Phenomena of Blowup and Global Existence of the Solution to a Nonlinear Schrödinger Equation

Abstract

We consider the following Cauchy problem: −𝑖𝑢_𝑡=Δ𝑢−𝑉(𝑥)𝑢+𝑓(𝑥,|𝑢|²)𝑢+(𝑊(𝑥)⋆|𝑢|²)𝑢,𝑥∈ℝ^𝑁,𝑡>0,𝑢(𝑥,0)=𝑢₀(𝑥),𝑥∈ℝ^𝑁, where 𝑉(𝑥) and 𝑊(𝑥) are real-valued potentials and 𝑉(𝑥)≥0 and 𝑊(𝑥) is even, 𝑓(𝑥,|𝑢|²) is measurable in 𝑥 and continuous in |𝑢|², and 𝑢₀(𝑥) is a complex-valued function of 𝑥. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem.

1. Introduction

In this paper, we consider the following Cauchy problem: −𝑖𝑢_𝑡=Δ𝑢−𝑉(𝑥)𝑢+𝑓𝑥,|𝑢|²𝑢+𝑊(𝑥)⋆|𝑢|²𝑢,𝑥∈ℝ^𝑁𝑢(),𝑡>0,𝑥,0=𝑢₀(𝑥)∈Σ,𝑥∈ℝ^𝑁, (1) where 𝑉(𝑥) and 𝑊(𝑥) are real-valued potentials, 𝑉(𝑥)≥0 and 𝑊(𝑥) is even, 𝑓(𝑥,|𝑢|²) is measurable in 𝑥 and continuous in |𝑢|², 𝑊(𝑥)⋆|𝑢|²𝑢(𝑥)=_ℝ𝑁𝑊𝑥−𝑦𝑢𝑦²𝑑𝑦𝑢(𝑥), (2)𝑢₀(𝑥) is a complex-valued function of 𝑥, and Σ is the Hilbert space: Σ=𝑢∈𝐻¹ℝ^𝑁,_ℝ𝑁𝑉(𝑥)|𝑢|²𝑑𝑥<+∞, (3) with the inner product 𝜑,𝜓=_ℝ𝑁𝜑𝜓+∇𝜑⋅∇𝜓+𝑉(𝑥)𝜑𝜓𝑑𝑥 (4) and the norm ‖𝑢‖²_Σ=_ℝ𝑁|𝑢|²+|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥. (5) Model (1) appears in the theory of Bose-Einstein condensation, nonlinear optics and theory of water waves (see [1, 2]).

For convenience, denote 1/(𝑁−2)⁺=+∞ when 𝑁=1,2 and (𝑁−2)⁺=𝑁−2 when 𝑁≥3. We also give some assumptions on 𝑉(𝑥), 𝑓(𝑥,𝑠), and 𝑊(𝑥) as follows.(V1)𝑉(𝑥)≥0 and 𝑉(𝑥)∈𝐿^𝑟(ℝ^𝑁)+𝐿^∞(ℝ^𝑁) for 𝑟≥1, 𝑟>𝑁/2.(V2)𝑉(𝑥)≥0, 𝑉(𝑥)∈𝐒^𝑐₁, and |𝐷^𝛼𝑉| is bounded for all |𝛼|≥2. Here 𝐒^𝑐₁ is the complementary set of 𝐒₁={𝑉(𝑥)satisfies(𝑉1)}.(f1)𝑓(𝑥,𝑠)∶ℝ^𝑁×ℝ→ℝ is measurable in 𝑥 and continuous in |𝑢|² with 𝑓(𝑥,0)=0.Assume that, for every 𝑘>0, there exists 𝐿(𝑘)<+∞ such that |𝑓(𝑥,𝑠₁)−𝑓(𝑥,𝑠₂)|≤𝐿(𝑘)|𝑠₁−𝑠₂| for all 0≤𝑠₁<𝑠₂<𝑘. Here 𝐿(𝑘)𝐿(𝑘)∈𝐶([0,∞)),if𝑁=1,≤𝐶1+𝑘^𝛼2with0≤𝛼<(𝑁−2)⁺,if𝑁≥2. (6)(W1)𝑊(𝑥) is even and 𝑊(𝑥)∈𝐿^𝑞(ℝ^𝑁)+𝐿^∞(ℝ^𝑁) for some 𝑞≥1, 𝑞>𝑁/4.

First, we consider the local well-posedness of (1). We have a proposition as follows.

Proposition 1 (local existence result). Assume that (𝑓1) and (𝑊1) are true, 𝑉(𝑥) satisfies (𝑉1) or (𝑉2), and 𝑢₀∈Σ. Then there exists a unique solution 𝑢 of (1) on a maximal time interval [0,𝑇_max) such that 𝑢∈𝐶(Σ;[0,𝑇_max)) and either 𝑇_max=+∞ or else 𝑇_max<+∞,lim_{𝑡→𝑇max}‖𝑢(⋅,𝑡)‖_Σ=+∞. (7)

Definition 2. If 𝑢∈𝐶(Σ;[0,𝑇)) with 𝑇=∞, we say that the solution 𝑢 of (1) exists globally. If 𝑢∈𝐶(Σ;[0,𝑇)) with 𝑇<+∞ and lim_𝑡→𝑇‖𝑢(⋅,𝑡)‖_Σ→+∞, we say that the solution 𝑢 of (1) blows up in finite time.

This paper is directly motivated by [1, 3–5]. Since Cazevave established some results on blowup and global existence of the solutions to (1) with (V1), (f1), and (W1) in [1], we are interested in the problems such as “What are the results about the blowup and global existence of the solutions to (1) with (V2), (f1), and (W1)?” On the other hand, since Gan et al. had established some sharp thresholds for global existence and blowup of the solution to the related problems to (1) (see [3–5] and the references therein), it is a natural way to consider the sharp threshold for global existence and blowup of the solution to (1).

About the topic of global existence and blowup in finite time, there are many results on the special cases of (1). We will recall some results on the following Cauchy problem: −𝑖𝑢_𝑡=Δ𝑢+𝑓|𝑢|²𝑢,𝑥∈ℝ^𝑁𝑢(),𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈ℝ^𝑁. (8) In [6], Glassey established some blowup results for (8). In [7], Berestyki and Cazenave established the sharp threshold for blowup of (8) with supercritical nonlinearity by considering a constrained variational problem. In [8], Weinstein presented a relationship between the sharp criterion for the global solution of (8) and the best constant in the Gagliardo-Nirenberg inequality. In [9], Cazenave and Weisseler established the local existence and uniqueness of the solution to (8) with 𝑓(|𝑢|²)𝑢=|𝑢|^4/𝑁𝑢. Very recently, Tao et al. in [10] studied the Cauchy problem (8) with 𝑓(|𝑢|²)𝑢=𝜇|𝑢|^𝑝1𝑢+𝜈|𝑢|^𝑝2𝑢, where 𝜇 and 𝜈 are real numbers, 0<𝑝₁<𝑝₂<4/(𝑁−2) with 𝑁≥3, and established the results on local and global well-posedness, asymptotic behavior (scattering), and finite time blowup under some assumptions. Other sharp thresholds were established by Chen et al. in [11, 12]. The following Cauchy problem −𝑖𝑢_𝑡=12Δ𝑢−|𝑥|²𝑢+𝐾(𝑥)|𝑢|^𝑝𝑢+𝑄(𝑥)|𝑢|^𝑞𝑢,𝑥∈ℝ^𝑁𝑢(),𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈ℝ^𝑁 (9) is also a special case of (1), where 0<𝑝<𝑞<4/(𝑁−2) with 𝑁≥3. In [2], Oh obtained the local well-posedness and global existence results of (9) under some conditions. In [3, 5], Gan et al. and Zhang, respectively, established the sharp thresholds for the global existence and blowup of the solutions to (9) under some conditions. In [4], Gan et al. dealt with −𝑖𝑢_𝑡=12Δ𝑢+𝑎|𝑢|^𝑝𝑢+𝐸₁|𝑢|²𝑢,𝑥∈ℝ^𝑁𝑢(),𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈ℝ^𝑁, (10) with 𝐸₁(𝜉) a singular integral operator, where 0<𝑝<4/(𝑁−2) with 𝑁≥3. They got the sharp threshold for global existence and blowup of the solution to (10) and the instability of the wave solutions. Very recently, Miao et al. also obtained some results on the blowup and global existence of the solution to a Hartree equation (see [13–15]). Naturally, we want to establish some new sharp thresholds for global existence and blowup of the solution to (1) in this paper and generalize these results above. Although the methods of our paper are inspired by the references above, our results, which will be stated in Section 2, are new and cover theirs.

This paper is organized as follows. In Section 2, we will recall some results of [1] and state our main results; then we will prove Proposition 1 and give some other properties. In Section 3, we will prove Theorems 3 and 4. In Section 4, we establish the sharp threshold for (1) with 𝑉(𝑥)≡0. In Section 5, we will prove Theorem 7.

2. Our Main Results

Now we will introduce some notations. Denote 𝐹𝑥,|𝑢|²=^|𝑢|2₀𝑓()𝐺|𝑢|𝑥,𝑠𝑑𝑠,₂=14_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²ℎ(𝑢)(𝑥)|𝑢|𝑑𝑥,=−𝑉𝑢+𝑓𝑥,²𝑢+𝑊(𝑥)⋆|𝑢|²𝐻(𝑢)1𝑢,=−2_ℝ^𝑁𝑉(𝑥)|𝑢|²1𝑑𝑥+2_ℝ^𝑁𝐹𝑥,|𝑢|²+1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥, (11)(12)(13)

mass (𝐿² norm) 𝑀(𝑢)∶=_ℝ𝑁|𝑢(𝑥,𝑡)|²𝑑𝑥^1/2, (14)

energy 𝐸(𝑢)1∶=2_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−1𝑑𝑥2_ℝ^𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥. (15)

In [1], Cazenave obtained some sufficient conditions on blowup and global existence of the solution to (1) with (V1), (f1), and (W1). The following two theorems can be looked at as the parallel results to Corollary 6.1.2 and Theorem 6.5.4 of [1], respectively.

