Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity

Abstract

We study the following nonlinear Schrödinger equation −Δ𝑢+𝑉(𝑥)𝑢=𝐾(𝑥)𝑓(𝑢),𝑥∈ℝ^𝑁,𝑢∈𝐻¹(ℝ^𝑁), where the potential 𝑉(𝑥) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem (𝒫) under a Nahari type condition. Furthermore, if 𝑉(𝑥),𝐾(𝑥) are radically symmetric with respect to 𝑥∈ℝ^𝑁, it is shown that problem (𝒫) has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if 𝑓(𝑢)=𝑢^𝑝, then the growth restriction 𝜎≤𝑝≤𝑁+2/𝑁−2 in Ambrosetti et al. (2005) can be relaxed to 𝜎≤𝑝≤𝑁+2/𝑁−2, where 𝜎<𝜎 if 0<𝛽<𝛼<2.

1. Introduction

The motivation of the paper is concerned with the existence of standing waves of the following nonlinear Schrödinger equation: 𝑖𝜕𝜓𝜕𝑡=−Δ_𝑥𝜓+𝑉𝜓−𝑓𝑥,𝜓,(𝑡,𝑥)∈ℝ₊×ℝ^𝑁,(1) where 𝑖 is the imaginary unit, 𝑉 is a real function on ℝ^𝑁, 𝜓(𝑥,𝑡)∶ℝ^𝑁×[0,+∞)→ℂ, and 𝑓 is supposed to satisfy that 𝑓(𝑥,𝛼𝑒^𝑖𝜃)=𝑓(𝑥,𝛼)𝑒^𝑖𝜃 for all 𝛼,𝜃∈ℝ. Problem (1) arises in many applications. For example, in some problems arising in nonlinear optics, in plasma physics, and in condensed matter physics, the presence of many particles leads one to consider nonlinear terms which simulates the interaction effect among them.

For problem (1), we are interested in looking for a stationary solution; that is, 𝜓(𝑥,𝑡)=𝑢(𝑥)𝑒^𝑖𝜔𝑡 with 𝑢(𝑥)>0 in ℝ^𝑁 and 𝜔>0 (the frequency); then it is not difficult to see that 𝑢 must satisfy −Δ𝑢+𝑉(𝑥)𝑢=𝑓(𝑥,𝑢),𝑥∈ℝ^𝑁,𝑢∈𝐻¹ℝ^𝑁.(2) Here and below, 𝑁≥3. Variational approach to (2) was initiated by Rabinowtiz [1], and since then several authors have studied (2) under different assumptions on 𝑉(𝑥) and the nonlinearity. If 𝑉(𝑥) is positive and bounded away from zero, then, by the well-known concentration compactness principle [2, 3], it is shown that there is bound states for problem (2); we mention here the work by Jeanjean and Tanaka [4, 5], Liu and Wang [6], Li et al. [7], Zhu [8], and the references therein.

If the potential 𝑉(𝑥) decays to zero at infinity, the methods used in the proceeding papers cannot be employed because the variational theory in 𝐻¹(ℝ^𝑁) cannot be used here. The earlier work on (2) we know of where 𝑉(𝑥) decays at infinity, is that by Ambrosetti et al. [9]; the authors proved that problem (2) has bounded states for 𝑓(𝑥,𝑢)=𝐾(𝑥)𝑢^𝑝 with 𝑎1+|𝑥|^𝛼(𝑥)(𝑥)≤𝑘≤𝑉≤𝐴,0<𝐾1+|𝑥|^𝛽,𝜎<𝑝<𝑁+2,𝑁−2(3) where 𝜎=𝑁+2−𝑁−24𝛽𝛼(𝑁−2),0<𝛽<𝛼,1,otherwise.(4) Following [9], by requiring some further assumptions on 𝑉(𝑥),𝐾(𝑥), in [10], the authors showed that there exist bound states of equation −𝜖²Δ𝑢+𝑉(𝑥)𝑢=𝐾(𝑥)𝑢^𝑝, 𝑥∈ℝ^𝑁, 𝑢>0, for all 𝑝 satisfying 1<𝑝<(𝑁+2)/(𝑁−2), provided that 𝜖 is sufficiently small. Motivated by the works [9, 10], in paper [11], the authors extended the results to potentials 𝑉(𝑥) that can both vanish and decay to zero at infinity. And since then, there are many papers on problem (2) with potential 𝑉(𝑥) vanishing at infinity; see, for example, [12–16].

