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

Authors: Xiaowei An; Desheng Li; Xianfa Song
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider the following Cauchy problem:-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN, 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.


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## Body

## 1. Introduction

In this paper, we consider the following Cauchy problem:(1)-iut=Δu-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x)∈Σ,x∈ℝN,
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,
(2)(W(x)⋆|u|2)u(x)=(∫ℝN‍W(x-y)|u(y)|2dy)u(x),u0(x) is a complex-valued function of x, and Σ is the Hilbert space:
(3)Σ={u∈H1(ℝN),∫ℝNV(x)|u|2dx<+∞},
with the inner product
(4)〈φ,ψ〉=∫ℝN[φψ-+∇φ·∇ψ-+V(x)φψ-]dx
and the norm
(5)∥u∥Σ2=∫ℝN(|u|2+|∇u|2+V(x)|u|2)dx.
Model (1) appears in the theory of Bose-Einstein condensation, nonlinear optics and theory of water waves (see [1, 2]).For convenience, denote1/(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)∈Lr(ℝN)+L∞(ℝN) for r≥1, r>N/2.(V2)
V(x)≥0, V(x)∈S1c, and |DαV| is bounded for all |α|≥2. Here S1c is the complementary set of S1={V(x)satisfies(V1)}.(f1)
f(x,s):ℝN×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
(6)L(k)∈C([0,∞)),ifN=1,L(k)≤C(1+kα)with0≤α<2(N-2)+,ifN≥2.(W1)
W(x) is even and W(x)∈Lq(ℝN)+L∞(ℝN) 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,Tmax) such that u∈C(Σ;[0,Tmax)) and either Tmax=+∞ or else
(7)Tmax<+∞,limt→Tmax∥u(·,t)∥Σ=+∞.Definition 2.
Ifu∈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 limt→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:
(8)-iut=Δu+f(|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN.
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
(9)-iut=12Δu-|x|2u+K(x)|u|pu+Q(x)|u|qu,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN
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
(10)-iut=12Δu+a|u|pu+E1(|u|2)u,aaaaaaaaaaaaaaax∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,
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 Section2, 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(11)F(x,|u|2)=∫0|u|2‍f(x,s)ds,G(|u|2)=14∫ℝN‍(W(x)⋆|u|2)|u|2dx,(12)h(u)=-V(x)u+f(x,|u|2)u+(W(x)⋆|u|2)u,(13)H(u)=-12∫ℝN‍V(x)|u|2dx+12∫ℝN‍F(x,|u|2)dx+14∫ℝN‍(W(x)⋆|u|2)|u|2dx,mass (L2 norm)
(14)M(u)≔(∫ℝN‍|u(x,t)|2dx)1/2,energy(15)E(u)≔12∫ℝN‍(|∇u|2+V(x)|u|2)dx-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍(W(x)⋆|u|2)|u|2dx.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 thatu0∈Σ, (V2) and (f1) are true, and
(16)W+(x)∈Lq(ℝN)+L∞(ℝN)
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,
(17)∥u(·,t)∥Σ<+∞∀0<t<+∞.Theorem 4 (blowup in finite time).
Assume thatu0∈Σ and |x|u0∈L2(ℝN), (V2), (f1), and (W1) are true. Suppose further that
(18)(N+2)F(x,|u|2)-N|u|2f(x,|u|2)≤0,(19)2V(x)+(x·∇V(x))≥0a.e.,(20)2W(x)+(x·∇W(x))≤0a.e.
If (1) E(u0)<0 or (2) E(u0)=0 and ℑ∫ℝN‍(x·∇u0)u-0dx<0, then the solution of (1) will blow up in finite time. That is, there exists Tmax<∞ such that
(21)limt→Tmax∥u(·,t)∥Σ=∞.Denote(22)Q(u):=2∫ℝN‍|∇u|2dx-∫ℝN‍(x·∇V(x))|u|2dx+N∫ℝN‍[F(x,|u|2)-|u|2f(x,|u|2)]dx+12∫ℝN‍((x·∇W(x))⋆|u|2)|u|2dx.We will establish the first type of sharp threshold as follows.Theorem 5 (sharp threshold I).
Assume thatV(x)≡0 and W(x)∈Lq(ℝN) 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
(23)lF(x,|u|2)≤|u|2f(x,|u|2)-F(x,|u|2)≤c1|u|2p1+2+c2|u|2p2+2,(24)NlW(x)+(x·∇W(x))≤0≤c3W(x)+(x·∇W(x)).
Let ω be a positive constant satisfying
(25)dI≔inf{u∈Σ∖{0};Q(u)=0}(ω∥u∥22+E(u))>0,
where Q(u) is defined by (22). Suppose that u0∈H1(ℝN) satisfies
(26)ω∥u0∥22+E(u0)<dI.
Then(1)
ifQ(u0)>0, the solution of (1) exists globally;(2)
ifQ(u0)<0, |x|u0∈L2(ℝN), and ℑ∫ℝN‍(x·∇u0)u-0dx<0, the solution of (1) blows up in finite time.Remark 6.
Theorem5 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:
(27)Iω(u)=ω∥u∥22+E(u),(28)Sω(u)=2ω∥u∥22+∫ℝN‍{|∇u|2+V(x)|u|2-f(x,|u|2)|u|2aaaaa-(W(x)⋆|u|2)|u|2}dx.Denote the Nehari manifold(29)𝒩≔{u∈Σ∖{0},Sω(u)=0}
and cross-manifold
(30)𝒞ℳ≔{u∈Σ∖{0},Sω(u)<0,Q(u)=0}.
Define
(31)d𝒩≔inf𝒩Iω(u),(32)dℳ≔inf𝒞ℳIω(u),(33)dII≔min(d𝒩,dℳ).In Section5, we will prove that dII is always positive. Therefore, it is reasonable to define the following cross-manifold:
(34)𝒦≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)<0,Q(u)<0}.We give the second type of sharp threshold as follows.Theorem 7 (sharp threshold II).