Theorem 3 (global existence). Assume that 𝑢₀∈Σ, (𝑉2) and (𝑓1) are true, and 𝑊⁺(𝑥)∈𝐿^𝑞ℝ^𝑁+𝐿^∞ℝ^𝑁 (16) for some 𝑞≥1, 𝑞≥𝑁/2 (and 𝑞>1 if 𝑁=2). Here 𝑊⁺=max(𝑊(𝑥),0). Suppose further that there exist constants 𝑐₁ and 𝑐₂ such that 𝐹(𝑥,|𝑢|²)≤𝑐₁|𝑢|²+𝑐₂|𝑢|^2𝑝+2 with 0<𝑝<2/𝑁. Then the solution of (1) exists globally. That is, ‖𝑢(⋅,𝑡)‖_Σ<+∞∀0<𝑡<+∞. (17)

Theorem 4 (blowup in finite time). Assume that 𝑢₀∈Σ and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), (𝑉2), (𝑓1), and (𝑊1) are true. Suppose further that (𝑁+2)𝐹𝑥,|𝑢|²−𝑁|𝑢|²𝑓𝑥,|𝑢|²(𝑥)+((𝑥(𝑥)+((𝑥≤0,2𝑉𝑥⋅∇𝑉))≥0a.e.,2𝑊𝑥⋅∇𝑊))≤0a.e. (18)(19)(20) If (1) 𝐸(𝑢₀)<0 or (2) 𝐸(𝑢₀)=0 and ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, then the solution of (1) will blow up in finite time. That is, there exists 𝑇_max<∞ such that lim_{𝑡→𝑇max}‖𝑢(⋅,𝑡)‖_Σ=∞. (21)

Denote 𝑄(𝑢)∶=2_ℝ𝑁|∇𝑢|²𝑑𝑥−_ℝ𝑁(𝑥⋅∇𝑉(𝑥))|𝑢|²𝑑𝑥+𝑁_ℝ𝑁𝐹𝑥,|𝑢|²−|𝑢|²𝑓𝑥,|𝑢|²+1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥. (22)

We will establish the first type of sharp threshold as follows.

Theorem 5 (sharp threshold I). Assume that 𝑉(𝑥)≡0 and 𝑊(𝑥)∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2. Suppose further that 𝑓(𝑥,0)=0 and there exist constants 𝑐₁,𝑐₂,𝑐₃>0 and 2/𝑁<𝑝₁,𝑝₂, 𝑙<2/(𝑁−2)⁺ such that 𝑙𝐹𝑥,|𝑢|²≤|𝑢|²𝑓𝑥,|𝑢|²−𝐹𝑥,|𝑢|²≤𝑐₁|𝑢|^2𝑝1+2+𝑐₂|𝑢|^2𝑝2+2,𝑁𝑙𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥))≤0≤𝑐₃𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥)). (23)(24) Let 𝜔 be a positive constant satisfying 𝑑_𝐼∶=inf_{{𝑢∈Σ⧵{0};𝑄(𝑢)=0}}𝜔‖𝑢‖²₂+𝐸(𝑢)>0, (25) where 𝑄(𝑢) is defined by (22). Suppose that 𝑢₀∈𝐻¹(ℝ^𝑁) satisfies 𝜔𝑢₀²₂+𝐸𝑢₀<𝑑_𝐼. (26)
Then(1)if 𝑄(𝑢₀)>0, the solution of (1) exists globally;(2)if 𝑄(𝑢₀)<0, |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), and ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, the solution of (1) blows up in finite time.

Remark 6. Theorem 5 is only suitable for (1) with 𝑉(𝑥)≡0. To establish the sharp threshold for (1) with 𝑉(𝑥)≠0, we will construct a type of cross-constrained variational problem and establish some cross-invariant manifolds. First, we introduce some functionals as follows: 𝐼_𝜔(𝑢)=𝜔‖𝑢‖²₂(𝑢),𝑆+𝐸_𝜔(𝑢)=2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝑓𝑥,|𝑢|²|𝑢|²−𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥. (27)(28)

Denote the Nehari manifold 𝒩∶=𝑢∈Σ⧵{0},𝑆_𝜔(𝑢)=0 (29) and cross-manifold 𝒞ℳ∶=𝑢∈Σ⧵{0},𝑆_𝜔(𝑢)<0,𝑄(𝑢)=0. (30) Define 𝑑_𝒩∶=inf_𝒩𝐼_𝜔(𝑢),𝑑_ℳ∶=inf_𝒞ℳ𝐼_𝜔(𝑢),𝑑_𝐼𝐼∶=min𝑑_𝒩,𝑑_ℳ. (31)(32)(33)

In Section 5, we will prove that 𝑑_𝐼𝐼 is always positive. Therefore, it is reasonable to define the following cross-manifold: 𝒦∶=𝑢∈Σ⧵{0}∶𝐼_𝜔(𝑢)<𝑑_𝐼𝐼,𝑆_𝜔(𝑢)<0,𝑄(𝑢)<0. (34)

We give the second type of sharp threshold as follows.

Theorem 7 (sharp threshold II). Assume that (f1), (W1), and (23). Suppose that 𝑊(𝑥)≥0,𝑁𝑙𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥))≤0 (35) and there exists a positive constant 𝑐 such that 𝑁𝑙𝑉(𝑥)+(𝑥⋅∇𝑉(𝑥))≥𝑐𝑉(𝑥)≥0 (36) with the same 𝑙 in (23). Assume further that the function 𝑓(𝑥,|𝑢|²) satisfies 𝑓(𝑥,0)=0 and 𝑓𝑥,|𝑢|²≤𝑓𝑥,𝑘²|𝑢|²,𝑓^′_𝑠𝑥,𝑘²|𝑢|²≤𝑓^′_𝑠𝑥,|𝑢|²,𝐹𝑥,𝑘²|𝑢|²−𝑘²|𝑢|²𝑓𝑥,𝑘²|𝑢|²≤𝑘²𝐹𝑥,|𝑢|²−|𝑢|²𝑓𝑥,|𝑢|² (37)(38) for 𝑘>1. Here 𝑓^′_𝑠(𝑥,𝑧) is the value of the partial derivative of 𝑓(𝑥,𝑠) with respect to 𝑠 at the point (𝑥,𝑧). If 𝑢₀∈Σ and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁) with 𝐼_𝜔(𝑢₀)=𝜔‖𝑢₀‖²₂+𝐸(𝑢₀)<𝑑_𝐼𝐼, then the solution of (1) blows up in finite time if and only if 𝑢₀∈𝒦.

Remark 8. (1) 𝑓(𝑥,|𝑢|²)≤𝑓(𝑥,𝑘²|𝑢|²) implies that 𝑘²𝐹(𝑥,|𝑢|²)≤𝐹(𝑥,𝑘²|𝑢|²) for 𝑘>1.
(2) The blowup of solution to (1) will benefit from the role of the potential 𝑉 if 𝑉(𝑥)≥0. In some cases, the blowup of the solution to (1) can be delayed or prevented by the role of potential (see [16] and the references therein).

In the sequel, we use 𝐶 and 𝑐 to denote various finite constants; their exact values may vary from line to line.

First, we will give the proof of Proposition 1.

Proof of Proposition 1. If (V1) is true, then there exist 𝑉₁(𝑥)∈𝐿^𝑟(ℝ^𝑁) with 𝑟≥1, 𝑟>𝑁/2, and 𝑉₂(𝑥)∈𝐿^∞(ℝ^𝑁) such that 𝑉(𝑥)=𝑉₁(𝑥)+𝑉₂(𝑥). (39) Noticing that 0<2𝑟/(𝑟−1)<2𝑁/(𝑁−2), using Hölder’s and Sobolev’s inequalities, we have _ℝ𝑁𝑉(𝑥)|𝑢|²𝑑𝑥=_ℝ𝑁𝑉₁(𝑥)|𝑢|²𝑑𝑥+_ℝ𝑁𝑉₂(𝑥)|𝑢|²≤𝑑𝑥_ℝ𝑁𝑉₁(𝑥)^𝑟𝑑𝑥^1/𝑟×_ℝ𝑁|𝑢|^{2𝑟/(𝑟−1)}𝑑𝑥^{(𝑟−1)/𝑟}+𝐶_ℝ𝑁|𝑢|²𝑑𝑥≤𝐶_ℝ𝑁|∇𝑢|²𝑑𝑥+𝐶_ℝ𝑁|𝑢|²𝑑𝑥 (40) for any 𝑢∈𝐻¹(ℝ^𝑁). Consequently, we have ‖𝑢‖_𝐻¹≤‖𝑢‖_Σ≤𝐶‖𝑢‖_𝐻¹. (41) By the results of Theorem 3.3.1 in [1], we have the local well-posedness result of (1) in Σ.
If (V2), (f1), and (W1) are true, similar to the proof of Theorem 3.5 in [2], we can establish the local well-posedness result of (1) in Σ. Roughly, we only need to replace |𝑢|^𝑝+1𝑢 by 𝑓(𝑥,|𝑢|²)𝑢+(𝑊(𝑥)⋆|𝑢|²)𝑢 in the proof, and we can obtain similar results under the assumptions of (V2), (f1), and (W1).