In this paper, more precisely we will focus on the following model equation: −Δ𝑢+𝑉(𝑥)𝑢=𝐾(𝑥)𝑓(𝑢),𝑥∈ℝ^𝑁,𝑢∈𝐻¹ℝ^𝑁.(5) To our best knowledge, it seems that there are few results on problem (5), where 𝑓(𝑢) does not satisfy (𝐴𝑅) condition; that is, for some 𝜃>0, 0≤𝐹(𝑢)≜^𝑢₀𝑓(𝑡)1𝑑𝑡≤𝑓(𝑢)()2+𝜃𝑢∀𝑢∈ℝ,𝐴𝑅 and simultaneously 𝑉(𝑥) decays to zero at infinity.

The main aim of the paper is to extend the result of [9] to problem (5) with much more general classes of 𝑓(𝑢). Moreover, if 𝑉(𝑥),𝐾(𝑥) are radically symmetric with respect to 𝑥∈ℝ^𝑁, problem (5) will be proved to also have a positive solution with some more general growth conditions of the nonlinearity. Particularly, the result can exactly extend the growth restriction of the special case of 𝑓(𝑢)=𝑢^𝑝 to a new one, and the range of which is bigger than the usual 𝜎≤𝑝≤(𝑁+2)/(𝑁−2) in [9]; for more details please see Theorem 6.

Throughout the paper, we make the following assumptions on 𝑉(𝑥), 𝐾(𝑥), and 𝑓(𝑡).(𝑉)𝑉∶ℝ^𝑁→ℝ is continuous and there exist 𝑎,𝐴,𝛼>0 such that 𝑎1+|𝑥|^𝛼≤𝑉(𝑥)≤𝐴.(6)(𝐾)𝐾∶ℝ^𝑁→ℝ is smooth and there exist 𝛽,𝑘>0 such that (𝑥)≤𝑘0<𝐾1+|𝑥|^𝛽.(7)(𝐹₁)𝑓_𝑡(𝑡) is a Caratheodory function and there exists 𝐶>0 such that 𝑓_𝑡(𝑡)≤𝐶1+|𝑡|^2∗−2,lim_|𝑡|→∞𝑓(𝑡)|𝑡|^2∗−1=0.(8)(𝐹₂)lim_|𝑡|→0(𝑓(𝑡)/|𝑡|)=0 and lim_|𝑡|→∞(𝐹(𝑡)/𝑡²)=∞.(𝐹₃)𝑓(𝑡)/|𝑡| is nondecreasing in 𝑡∈ℝ⧵{0}.

Remark 1. If 𝛽>𝛼>0, then liminf_|𝑥|→∞(𝑉(𝑥)/𝐾(𝑥))=+∞, which implies that 𝐾(𝑥)≤𝐶𝑉(𝑥).

Throughout this paper, we define the following weighted Sobolev space: 𝐻=𝑢∈𝒟^1,2ℝ^𝑁∶_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥<∞.(9) Clearly, 𝐻¹(ℝ^𝑁)⊂𝐻. 𝐻 is a Hilbert space with norm and scalar product, respectively, ‖𝑢‖=_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥^1/2,(𝑢,𝑣)=_ℝ𝑁(∇𝑢∇𝑣+𝑉(𝑥)𝑢𝑣)𝑑𝑥.(10) Denote by 𝐿^𝑞_𝐾 the weighted space of measurable 𝑢∶ℝ^𝑁→ℝ such that |𝑢|_𝑞,𝐾=_ℝ𝑁𝐾(𝑥)|𝑢(𝑥)|^𝑞𝑑𝑥^1/𝑞<∞.(11)𝐻 and 𝐿^𝑞_𝐾 are particular cases of weighted spaces in discussions in [17], where the following result is proved.

Proposition 2. Let 𝑁≥3, and suppose that (𝑉), (𝐾) hold with 𝛼∈(0,2) and 𝛽>0. Then 𝐻↪𝐿_𝐾^𝑝+1,(12) where 𝜎≤𝑝≤(𝑁+2)/(𝑁−2), and there is 𝐶>0 such that |𝑢|_𝑝+1,𝐾≤𝐶‖𝑢‖,∀𝑢∈𝐻.(13) Furthermore, the embedding of 𝐻 into 𝐿_𝐾^𝑝+1 is compact if 𝑝∈(𝜎,(𝑁+2)/(𝑁−2)).

