12𝑣_𝑛²−12𝑏_Ω𝑄𝜏_𝑛𝑧₀+𝑣_𝑛²−𝑑𝑥_Ω𝑄𝐺𝑢_𝑛𝑢_𝑛²𝜆𝑑𝑥−𝑞𝑢_𝑛^2ℝ𝑁𝑄𝑢_𝑛^𝑞≤1𝑑𝑥2𝜏²_𝑛𝑧₀²+12𝑣_𝑛²−12𝑏_Ω𝑄𝜏_𝑛𝑧₀+𝑣_𝑛²−𝑑𝑥_Ω𝑄𝐺𝑢_𝑛𝑢_𝑛²𝑑𝑥. (36) Clearly, |𝐺(𝑢)|≤𝑐₀𝑢², for some 𝑐₀>0 and 𝐺(𝑢)/𝑢²→0, as |𝑢|→∞. Since 𝜏_𝑛→𝜏, in ℝ, 𝑣_𝑛→𝑣 in 𝑋_𝑚, then 𝜏_𝑛𝑧₀+𝑣_𝑛→𝑢^′=𝜏𝑧₀+𝑣, in 𝐿²(ℝ^𝑁;𝑄). It is easy to see from the Lebesgue dominated converge theorem that _Ω𝑄𝐺𝑢_𝑛𝑢_𝑛²𝑑𝑥=_Ω𝑄𝐺𝑢_𝑛𝑢²_𝑛𝜏²_𝑛+𝑣²_𝑛𝑑𝑥⟶0. (37) Hence 0≤(1/2)𝜏²‖𝑧₀‖²+(1/2)‖𝑣‖²−(1/2)𝑏∫_Ω𝑄|𝑡𝑧₀+𝑣|²𝑑𝑥<0; this is impossible.

3. Proof of Main Results

Proof of Theorem 1. Firstly, we will prove that, for any fixed 𝜆, the functionals 𝐼^±_𝜆 have a local minimizer, respectively; then problem (P)_𝜆 has two nontrivial solutions: one is nonnegative; the other one is nonpositive.

Similar to [25], for 𝜌⁺_𝜆>0 given by Lemma 12(i), define 𝐵_𝜌⁺_𝜆=𝑢∈𝑋∣‖𝑢‖≤𝜌⁺_𝜆,𝜕𝐵_𝜌⁺_𝜆=𝑢∈𝑋∣‖𝑢‖=𝜌⁺_𝜆 (38) and 𝐵_𝜌⁺_𝜆 is a complete metric space with the distance dist(𝑢,𝑣)=‖𝑢−𝑣‖for𝑢,𝑣∈𝐵_𝜌⁺_𝜆. (39) By Lemma 12, we have that 𝐼⁺_𝜆(𝑢)≥𝛽⁺_𝜆>0,𝑢∈𝜕𝐵_𝜌⁺_𝜆. (40) Clearly, 𝐼⁺_𝜆∈𝐶¹(𝐵_𝜌⁺_𝜆,ℝ); hence 𝐼⁺_𝜆 is lower semicontinuous and bounded from below on 𝐵_𝜌⁺_𝜆. Let 𝑐¹_𝜆=inf_{𝑢∈𝐵𝜌+𝜆}𝐼⁺_𝜆(𝑢). (41) By the definition of 𝐼⁺_𝜆, we can easily claim that 𝑐¹_𝜆<0. Indeed, since 𝑞<2 if 𝑡>0 is small enough, 𝐼⁺_𝜆𝑡𝜙₁=𝑡²2𝜙₁²−𝜆𝑡^𝑞𝑞_ℝ^𝑁𝑄(𝑥)𝜙₁^𝑞−𝑑𝑥_ℝ𝑁𝑄(𝑥)𝐹𝑡𝜙₁𝑑𝑥≤𝑡²𝜙₁²−𝜆𝑡^𝑞𝑞_ℝ^𝑁𝑄(𝑥)𝜙₁^𝑞𝑑𝑥by𝑓₃<0. (42) By Lemma 9, for any 𝑛>0, there exists a 𝑢_𝑛 such that 𝑐¹_𝜆≤𝐼⁺_𝜆𝑢_𝑛≤𝑐¹_𝜆+1𝑛,𝐼⁺_𝜆(𝑤)≥𝐼⁺_𝜆𝑢_𝑛−1𝑛𝑢_𝑛−𝑤,for𝑤∈𝐵_𝜌⁺_𝜆. (43) Then, ‖𝑢_𝑛‖<𝜌⁺_𝜆 for 𝑛≥1 large enough. Otherwise, if ‖𝑢_𝑛‖=𝜌⁺_𝜆 for infinitely many 𝑛, without loss of generality, we may assume that ‖𝑢_𝑛‖=𝜌⁺_𝜆 for all 𝑛∈ℕ, and it follows from (40) that 𝐼⁺_𝜆𝑢_𝑛≥𝛽⁺_𝜆>0. (44) Let 𝑛→∞ and combine (43); we can get that 0<𝛽⁺_𝜆≤𝑐¹_𝜆<0. This is a contradiction.