Noticing that ℑℎ(𝑢)𝑢=0 and ℎ(𝑢)=𝐻^′(𝑢), following the method of [6] and the discussion in Chapter 3 of [1], one can obtain the conservation of mass and energy. We give the following proposition without proof.

Proposition 9. Assume that 𝑢(𝑥,𝑡) is a solution of (1). Then 𝑀(𝑢)=_ℝ𝑁|𝑢(𝑥,𝑡)|²𝑑𝑥^1/2=_ℝ𝑁𝑢₀(𝑥)²𝑑𝑥^1/2=𝑀𝑢₀,𝐸(𝑢)=12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝐹𝑥,|𝑢|²|𝑢|𝑑𝑥−𝐺²=𝐸𝑢₀ (42) for any 0≤𝑡<𝑇_max.

We will recall some results on blowup and global existence of the solution to (1) with (V1), (f1), and (W1).

Theorem A (Corollary 6.12 of [1]). Assume that (𝑉1), (𝑓1), and (16). Suppose that there exist 𝐴≥0 and 0≤𝑝<2/𝑁 such that 𝐹|𝑢|²≤𝐴|𝑢|²1+|𝑢|^2𝑝. (43) Then the maximal strong 𝐻¹-solution of (1) is global and sup{‖𝑢‖_𝐻¹∶𝑡∈ℝ}<∞ for every 𝑢₀∈𝐻¹(ℝ^𝑁).

Theorem B (Theorem 6.54 of [1]). Assume that (𝑉1), (𝑓1), (𝑊1), and (18)–(20). If 𝑢₀∈𝐻¹(ℝ^𝑁), |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), and 𝐸(𝑢₀)<0, then the 𝐻¹-solution of (1) will blow up in finite time.

Let 𝐽(𝑡)=∫_ℝ𝑁|𝑥|²|𝑢|²𝑑𝑥. After some elementary computations, we obtain 𝐽^′(𝑡)=4ℑ_ℝ𝑁(𝑥⋅∇𝑢)𝑢𝑑𝑥,𝐽^′′(𝑡)=4𝑄(𝑢). (44)

We have the following proposition.

Proposition 10. Assume that 𝑢(𝑥,𝑡) is a solution of (1) with 𝑢₀∈Σ and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁). Then the solution to (1) will blow up in finite time if either(1)there exists a constant 𝑐<0 such that 𝐽^′′(𝑡)=4𝑄(𝑢)≤𝑐<0 or(2)𝐽^′′(𝑡)=4𝑄(𝑢)≤0 and 𝐽^′(0)=ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0.

Proof. Since 𝑢₀∈Σ and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), we have 𝐽^′(0)<4_ℝ𝑁𝑥𝑢₀∇𝑢₀𝑑𝑥≤8_ℝ𝑁∇𝑢₀²+𝑥𝑢₀²𝑑𝑥<+∞. (45)
(1) If 𝐽^′′(𝑡)≤𝑐<0, integrating it from 0 to 𝑡, we get 𝐽^′(𝑡)<𝑐𝑡+𝐽^′(0). Since 𝑐<0, we know that there exists a 𝑡₀≥max(0,𝐽^′(0)/−𝑐) such that 𝐽^′(𝑡)<𝐽^′(𝑡₀)<0 for 𝑡>𝑡₀. On the other hand, we have 0≤𝐽(𝑡)=𝐽𝑡₀+^𝑡_𝑡0𝐽^′(𝑠)𝑑𝑠<𝐽𝑡₀+𝐽^′𝑡₀𝑡−𝑡₀, (46) which implies that there exists a 𝑇_max<+∞ satisfying lim_{𝑡→𝑇max}𝐽(𝑡)=0. (47) Using the inequality 𝑔²₂≤2𝑁∇𝑔₂𝑥𝑔₂if𝑔∈𝐻¹ℝ^𝑁,𝑥𝑔∈𝐿²ℝ^𝑁 (48) and noticing that ‖𝑢(⋅,𝑡)‖₂=‖𝑢₀‖₂, we have lim_{𝑡→𝑇max}‖∇𝑢‖₂=+∞. (49) Consequently, lim_{𝑡→𝑇max}‖𝑢‖_Σ=+∞. (50)
(2) Similar to (46), we can get 0≤𝐽(𝑡)≤𝐽(0)+𝐽^′(0)𝑡, (51) which implies that the solution will blow up in a finite time 𝑇_max≤𝐽(0)/−𝐽^′(0).

3. The Sufficient Conditions on Global Existence and Blowup in Finite Time

In this section, we will prove Theorems 3 and 4, which give some sufficient conditions on global existence and blowup of the solution to (1).

We would like to give some examples of 𝑉(𝑥), 𝑓(𝑥,|𝑢|²), and 𝑊(𝑥). It is easy to verify that they satisfy the conditions of Theorem 3.

Example 11. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=𝑒^{−𝜋|𝑥|2}, and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝 with 𝑏 a real constant and 0<𝑝<2/𝑁.

Example 12. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=|𝑥|²/(1+|𝑥|²), and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝ln(1+|𝑢|²) with 𝑏 a real constant and 0<𝑝<2/𝑁.

Proof of Theorem 3. Letting 𝑊⁺(𝑥)=𝑊₁(𝑥)+𝑊₂(𝑥), where 𝑊₁∈𝐿^∞(ℝ^𝑁) and 𝑊₂∈𝐿^𝑞(ℝ^𝑁) with 𝑞>𝑁/2, using Hölder’s and Young’s inequalities, we obtain _ℝ𝑁𝑊₂(𝑥)⋆(𝑢𝑣)𝑤𝑧𝑑𝑥≤𝑊_2𝐿𝑞‖𝑢‖_𝐿^𝑟‖𝑣‖_𝐿^𝑟‖𝑤‖_𝐿^𝑟‖𝑧‖_𝐿^𝑟 (52) with 𝑟=4𝑞/(2𝑞−1). Specifically, we have _ℝ𝑁𝑊₂(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥≤𝑊_2𝐿𝑞‖𝑢‖⁴_𝐿^𝑟. (53) Using (53) and Gagliardo-Nirenberg’s inequality, we get 14_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²≤𝑊𝑑𝑥_1𝐿∞‖𝑢‖⁴_𝐿2+𝑊_2𝐿𝑞‖𝑢‖⁴_{𝐿4𝑞/(2𝑞−1)}≤𝑊_1𝐿∞‖𝑢‖⁴_𝐿2+𝐶𝑊_2𝐿𝑞‖∇𝑢‖_𝐿^𝑁/𝑞₂‖𝑢‖_𝐿^{(4𝑞−𝑁)/𝑞}₂. (54) Using Young’s inequality, from (54), we have 𝐶𝑊_2𝐿𝑞‖∇𝑢‖_𝐿^𝑁/𝑞₂‖𝑢‖_𝐿^{(4𝑞−𝑁)/𝑞}₂≤𝜀‖∇𝑢‖²_𝐿2+𝐶𝜀,𝑊_2𝐿𝑞‖𝑢‖_𝐿^{(8𝑞−2𝑁)/(2𝑞−𝑁)}₂ (55) for some 𝜀>0. Noticing that 𝐹(𝑥,|𝑢|²)≤𝑐₁|𝑢|²+𝑐₂|𝑢|^2𝑝+2, using Gagliardo-Nirenberg’s inequality and (55) with 𝜀=1/4, we get 𝐸𝑢₀=12_ℝ^𝑁|∇𝑢₀|²+𝑉(𝑥)|𝑢₀|²−𝐹𝑥,𝑢₀²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆𝑢₀²𝑢₀²=1𝑑𝑥2_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²≥1𝑑𝑥2_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝑐₁|𝑢|²−𝑐₂|𝑢|^2𝑝+2−𝑊𝑑𝑥_1𝐿∞‖𝑢‖⁴_𝐿2−𝐶𝑊_2𝐿𝑞‖∇𝑢‖_𝐿^𝑁/𝑞₂‖𝑢‖_𝐿^{(4𝑞−𝑁)/𝑞}₂≥12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝑐₁|𝑢|²𝑑𝑥−𝑐₂𝐶_𝑁ℝ𝑁|∇𝑢|²𝑑𝑥^𝑝𝑁/2_ℝ𝑁|𝑢|²𝑑𝑥^{(2+𝑝(2−𝑁))/2}−𝑊_1𝐿∞‖𝑢‖⁴_𝐿2−14‖∇𝑢‖²_𝐿2−𝐶‖𝑢‖_𝐿^{(8𝑞−2𝑁)/(2𝑞−𝑁)}₂. (56) Since ‖𝑢‖₂=‖𝑢₀‖₂, from (56), we can obtain 4𝐸𝑢₀+𝐶𝑢₀²_𝐿2+𝐶𝑢₀⁴_𝐿2+𝐶𝑢_0𝐿^{(8𝑞−2𝑁)/(2𝑞−𝑁)}₂≥_ℝ𝑁𝑉(𝑥)|𝑢|²𝑑𝑥+_ℝ𝑁|∇𝑢|²×𝑑𝑥1−𝑐_ℝ𝑁|∇𝑢|²𝑑𝑥^{(𝑝𝑁/2)−1}. (57) Since 𝑝<2/𝑁 means that (𝑝𝑁/2)−1<0, (57) implies that ‖𝑢‖²_Σ is always controlled by 4𝐸(𝑢₀)+𝐶‖𝑢₀‖²_𝐿2+𝐶‖𝑢₀‖⁴_𝐿2+𝐶‖𝑢₀‖_𝐿^{(8−2𝑁)/(2𝑞−𝑁)}₂. That is, the solution of (1) exists globally.