Remark 3. For the case of 𝛽≥𝛼>0, 𝜎=1, and consequently 𝐻↪𝐿_𝐾^𝑝+1, for all1≤𝑝≤(𝑁+2)/(𝑁−2), and also the embedding is compact if 1<𝑝<(𝑁+2)/(𝑁−2).

Furthermore, we define the energy functional associated with problem (5) by 𝐼(𝑢)=12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝑑𝑥_ℝ𝑁𝐾(𝑥)𝐹(𝑢)𝐹(𝑢)=𝑑𝑥,_𝑢0𝑓(𝑡)𝑑𝑡.(14) By Proposition 2, 𝐼 is well defined on 𝐻 and 𝐼∈𝐶¹(𝐻,ℝ) with 𝐼^′(𝑢),𝑣=_ℝ𝑁((𝑥))−∇𝑢∇𝑣+𝑉𝑢𝑣𝑑𝑥_ℝ𝑁𝐾(𝑥)𝑓(𝑢)𝑣𝑑𝑥∀𝑣∈𝐻.(15)

Definition 4. A function 𝑢∈𝐻 is said to be a solution of problem (5) provided that 𝑢≢0 satisfies _ℝ𝑁(𝑥)−∇𝑢∇𝜑+𝑉𝑢𝜑𝑑𝑥_ℝ𝑁𝐾(𝑥)𝑓(𝑢)𝜑𝑑𝑥=0∀𝜑∈𝐻.(16)

Notation. Hereafter we use the following notation.(i)2^∗=2𝑁/𝑁−2 if 𝑁≥3 and for some 2<𝑞<2^∗=∞ if 𝑁=2.(ii)𝐿^𝑞(Ω), 1≤𝑞≤+∞, Ω⊆ℝ^𝑁, denotes a Lebesgue space; the norm in 𝐿^𝑞(Ω) is denoted by |𝑢|_𝑞,Ω.(iii)For any 𝑅>0 and for any 𝑦∈ℝ^𝑁, 𝐵_𝑅(𝑦) denotes the ball of radius 𝑅 centered at 𝑦.(iv)𝐶,𝐶_𝑖 are various positive constants.(v)𝑜(1) denotes the quantity which tends to zero as 𝑛→∞.

Our main results are the following.

Theorem 5. Let (𝐹₁)--(𝐹₃) and (𝑉),(𝐾) hold with 𝛽>𝛼, 𝛼∈(0,2). Then problem (5) has a ground state 𝑢∈𝐻¹(ℝ^𝑁).

If 𝑉(𝑥),𝐾(𝑥) are radically symmetric with respect to 𝑥∈ℝ^𝑁 and 𝑓(𝑡) satisfies the following conditions:(𝐹₄)𝑓∈𝐶(ℝ,ℝ⁺), 𝑓(𝑡)≡0 for all 𝑡≤0;(𝐹₅)𝑓(𝑡)/𝑡 is nondecreasing with respect to 𝑡>0;(𝐹₆)lim_𝑡→0(𝑓(𝑡)/𝑡^𝑝)=𝑙₁∈[0,∞); lim_𝑡→∞(𝑓(𝑡)/𝑡^𝑞)=𝑙₂∈(0,∞), where 𝑞^∗−1<𝑝≤𝑞<𝑁+2/𝑁−2, and 𝑞^∗=4𝑁−1−𝛽+2𝛼2(𝑁−1)−𝛼,0<𝛽<𝛼<2,2,otherwise,(17)then we have the following.

Theorem 6. Assume that 𝑉(𝑥), 𝐾(𝑥) are radically symmetric with respect to 𝑥∈ℝ^𝑁, and let (𝑉), (𝐾), and (𝐹₄)--(𝐹₆) hold. Then problem (5) has a positive solution 𝑢∈𝐻¹_𝑟(ℝ^𝑁).

Remark 7. If 𝑓(𝑢)=𝑢^𝑘, then the growth restriction 𝜎≤𝑘≤𝑁+2/𝑁−2 in [9] can be relaxed to 𝑞^∗−1≤𝑘≤𝑁+2/𝑁−2, where 𝑞^∗−1<𝜎 if 0<𝛽<𝛼<2.

2. Variational Setting and Some Preliminaries

In this section, we describe the variational framework for the study of the critical points of the functional defined in (14).