Assume that (f1), (W1), and (23). Suppose that
(35)W(x)≥0,NlW(x)+(x·∇W(x))≤0
and there exists a positive constant c such that
(36)NlV(x)+(x·∇V(x))≥cV(x)≥0
with the same l in (23). Assume further that the function f(x,|u|2) satisfies f(x,0)=0 and
(37)f(x,|u|2)≤f(x,k2|u|2),fs′(x,k2|u|2)≤fs′(x,|u|2),(38)F(x,k2|u|2)-k2|u|2f(x,k2|u|2)≤k2[F(x,|u|2)-|u|2f(x,|u|2)]
for k>1. Here fs′(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(ℝN) with Iω(u0)=ω∥u0∥22+E(u0)<dII, 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 useC and c to denote various finite constants; their exact values may vary from line to line.First, we will give the proof of Proposition1.Proof of Proposition1.
If (V1) is true, then there existV1(x)∈Lr(ℝN) with r≥1, r>N/2, and V2(x)∈L∞(ℝN) such that
(39)V(x)=V1(x)+V2(x).
Noticing that 0<2r/(r-1)<2N/(N-2), using Hölder’s and Sobolev’s inequalities, we have
(40)∫ℝNV(x)|u|2dx=∫ℝNV1(x)|u|2dx+∫ℝNV2(x)|u|2dx≤(∫ℝN‍|V1(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
for any u∈H1(ℝN). Consequently, we have
(41)∥u∥H1≤∥u∥Σ≤C∥u∥H1.
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 thatℑh(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 thatu(x,t) is a solution of (1). Then
(42)M(u)=(∫ℝN‍|u(x,t)|2dx)1/2=(∫ℝN‍|u0(x)|2dx)1/2=M(u0),E(u)=12∫ℝN‍{|∇u|2+V(x)|u|2-F(x,|u|2)}dx-G(|u|2)=E(u0)
for any 0≤t<Tmax.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
(43)F(|u|2)≤A|u|2(1+|u|2p).
Then the maximal strong H1-solution of (1) is global and sup{∥u∥H1:t∈ℝ}<∞ for every u0∈H1(ℝN).Theorem B (Theorem 6.54 of [1]).
Assume that(V1), (f1), (W1), and (18)–(20). If u0∈H1(ℝN), |x|u0∈L2(ℝN), and E(u0)<0, then the H1-solution of (1) will blow up in finite time.LetJ(t)=∫ℝN‍|x|2|u|2dx. After some elementary computations, we obtain
(44)J′(t)=4ℑ∫ℝN(x·∇u)u-dx,J′′(t)=4Q(u).We have the following proposition.Proposition 10.
Assume thatu(x,t) is a solution of (1) with u0∈Σ and |x|u0∈L2(ℝN). Then the solution to (1) will blow up in finite time if either(1)
there exists a constantc<0 such that J′′(t)=4Q(u)≤c<0 or(2)
J′′(t)=4Q(u)≤0 and J′(0)=ℑ∫ℝN‍(x·∇u0)u-0dx<0.Proof.
Sinceu0∈Σ and |x|u0∈L2(ℝN), we have
(45)|J′(0)|<4∫ℝN‍|xu-0||∇u0|dx≤8∫ℝN‍(|∇u0|2+|xu0|2)dx<+∞.
(1) IfJ′′(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
(46)0≤J(t)=J(t0)+∫t0t‍J′(s)ds<J(t0)+J′(t0)(t-t0),
which implies that there exists a Tmax<+∞ satisfying
(47)limt→TmaxJ(t)=0.
Using the inequality
(48)∥g∥22≤2N∥∇g∥2∥xg∥2ifg∈H1(ℝN),xg∈L2(ℝN)
and noticing that ∥u(·,t)∥2=∥u0∥2, we have
(49)limt→Tmax∥∇u∥2=+∞.
Consequently,
(50)limt→Tmax∥u∥Σ=+∞.
(2) Similar to (46), we can get
(51)0≤J(t)≤J(0)+J′(0)t,
which implies that the solution will blow up in a finite time Tmax≤J(0)/-J′(0).


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

In this section, we will prove Theorems3 and 4, which give some sufficient conditions on global existence and blowup of the solution to (1).We would like to give some examples ofV(x), f(x,|u|2), and W(x). It is easy to verify that they satisfy the conditions of Theorem 3.Example 11.
Consider thatV(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 thatV(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 Theorem3.
LettingW+(x)=W1(x)+W2(x), where W1∈L∞(ℝN) and W2∈Lq(ℝN) with q>N/2, using Hölder’s and Young’s inequalities, we obtain
(52)∫ℝN‍(W2(x)⋆(uv))wzdx≤∥W2∥Lq∥u∥Lr∥v∥Lr∥w∥Lr∥z∥Lr
with r=4q/(2q-1). Specifically, we have
(53)∫ℝN‍(W2(x)⋆|u|2)|u|2dx≤∥W2∥Lq∥u∥Lr4.
Using (53) and Gagliardo-Nirenberg’s inequality, we get
(54)14∫ℝN‍(W(x)⋆|u|2)|u|2dx≤∥W1∥L∞∥u∥L24+∥W2∥Lq∥u∥L4q/(2q-1)4≤∥W1∥L∞∥u∥L24+C∥W2∥Lq∥∇u∥L2N/q∥u∥L2(4q-N)/q.
Using Young’s inequality, from (54), we have
(55)C∥W2∥Lq∥∇u∥L2N/q∥u∥L2(4q-N)/q≤ε∥∇u∥L22+C(ε,∥W2∥Lq)∥u∥L2(8q-2N)/(2q-N)
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
(56)E(u0)=12(∫ℝN‍{|∇u0|2+V(x)|u0|2-F(x,|u0|2)}dx)-14∫ℝN‍(W(x)⋆|u0|2)|u0|2dx=12(∫ℝN‍{|∇u|2+V(x)|u|2-F(x,|u|2)}dx)-14∫ℝN‍(W(x)⋆|u|2)|u|2dx≥12(∫ℝN‍{|∇u|2+V(x)|u|2-c1|u|2-c2|u|2p+2}dx)-∥W1∥L∞∥u∥L24-C∥W2∥Lq∥∇u∥L2N/q∥u∥L2(4q-N)/q≥12(∫ℝN‍{|∇u|2+V(x)|u|2-c1|u|2}dx)-c2CN(∫ℝN‍|∇u|2dx)pN/2(∫ℝN‍|u|2dx)(2+p(2-N))/2-∥W1∥L∞∥u∥L24-14∥∇u∥L22-C∥u∥L2(8q-2N)/(2q-N).