We would like to give some examples of 𝑉(𝑥), 𝑊(𝑥), and 𝑓(𝑥,|𝑢|²). It is easy to verify that they satisfy the conditions of Theorem 4.

Example 13. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=|𝑥|⁻², and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝 with 𝑏>0 and 𝑝>2/𝑁 with 𝑁≥3.

Example 14. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=|𝑥|⁻², and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝ln(1+|𝑢|²) with 𝑏>0 and 𝑝≥2/𝑁 with 𝑁≥3.

Proof of Theorem 4. Set 𝑦(𝑡)=𝐽^′(𝑡)=4ℑ_ℝ𝑁(𝑥⋅∇𝑢)𝑢𝑑𝑥. (58) Using (18)–(20), we have 𝑦^′(𝑡)=8_ℝ𝑁|∇𝑢|²𝑑𝑥−4_ℝ𝑁(𝑥⋅∇𝑉(𝑥))|𝑢|²𝑑𝑥+4𝑁_ℝ𝑁𝐹𝑥,|𝑢|²−|𝑢|²𝑓𝑥,|𝑢|²𝑑𝑥+2_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²(𝑢)𝑑𝑥=16𝐸+4_ℝ𝑁[−2𝑉(𝑥)−(𝑥⋅∇𝑉(𝑥))]|𝑢|²+(𝑁+2)𝐹𝑥,|𝑢|²−𝑁|𝑢|²𝑓𝑥,|𝑢|²𝑑𝑥+2_ℝ𝑁{2𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥))}⋆|𝑢|²|𝑢|²(𝑢)𝑢𝑑𝑥≤16𝐸=16𝐸₀<0. (59) From (58) and (59), we obtain ‖𝑥𝑢(𝑥,𝑡)‖²_𝐿2≤𝑥𝑢₀²_𝐿2+4𝑡ℑ_ℝ𝑁𝑢₀𝑥⋅∇𝑢₀𝑑𝑥+8𝑡²𝐸𝑢₀. (60) Since ‖𝑥𝑢(𝑥,𝑡)‖²_𝐿2≥0, whether (1) or (2), (60) will be absurd for 𝑡>0 large enough. Therefore, the solution of (1) will blow up in finite time.

4. The Sharp Threshold for Global Existence and Blowup of the Solution to (1) with 𝑉(𝑥)≡0 and 𝑊∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2

In this section, we will establish the sharp threshold for global existence and blowup of the solution to (1) with 𝑉(𝑥)≡0 and 𝑊∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2.

Before giving the proof of Theorem 5, we would like to give some examples of 𝑓(𝑥,|𝑢|²) and 𝑊(𝑥). It is easy to verify that they satisfy the conditions of Theorem 5.

Example 15. Consider that 𝑊(𝑥)≡0, 𝑓(𝑥,|𝑢|²)=𝑐|𝑢|^2𝑞1+𝑑|𝑢|^2𝑞2 with 𝑐<0, 𝑑>0 and 𝑞₂>2/𝑁, 𝑞₂>𝑞₁>0.

Example 16. Consider that 𝑊(𝑥)≡0, 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝ln(1+|𝑢|²) with 𝑏>0 and 𝑝>2/𝑁.

Example 17. Let 𝑓(𝑥,|𝑢|²) be one of those in Examples 15 and 16. Let 𝑊(𝑥)=1|𝑥|^𝑁𝑙,|𝑥|𝜑(𝑥)|𝑥|1≤1,,1≤≤2,|𝑥|^𝐾,|𝑥|≥2, (61) where 2<𝑁𝑙<𝑁/𝑞<𝐾 and 𝜑(𝑥) satisfies 𝑁𝑙𝜑(𝑥)+𝑥⋅∇𝜑≤0≤𝑐₃𝜑(𝑥)+𝑥⋅∇𝜑 (62) when 1≤|𝑥|≤2 and makes 𝑊(𝑥) smooth. Obviously, 𝑊∈𝐿^𝑞(ℝ^𝑁).

Proof of Theorem 5. We will proceed in four steps.
Step  1. We will prove 𝑑_𝐼>0. 𝑢∈𝐻¹(ℝ^𝑁)⧵{0} and 𝑄(𝑢)=0 mean that 2_ℝ𝑁|∇𝑢|²𝑑𝑥=𝑁_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²−𝐹𝑥,|𝑢|²−1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²≤𝑁()𝑑𝑥𝑙+1𝑙_ℝ^𝑁𝑐₁|𝑢|^2𝑝1+2+𝑐₂|𝑢|^2𝑝2+2𝑑𝑥+𝐶_ℝ𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²‖𝑢‖𝑑𝑥≤𝐶^2𝑝1+2_2𝑝1+2+𝐶‖𝑢‖^2𝑝2+2_2𝑝2+2+𝐶‖𝑊‖_𝐿^𝑞‖𝑢‖⁴_{𝐿4𝑞/(2𝑞−1)}. (63) Using Gagliardo-Nirenberg’s and Hölder’s inequalities, we can get 2≤𝐶(‖∇𝑢‖²₂)^𝑁𝑝1/2(‖𝑢‖²₂)^{𝑝1+1−(𝑁𝑝1/2)}+𝐶(‖∇𝑢‖²₂)^𝑁𝑝2/2(‖𝑢‖²₂)^{𝑝2+1−(𝑁𝑝2)/2}+𝐶(‖∇𝑢‖²₂)^𝑁/2𝑞(‖𝑢‖²₂)^{(4𝑞−𝑁)/2𝑞}≤𝐶‖∇𝑢‖²₂+‖𝑢‖²₂^𝑝1+1+‖∇𝑢‖²₂+‖𝑢‖²₂^𝑝2+1+‖∇𝑢‖²₂+‖𝑢‖²₂². (64) That is, ‖∇𝑢‖²₂+‖𝑢‖²₂≥𝐶>0 (65) if 𝑄(𝑢)=0 and 𝑢∈𝐻¹(ℝ^𝑁)⧵{0}.
On the other hand, if 𝑄(𝑢)=0, we have 2_ℝ𝑁|∇𝑢|²𝑑𝑥=𝑁_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²−𝐹𝑥,|𝑢|²−1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥≥𝑁𝑙_ℝ𝑁𝐹𝑥,|𝑢|²+𝑑𝑥𝑁𝑙2_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥; (66) that is, −12_ℝ^𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²1𝑑𝑥≥−𝑁𝑙_ℝ^𝑁|∇𝑢|²𝑑𝑥. (67) Using (67), we can obtain 𝜔‖𝑢‖²₂+𝐸(𝑢)=𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²−1𝑑𝑥2_ℝ^𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²‖𝑢‖𝑑𝑥≥𝜔²₂+12−1𝑁𝑙_ℝ^𝑁|∇𝑢|²1𝑑𝑥≥min𝜔,2−1×‖‖𝑁𝑙∇𝑢²₂+‖𝑢‖²₂≥𝐶>0 (68) from (65). Hence 𝑑_𝐼>0. (69)
Step  2. Denote 𝐾₊=𝑢∈𝐻¹ℝ^𝑁⧵{0},𝑄(𝑢)>0,𝜔‖𝑢‖²₂+𝐸(𝑢)<𝑑_𝐼,𝐾₋=𝑢∈𝐻¹ℝ^𝑁⧵{0},𝑄(𝑢)<0,𝜔‖𝑢‖²₂+𝐸(𝑢)<𝑑_𝐼. (70) We will prove that 𝐾₊ and 𝐾₋ are invariant sets of (1) with 𝑉(𝑥)≡0 and 𝑊∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2. That is, we need to show that 𝑢(⋅,𝑡)∈𝒦 for all 𝑡∈(0,𝑇_max) if 𝑢₀∈𝐾₊. Since ‖𝑢‖₂ and 𝐸(𝑢) are conservation quantities for (1), we have 𝑢(⋅,𝑡)∈𝐻¹ℝ^𝑁⧵{0},𝜔‖𝑢(⋅,𝑡)‖²₂+𝐸(𝑢(⋅,𝑡))<𝑑_𝐼 (71) for all 𝑡∈(0,𝑇_max) if 𝑢₀∈𝐾₊. We need to prove that 𝑄(𝑢(⋅,𝑡))>0. Otherwise, assume that there exists a 𝑡₁∈(0,𝑇_max) satisfying 𝑄(𝑢(⋅,𝑡₁))=0 by the continuity. Note that (71) implies 𝜔‖𝑢(⋅,𝑡)‖²₂+𝐸𝑢⋅,𝑡₁<𝑑_𝐼. (72) However, the inequality above and 𝑄(𝑢(⋅,𝑡₁))=0 are contradictions to the definition of 𝑑_𝐼. Therefore, 𝑄(𝑢(⋅,𝑡))>0. Consequently, (71) and 𝑄(𝑢(⋅,𝑡))>0 imply that 𝑢(⋅,𝑡)∈𝐾₊. That is, 𝐾₊ is an invariant set of (1) with 𝑉(𝑥)≡0 and 𝑊∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2. Similarly, we can prove that 𝐾₋ is also an invariant set of (1) with 𝑉(𝑥)≡0 and 𝑊∈𝐿^𝑞(ℝ^𝑁) with 𝑁/4<𝑞<𝑁/2.
Step  3. Assume that 𝑄(𝑢₀)>0 and 𝜔‖𝑢₀‖²₂+𝐸(𝑢₀)<𝑑_𝐼. By the results of Step 2, we have 𝑄(𝑢(⋅,𝑡))>0 and 𝜔‖𝑢(⋅,𝑡)‖²₂+𝐸(𝑢(⋅,𝑡))<𝑑_𝐼. That is, −2‖∇𝑢(⋅,𝑡)‖²₂<−𝑁_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²−𝐹𝑥,|𝑢|²+1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥<−𝑁𝑙_ℝ𝑁𝐹𝑥,|𝑢|²−𝑑𝑥𝑁𝑙2_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑑𝑥,_𝐼>𝜔‖𝑢(⋅,𝑡)‖²₂+12‖∇𝑢(⋅,𝑡)‖²₂−12_ℝ^𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥. (73) The two inequalities imply that 𝜔‖𝑢(⋅,𝑡)‖²₂+12−1‖(𝑁𝑙∇𝑢⋅,𝑡)‖²₂<𝑑_𝐼, (74) which means that ‖𝑢(⋅,𝑡)‖_𝐻¹_(ℝ^𝑁₎<∞; (75) that is, the solution exists globally.
Step  4. Assume that 𝑄(𝑢₀)<0 and 𝜔‖𝑢₀‖²₂+𝐸(𝑢₀)<𝑑_𝐼. By the results of Step 2, we obtain 𝑄(𝑢(⋅,𝑡))<0 and 𝜔‖𝑢(⋅,𝑡)‖²₂+𝐸(𝑢(⋅,𝑡))<𝑑_𝐼. Hence we get 𝐽^′′(𝑡)=4𝑄(𝑢)<0,𝐽^′(0)=4ℑ_ℝ𝑁𝑥⋅∇𝑢₀𝑢₀𝑑𝑥<0. (76) By the results of Proposition 10, the solution will blow up in finite time.