Set 𝑐∶=inf{𝐼(𝑢)∶𝑢∈𝐻},(18) where 𝒩={𝑢∈𝐻⧵{0}∶𝛾(𝑢)≜⟨𝐼^′(𝑢),𝑢⟩=0}. First, it is necessary to show that 𝑐 is a positive number. Now we give the following two lemmas.

Lemma 8. Suppose that (𝐹₁)--(𝐹₃) hold. Then for each 𝑢∈𝐻⧵{0}, there exists a unique 𝑡=𝑡(𝑢)>0 such that 𝑡(𝑢)𝑢∈𝒩 and max_𝑡≥0𝐼(𝑡𝑢)=𝐼(𝑡(𝑢)𝑢).

Proof. The proof of this lemma is similar to the case of assuming (𝐴𝑅) condition, which can be found in [18], so we omit it here.

Lemma 9. 𝑐>0.

Proof. Let 𝑐₁∶=inf_{{0}𝑢∈ℋ⧵}max_𝑡≥0𝐼(𝑡𝑢),𝑐₂∶=inf_𝛾∈Γmax_[]𝑡∈0,1𝐼𝛾(𝑡),(19) where Γ∶={𝛾∈𝐶(([0,1]),𝐻)∶𝛾(0)=0,𝐼(𝛾(1))<0}. By Lemma 8, it is easy to see 𝑐=𝑐₁.
For any 𝑢∈𝐻⧵{0}, by (𝐹₂), there exists 𝑡>0 large enough such that 𝐼(𝑡𝑢)<0. Let 𝛾(𝑠)=𝑠𝑡𝑢; then 𝛾(𝑠)∈Γ, and 𝑐₂≤𝐼𝛾(𝑠)=𝐼(𝑠𝑡𝑢)≤max_𝑡≥0𝐼(𝑡𝑢),(20) so 𝑐₁≥𝑐₂. From (𝐹₁), (𝐹₂), there is 𝜌>0 small enough such that inf_{‖𝑢‖=𝜌}𝐼(𝑢)>𝛼>0. Since 𝛾(0)=0, for large enough 𝑡>0, ‖𝛾(1)‖>𝜌, there exists 𝑠₀∈[0,1] such that ‖𝛾(𝑠₀)‖=𝜌, so 𝑐₂≥𝐼𝛾𝑠₀≥inf_{‖𝑢‖=𝜌}𝐼(𝑢)>𝛼>0.(21) Thus, 𝑐=𝑐₁≥𝑐₂>0.

Lemma 10. Let {𝑢_𝑛} be a minimizing sequence of 𝑐 defined in (18). Then(i)there is 𝜌>0 such that liminf_𝑛→∞‖𝑢_𝑛‖≥𝜌>0,(ii){𝑢_𝑛} is bounded in 𝐻,(iii)for a subsequence, {𝑢_𝑛} converges weakly to 𝑢≢0.

Proof. (i) By (𝐹₁), (𝐹₂), for any 𝜖>0, there exists 𝐶_𝜖>0 such that 𝑓(𝑡)≤𝜖|𝑡|+|𝑡|^2∗−1+𝐶_𝜖|𝑡|^𝑞−1,𝑞∈2,2^∗.(22) From 𝛾(𝑢_𝑛)=0, 𝐻↪𝐿^𝑞_𝐾(ℝ^𝑁), it is easy to get that liminf_𝑛→∞‖𝑢_𝑛‖≥𝜌>0.
(ii) If {𝑢_𝑛} is not bounded, we define 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖, so ‖𝑣_𝑛‖=1. Passing to a subsequence we have 𝑣_𝑛⇀𝑣∈𝐻,𝑣_𝑛⟶𝑣∈𝐿^𝑞_𝐾ℝ^𝑁,2<𝑞<2^∗,𝑣_𝑛⟶𝑣a.e.onℝ^𝑁.(23)
If 𝑣≢0, we have 12−_ℝ𝑁𝐾(𝑥)𝐹𝑢_𝑛𝑢_𝑛²𝑣_𝑛²(1)𝑑𝑥=𝑐+𝑜𝑢_𝑛²>0.(24) By Fatou’s lemma and (𝐹₂), we have a contradiction as follows: 12≥liminf_{𝑛→∞ℝ𝑁}𝐾(𝑥)𝐹𝑢_𝑛𝑢_𝑛²𝑣_𝑛²≥𝑑𝑥_ℝ𝑁liminf_𝑛→∞𝐾(𝑥)𝐹𝑢_𝑛𝑢_𝑛²𝑣_𝑛²𝑑𝑥=∞.(25) If 𝑣≡0, by Proposition 2, 𝑣_𝑛⟶0 in 𝐿^𝑞_𝐾(ℝ^𝑁). By (22) and for any 𝑅>0 which is determined later, we get _ℝ𝑁𝐾(𝑥)𝐹𝑅𝑣_𝑛𝑑𝑥⟶0as𝑛⟶∞.(26) By Lemma 8, 𝐼(𝑡𝑢_𝑛)≤𝐼(𝑢_𝑛) for any 𝑡>0; then we have 𝑐+𝑜(1)⟵𝐼𝑢_𝑛≥𝐼𝑅𝑣_𝑛=12𝑅²−_ℝ𝑁𝐾(𝑥)𝐹𝑅𝑣_𝑛⟶12𝑅²−𝑜(1),(27) which is a contradiction if we take 𝑅>(2𝑐)^1/2. Thus {𝑢_𝑛} is bounded in 𝐻.
(iii) We can assume {𝑢_𝑛} weakly converges to 𝑢. If 𝑢≡0, then by (i) and noting the fact that 𝐻↪𝐿^𝑞_𝐾, 𝑞∈(2,2^∗) is compact, we have a contradiction as follows: 0<𝜌²≤𝑢_𝑛²=_ℝ𝑁𝐾(𝑥)𝑓𝑢_𝑛𝑢_𝑛𝑑𝑥⟶0.(28)