We prove now that 𝐼_𝜆^+′(𝑢_𝑛)→0, as 𝑛→∞. In fact, for any 𝑢∈𝑋 with ‖𝑢‖=1, let 𝑤_𝑛=𝑢_𝑛+𝑡𝑢 and, for any fixed 𝑛≥1, we have ‖𝑤_𝑛‖≤‖𝑢_𝑛‖+𝑡<𝜌⁺_𝜆 if 𝑡>0 is small enough. So it follows from (43) that 𝐼⁺_𝜆𝑢_𝑛+𝑡𝑢≥𝐼⁺_𝜆𝑢_𝑛−𝑡𝑛‖𝑢‖. (45) That is, 𝐼⁺_𝜆𝑢_𝑛+𝑡𝑢−𝐼⁺_𝜆𝑢_𝑛𝑡1≥−𝑛‖𝑢‖1=−𝑛. (46) Let 𝑡→0; we see that ⟨𝐼_𝜆^+′(𝑢_𝑛),𝑢⟩≥−1/𝑛, and this gives 𝐼_𝜆^+′𝑢_𝑛<1,𝑢𝑛forany𝑢∈𝑋with‖𝑢‖=1. (47) So, 𝐼_𝜆^+′(𝑢_𝑛)→0, as 𝑛→∞, and, by (43), 𝐼⁺_𝜆(𝑢_𝑛)→𝑐¹_𝜆<0, as 𝑛→∞. Then, for any given 𝜆, {𝑢_𝑛} is a bounded (PS)_𝑐¹_𝜆 sequence of 𝐼⁺_𝜆. By the compactness of Sobolev embedding Lemma 6 and a standard procedure, we see that there exists 𝑢¹_𝜆∈𝑋 such that 𝐼_𝜆^+′(𝑢¹_𝜆)=0. Since 𝐼_𝜆^+′𝑢¹_𝜆𝑢_𝜆¹⁻=0, (48) this implies that 𝑢¹_𝜆≥0. That is, 𝑢¹_𝜆 is a nontrivial solution of problem (P)_𝜆. For the case 𝐼⁻_𝜆, by the same argument, we can get that problem (P)_𝜆 has another nontrivial solution, which is nonpositive.

Secondly, we will prove that there exists 𝜆₁>0 such that, for 𝜆∈(0,𝜆₁), problem (P)_𝜆 enjoys two mountain pass solutions.

Define 𝑐_𝜆^2±=inf_𝛾∈Γmax_0≤𝑡≤1𝐼^±_𝜆𝛾(𝑡), (49) where Γ∶={𝛾∈𝐶([0,1],𝑋)∶𝛾(0)=0,𝛾(1)=𝑒^±_𝜆}.

By Lemmas 10 and 12, we only need to prove that 𝐼^±_𝜆 satisfies (𝐶)_𝑐𝜆^2± condition.

Lemma 14. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies (𝑓₁), (𝑓₂) with 𝜇₁<𝑏, (𝑓₃), and (𝑓₄) then, for any fixed 𝜆>0, the functional 𝐼^±_𝜆 satisfies the (𝐶)_𝑐𝜆^2± condition.