As a corollary of Theorem 5, we obtain the sharp threshold for global existence and blowup of the solution of (8) as follows.

Corollary 18. Assume that 𝑓(𝑥,0)=0 and (23). Let 𝜔 be a positive constant satisfying 𝑑^′_𝐼∶=inf_{{0}𝑢∈Σ⧵;𝑄1(𝑢)=0}𝜔‖𝑢‖²₂+𝐸(𝑢)>0. (77) Here 𝑄₁(𝑢)∶=2_ℝ𝑁|∇𝑢|²𝑑𝑥+𝑁_ℝ𝑁𝐹𝑥,|𝑢|²−|𝑢|²𝑓𝑥,|𝑢|²𝑑𝑥. (78)
Suppose that 𝑢₀∈𝐻¹(ℝ^𝑁) satisfies 𝜔𝑢₀²₂+𝐸𝑢₀<𝑑^′_𝐼. (79)
Then(1)if 𝑄₁(𝑢₀)>0, the solution of (8) exists globally;(2)if 𝑄₁(𝑢₀)<0, |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), and ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, the solution of (8) blows up in finite time.

Remark 19. In Theorem 1.5 of [10], Tao et al. proved the following.
Assume that 𝑢(𝑥,𝑡) is a solution of (8) with 𝑓(𝑥,|𝑢|²)𝑢=𝜇|𝑢|^𝑝1𝑢+𝜈|𝑢|^𝑝2𝑢, where 𝜇>0, 𝜈>0, 4/𝑁≤𝑝₁<𝑝₂≤4/(𝑁−2) with 𝑁≥3, ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), and 𝐸(𝑢₀)<0. Then blowup occurs.
Corollary 18 covers the result above under some conditions. In fact, if 𝑓(𝑥,|𝑢|²)𝑢=𝜇|𝑢|^𝑝1𝑢+𝜈|𝑢|^𝑝2𝑢, then 𝑄₁(𝑢)=4𝐸(𝑢)−𝑁𝑝₁−4𝜇𝑝₁‖𝑢‖+2^{𝑝1𝑝+21+2}−𝑁𝑝₂−4𝜈𝑝₂‖𝑢‖+2^{𝑝2𝑝+22+2}≤𝐸(𝑢); (80) hence 𝐸(𝑢₀)<0 implies that 𝑄₁(𝑢₀)<0. That is, our blowup condition is weaker than theirs. On the other hand, our conclusion is still true if 0<𝐸(𝑢₀)<𝑑^′_𝐼−𝜔‖𝑢₀‖²₂ with 𝑄₁(𝑢₀)<0, ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁). In other words, our result is stronger than theirs if 𝜔‖𝑢₀‖²₂+𝐸(𝑢₀)<𝑑^′_𝐼 with 𝑄₁(𝑢₀)<0, ℑ∫_ℝ𝑁(𝑥⋅∇𝑢₀)𝑢₀𝑑𝑥<0, and |𝑥|𝑢₀∈𝐿²(ℝ^𝑁).

5. Sharp Threshold for the Blowup and Global Existence of the Solution to (1)

Theorem 7 is inspired by [5], but it extends the results to more general case. We need subtle estimates and more sophisticated analysis in the proof.

First, we would like to give some examples of 𝑉(𝑥), 𝑓(𝑥,|𝑢|²), and 𝑊(𝑥). It is easy to verify that they satisfy the conditions of Theorem 7.

Example 20. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=𝑎|𝑥|^−𝐾 with 2<𝑁𝑙<𝐾<𝑁/𝑞<4 for 𝑥∈ℝ^𝑁 and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝1+𝑐|𝑢|^2𝑝2 with 𝑎≥0, 𝑏>0, 𝑐>0, and 𝑝₂>𝑝₁>2/𝑁.

Example 21. Consider that 𝑉(𝑥)=|𝑥|², 𝑊(𝑥)=𝑎|𝑥|^−𝐾 with 2<𝑁𝑙<𝐾<𝑁/𝑞<4 for 𝑥∈ℝ^𝑁 and 𝑓(𝑥,|𝑢|²)=𝑐|𝑢|^2𝑞1+𝑑|𝑢|^2𝑞2 with 𝑎≥0, 𝑐 is a real number, 𝑑>0, and 𝑞₂>2/𝑁, 𝑞₂>𝑞₁>0.

Example 22. Consider that 𝑉(𝑥)=|𝑥|²/(1+|𝑥|²), 𝑊(𝑥)=𝑎|𝑥|^−𝐾 with 2<𝑁𝑙<𝐾<𝑁/𝑞<4 for 𝑥∈ℝ^𝑁 and 𝑓(𝑥,|𝑢|²)=𝑏|𝑢|^2𝑝ln(1+|𝑢|²) with 𝑎≥0, 𝑏>0, and 𝑝>2/𝑁.

5.1. Some Invariant Manifolds

In this subsection, we will prove that 𝑑_𝒩,𝑑_ℳ,𝑑_𝐼𝐼>0 and construct some invariant manifolds.

Proposition 23. Assume that the conditions of Theorem 7 hold. Then 𝑑_𝒩>0.

Proof. Assume that 𝑢∈Σ⧵{0} satisfying 𝑆_𝜔(𝑢)=0. Using Gagliardo-Nirenberg’s and Young’s inequalities, we have 2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²=𝑑𝑥_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²+𝑑𝑥_ℝ𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²≤𝑑𝑥𝑙+1𝑙_ℝ^𝑁𝑐₁|𝑢|^2𝑝1+2+𝑐₂|𝑢|^2𝑝2+2+𝑊𝑑𝑥_1𝐿∞‖𝑢‖⁴₂+𝑊_2𝐿𝑞‖𝑢‖⁴_{𝐿4𝑞/(2𝑞−1)}≤𝐶(‖∇𝑢‖²₂)^𝑁𝑝1/2(‖𝑢‖²₂)^{𝑝1+1−(𝑁𝑝1/2)}+𝐶(‖∇𝑢‖²₂)^𝑁𝑝2/2(‖𝑢‖²₂)^{𝑝2+1−(𝑁𝑝2/2)}+𝑊_1𝐿∞‖𝑢‖⁴₂+𝐶𝑊_2𝐿𝑞‖∇𝑢‖₂^𝑁/𝑞‖𝑢‖₂^{(4𝑞−𝑁)/𝑞}≤𝐶(‖∇𝑢‖²₂)^𝑁𝑝1/2(‖𝑢‖²₂)^{𝑝1+1−(𝑁𝑝1/2)}+𝐶(‖∇𝑢‖²₂)^𝑁𝑝2/2(‖𝑢‖²₂)^{𝑝2+1−(𝑁𝑝2/2)}+𝐶‖𝑢‖⁴₂+‖∇𝑢‖⁴₂+𝐶𝑊_2𝐿𝑞‖𝑢‖⁴₂. (81) Using Hölder’s inequality, from (81), we can obtain 2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²‖𝑢‖𝑑𝑥≤𝐶2𝜔²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥^𝑝1+1+𝐶2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥^𝑝2+1+𝐶2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥². (82) Equation (82) implies that 2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥≥𝐶>0 (83) for some positive constant 𝐶.
On the other hand, if 𝑆_𝜔(𝑢)=0, we get 𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²=1𝑑𝑥2_ℝ^𝑁𝑓𝑥,|𝑢|²|𝑢|²+1𝑑𝑥2_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²(1𝑑𝑥≥min𝑙+1,2)2_ℝ^𝑁𝐹𝑥,|𝑢|²+1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥. (84) From (84), we obtain 𝐼_𝜔(𝑢)=𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝐹𝑥,|𝑢|²|𝑢|𝑑𝑥−𝐺²𝑙≥min2(),1𝑙+14×2𝜔‖𝑢‖²₂+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥≥𝐶>0. (85)
Consequently, 𝑑_𝒩=inf_𝒩𝐼_𝜔(𝑢)>𝐶>0. (86)

Now, we will give some properties of 𝐼_𝜔(𝑢), 𝑆_𝜔(𝑢), and 𝑄(𝑢). We have a proposition as follows.