3. Proof of the Main Results

The aim of this section is to prove Theorems 5 and 6. For Theorem 5, we will take two steps; the first is to show the existence of nonzero critical point 𝑢∈𝐻 of 𝐼, and the second is to prove the critical point 𝑢 is a bound state; that is 𝑢∈𝐿²(ℝ^𝑁). To prove Theorem 6, we use a weighted Sobolev embedding theorem, which is based upon the results discussed in [19].

Proof of Theorem 5.  
Step  1. Let {𝑢_𝑛}⊂𝒩 be the sequence minimizing for 𝑐 given in (18). By Lemma 10, {𝑢_𝑛} is bounded in 𝐻, and there is some 0≢𝑢∈𝐻 such that 𝑢⇀_𝑛^𝑛𝑢weaklyin𝐻,(29) and by Proposition 2 and Remark 3, we get 𝑢_𝑛⟶𝑢 in 𝐿^𝑞_𝐾(ℝ^𝑁), 𝑞∈(2,2^∗), and consequently _ℝ𝑁𝐾(𝑥)𝐹𝑢_𝑛𝑑𝑥⟶_ℝ𝑁𝐾(𝑥)𝐹(𝑢)𝑑𝑥as𝑛⟶∞.(30) Using the lower semicontinuity, we have 𝐼(𝑢)≤𝑐. If 𝑢∈𝒩, we have 𝐼(𝑢)=𝑐. If 𝑢∉𝒩, by Lemma 8, there exists 𝑡=𝑡(𝑢)>0 such that 𝑡(𝑢)𝑢∈𝒩; then 𝑐≤𝐼(𝑡(𝑢)𝑢)≤liminf_𝑛→∞𝐼𝑡(𝑢)𝑢_𝑛≤liminf_𝑛→∞𝐼𝑢_𝑛=𝑐.(31) Since 𝒩 is smooth, the minimizer is a critical point of 𝐼.
Next we will show that the solution found above in 𝐻 belongs indeed to 𝐿²(ℝ^𝑁). For this purpose, we require some preliminary decay estimates on 𝑢, which are essentially motivated by [9]. However, we should mention that these proofs are partly different from those in [9]. First we will give the following proposition, which is a special case of Proposition 11 in [9].

Proposition 11. Let 0<𝛼<2 and 2<𝑞<2^∗. Then for all 𝜖>0 there exists 𝑅>0 such that for all 𝑅>𝑅_|𝑥|>𝑅𝐾(𝑥)|𝑢(𝑥)|^𝑞𝑑𝑥≤𝜖_|𝑥|>𝑅|∇𝑢(𝑥)|²+𝑉(𝑥)𝑢²(𝑥)𝑑𝑥^𝑞/2,∀𝑢∈𝐻.(32)