Since ∥u∥2=∥u0∥2, from (56), we can obtain
(57)4E(u0)+C∥u0∥L22+C∥u0∥L24+C∥u0∥L2(8q-2N)/(2q-N)≥∫ℝN‍V(x)|u|2dx+∫ℝN‍|∇u|2dx×(1-c{∫ℝN‍|∇u|2dx}(pN/2)-1).
Since p<2/N means that (pN/2)-1<0, (57) implies that ∥u∥Σ2 is always controlled by 4E(u0)+C∥u0∥L22+C∥u0∥L24+C∥u0∥L2(8-2N)/(2q-N). That is, the solution of (1) exists globally.We would like to give some examples ofV(x), W(x), and f(x,|u|2). It is easy to verify that they satisfy the conditions of Theorem 4.Example 13.
Consider thatV(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 thatV(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 Theorem4.
Set(58)y(t)=J′(t)=4ℑ∫ℝN‍(x·∇u)u-dx.
Using (18)–(20), we have
(59)y′(t)=8∫ℝN‍|∇u|2dx-4∫ℝN‍(x·∇V(x))|u|2dx+4N∫ℝN‍[F(x,|u|2)-|u|2f(x,|u|2)]dx+2∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx=16E(u)+4∫ℝN‍([-2V(x)-(x·∇V(x))]|u|2+[(N+2)F(x,|u|2)-N|u|2f(x,|u|2)])dx+2∫ℝN‍[{2W(x)+(x·∇W(x))}⋆|u|2]|u|2dx≤16E(u)=16E(u0)<0.
From (58) and (59), we obtain
(60)∥xu(x,t)∥L22≤∥xu0∥L22+4tℑ∫ℝN‍u-0(x·∇u0)dx+8t2E(u0).
Since ∥xu(x,t)∥L22≥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∈Lq(ℝN) 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∈Lq(ℝN) with N/4<q<N/2.Before giving the proof of Theorem5, 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 thatW(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 thatW(x)≡0, f(x,|u|2)=b|u|2pln(1+|u|2) with b>0 and p>2/N.Example 17.
Letf(x,|u|2) be one of those in Examples 15 and 16. Let
(61)W(x)={1|x|Nl,|x|≤1,φ(x),1≤|x|≤2,1|x|K,|x|≥2,
where 2<Nl<N/q<K and φ(x) satisfies
(62)Nlφ(x)+(x·∇φ)≤0≤c3φ(x)+(x·∇φ)
when 1≤|x|≤2 and makes W(x) smooth. Obviously, W∈Lq(ℝN).Proof of Theorem5.
We will proceed in four steps.
Step  1. We will prove dI>0. u∈H1(ℝN)∖{0} and Q(u)=0 mean that
(63)2∫ℝN‍|∇u|2dx=N∫ℝN‍[|u|2f(x,|u|2)-F(x,|u|2)]dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≤N(l+1)l∫ℝN‍[c1|u|2p1+2+c2|u|2p2+2]dx+C∫ℝN‍(W(x)⋆|u|2)|u|2dx≤C∥u∥2p1+22p1+2+C∥u∥2p2+22p2+2+C∥W∥Lq∥u∥L4q/(2q-1)4.
Using Gagliardo-Nirenberg’s and Hölder’s inequalities, we can get
(64)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∥22)2}.
That is,
(65)∥∇u∥22+∥u∥22≥C>0
if Q(u)=0 and u∈H1(ℝN)∖{0}.
On the other hand, ifQ(u)=0, we have
(66)2∫ℝN‍|∇u|2dx=N∫ℝN‍[|u|2f(x,|u|2)-F(x,|u|2)]dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≥Nl∫ℝN‍F(x,|u|2)dx+Nl2∫ℝN‍{W(x)⋆|u|2}|u|2dx;
that is,
(67)-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍{W(x)⋆|u|2}|u|2dx≥-1Nl∫ℝN‍|∇u|2dx.
Using (67), we can obtain
(68)ω∥u∥22+E(u)=ω∥u∥22+12∫ℝN‍|∇u|2dx-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍{W(x)⋆|u|2}|u|2dx≥ω∥u∥22+(12-1Nl)∫ℝN‍|∇u|2dx≥min{ω,(12-1Nl)}×(∥∇u∥22+∥u∥22)≥C>0
from (65). Hence
(69)dI>0.
Step  2. Denote
(70)K+={u∈H1(ℝN)∖{0},Q(u)>0,ω∥u∥22+E(u)<dI},K-={u∈H1(ℝN)∖{0},Q(u)<0,ω∥u∥22+E(u)<dI}.
We will prove that K+ and K- are invariant sets of (1) with V(x)≡0 and W∈Lq(ℝN) with N/4<q<N/2. That is, we need to show that u(·,t)∈𝒦 for all t∈(0,Tmax) if u0∈K+. Since ∥u∥2 and E(u) are conservation quantities for (1), we have
(71)u(·,t)∈H1(ℝN)∖{0},ω∥u(·,t)∥22+E(u(·,t))<dI
for all t∈(0,Tmax) if u0∈K+. We need to prove that Q(u(·,t))>0. Otherwise, assume that there exists a t1∈(0,Tmax) satisfying Q(u(·,t1))=0 by the continuity. Note that (71) implies
(72)ω∥u(·,t)∥22+E(u(·,t1))<dI.
However, the inequality above and Q(u(·,t1))=0 are contradictions to the definition of dI. 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∈Lq(ℝN) 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∈Lq(ℝN) with N/4<q<N/2.