Proposition 24. Assume that 𝑄(𝑢) and 𝑆_𝜔(𝑢) are defined by (22) and (28). Then one has the following.(i)There at least exists a 𝑤^⋆∈Σ⧵{0} such that 𝑆_𝜔𝑤^⋆=0,𝑄𝑤^⋆=0. (87)(ii)There at least exists a 𝑢^∗∈Σ⧵{0} such that 𝑆_𝜔𝑢^∗<0,𝑄𝑢^∗=0. (88)

Proof. (i) Noticing the assumptions on 𝑉(𝑥), 𝑊(𝑥), and 𝑓(𝑥,|𝑢|²), similar to the proof of Theorem 1.7 in [17], it is easy to prove that there exists a 𝑤^⋆∈Σ⧵{0} satisfying 2𝜔𝑤^⋆+𝑉(𝑥)𝑤^⋆−Δ𝑤^⋆=𝑓𝑥,𝑤^⋆2𝑤^⋆+𝑊(𝑥)⋆𝑤^⋆2𝑤^⋆inℝ^𝑁. (89) Multiplying (89) by 𝑤^⋆ and integrating over ℝ^𝑁 by part, we can get 𝑆_𝜔(𝑤^⋆)=0.
Multiplying (89) by (𝑥⋅∇𝑤^⋆) and integrating over ℝ^𝑁 by part, we obtain Pohozaev’s identity: 𝑁𝜔𝑤^⋆2₂+𝑁−22_ℝ^𝑁∇𝑤^⋆2+𝑁𝑑𝑥2_ℝ^𝑁𝑉(𝑥)𝑤^⋆2+1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑉(𝑥))𝑤^⋆2=𝑁𝑑𝑥2_ℝ^𝑁𝐹𝑥,𝑤^⋆2+𝑁𝑑𝑥2_ℝ^𝑁𝑊(𝑥)⋆𝑤^⋆2𝑤^⋆2+1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^⋆2𝑤^⋆2𝑑𝑥. (90) From 𝑆_𝜔(𝑤^⋆)=0 and (90), we can get 𝑄(𝑤^⋆)=0.
(ii) Letting 𝑣_𝑘,𝜆(𝑥)=𝑘𝑤^⋆(𝜆𝑥) for 𝑘>0 and 𝜆>0, we can obtain 𝑆_𝜔𝑣_𝑘,𝜆=2𝜔𝑘²_ℝ𝑁𝑤^⋆(𝜆𝑥)²𝑑𝑥+𝑘²_ℝ𝑁∇𝑤^⋆(𝜆𝑥)²𝑑𝑥+𝑘²_ℝ𝑁𝑉(𝑥)𝑤^⋆(𝜆𝑥)²𝑑𝑥−𝑘²_ℝ𝑁𝑤^⋆(𝜆𝑥)²𝑓𝑥,𝑘²𝑤^⋆(𝜆𝑥)²𝑑𝑥−𝑘⁴_ℝ𝑁𝑊(𝑥)⋆𝑤^⋆(𝜆𝑥)²|𝑤^⋆(𝜆𝑥)|²𝑄𝑣𝑑𝑥,_𝑘,𝜆=2𝑘²_ℝ𝑁|∇𝑤^⋆(𝜆𝑥)|²𝑑𝑥−𝑘²_ℝ𝑁(𝑥⋅∇𝑉(𝑥))𝑤^⋆(𝜆𝑥)²𝑑𝑥−𝑁_ℝ𝑁𝑘²𝑤^⋆(𝜆𝑥)²𝑓𝑥,𝑘²𝑤^⋆(𝜆𝑥)²−𝐹𝑥,𝑘²𝑤^⋆(𝜆𝑥)²+𝑘𝑑𝑥₄2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^⋆(𝜆𝑥)²×𝑤^⋆(𝜆𝑥)²𝑑𝑥. (91)(92) Looking at 𝑆_𝜔(𝑣_𝑘,𝜆) and 𝑄(𝑣_𝑘,𝜆) as the functions of (𝑘,𝜆), setting 𝑔(𝑘,𝜆)=𝑆_𝜔(𝑣_𝑘,𝜆) and 𝜂(𝑘,𝜆)=𝑄(𝑣_𝑘,𝜆), we get that 𝑔(1,1)=0 and 𝜂(1,1)=0. We want to prove that there exists a pair of (𝑘,𝜆) such that 𝑔(𝑘,𝜆)=𝑆_𝜔(𝑣_𝑘,𝜆)<0 and 𝜂(𝑘,𝜆)=𝑄(𝑣_𝑘,𝜆)=0. Since 𝜂(1,1)=0, we know that the image of 𝜂(𝑘,𝜆) and the plane 𝜂=0 intersect in the space of (𝑘,𝜆,𝜂) and form a curve 𝜂(𝑘,𝜆)=0. Hence there exist many positive real number pairs (𝑘,𝜆) relying on 𝑤^⋆ such that 𝑄(𝑣_𝑘,𝜆)=0 near (1,1) with 𝑘>1. On the other hand, under the assumptions of 𝑉(𝑥) and 𝑊(𝑥), it is easy to see that 𝑔(𝑘,1)<0 for any 𝑘>1. By the continuity, we can choose that a pair of (𝑘,𝜆) near (1,1) with 𝑘>1 satisfies both 𝑄(𝑣_𝑘,𝜆)=0 and 𝑆_𝜔(𝑣_𝑘,𝜆)<0. Letting 𝑢^∗=𝑣_𝑘,𝜆 for this (𝑘,𝜆), we get that 𝑆_𝜔(𝑢^∗)<0 and 𝑄(𝑢^∗)=0.

Proposition 24 means that 𝒞ℳ is not empty and 𝑑_ℳ is well defined. Moreover, we have the following.

Proposition 25. Assume that the conditions of Theorem 7 hold. Then 𝑑_ℳ>0.

Proof. 𝑢∈Σ⧵{0} and 𝑆_𝜔(𝑢)<0 imply that 2𝜔_ℝ𝑁|𝑢|²𝑑𝑥+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²<𝑑𝑥_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²+𝑑𝑥_ℝ𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²≤𝑑𝑥𝑙+1𝑙_ℝ^𝑁𝑐₁|𝑢|^2𝑝1+2+𝑐₂|𝑢|^2𝑝2+2+𝑊𝑑𝑥_1𝐿∞‖𝑢‖⁴_𝐿2+𝐶𝑊_1𝐿𝑞‖∇𝑢‖_𝐿^𝑁/𝑞₂‖𝑢‖_𝐿^{(4𝑞−𝑁)/𝑞}₂. (93) Similar to (81) and (82), from (93), we have 2𝜔_ℝ𝑁|𝑢|²𝑑𝑥+_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥≥𝐶>0. (94) On the other hand, if 𝑄(𝑢)=0, we have 2_ℝ𝑁|∇𝑢|²𝑑𝑥−_ℝ𝑁(𝑥⋅∇𝑉(𝑥))|𝑢|²𝑑𝑥=𝑁_ℝ𝑁|𝑢|²𝑓𝑥,|𝑢|²−𝐹𝑥,|𝑢|²−1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥≥𝑁𝑙_ℝ𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥; (95) that is, −12_ℝ^𝑁𝐹𝑥,|𝑢|²+1𝑑𝑥4𝑁𝑙_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²1𝑑𝑥≥−𝑁𝑙_ℝ^𝑁|∇𝑢|²1𝑑𝑥+2𝑁𝑙_ℝ^𝑁(𝑥⋅∇𝑉(𝑥))|𝑢|²𝑑𝑥. (96) Using (23), (35), (36), (94), and (96), we can get 𝐼_𝜔(𝑢)=𝜔_ℝ𝑁|𝑢|²+1𝑑𝑥2_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥≥𝜔_ℝ𝑁|𝑢|²𝑑𝑥+𝑁𝑙−22𝑁𝑙_ℝ^𝑁|∇𝑢|²+1𝑑𝑥2𝑁𝑙_ℝ^𝑁[𝑁𝑙𝑉(𝑥)+(𝑥⋅∇𝑉(𝑥))]|𝑢|²−1𝑑𝑥4𝑁𝑙_ℝ^𝑁[𝑁𝑙𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥))]⋆|𝑢|²×|𝑢|²𝑑𝑥≥𝐶2𝜔_ℝ𝑁|𝑢|²+𝑑𝑥_ℝ𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥≥𝐶>0. (97) Consequently, 𝑑_ℳ=inf_𝒞ℳ𝐼_𝜔(𝑢)>𝐶>0. (98)

By the conclusions of Proposition 23 and Proposition 25, we have 𝑑_𝐼𝐼=min𝑑_ℳ,𝑑_𝒩>0. (99)

Now we define the following manifolds: 𝒦∶=𝑢∈Σ⧵{0}∶𝐼_𝜔(𝑢)<𝑑_𝐼𝐼,𝑆_𝜔(𝑢)(𝑢),𝒦<0,𝑄<0₊∶=𝑢∈Σ⧵{0}∶𝐼_𝜔(𝑢)<𝑑_𝐼𝐼,𝑆_𝜔(𝑢)(𝑢),ℛ<0,𝑄>0₊∶=𝑢∈Σ⧵{0}∶𝐼_𝜔(𝑢)<𝑑_𝐼𝐼,𝑆_𝜔(𝑢)>0. (100)(101)(102) The following proposition will show some properties of 𝒦, 𝒦₊, and ℛ₊.