Proof. Here, we only prove the case for 𝐼⁺_𝜆.
For every (C)_𝑐𝜆²⁺ sequence {𝑢_𝑛}, 𝐼⁺_𝜆𝑢_𝑛⟶𝑐_𝜆²⁺,as𝑛⟶+∞,1+𝑢_𝑛𝐼_𝜆^+′𝑢_𝑛⟶0,as𝑛⟶∞. (50)(51) We claim that the sequence {𝑢_𝑛} is bounded in 𝑋. Seeking a contradiction, we suppose that ‖𝑢_𝑛‖→∞. Let 𝑧_𝑛=𝑢_𝑛/‖𝑢_𝑛‖; up to a subsequence, we get that 𝑧_𝑛⇀𝑧 in 𝑋, 𝑧_𝑛→𝑧 in 𝐿^𝑠(ℝ^𝑁;𝑄), 2_∗<𝑠<2𝑁/(𝑁−2), 𝑧_𝑛(𝑥)→𝑧(𝑥) a.e. 𝑥∈ℝ^𝑁.We claim that 𝑧≠0. Otherwise, 𝑧=0, since by (51) 𝑜(1)=𝐼_𝜆^+′𝑢_𝑛,𝑢_𝑛=𝑢_𝑛²−𝜆_ℝ𝑁𝑄𝑢⁺_𝑛(𝑥)^𝑞𝑑𝑥−_ℝ𝑁𝑄𝑓𝑢⁺_𝑛(𝑥)𝑢⁺_𝑛(𝑥)𝑑𝑥. (52) Dividing ‖𝑢_𝑛‖² in both sides of (52), we get that 𝑜(1)=1−_ℝ𝑁𝑄𝑓𝑢⁺_𝑛𝑢⁺_𝑛𝑢_𝑛²𝑑𝑥. (53)(𝑓₁), (𝑓₂), and (𝑓₄) with 𝜇₁<𝑏<+∞ imply that there exists 𝐶>0, such that 𝑓𝑢⁺_𝑛≤𝐶𝑢⁺_𝑛. (54) Combining (53) and (54), we have 1=_ℝ𝑁𝑄𝑓𝑢⁺_𝑛𝑢⁺_𝑛𝑢_𝑛²𝑑𝑥+𝑜(1)≤𝐶_ℝ𝑁𝑄(𝑥)𝑧⁺_𝑛²𝑑𝑥+𝑜(1). (55) Letting 𝑛→∞, we get a contradiction. Thus, 𝑧≠0 in 𝑋.
Set 𝑃_𝑛(𝑥)=𝑓𝑢_𝑛(𝑥)𝑢_𝑛(𝑥),for𝑥∈ℝ^𝑁,𝑢_𝑛(𝑥)>0,0,for𝑥∈ℝ^𝑁,𝑢_𝑛(𝑥)≤0. (56) From 𝐼_𝜆^+′(𝑢_𝑛)=𝑜(1), _ℝ𝑁∇𝑢_𝑛∇𝜙𝑑𝑥+_ℝ𝑁𝑉(𝑥)𝑢_𝑛𝜙𝑑𝑥−𝜆_ℝ𝑁𝑄(𝑥)𝑢⁺_𝑛^𝑞−1−𝜙𝑑𝑥_ℝ𝑁𝑄(𝑥)𝑓𝑢⁺_𝑛𝜙𝑑𝑥=𝑜(1), (57) for all 𝜙∈𝐶^∞_0,𝑟(ℝ^𝑁). Dividing ‖𝑢_𝑛‖ in both sides of the above equality, there holds _ℝ𝑁∇𝑧_𝑛∇𝜙𝑑𝑥+_ℝ𝑁𝑉(𝑥)𝑧_𝑛𝜙𝑑𝑥−_ℝ𝑁𝑄(𝑥)𝑃_𝑛(𝑥)𝑧⁺_𝑛(1).𝜙𝑑𝑥=𝑜 (58) By (54), |𝑃_𝑛(𝑥)|≤𝐶 for 𝑥∈ℝ^𝑁. Then we have _{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)=0}supp𝜙}𝑄𝑃_𝑛𝑧⁺_𝑛𝜙𝑑𝑥≤𝐶_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)=0}supp𝜙}𝑄𝑧⁺_𝑛𝜙(1)𝑑𝑥=𝑜+𝐶_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)=0}supp𝜙}𝑄𝑧⁺𝜙𝑑𝑥=𝑜(1). (59) On the other hand, since 𝑧⁺_𝑛(𝑥)→𝑧⁺(𝑥) for a.e.𝑥∈ℝ^𝑁, we have lim_𝑛→∞𝑢⁺_𝑛(𝑥)=+∞ for a.e. 𝑥∈{𝑥∈ℝ^𝑁∣𝑧⁺(𝑥)>0}, which implies that lim_𝑛→∞𝑃_𝑛(𝑥)=𝑏, for a.e.𝑥∈{𝑥∈ℝ^𝑁∣𝑧⁺(𝑥)>0}. Besides |𝑃_𝑛(𝑥)|≤𝐶, for a.e. 