Step  3. Assume that Q(u0)>0 and ω∥u0∥22+E(u0)<dI. By the results of Step 2, we have Q(u(·,t))>0 and ω∥u(·,t)∥22+E(u(·,t))<dI. That is,
(73)-2∥∇u(·,t)∥22<-N∫ℝN‍[|u|2f(x,|u|2)-F(x,|u|2)]dx+12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx<-Nl∫ℝN‍F(x,|u|2)dx-Nl2∫ℝN‍{W(x)⋆|u|2}|u|2dx,dI>ω∥u(·,t)∥22+12∥∇u(·,t)∥22-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍{W(x)⋆|u|2}|u|2dx.
The two inequalities imply that
(74)ω∥u(·,t)∥22+(12-1Nl)∥∇u(·,t)∥22<dI,
which means that
(75)∥u(·,t)∥H1(ℝN)<∞;
that is, the solution exists globally.
Step  4. Assume that Q(u0)<0 and ω∥u0∥22+E(u0)<dI. By the results of Step 2, we obtain Q(u(·,t))<0 and ω∥u(·,t)∥22+E(u(·,t))<dI. Hence we get
(76)J′′(t)=4Q(u)<0,J′(0)=4ℑ∫ℝN‍(x·∇u0)u-0dx<0.
By the results of Proposition 10, the solution will blow up in finite time.As a corollary of Theorem5, we obtain the sharp threshold for global existence and blowup of the solution of (8) as follows.Corollary 18.
Assume thatf(x,0)=0 and (23). Let ω be a positive constant satisfying
(77)dI′≔inf{u∈Σ∖{0};Q1(u)=0}(ω∥u∥22+E(u))>0.
Here
(78)Q1(u)≔2∫ℝN‍|∇u|2dx+N∫ℝN‍[F(x,|u|2)-|u|2f(x,|u|2)]dx.
Suppose thatu0∈H1(ℝN) satisfies
(79)ω∥u0∥22+E(u0)<dI′.
Then(1)
ifQ1(u0)>0, the solution of (8) exists globally;(2)
ifQ1(u0)<0, |x|u0∈L2(ℝN), and ℑ∫ℝN‍(x·∇u0)u-0dx<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 thatu(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)u-0dx<0, |x|u0∈L2(ℝN), and E(u0)<0. Then blowup occurs.
Corollary18 covers the result above under some conditions. In fact, if f(x,|u|2)u=μ|u|p1u+ν|u|p2u, then
(80)Q1(u)=4E(u)-(Np1-4)μ(p1+2)∥u∥p1+2p1+2-(Np2-4)ν(p2+2)∥u∥p2+2p2+2≤E(u);
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)<dI′-ω∥u0∥22 with Q1(u0)<0, ℑ∫ℝN‍(x·∇u0)u-0dx<0, and |x|u0∈L2(ℝN). In other words, our result is stronger than theirs if ω∥u0∥22+E(u0)<dI′ with Q1(u0)<0, ℑ∫ℝN‍(x·∇u0)u-0dx<0, and |x|u0∈L2(ℝN).


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

Theorem7 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 ofV(x), f(x,|u|2), and W(x). It is easy to verify that they satisfy the conditions of Theorem 7.Example 20.
Consider thatV(x)=|x|2, W(x)=a|x|-K with 2<Nl<K<N/q<4 for x∈ℝN 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 thatV(x)=|x|2, W(x)=a|x|-K with 2<Nl<K<N/q<4 for x∈ℝN 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 thatV(x)=|x|2/(1+|x|2), W(x)=a|x|-K with 2<Nl<K<N/q<4 for x∈ℝN 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 thatd𝒩,dℳ,dII>0 and construct some invariant manifolds.Proposition 23.
Assume that the conditions of Theorem7 hold. Then d𝒩>0.Proof.
Assume thatu∈Σ∖{0} satisfying Sω(u)=0. Using Gagliardo-Nirenberg’s and Young’s inequalities, we have
(81)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx=∫ℝN‍|u|2f(x,|u|2)dx+∫ℝN‍(W(x)⋆|u|2)|u|2dx≤l+1l∫ℝN‍[c1|u|2p1+2+c2|u|2p2+2]dx+∥W1∥L∞∥u∥24+∥W2∥Lq∥u∥L4q/(2q-1)4≤C(∥∇u∥22)Np1/2(∥u∥22)p1+1-(Np1/2)+C(∥∇u∥22)Np2/2(∥u∥22)p2+1-(Np2/2)+∥W1∥L∞∥u∥24+C∥W2∥Lq∥∇u∥2N/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∥24+∥∇u∥24+C(∥W2∥Lq)∥u∥24.
Using Hölder’s inequality, from (81), we can obtain
(82)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx≤C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)p1+1+C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)p2+1+C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)2.
Equation (82) implies that
(83)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx≥C>0
for some positive constant C.
On the other hand, ifSω(u)=0, we get
(84)ω∥u∥22+12∫ℝN‍(|∇u|2+V(x)|u|2)dx=12∫ℝN‍f(x,|u|2)|u|2dx+12∫ℝN‍(W(x)⋆|u|2)|u|2dx≥min(l+1,2)(12∫ℝN‍F(x,|u|2)dx+14∫ℝN‍(W(x)⋆|u|2)|u|2dx).
From (84), we obtain
(85)Iω(u)=ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-G(|u|2)≥min(l2(l+1),14)×(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)≥C>0.
Consequently,(86)d𝒩=inf𝒩Iω(u)>C>0.Now, we will give some properties ofIω(u), Sω(u), and Q(u). We have a proposition as follows.Proposition 24.
Assume thatQ(u) and Sω(u) are defined by (22) and (28). Then one has the following.(i)
There at least exists aw⋆∈Σ∖{0} such that
(87)Sω(w⋆)=0,Q(w⋆)=0.(ii)
There at least exists au*∈Σ∖{0} such that
(88)Sω(u*)<0,Q(u*)=0.Proof.
(i) Noticing the assumptions onV(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
(89)2ωw⋆+V(x)w⋆-Δw⋆=f(x,|w⋆|2)w⋆+(W(x)⋆|w⋆|2)w⋆inℝN.
Multiplying (89) by w⋆ and integrating over ℝN by part, we can get Sω(w⋆)=0.