Proposition 26. Assume that the conditions of Theorem 7 hold. Then(i)𝒦, 𝒦₊, and ℛ₊ are not empty;(ii)𝒦, 𝒦₊, and ℛ₊ are invariant manifolds of (1).

Proof. (i) In order to prove 𝒦 is not empty, we only need to find that there at least exists a 𝑤∈𝒦. For 𝑤^⋆∈Σ⧵{0} satisfying 𝑆_𝜔(𝑤^⋆)=0 and 𝑄(𝑤^⋆)=0, letting 𝑤_𝜌=𝜌𝑤^⋆ for 𝜌>0, we have 𝑆_𝜔𝑤_𝜌=𝜌²_ℝ𝑁2𝜔|𝑤^⋆|²+|∇𝑤^⋆|²+𝑉(𝑥)|𝑤^⋆|²−𝑑𝑥_ℝ𝑁𝜌²𝑤^⋆2𝑓𝑥,𝜌²𝑤^⋆2𝑑𝑥−𝜌⁴_ℝ𝑁𝑊(𝑥)⋆𝑤^⋆2𝑤^⋆2𝑄𝑤𝑑𝑥,_𝜌=𝜌²_ℝ𝑁2∇𝑤^⋆2−(𝑥⋅∇𝑉(𝑥))𝑤^⋆2𝑑𝑥+𝑁_ℝ𝑁𝐹𝑥,𝜌²𝑤^⋆2−𝜌²𝑤^⋆2𝑓𝑥,𝜌²𝑤^⋆2+1𝑑𝑥2𝜌⁴_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^⋆2𝑤^⋆2𝐼𝑑𝑥,_𝜔𝑢_𝜌=12𝜌²_ℝ𝑁2𝜔|𝑤^⋆|²+|∇𝑤^⋆|²+𝑉(𝑥)𝑤^⋆2−1𝑑𝑥2_ℝ^𝑁𝐹𝑥,𝜌²𝑤^⋆2−1𝑑𝑥4𝜌⁴_ℝ𝑁𝑊(𝑥)⋆𝑤^⋆2𝑤^⋆2𝑑𝑥. (103)
Since 𝑓(𝑥,|𝑤^∗|²)<𝑓(𝑥,𝜌²|𝑤^∗|²) and 𝜌²𝐹(𝑥,|𝑤^∗|²)<𝐹(𝑥,𝜌²|𝑤^∗|²) for 𝜌>1 and from (38), we can obtain 𝑆_𝜔𝑤_𝜌<𝜌²𝑆_𝜔𝑤^⋆=0,𝑄𝑤_𝜌<𝜌²𝑄𝑤^⋆=0 (104) for any 𝜌>1. Noticing 𝑑_𝐼𝐼>0, we also can choose 𝜌>1 closing to 1 enough such that 𝐼_𝜔𝑤_𝜌<𝜌²𝐼_𝜔𝑤^⋆<𝑑_𝐼𝐼. (105) Equations (104) and (105) mean that 𝑤_𝜌∈𝒦. That is, 𝒦 is not empty.
Similar to (104), we can obtain 𝑆_𝜔𝑤_𝜌>𝜌²𝑆_𝜔𝑤^⋆=0 (106) for any 0<𝜌<1. Noticing 𝑑_𝐼𝐼>0, we also can choose 0<𝜌<1 closing to 1 enough such that 𝐼_𝜔(𝑤_𝜌)<𝑑_𝐼𝐼 by continuity, which implies that 𝑤_𝜌∈ℛ₊. That is, ℛ₊ is not empty.
For 𝑤^∗∈Σ satisfying 𝑆_𝜔(𝑤^∗)<0 and 𝑄(𝑤^∗)=0, letting 𝑤_𝜎=𝜎𝑤^∗ for 𝜎>0, we have 𝑄𝑤_𝜎=𝜎²_ℝ𝑁2∇𝑤^∗2−(𝑥⋅∇𝑉(𝑥))𝑤^∗2−𝑑𝑥_ℝ𝑁𝑁𝜎²𝑤^∗2𝑓𝑥,𝜎²𝑤^∗2−𝐹𝑥,𝜎²𝑤^∗2+1𝑑𝑥2𝜎⁴_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑆𝑑𝑥,_𝜔𝑤_𝜎=𝜎²_ℝ𝑁2𝜔𝑤^∗2+∇𝑤^∗2+𝑉(𝑥)𝑤^∗2−𝑑𝑥_ℝ𝑁𝜎²𝑤^∗2𝑓𝑥,𝜎²𝑤^∗2𝑑𝑥−𝜎⁴_ℝ𝑁𝑊(𝑥)⋆𝑤^∗2𝑤^∗2𝐼𝑑𝑥,_𝜔𝑤_𝜎=12𝜎²_ℝ𝑁2𝜔𝑤^∗2+∇𝑤^∗2+𝑉(𝑥)𝑤^∗2−1𝑑𝑥2_ℝ^𝑁𝐹𝑥,𝜎²𝑤^∗2−1𝑑𝑥4𝜎⁴_ℝ𝑁𝑊(𝑥)⋆𝑤^∗2𝑤^∗2𝑑𝑥. (107)
Since 𝜙(𝜎)=𝑄(𝑤_𝜎) is a smooth function of 𝜎 and 𝑄(𝑤^∗)=0, we have 𝜙(1)=0. If 𝜙^′(1)≠0, then there exists a 𝜎₀>0 such that 𝑄(𝑢_𝜎)=𝜙(𝜎)>0 for 𝜎∈(1,𝜎₀) if 𝜎₀>1 (or 𝜎∈(𝜎₀,1) if 𝜎₀<1). By continuity, we can choose such 𝜎₀ closing to 1 enough such that 𝑆_𝜔(𝑤_𝜎)<0 and 𝐼_𝜔(𝑤_𝜎)<𝑑_𝐼𝐼 for 𝜎∈(1,𝜎₀) if 𝜎₀>1 (or 𝜎∈(𝜎₀,1) if 𝜎₀<1). That is, 𝑤_𝜎∈𝒦₊ and 𝒦₊ is not empty.
If 𝜙^′(1)=0, from 𝜙(1)=0 and 𝜙^′(1)=0, we can, respectively, obtain −𝑁_ℝ𝑁𝑤^∗2𝑓𝑥,𝑤^∗2−𝐹𝑥,𝑤^∗2𝑑𝑥=−𝑁_ℝ𝑁𝑤^∗4𝑓^′_𝑠𝑥,𝑤^∗2+1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑄𝑤𝑑𝑥,_∗=_ℝ𝑁2∇𝑤^∗2−(𝑥⋅∇𝑉(𝑥))𝑤^∗2−𝑁𝑤^∗4𝑓^′_𝑠𝑥,𝑤^∗2+𝑑𝑥_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑑𝑥. (108)(109)
Letting 𝑤_𝜎=𝜎𝑤^∗, we have 𝑄𝑤_𝜎=𝜎²_ℝ𝑁2∇𝑤^∗2−(𝑥⋅∇𝑉(𝑥))𝑤^∗2−𝑁𝑤^∗4𝑓^′_𝑠𝑥,𝜎²𝑤^∗2𝑑𝑥+𝜎⁴_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑑𝑥>𝜎²_ℝ𝑁2∇𝑤^∗2−(𝑥⋅∇𝑉(𝑥))𝑤^∗2−𝑁𝑤^∗4𝑓^′_𝑠𝑥,𝑤^∗2𝑑𝑥+𝜎⁴_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑑𝑥=𝜎²𝑄𝑤^∗+𝜎⁴−𝜎²×_ℝ𝑁(𝑥⋅∇𝑊(𝑥))⋆𝑤^∗2𝑤^∗2𝑑𝑥>0 (110) for 0<𝜎<1. By continuity, we can choose such 𝜎 closing to 1 enough such that 𝑆_𝜔(𝑤_𝜎)<0 and 𝐼_𝜔(𝑤_𝜎)<𝑑_𝐼𝐼. That is to say, 𝑤_𝜎∈𝒦₊ and 𝒦₊ is not empty.
(ii) In order to prove that 𝒦 is the invariant manifold of (1), we need to show that, if 𝑢₀∈𝒦, then solution 𝑢(𝑥,𝑡) of (1) satisfies 𝑢(𝑥,𝑡)∈𝒦 for any 𝑡∈[0,𝑇).
Assume that 𝑢(𝑥,𝑡) is a solution of (1) with 𝑢₀∈𝒦. Then we can obtain 𝐼_𝜔(𝑢(⋅,𝑡))=𝐸(𝑢(⋅,𝑡))+𝜔‖𝑢(⋅,𝑡)‖²₂=𝐸𝑢₀+𝜔𝑢₀²₂=𝐼_𝜔𝑢₀<𝑑_𝐼𝐼 (111) for 𝑡∈[0,𝑇). Next we prove that 𝑆_𝜔(𝑢(⋅,𝑡))<0 for 𝑡∈[0,𝑇). Otherwise, by continuity, there exists a 𝑡₀∈(0,𝑇) such that 𝑆_𝜔(𝑢(⋅,𝑡₀))=0 because of 𝑆_𝜔(𝑢₀)<0. Since ‖𝑢(⋅,𝑡)‖²₂=‖𝑢₀‖²₂ and 𝑢₀∈Σ⧵{0}, it is easy to see that 𝑢(⋅,𝑡₀)∈Σ⧵{0}. By the definitions of 𝑑_𝒩 and 𝑑_𝐼𝐼, we know that 𝐼_𝜔(𝑢(⋅,𝑡₀))≥𝑑_𝒩≥𝑑_𝐼𝐼, which is a contradiction to 𝐼_𝜔(𝑢(⋅,𝑡))<𝑑_𝐼𝐼 for 𝑡∈[0,𝑇). Hence 𝑆_𝜔(𝑢(⋅,𝑡))<0 for all 𝑡∈[0,𝑇).
Now we only need to prove that 𝑄(𝑢(⋅,𝑡))<0 for 𝑡∈[0,𝑇). Otherwise, since 𝑄(𝑢₀)<0, there exists a 𝑡₁∈(0,𝑇) such that 𝑄(𝑢(⋅,𝑡₁))=0 by continuity. 𝑆_𝜔(𝑢(⋅,𝑡₁))<0 means that 𝑢(⋅,𝑡₁)∈𝒞ℳ. By the definitions of 𝑑_ℳ and 𝑑_𝐼𝐼, we obtain 𝐼_𝜔(𝑢(⋅,𝑡₁))≥𝑑_ℳ≥𝑑_𝐼𝐼, which is a contradiction to 𝐼_𝜔(𝑢(⋅,𝑡))<𝑑_𝐼𝐼 for 𝑡∈[0,𝑇). Hence 𝑄(𝑢(⋅,𝑡))<0 for all 𝑡∈[0,𝑇).
By the discussions above, we know that 𝑢(𝑥,𝑡)∈𝒦 for any 𝑡∈[0,𝑇) if 𝑢₀∈𝒦, which means that 𝒦 is the invariant manifold of (1).
Similarly, we can prove that 𝒦₊, and ℛ₊ are also invariant manifolds of (1).