Lemma 12. Suppose that (𝑉), (𝐾), (𝐹₁), and (𝐹₂) hold. Let 𝛽>𝛼, 𝛼∈(0,2) and let 𝑢 be the critical point of 𝐼 on 𝐻. Then there exist 𝑡₀,𝛿₁∈(0,1) and 𝑛(𝛼)>0 such that, for all 𝑛≥𝑛(𝛼), _Ω𝑛+1|∇𝑢|²+𝑉(𝑥)|𝑢|²≤𝑑𝑥1−𝑡₀+𝛿₁𝑡₀×_Ω𝑛|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥,(33) where Ω_𝑛=ℝ^𝑁⧵𝐵_𝑅𝑛,𝑅_𝑛=𝑏^𝑛1,𝑏=1−𝑡₀^1/4.(34)

Proof. Since 𝛽>𝛼, there is 𝛿∈(0,1) such that 𝛽>𝛼+𝛿. Let 𝛾=𝛼+𝛿; then 𝛾∈(0,3), so there exists 𝛿₁∈(0,1) such that (𝛾+1)/4<1−𝛿₁.
Noting that lim_𝑡→0+ln(1−𝑡)+𝛿₁𝑡(1/4)ln(1/(1−𝑡))=−41−𝛿₁<−𝛾+1,(35) we may choose 𝑡₀∈(0,1) such that ln1−𝑡₀+𝛿₁𝑡₀(1/4)ln1/1−𝑡₀<−𝛾+1.(36) Let 𝑅_𝑛 be given by (34), and let 𝜒_𝑛(𝑥) be a cut-off function such that 𝜒_𝑛(𝑥)=0,if0≤|𝑥|≤𝑅_𝑛,1,if|𝑥|≥𝑅_𝑛+1.(37) Moreover, there exists 𝐴₀>0, independent of 𝑛, such that |∇𝜒_𝑛(𝑥)|≤𝐴₀/(𝑅_𝑛+1−𝑅_𝑛) for all 𝑥∈ℝ^𝑁. Noting that 𝛼∈(0,2) and 𝑏>1, we see that lim_𝑛→∞1+𝑏^𝛼(𝑛+1)𝑏^2(𝑛+1)𝐴₀𝑏𝑏−1²=0.(38) So there exists 𝑛(𝛼)>0 such that, for all 𝑛≥𝑛(𝛼), 1+𝑏^𝛼(𝑛+1)𝑏^2(𝑛+1)𝐴₀𝑏𝑏−1²≤2𝛿₁𝑡₀²𝑎,(39) where 𝑎 is given by (𝑉). From (39) and (𝑉), for 𝑅_𝑛≤|𝑥|≤𝑅_𝑛+1, we have ∇𝜒_𝑛(𝑥)²≤𝐴²₀𝑅_𝑛+1−𝑅_𝑛⁻²=𝑏^−2(𝑛+1)𝐴₀𝑏𝑏−1²≤𝑎1+𝑅_𝑛+1^𝛼2𝛿1−𝜂₀²≤𝑉(𝑥)2𝛿₁𝑡₀².(40) Then by the definition of 𝜒_𝑛(𝑥), for all 𝑛≥𝑛(𝛼) and 𝑥∈ℝ^𝑁, ∇𝜒_𝑛(𝑥)²≤𝑉(𝑥)2𝛿₁𝑡₀².(41)
Since ⟨𝐼^′(𝑢),𝑢𝜒_𝑛⟩=0, by (𝐹₁), (𝐹₂), and Young inequality, for any 𝜖>0, 𝜖₁>0, we have _ℝ𝑁𝜒_𝑛|∇𝑢|²+𝑉(𝑥)|𝑢|²=𝑑𝑥_ℝ𝑁𝐾(𝑥)𝑓(𝑢)𝑢𝜒_𝑛𝑑𝑥−_ℝ𝑁∇𝑢∇𝜒_𝑛≤𝑢𝑑𝑥_Ω𝑛𝐾(𝑥)𝜖|𝑢|²+|𝑢|^2∗+𝐶_𝜖|𝑢|^𝑞+𝑑𝑥_Ω𝑛𝜖₁|∇𝑢|²𝑑𝑥+_Ω𝑛14𝜖₁∇𝜒_𝑛²|𝑢|²𝑑𝑥,(42) where 𝑞∈(2,2^∗). Taking 𝜖₁=𝛿₁𝑡₀, then by (41), (42), for all 𝑛≥𝑛(𝛼), _Ω𝑛+1|∇𝑢|²+𝑉(𝑥)|𝑢|²≤𝑑𝑥_Ω𝑛𝜒_𝑛|∇𝑢|²+𝑉(𝑥)|𝑢|²≤𝑑𝑥_Ω𝑛𝐾(𝑥)𝜖|𝑢|²+|𝑢|^2∗+𝐶_𝜖|𝑢|^𝑞𝑑𝑥+𝛿₁𝑡_0Ω𝑛|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥.(43) Suppose that ‖𝑢‖²≤Γ, Γ>0; then, from Proposition 11 and Remark 3, for the above 𝜖>0 given and 𝑅 sufficiently large, we deduce that _Ω𝑛+1|∇𝑢|²+𝑉(𝑥)|𝑢|²≤𝑑𝑥𝜖+𝜖Γ^(2∗−2)/2+𝜖𝐶_𝜖Γ^(𝑞−2)/2+𝛿₁𝑡₀×_Ω𝑛|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥.(44) Choosing sufficiently small 𝜖>0 (and hence for 𝑅 large) such that 𝜖+𝜖Γ^(2∗−2)/2+𝜖𝐶_𝜖Γ^(𝑞−2)/2<1−𝑡₀, we obtain the assertion.