Multiplying (89) by (x·∇w⋆) and integrating over ℝN by part, we obtain Pohozaev’s identity:
(90)Nω∥w⋆∥22+N-22∫ℝN‍|∇w⋆|2dx+N2∫ℝN‍V(x)|w⋆|2dx+12∫ℝN‍(x·∇V(x))|w⋆|2dx=N2∫ℝN‍F(x,|w⋆|2)dx+N2∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx+12∫ℝN‍{(x·∇W(x))⋆|w⋆|2}|w⋆|2dx.
From Sω(w⋆)=0 and (90), we can get Q(w⋆)=0.
(ii) Lettingvk,λ(x)=kw⋆(λx) for k>0 and λ>0, we can obtain
(91)Sω(vk,λ)=2ωk2∫ℝN‍|w⋆(λx)|2dx+k2∫ℝN‍|∇w⋆(λx)|2dx+k2∫ℝN‍V(x)|w⋆(λx)|2dx-k2∫ℝN‍|w⋆(λx)|2f(x,k2|w⋆(λx)|2)dx-k4∫ℝN‍(W(x)⋆|w⋆(λx)|2)|w⋆(λx)|2dx,(92)Q(vk,λ)=2k2∫ℝN‍|∇w⋆(λx)|2dx-k2∫ℝN‍(x·∇V(x))|w⋆(λx)|2dx-N∫ℝN‍[k2|w⋆(λx)|2f(x,k2|w⋆(λx)|2)]-F(x,k2|w⋆(λx)|2)dx+k42∫ℝN‍((x·∇W(x))⋆|w⋆(λx)|2)×|w⋆(λx)|2dx.
Looking at Sω(vk,λ) and Q(vk,λ) as the functions of (k,λ), setting g(k,λ)=Sω(vk,λ) and η(k,λ)=Q(vk,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)=Sω(vk,λ)<0 and η(k,λ)=Q(vk,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(vk,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(vk,λ)=0 and Sω(vk,λ)<0. Letting u*=vk,λ for this (k,λ), we get that Sω(u*)<0 and Q(u*)=0.Proposition24 means that 𝒞ℳ is not empty and dℳ is well defined. Moreover, we have the following.Proposition 25.
Assume that the conditions of Theorem7 hold. Then dℳ>0.Proof.
u
∈
Σ
∖
{
0
} and Sω(u)<0 imply that
(93)2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx<∫ℝN‍|u|2f(x,|u|2)dx+∫ℝN‍(W(x)⋆|u|2)|u|2dx≤l+1l∫ℝN‍[c1|u|2p1+2+c2|u|2p2+2]dx+∥W1∥L∞∥u∥L24+C∥W1∥Lq∥∇u∥L2N/q∥u∥L2(4q-N)/q.
Similar to (81) and (82), from (93), we have
(94)2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx≥C>0.
On the other hand, if Q(u)=0, we have
(95)2∫ℝN‍|∇u|2dx-∫ℝN‍(x·∇V(x))|u|2dx=N∫ℝN‍[|u|2f(x,|u|2)-F(x,|u|2)]dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≥Nl∫ℝN‍F(x,|u|2)dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx;
that is,
(96)-12∫ℝN‍F(x,|u|2)dx+14Nl∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≥-1Nl∫ℝN‍|∇u|2dx+12Nl∫ℝN‍(x·∇V(x))|u|2dx.
Using (23), (35), (36), (94), and (96), we can get
(97)Iω(u)=ω∫ℝN‍|u|2dx+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-14∫ℝN‍(W(x)⋆|u|2)|u|2dx≥ω∫ℝN‍|u|2dx+Nl-22Nl∫ℝN‍|∇u|2dx+12Nl∫ℝN‍[NlV(x)+(x·∇V(x))]|u|2dx-14Nl∫ℝN‍{[NlW(x)+(x·∇W(x))]⋆|u|2}×|u|2dx≥C(2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx)≥C>0.
Consequently,
(98)dℳ=inf𝒞ℳIω(u)>C>0.By the conclusions of Proposition23 and Proposition 25, we have
(99)dII=min{dℳ,d𝒩}>0.Now we define the following manifolds:(100)𝒦≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)<0,Q(u)<0},(101)𝒦+≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)<0,Q(u)>0},(102)ℛ+≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)>0}.
The following proposition will show some properties of 𝒦, 𝒦+, and ℛ+.Proposition 26.
Assume that the conditions of Theorem7 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 prove𝒦 is not empty, we only need to find that there at least exists a w∈𝒦. For w⋆∈Σ∖{0} satisfying Sω(w⋆)=0 and Q(w⋆)=0, letting wρ=ρw⋆ for ρ>0, we have
(103)Sω(wρ)=ρ2∫ℝN‍{2ω|w⋆|2+|∇w⋆|2+V(x)|w⋆|2}dx-∫ℝN‍ρ2|w⋆|2f(x,ρ2|w⋆|2)dx-ρ4∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx,Q(wρ)=ρ2∫ℝN‍(2|∇w⋆|2-(x·∇V(x))|w⋆|2)dx+N∫ℝN‍[F(x,ρ2|w⋆|2)-ρ2|w⋆|2f(x,ρ2|w⋆|2)]dx+12ρ4∫ℝN‍{(x·∇W(x))⋆|w⋆|2}|w⋆|2dx,Iω(uρ)=12ρ2∫ℝN‍{2ω|w⋆|2+|∇w⋆|2+V(x)|w⋆|2}dx-12∫ℝN‍F(x,ρ2|w⋆|2)dx-14ρ4∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx.
Sincef(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
(104)Sω(wρ)<ρ2Sω(w⋆)=0,Q(wρ)<ρ2Q(w⋆)=0
for any ρ>1. Noticing dII>0, we also can choose ρ>1 closing to 1 enough such that
(105)Iω(wρ)<ρ2Iω(w⋆)<dII.
Equations (104) and (105) mean that wρ∈𝒦. That is, 𝒦 is not empty.
Similar to (104), we can obtain
(106)Sω(wρ)>ρ2Sω(w⋆)=0
for any 0<ρ<1. Noticing dII>0, we also can choose 0<ρ<1 closing to 1 enough such that Iω(wρ)<dII by continuity, which implies that wρ∈ℛ+. That is, ℛ+ is not empty.