Remark 27. By the definitions of 𝑑_𝐼𝐼, 𝑑_𝒩, 𝑑_ℳ, 𝒦, 𝒦₊, and ℛ₊, it is easy to see that 𝑢∈Σ⧵{0}∶𝐼_𝜔(𝑢)<𝑑_𝐼𝐼=𝒦∪𝒦₊∪ℛ₊. (112)

5.2. Proof of Theorem 7

Proof of Theorem 7 depends on the following two lemmas.

Lemma 28. Assume that the conditions of Theorem 7 hold. Then the solutions of (1) with 𝑢₀∈𝒦 will blow up in finite time.

Proof. Since 𝑢₀∈𝒦 and 𝒦 is the invariant manifold of (1), we have 𝑄(𝑢(𝑥,𝑡))<0, 𝑆_𝜔(𝑢(𝑥,𝑡))<0, and 𝐼_𝜔(𝑢(𝑥,𝑡))<𝑑_𝐼𝐼.
Under the conditions of Theorem 7, we have 𝐽^′′(𝑡)=4𝑄(𝑢)<0 and 𝐽^′(0)<0. By the results of Proposition 10, the solution 𝑢(𝑥,𝑡) will blow up in finite time. The conclusion of this lemma is true.

On the other hand, we have a parallel result on global existence.

Lemma 29. Assume that the conditions of Theorem 7 hold. If 𝑢₀∈𝒦₊ or 𝑢₀∈ℛ₊, then the solutions of (1) exist globally.

Proof. Case  1. Assume that 𝑢(𝑥,𝑡) is a solution of (1) with 𝑢₀∈𝒦₊. Since 𝒦₊ is an invariant manifold of (1), we know that 𝑢(⋅,𝑡)∈𝒦₊, which means that 𝐼_𝜔(𝑢(⋅,𝑡))<𝑑_𝐼𝐼 and 𝑄(𝑢(⋅,𝑡))>0. 𝑄(𝑢(⋅,𝑡))>0 and (23) imply that 2_ℝ𝑁|∇𝑢|²𝑑𝑥−_ℝ𝑁(𝑥⋅∇𝑉(𝑥))|𝑢|²𝑑𝑥≥𝑁𝑙_ℝ𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥2_ℝ^𝑁(𝑥⋅∇𝑊(𝑥))⋆|𝑢|²|𝑢|²𝑑𝑥. (113) By the definition of 𝐼_𝜔(𝑢) and using (113), we have 𝑑_𝐼𝐼>𝐼_𝜔(𝑢(⋅,𝑡))=𝜔_ℝ𝑁|𝑢|²+1𝑑𝑥2_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−1𝑑𝑥2_ℝ^𝑁𝐹𝑥,|𝑢|²−1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥≥𝜔_ℝ𝑁|𝑢|²𝑑𝑥+𝑁𝑙−22𝑁𝑙_ℝ^𝑁|∇𝑢|²+𝑑𝑥_ℝ𝑁𝑁𝑙𝑉(𝑥)+(𝑥⋅∇𝑉(𝑥))|𝑢|2𝑁𝑙²−1𝑑𝑥4𝑁𝑙_ℝ^𝑁(𝑥)+((𝑥⋆|𝑢|[𝑁𝑙𝑊𝑥⋅∇𝑊))]₂|𝑢|²𝑑𝑥≥𝐶_ℝ𝑁|𝑢|²𝑑𝑥+_ℝ𝑁|∇𝑢|²𝑑𝑥+_ℝ𝑁𝑉(𝑥)|𝑢|²𝑑𝑥. (114) Equation (114) means that 𝑢(𝑥,𝑡) exists globally.
Case  2. Assume that 𝑢(𝑥,𝑡) is a solution of (1) with 𝑢₀∈ℛ₊. Since ℛ₊ is also an invariant manifold of (1), we know that 𝑢(𝑥,𝑡),∈ℛ₊, which means that 𝐼_𝜔(𝑢(⋅,𝑡))<𝑑_𝐼𝐼 and 𝑆_𝜔(𝑢(⋅,𝑡))>0. Since 𝑆_𝜔(𝑢)>0, we can get 𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²>1𝑑𝑥2_ℝ^𝑁𝑓𝑥,|𝑢|²|𝑢|²+1𝑑𝑥2_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²(1𝑑𝑥≥min𝑙+1,2)2_ℝ^𝑁𝐹𝑥,|𝑢|²+1𝑑𝑥4_ℝ^𝑁𝑊(𝑥)⋆|𝑢|²|𝑢|²𝑑𝑥. (115) From (115), we can obtain 𝐼_𝜔(𝑢)=𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²−𝐹𝑥,|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑙≥min(),1𝑙+12×𝜔‖𝑢‖²₂+12_ℝ^𝑁|∇𝑢|²+𝑉(𝑥)|𝑢|²𝑑𝑥. (116) Equation (116) implies that the solution 𝑢(𝑥,𝑡) exists globally.

Proof of Theorem 7. By the results of Lemmas 28 and 29, we know that Theorem 7 is right.

As a corollary of Theorem 7, we obtain a sharp threshold for the blowup in finite time and global existence of the solution of (9) as follows.

Corollary 30. Assume that 𝑓(𝑥,|𝑢|²)≡0, 𝑉(𝑥)≡0, 𝑊(𝑥)>0 for all 𝑥∈ℝ^𝑁, 𝑊(𝑥) is even, and 𝑊(𝑥)∈𝐿^∞(ℝ^𝑁)+𝐿^𝑞(ℝ^𝑁) with some 𝑞>𝑁/4. Suppose further that there exists 𝑙 satisfying 2<𝑁𝑙 and 𝑁𝑙𝑊(𝑥)+(𝑥⋅∇𝑊(𝑥))≤0. (117) If 𝑢₀∈𝐻¹(ℝ^𝑁), |𝑥|𝑢₀∈𝐿²(ℝ^𝑁), and 𝐼_𝜔(𝑢₀)=𝜔‖𝑢₀‖²₂+𝐸(𝑢₀)<𝑑_𝐼𝐼, then the solution of (9) blows up in finite time if and only if 𝑢₀∈𝒦.

Remark 31. A typical example is −𝑖𝑢_𝑡=Δ𝑢+|𝑥|^−𝐾⋆|𝑢|²𝑢,𝑥∈ℝ^𝑁𝑢(),𝑡>0,𝑥,0=𝑢₀(𝑥),𝑥∈ℝ^𝑁, (118) which is also a special case of (1) with 𝑉(𝑥)≡0, 𝑓(𝑥,|𝑢|²)≡0, and 𝑊(𝑥)=|𝑥|^−𝐾 with 2<𝑁𝑙<𝐾<𝑁/𝑞<4. Letting 𝑊(𝑥)=𝑊₁(𝑥)+𝑊₂(𝑥) with 𝑊₁(𝑥)=|𝑥||𝑥|0,≤1,_−𝐾,|𝑥|𝑊>1,₂(𝑥)=|𝑥|^−𝐾,|𝑥||𝑥|≤1,0,>1, (119)(120) we can see that 𝑊₁(𝑥)∈𝐿^∞(ℝ^𝑁) and 𝑊₂(𝑥)∈𝐿^𝑞(ℝ^𝑁) with some 𝑁/4<𝑞<𝑁/2. Corollary 30 gives the sharp threshold for blowup and global existence of the solution to (118).

Acknowledgments

The authors are grateful to the referees for their helpful comments. In addition, the second author is supported by the National Natural Science Foundation of China, Grant 11071185 and Natural Science Foundation of Tianjin (09JCYBJC01800). The third author is supported by the National Natural Science Foundation of China, Grant 11071237.