𝑥∈ℝ^𝑁. Using the Lebesgue’s Dominated Convergence theorem, we obtain that _{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)−𝑏𝑧⁺_𝑛≤𝜙𝑑𝑥_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)−𝑏𝑧⁺_𝑛𝜙≤𝑑𝑥_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)−𝑏²𝜙𝑑𝑥^1/2×_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑧⁺_𝑛²𝜙𝑑𝑥^1/2≤𝐶_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)−𝑏²𝜙𝑑𝑥^1/2=𝑜(1). (60) By (59) and (60), _ℝ𝑁𝑄𝑃_𝑛(𝑥)𝑧⁺_𝑛=𝜙𝑑𝑥_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)=0}supp𝜙}𝑄𝑃_𝑛(𝑥)𝑧⁺_𝑛𝜙𝑑𝑥+_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)𝑧⁺_𝑛(1)+𝜙𝑑𝑥=𝑜_{{𝑥∈ℝ𝑁∣𝑧+⋂(𝑥)>0}supp𝜙}𝑄𝑃_𝑛(𝑥)𝑧⁺_𝑛(1)𝜙𝑑𝑥=𝑜+𝑏_ℝ𝑁𝑄𝑧⁺𝜙𝑑𝑥. (61) Combining (58) and (61), letting 𝑛→∞, there holds _ℝ𝑁∇𝑧∇𝜙+𝑉𝑧𝜙𝑑𝑥=𝑏_ℝ𝑁𝑄𝑧⁺𝜙𝑑𝑥. (62) We claim that meas{𝑥∈ℝ^𝑁,𝑧⁺(𝑥)≠0}>0. Otherwise 𝑧⁺=0; taking 𝜙=𝑧 in (62), we have 𝑧=0, which is impossible. Taking 𝜙=𝑧⁻ in (62), then we can get 𝑧≥0. Moreover, by the Hopf’ Lemma, we also can get 𝑧>0 in ℝ^𝑁. Taking 𝜙=𝜙₁ in (62), we obtain _ℝ𝑁∇𝑧∇𝜙₁+𝑉𝑧𝜙₁𝑑𝑥=𝑏_ℝ𝑁𝑄𝑧⁺𝜙₁𝑑𝑥. (63) Since 𝜙₁>0 is the eigenfunction associated to 𝜇₁, and 𝑧≥0, we have _ℝ𝑁∇𝑧∇𝜙₁+𝑉𝑧𝜙₁𝑑𝑥=𝜇_1ℝ𝑁𝑄𝑧𝜙₁𝑑𝑥. (64) This is impossible, since 𝑏>𝜇₁. Then {𝑢_𝑛} is bounded in 𝑋. Since the embedding from 𝑋 into 𝐿^𝑠(ℝ^𝑁;𝑄),𝑠∈(2_∗,2𝑁/(𝑁−2)) is compact, there exists 𝑢²_𝜆, such that 𝑢_𝑛→𝑢²_𝜆 strongly in 𝑋, and 𝐼⁺_𝜆(𝑢²_𝜆)=𝑐_𝜆²⁺≥𝛽⁺₁>0, 𝐼_𝜆^+′(𝑢²_𝜆)=0.
Finally, since 𝐼_𝜆^+′(𝑢²_𝜆)𝑢_𝜆²⁻=0, then _ℝ𝑁∇𝑢²_𝜆∇𝑢_𝜆²⁻+𝑉(𝑥)𝑢²_𝜆𝑢_𝜆²⁻𝑑𝑥=𝜆_ℝ𝑁𝑄(𝑥)𝑢_𝜆^2+𝑞−1𝑢_𝜆²⁻+_ℝ𝑁𝑄(𝑥)𝑓𝑢_𝜆²⁺𝑢_𝜆²⁻𝑑𝑥=0. (65) We have 𝑢_𝜆²⁻=0, i.e., 𝑢²_𝜆≥0. Thus, 𝑢 is a nonnegative solution for problem (P)_𝜆. Similarly, for 𝐼⁻_𝜆(𝑢)=12‖𝑢‖²−𝜆𝑞_ℝ^𝑁𝑄|𝑢⁻|^𝑞𝑑𝑥−_ℝ𝑁𝑄𝐹(𝑢⁻)𝑑𝑥, (66) we can also get a nonpositive solution for problem (P)_𝜆.
Thus, problem (P)_𝜆 has at least four nontrivial solutions. The proof of Theorem 1 is complete.