Forw*∈Σ satisfying Sω(w*)<0 and Q(w*)=0, letting wσ=σw* for σ>0, we have
(107)Q(wσ)=σ2∫ℝN‍(2|∇w*|2-(x·∇V(x))|w*|2)dx-∫ℝN‍N[σ2|w*|2f(x,σ2|w*|2)-F(x,σ2|w*|2)]dx+12σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx,Sω(wσ)=σ2∫ℝN‍{2ω|w*|2+|∇w*|2+V(x)|w*|2}dx-∫ℝN‍σ2|w*|2f(x,σ2|w*|2)dx-σ4∫ℝN‍(W(x)⋆|w*|2)|w*|2dx,Iω(wσ)=12σ2∫ℝN‍{2ω|w*|2+|∇w*|2+V(x)|w*|2}dx-12∫ℝN‍F(x,σ2|w*|2)dx-14σ4∫ℝN‍(W(x)⋆|w*|2)|w*|2dx.
Sinceϕ(σ)=Q(wσ) 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(uσ)=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 Sω(wσ)<0 and Iω(wσ)<dII for σ∈(1,σ0) if σ0>1 (or σ∈(σ0,1) if σ0<1). That is, wσ∈𝒦+ and 𝒦+ is not empty.
Ifϕ′(1)=0, from ϕ(1)=0 and ϕ′(1)=0, we can, respectively, obtain
(108)-N∫ℝN‍[|w*|2f(x,|w*|2)-F(x,|w*|2)]dx=-N∫ℝN‍|w*|4fs′(x,|w*|2)dx+12∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx,(109)Q(w*)=∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,|w*|2)∫ℝN‍)dx+∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx.
Lettingwσ=σw*, we have
(110)Q(wσ)=σ2∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,σ2|w*|2)∫ℝN‍)dx+σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx>σ2∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,|w*|2)∫ℝN‍)dx+σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx=σ2Q(w*)+(σ4-σ2)×∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx>0
for 0<σ<1. By continuity, we can choose such σ closing to 1 enough such that Sω(wσ)<0 and Iω(wσ)<dII. That is to say, wσ∈𝒦+ and 𝒦+ is not empty.
(ii) In order to prove that𝒦 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 thatu(x,t) is a solution of (1) with u0∈𝒦. Then we can obtain
(111)Iω(u(·,t))=E(u(·,t))+ω∥u(·,t)∥22=E(u0)+ω∥u0∥22=Iω(u0)<dII
for t∈[0,T). Next we prove that Sω(u(·,t))<0 for t∈[0,T). Otherwise, by continuity, there exists a t0∈(0,T) such that Sω(u(·,t0))=0 because of Sω(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 d𝒩 and dII, we know that Iω(u(·,t0))≥d𝒩≥dII, which is a contradiction to Iω(u(·,t))<dII for t∈[0,T). Hence Sω(u(·,t))<0 for all t∈[0,T).
Now we only need to prove thatQ(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. Sω(u(·,t1))<0 means that u(·,t1)∈𝒞ℳ. By the definitions of dℳ and dII, we obtain Iω(u(·,t1))≥dℳ≥dII, which is a contradiction to Iω(u(·,t))<dII for t∈[0,T). Hence Q(u(·,t))<0 for all t∈[0,T).
By the discussions above, we know thatu(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 that𝒦+, and ℛ+ are also invariant manifolds of (1).Remark 27.
By the definitions ofdII, d𝒩, dℳ, 𝒦, 𝒦+, and ℛ+, it is easy to see that
(112){u∈Σ∖{0}:Iω(u)<dII}=𝒦∪𝒦+∪ℛ+.


### 5.2. Proof of Theorem7

Proof of Theorem7 depends on the following two lemmas.Lemma 28.
Assume that the conditions of Theorem7 hold. Then the solutions of (1) with u0∈𝒦 will blow up in finite time.Proof.
Sinceu0∈𝒦 and 𝒦 is the invariant manifold of (1), we have Q(u(x,t))<0, Sω(u(x,t))<0, and Iω(u(x,t))<dII.
Under the conditions of Theorem7, 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 Theorem7 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 Iω(u(·,t))<dII and Q(u(·,t))>0. Q(u(·,t))>0 and (23) imply that
(113)2∫ℝN‍|∇u|2dx-∫ℝN‍(x·∇V(x))|u|2dx≥Nl∫ℝN‍F(x,|u|2)dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx.
By the definition of Iω(u) and using (113), we have
(114)dII>Iω(u(·,t))=ω∫ℝN‍|u|2dx+12∫ℝN‍[|∇u|2+V(x)|u|2]dx-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍(W(x)⋆|u|2)|u|2dx≥ω∫ℝN‍|u|2dx+Nl-22Nl∫ℝN‍|∇u|2dx+∫ℝN‍NlV(x)+(x·∇V(x))2Nl|u|2dx-14Nl∫ℝN‍{∫ℝN‍[NlW(x)+(x·∇W(x))]⋆|u|2∫ℝN‍[NlW(x)+(x·∇W(x))]}|u|2dx≥C(∫ℝN‍|u|2dx+∫ℝN‍|∇u|2dx+∫ℝN‍V(x)|u|2dx).
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 Iω(u(·,t))<dII and Sω(u(·,t))>0. Since Sω(u)>0, we can get
(115)ω∥u∥22+12∫ℝN‍(|∇u|2+V(x)|u|2)dx>12∫ℝN‍f(x,|u|2)|u|2dx+12∫ℝN‍(W(x)⋆|u|2)|u|2dx≥min(l+1,2)(12∫ℝN‍F(x,|u|2)dx+14∫ℝN‍(W(x)⋆|u|2)|u|2dx).
From (115), we can obtain
(116)Iω(u)=ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-G(|u|2)≥min(l(l+1),12)×(ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2]dx).
Equation (116) implies that the solution u(x,t) exists globally.Proof of Theorem7.
By the results of Lemmas28 and 29, we know that Theorem 7 is right.As a corollary of Theorem7, we obtain a sharp threshold for the blowup in finite time and global existence of the solution of (9) as follows.Corollary 30.