Lemma 13. Let 𝑛(𝛼) be given in Lemma 12. Then there exists 𝑅(𝛼)>𝑛(𝛼) such that, for any 𝜌>𝑅(𝛼), _|𝑥|>𝜌|∇𝑢|²+𝑉(𝑥)|𝑢|²≤‖𝑢‖𝑑𝑥₂exp𝐶𝛿₁,𝜂₀,𝜃ln𝜌,(45) where 𝐶(𝛿₁,𝑡₀,𝜃)<−(𝛾+1).

Proof. From (36), there is 𝜃∈(0,1) such that 𝐶𝛿₁,𝑡₀,𝜃∶=𝜃ln1−𝑡₀+𝛿₁𝑡₀(1/4)ln1/1−𝑡₀<−𝛾+1.(46) Since lim[((ln𝑅−ln𝑛(𝛼))/ln𝑏)−2]()=1𝜃ln𝑅/ln𝑏𝜃>1.(47) There exists a constant 𝑅(𝛼)>𝑛(𝛼) such that (𝛼)(𝛼)(𝛼)ln𝑅−ln𝑛>2ln𝑏,(48)ln𝜌−ln𝑛ln𝑏−2≥𝜃ln𝜌(𝛼).ln𝑏>0,∀𝜌>𝑅(49) Given 𝜌>𝑅(𝛼), there exist two integers 𝑛, 𝑛 such that 𝑅_𝑛≤𝑛(𝛼)≤𝑅_𝑛+1,𝑅𝑛₋₁≤𝜌≤𝑅𝑛.(50) Thus 𝑛−(𝛼)𝑛>ln𝜌−ln𝑛ln𝑏>2.(51) By Lemma 12, (49), and (51), we deduce that _|𝑥|>𝜌|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥≤_|𝑥|>𝑅𝑛₋₁|∇𝑢|²+𝑉(𝑥)|𝑢|²≤𝑑𝑥1−𝑡₀+𝛿₁𝑡₀𝑛_−𝑛−2×_{|𝑥|>𝑅𝑛+1}|∇𝑢|²+𝑉(𝑥)|𝑢|²≤‖𝑢‖𝑑𝑥₂expln1−𝑡₀+𝛿₁𝑡₀ln𝜌−ln𝑛(𝛼)≤‖𝑢‖ln𝑏−2₂exp𝐶𝛿₁,𝑡₀,𝜃ln𝜌.(52) That is, (45) holds.