Proof of Theorem 2. In order to prove Theorem 2, we firstly verify that the functional 𝐼_𝜆 enjoys the linking structure. This can be easily got form Lemma 13. In fact, for 𝑌=𝑋_𝑚, 𝑍=𝑋^⟂_𝑚, 𝑧∈span{𝜙_𝑚+1} with ‖𝑧‖=𝑟(𝜆), 𝑀∶={𝑢=𝑦+𝑡𝑧∣‖𝑢‖≤𝑅,𝑡≥0,𝑦∈𝑌},  𝑀₀∶={𝑢=𝑦+𝑡𝑧∣𝑦∈𝑌,‖𝑢‖=𝑅 and 𝑡≥0 or ‖𝑢‖≤𝑅 and 𝑡=0}, 𝑁∶={𝑢∈𝑍∣‖𝑢‖=𝑟(𝜆)}. Lemma 13 implies that there exists 𝜆^∗∗>0, such that, for 0<𝜆<𝜆^∗∗, inf_𝑢∈𝑁𝐼_𝜆(𝑢)>sup_𝑢∈𝑀0𝐼_𝜆(𝑢). (67) Define 𝑐_𝜆=∶inf_𝛾∈Γmax_𝑢∈𝑀𝐼_𝜆𝛾(𝑢),Γ∶=𝛾∈𝐶(𝑀,𝑋)∶𝛾_𝑀0=𝑖𝑑. (68) Next, we prove that the functional 𝐼_𝜆 satisfies the (C)_𝑐𝜆 condition.