Assume thatf(x,|u|2)≡0, V(x)≡0, W(x)>0 for all x∈ℝN, W(x) is even, and W(x)∈L∞(ℝN)+Lq(ℝN) with some q>N/4. Suppose further that there exists l satisfying 2<Nl and
(117)NlW(x)+(x·∇W(x))≤0.
If u0∈H1(ℝN), |x|u0∈L2(ℝN), and Iω(u0)=ω∥u0∥22+E(u0)<dII, then the solution of (9) blows up in finite time if and only if u0∈𝒦.Remark 31.
A typical example is(118)-iut=Δu+(|x|-K⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,
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
(119)W1(x)={0,|x|≤1,|x|-K,|x|>1,(120)W2(x)={|x|-K,|x|≤1,0,|x|>1,
we can see that W1(x)∈L∞(ℝN) and W2(x)∈Lq(ℝN) with some N/4<q<N/2. Corollary 30 gives the sharp threshold for blowup and global existence of the solution to (118).


## 5.1. Some Invariant Manifolds

In this subsection, we will prove thatd𝒩,dℳ,dII>0 and construct some invariant manifolds.Proposition 23.
Assume that the conditions of Theorem7 hold. Then d𝒩>0.Proof.
Assume thatu∈Σ∖{0} satisfying Sω(u)=0. Using Gagliardo-Nirenberg’s and Young’s inequalities, we have
(81)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx=∫ℝN‍|u|2f(x,|u|2)dx+∫ℝN‍(W(x)⋆|u|2)|u|2dx≤l+1l∫ℝN‍[c1|u|2p1+2+c2|u|2p2+2]dx+∥W1∥L∞∥u∥24+∥W2∥Lq∥u∥L4q/(2q-1)4≤C(∥∇u∥22)Np1/2(∥u∥22)p1+1-(Np1/2)+C(∥∇u∥22)Np2/2(∥u∥22)p2+1-(Np2/2)+∥W1∥L∞∥u∥24+C∥W2∥Lq∥∇u∥2N/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∥24+∥∇u∥24+C(∥W2∥Lq)∥u∥24.
Using Hölder’s inequality, from (81), we can obtain
(82)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx≤C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)p1+1+C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)p2+1+C(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)2.
Equation (82) implies that
(83)2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx≥C>0
for some positive constant C.
On the other hand, ifSω(u)=0, we get
(84)ω∥u∥22+12∫ℝN‍(|∇u|2+V(x)|u|2)dx=12∫ℝN‍f(x,|u|2)|u|2dx+12∫ℝN‍(W(x)⋆|u|2)|u|2dx≥min(l+1,2)(12∫ℝN‍F(x,|u|2)dx+14∫ℝN‍(W(x)⋆|u|2)|u|2dx).
From (84), we obtain
(85)Iω(u)=ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-G(|u|2)≥min(l2(l+1),14)×(2ω∥u∥22+∫ℝN‍[|∇u|2+V(x)|u|2]dx)≥C>0.
Consequently,(86)d𝒩=inf𝒩Iω(u)>C>0.Now, we will give some properties ofIω(u), Sω(u), and Q(u). We have a proposition as follows.Proposition 24.
Assume thatQ(u) and Sω(u) are defined by (22) and (28). Then one has the following.(i)
There at least exists aw⋆∈Σ∖{0} such that
(87)Sω(w⋆)=0,Q(w⋆)=0.(ii)
There at least exists au*∈Σ∖{0} such that
(88)Sω(u*)<0,Q(u*)=0.Proof.
(i) Noticing the assumptions onV(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
(89)2ωw⋆+V(x)w⋆-Δw⋆=f(x,|w⋆|2)w⋆+(W(x)⋆|w⋆|2)w⋆inℝN.
Multiplying (89) by w⋆ and integrating over ℝN by part, we can get Sω(w⋆)=0.
Multiplying (89) by (x·∇w⋆) and integrating over ℝN by part, we obtain Pohozaev’s identity:
(90)Nω∥w⋆∥22+N-22∫ℝN‍|∇w⋆|2dx+N2∫ℝN‍V(x)|w⋆|2dx+12∫ℝN‍(x·∇V(x))|w⋆|2dx=N2∫ℝN‍F(x,|w⋆|2)dx+N2∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx+12∫ℝN‍{(x·∇W(x))⋆|w⋆|2}|w⋆|2dx.
From Sω(w⋆)=0 and (90), we can get Q(w⋆)=0.
(ii) Lettingvk,λ(x)=kw⋆(λx) for k>0 and λ>0, we can obtain
(91)Sω(vk,λ)=2ωk2∫ℝN‍|w⋆(λx)|2dx+k2∫ℝN‍|∇w⋆(λx)|2dx+k2∫ℝN‍V(x)|w⋆(λx)|2dx-k2∫ℝN‍|w⋆(λx)|2f(x,k2|w⋆(λx)|2)dx-k4∫ℝN‍(W(x)⋆|w⋆(λx)|2)|w⋆(λx)|2dx,(92)Q(vk,λ)=2k2∫ℝN‍|∇w⋆(λx)|2dx-k2∫ℝN‍(x·∇V(x))|w⋆(λx)|2dx-N∫ℝN‍[k2|w⋆(λx)|2f(x,k2|w⋆(λx)|2)]-F(x,k2|w⋆(λx)|2)dx+k42∫ℝN‍((x·∇W(x))⋆|w⋆(λx)|2)×|w⋆(λx)|2dx.
Looking at Sω(vk,λ) and Q(vk,λ) as the functions of (k,λ), setting g(k,λ)=Sω(vk,λ) and η(k,λ)=Q(vk,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)=Sω(vk,λ)<0 and η(k,λ)=Q(vk,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(vk,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(vk,λ)=0 and Sω(vk,λ)<0. Letting u*=vk,λ for this (k,λ), we get that Sω(u*)<0 and Q(u*)=0.Proposition24 means that 𝒞ℳ is not empty and dℳ is well defined. Moreover, we have the following.Proposition 25.