Proof of Theorem 5.  
Step 2. Let 𝑢 be the weak solution found in 𝐻, and let 𝑟≥2, and |𝑦|≥2𝑟; then |𝑥|sup1+^𝛾𝑎𝑦^𝛾∶𝑥∈𝐵_𝑟𝑦≤1+𝑟+𝑦^𝛾𝑎𝑦^𝛾(𝑦≤sup1+3/2)^𝛾𝑎𝑦^𝛾∶𝑦≥4,(53) where 𝛾=𝛼+𝛿₁. Denote 𝐶₁𝛾(𝑦=sup1+3/2)^𝛾𝑎𝑦^𝛾∶𝑦=1≥44^𝛾𝑎+3^𝛾2^𝛾𝑎.(54) Then |𝑥|sup1+^𝛾𝑎∶𝑥∈𝐵_𝑟𝑦≤𝐶₁𝛾𝑦^𝛾.(55)
Since 𝐵_𝑟(𝑦)⊂{𝑥∈ℝ^𝑁∶|𝑥|≥|𝑦|/2}, 𝛾>𝛼, by Lemma 13 and (55), for all |𝑦|≥2𝑅(𝛼), we have _{𝐵𝑟(𝑦)}𝑢²𝑑𝑥≤_{𝐵𝑟(𝑦)}1+|𝑥|^𝛾𝑎1+|𝑥|^𝛼1+|𝑥|^𝛾𝑉(𝑥)𝑢²𝑑𝑥≤𝐶₁𝛾𝑦^𝛾_{𝐵𝑟(𝑦)}𝑉(𝑥)𝑢²𝑑𝑥≤𝐶₁𝛾𝑦^𝛾_{|𝑥|≥|𝑦|/2}𝑉(𝑥)𝑢²𝑑𝑥≤𝐶₁𝛾2^{−𝐶(𝛿1,𝑡0,𝜃)}𝑦^𝛾‖𝑢‖²𝑦^{𝐶(𝛿1,𝑡0,𝜃)}=𝐶₂‖𝑢‖²𝑦^{𝛾+𝐶(𝛿1,𝑡0,𝜃)},(56) where 𝐶₂=𝐶₁(𝛾)2^{−𝐶(𝛿1,𝑡0,𝜃)}.
With similar arguments of [15], let 𝑚∈ℕ⁺ with |𝑦_𝑖|≥2(𝑖=1⋯𝑚) be such that 𝐵₅⧵𝐵₂⊂^𝑚_𝑖=1𝐵₁𝑦_𝑖.(57)
Let 𝐾₀ denote a positive integral such that 2^𝐾0>𝑅(𝛼); by (56) we deduce that _|𝑥|≥2𝑢²𝑑𝑥≤^∞_{𝑘=02𝑘𝐵5⧵𝐵2}𝑢²≤𝑑𝑥_{𝑚∞𝑖=1𝑘=0𝐵2𝑘(2𝑘𝑦𝑖)}𝑢²≤𝑑𝑥_{𝑚𝐾𝑖=10−1𝑘=0𝐵2𝑘(2𝑘𝑦𝑖)}𝑢²𝑑𝑥+𝐶₂‖𝑢‖^2𝑚∞_{𝑖=1𝑘=𝐾0}2^𝑘𝑦_𝑖^{𝛾+𝐶(𝛿1,𝑡0,𝜃)}.(58) Since 𝛾+𝐶(𝛿₁,𝑡₀,𝜃)<−1, then the above inequality implies that ∫_|𝑥|≥2𝑢²𝑑𝑥<+∞ and 𝑢∈𝐿²(ℝ^𝑁). Thus, 𝑢∈𝐻¹(ℝ^𝑁) is a bound state of problem (5).

Before the proof of Theorem 6, let us introduce some functional spaces. Denote by ℋ the weighted Sobolev space ℋ∶=𝐷_𝑟^1,2ℝ^𝑁∶_ℝ𝑁|∇𝑢|²+𝑉(|𝑥|)𝑢²𝑑𝑥<∞,(59) where 𝒟_𝑟^1,2={𝑢∈𝐿^2∗(ℝ^𝑁)∶𝑢(𝑥)=𝑢(|𝑥|),∇𝑢∈𝐿²(ℝ^𝑁)}.

Denote by ℒ^𝑞_𝐾 the weighted space of measurable 𝑢∶ℝ^𝑁⟶ℝ such that |𝑢|_𝑞,𝐾=_ℝ𝑁𝐾(|𝑥|)|𝑢(𝑥)|^𝑞𝑑𝑥^1/𝑞<∞.(60)

Now we give the following embedding result, whose proof can be derived from Theorem 1 of [19].

Proposition 14. Let 𝑁≥3, and suppose that 𝑉(𝑥), 𝐾(𝑥) are radial, and the assumptions (𝑉), (𝐾) hold with 𝛼∈(0,2) and 𝛽>0. Then ℋ↪ℒ^𝑞_𝐾,(61) where 𝑞^∗≤𝑞≤2𝑁/(𝑁−2). Furthermore, the embedding is compact if 𝑞∈(𝑞^∗,2𝑁/(𝑁−2)).

Proof of Theorem 6. Based upon Proposition 14, the proof of Theorem 6 can be followed from some standard techniques; we leave the details to the readers.

Acknowledgment

This work was supported by NSFC Grant no. 11201083.