Lemma 15. Under the assumptions (𝑉), (𝑄), and (𝐴_𝑖), 𝑖=1,…,5, if 𝑓 satisfies the assumptions of Theorem 2, then, for any given 𝜆>0, the functional 𝐼_𝜆 satisfies the (C)_𝑐𝜆 condition.

Proof. For every (C)_𝑐𝜆 sequence {𝑢_𝑛}, 𝐼_𝜆𝑢_𝑛⟶𝑐_𝜆,as𝑢𝑛⟶+∞,1+_𝑛𝐼^′_𝜆𝑢_𝑛⟶0,as𝑛⟶+∞. (69) Here we just prove that {𝑢_𝑛} is bounded. Seeking a contradiction we suppose that ‖𝑢_𝑛‖→∞. Letting 𝑤_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, up to a subsequence, we get that 𝑤_𝑛⇀𝑤 in 𝑋, 𝑤_𝑛→𝑤 in 𝐿^𝑠(ℝ^𝑁;𝑄),  2_∗<𝑠<2𝑁/(𝑁−2), 𝑤_𝑛(𝑥)→𝑤(𝑥) a.e. 𝑥∈ℝ^𝑁.Now, we consider the two possible cases.
Case  1 (𝑤=0 in 𝑋). From 𝑜(1)=⟨𝐼^′_𝜆(𝑢_𝑛),𝑢_𝑛⟩, we have 𝑜(1)=𝑢_𝑛²−𝜆_ℝ𝑁𝑄𝑢_𝑛^𝑞𝑑𝑥−_ℝ𝑁𝑄𝑓𝑢_𝑛𝑢_𝑛𝑑𝑥. (70) Dividing ‖𝑢_𝑛‖² in both sides of the above equality, we get that 𝑜(1)=1−_ℝ𝑁𝑄𝑓𝑢_𝑛𝑢_𝑛𝑢_𝑛²𝑑𝑥. (71) Since (𝑓₁), (𝑓₅), and (𝑓₂) with 𝜇_𝑘+1<𝑏<+∞ imply that 𝑓𝑢_𝑛𝑢_𝑛≤𝐶𝑢_𝑛²,forsome𝐶>0, (72) combing (71) and (72), we have 1=_ℝ𝑁𝑄𝑓𝑢_𝑛𝑢_𝑛𝑢_𝑛²𝑑𝑥+𝑜(1)≤𝐶_ℝ𝑁𝑄𝑤_𝑛²𝑑𝑥+𝑜(1). (73) Let 𝑛→∞; we get a contradiction.
Case  2 (𝑤≠0 in 𝑋). From 𝐼_𝜆^+′(𝑢_𝑛)=𝑜(1), _ℝ𝑁∇𝑢_𝑛∇𝜙𝑑𝑥+_ℝ𝑁𝑉(𝑥)𝑢_𝑛𝜙𝑑𝑥−𝜆_ℝ𝑁𝑄(𝑥)𝑢_𝑛^𝑞−2𝑢_𝑛𝜙𝑑𝑥−_ℝ𝑁𝑄(𝑥)𝑓𝑢_𝑛(1),𝜙𝑑𝑥=𝑜 (74) for all 𝜙∈𝐶^∞_0,𝑟(ℝ^𝑁). Dividing ‖𝑢_𝑛‖ in both sides of the above equality, there holds _ℝ𝑁∇𝑤_𝑛∇𝜙𝑑𝑥+_ℝ𝑁𝑉(𝑥)𝑤_𝑛−𝜙𝑑𝑥_ℝ𝑁𝑄(𝑥)𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛𝜙𝑑𝑥=𝑜(1). (75) By (72), |𝑓(𝑢_𝑛)/𝑢_𝑛|≤𝐶 for 𝑥∈ℝ^𝑁. Then we have _{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)=0}supp𝜙}𝑄𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛𝜙𝑑𝑥≤𝐶_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)=0}supp𝜙}𝑄𝑤_𝑛𝜙(1)𝑑𝑥=𝑜+𝐶_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)=0}supp𝜙}𝑄𝑤𝜙𝑑𝑥=𝑜(1). (76) On the other hand, since 𝑤_𝑛(𝑥)→𝑤(𝑥) for a.e. 𝑥∈ℝ^𝑁, we have lim_𝑛→∞|𝑤_𝑛(𝑥)|=+∞ for a.e. 𝑥∈{𝑥∈ℝ^𝑁|𝑤(𝑥)≠0}, which implies that lim_𝑛→∞(𝑓(𝑢_𝑛)/𝑢_𝑛)=𝑏, for a.e. 𝑥∈{𝑥∈ℝ^𝑁|𝑤(𝑥)≠0}. Besides |𝑓(𝑢_𝑛)/𝑢_𝑛|≤𝐶, for a.e. 𝑥∈ℝ^𝑁. Using the Lebesgue’s Dominated Convergence theorem, we obtain that _{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)≠0}supp𝜙}𝑄𝑓𝑢_𝑛𝑢_𝑛−𝑏𝑤_𝑛≤𝜙𝑑𝑥_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)≠0}supp𝜙}𝑄𝑓𝑢_𝑛𝑢_𝑛−𝑏𝑤_𝑛𝜙≤𝑑𝑥_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)≠0}supp𝜙}𝑄𝑓𝑢_𝑛𝑢_𝑛−𝑏²𝜙𝑑𝑥^1/2×_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)≠0}supp𝜙}𝑄𝑤_𝑛²𝜙𝑑𝑥^1/2≤𝐶_{{𝑥∈ℝ𝑁⋂∣𝑤(𝑥)≠0}supp𝜙}𝑄𝑓𝑢_𝑛𝑢_𝑛−𝑏²𝜙𝑑𝑥^1/2=𝑜(1). (77)
By (76) and (77), _ℝ𝑁𝑄𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛=𝜙𝑑𝑥_{{𝑥∈ℝ𝑁∣𝑤(𝑥)=0}}𝑄𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛+𝜙𝑑𝑥_{{𝑥∈ℝ𝑁∣𝑤(𝑥)≠0}}𝑄𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛(1)+𝜙𝑑𝑥=𝑜_{{𝑥∈ℝ𝑁∣𝑤(𝑥)≠0}}𝑄𝑓𝑢_𝑛𝑢_𝑛𝑤_𝑛(1)𝜙𝑑𝑥=𝑜+𝑏_ℝ𝑁𝑄𝑤𝜙𝑑𝑥. (78) Combining (75) and (78) and letting 𝑛→∞, there holds _ℝ𝑁∇𝑤∇𝜙+𝑉𝑤𝜙𝑑𝑥=𝑏_ℝ𝑁𝑄𝑤𝜙𝑑𝑥. (79) It implies that 𝑏 is an eigenvalue which contradicts (𝑓₅). Thus, {𝑢_𝑛} is bounded in 𝑋. Since the embedding from 𝑋 into 𝐿^𝑠(ℝ^𝑁;𝑄),𝑠∈(2_∗,2𝑁/(𝑁−2)) is compact, there exists 𝑢_𝜆, such that 𝑢_𝑛→𝑢_𝜆 strongly in 𝑋, and 𝐼_𝜆(𝑢_𝜆)=𝑐_𝜆, 𝐼^′_𝜆(𝑢_𝜆)=0.

Remark 16. The sublinear term |𝑢|^𝑞−2𝑢 can be relaxed to more general type, and the function 𝑄 before the sublinear term and asymptotically linear term can also be different.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

The authors declare that the study was realized in collaboration with the same responsibility.

Acknowledgment

The authors are grateful to the anonymous referees for their helpful comments and suggestions.