Assume that the conditions of Theorem7 hold. Then dℳ>0.Proof.
u
∈
Σ
∖
{
0
} and Sω(u)<0 imply that
(93)2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx<∫ℝN‍|u|2f(x,|u|2)dx+∫ℝN‍(W(x)⋆|u|2)|u|2dx≤l+1l∫ℝN‍[c1|u|2p1+2+c2|u|2p2+2]dx+∥W1∥L∞∥u∥L24+C∥W1∥Lq∥∇u∥L2N/q∥u∥L2(4q-N)/q.
Similar to (81) and (82), from (93), we have
(94)2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx≥C>0.
On the other hand, if Q(u)=0, we have
(95)2∫ℝN‍|∇u|2dx-∫ℝN‍(x·∇V(x))|u|2dx=N∫ℝN‍[|u|2f(x,|u|2)-F(x,|u|2)]dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≥Nl∫ℝN‍F(x,|u|2)dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx;
that is,
(96)-12∫ℝN‍F(x,|u|2)dx+14Nl∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx≥-1Nl∫ℝN‍|∇u|2dx+12Nl∫ℝN‍(x·∇V(x))|u|2dx.
Using (23), (35), (36), (94), and (96), we can get
(97)Iω(u)=ω∫ℝN‍|u|2dx+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-14∫ℝN‍(W(x)⋆|u|2)|u|2dx≥ω∫ℝN‍|u|2dx+Nl-22Nl∫ℝN‍|∇u|2dx+12Nl∫ℝN‍[NlV(x)+(x·∇V(x))]|u|2dx-14Nl∫ℝN‍{[NlW(x)+(x·∇W(x))]⋆|u|2}×|u|2dx≥C(2ω∫ℝN‍|u|2dx+∫ℝN‍[|∇u|2+V(x)|u|2]dx)≥C>0.
Consequently,
(98)dℳ=inf𝒞ℳIω(u)>C>0.By the conclusions of Proposition23 and Proposition 25, we have
(99)dII=min{dℳ,d𝒩}>0.Now we define the following manifolds:(100)𝒦≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)<0,Q(u)<0},(101)𝒦+≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)<0,Q(u)>0},(102)ℛ+≔{u∈Σ∖{0}:Iω(u)<dII,Sω(u)>0}.
The following proposition will show some properties of 𝒦, 𝒦+, and ℛ+.Proposition 26.
Assume that the conditions of Theorem7 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 prove𝒦 is not empty, we only need to find that there at least exists a w∈𝒦. For w⋆∈Σ∖{0} satisfying Sω(w⋆)=0 and Q(w⋆)=0, letting wρ=ρw⋆ for ρ>0, we have
(103)Sω(wρ)=ρ2∫ℝN‍{2ω|w⋆|2+|∇w⋆|2+V(x)|w⋆|2}dx-∫ℝN‍ρ2|w⋆|2f(x,ρ2|w⋆|2)dx-ρ4∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx,Q(wρ)=ρ2∫ℝN‍(2|∇w⋆|2-(x·∇V(x))|w⋆|2)dx+N∫ℝN‍[F(x,ρ2|w⋆|2)-ρ2|w⋆|2f(x,ρ2|w⋆|2)]dx+12ρ4∫ℝN‍{(x·∇W(x))⋆|w⋆|2}|w⋆|2dx,Iω(uρ)=12ρ2∫ℝN‍{2ω|w⋆|2+|∇w⋆|2+V(x)|w⋆|2}dx-12∫ℝN‍F(x,ρ2|w⋆|2)dx-14ρ4∫ℝN‍(W(x)⋆|w⋆|2)|w⋆|2dx.
Sincef(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
(104)Sω(wρ)<ρ2Sω(w⋆)=0,Q(wρ)<ρ2Q(w⋆)=0
for any ρ>1. Noticing dII>0, we also can choose ρ>1 closing to 1 enough such that
(105)Iω(wρ)<ρ2Iω(w⋆)<dII.
Equations (104) and (105) mean that wρ∈𝒦. That is, 𝒦 is not empty.
Similar to (104), we can obtain
(106)Sω(wρ)>ρ2Sω(w⋆)=0
for any 0<ρ<1. Noticing dII>0, we also can choose 0<ρ<1 closing to 1 enough such that Iω(wρ)<dII by continuity, which implies that wρ∈ℛ+. That is, ℛ+ is not empty.
Forw*∈Σ satisfying Sω(w*)<0 and Q(w*)=0, letting wσ=σw* for σ>0, we have
(107)Q(wσ)=σ2∫ℝN‍(2|∇w*|2-(x·∇V(x))|w*|2)dx-∫ℝN‍N[σ2|w*|2f(x,σ2|w*|2)-F(x,σ2|w*|2)]dx+12σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx,Sω(wσ)=σ2∫ℝN‍{2ω|w*|2+|∇w*|2+V(x)|w*|2}dx-∫ℝN‍σ2|w*|2f(x,σ2|w*|2)dx-σ4∫ℝN‍(W(x)⋆|w*|2)|w*|2dx,Iω(wσ)=12σ2∫ℝN‍{2ω|w*|2+|∇w*|2+V(x)|w*|2}dx-12∫ℝN‍F(x,σ2|w*|2)dx-14σ4∫ℝN‍(W(x)⋆|w*|2)|w*|2dx.
Sinceϕ(σ)=Q(wσ) 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(uσ)=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 Sω(wσ)<0 and Iω(wσ)<dII for σ∈(1,σ0) if σ0>1 (or σ∈(σ0,1) if σ0<1). That is, wσ∈𝒦+ and 𝒦+ is not empty.
Ifϕ′(1)=0, from ϕ(1)=0 and ϕ′(1)=0, we can, respectively, obtain
(108)-N∫ℝN‍[|w*|2f(x,|w*|2)-F(x,|w*|2)]dx=-N∫ℝN‍|w*|4fs′(x,|w*|2)dx+12∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx,(109)Q(w*)=∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,|w*|2)∫ℝN‍)dx+∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx.
Lettingwσ=σw*, we have
(110)Q(wσ)=σ2∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,σ2|w*|2)∫ℝN‍)dx+σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx>σ2∫ℝN‍(∫ℝN‍2|∇w*|2-(x·∇V(x))|w*|2-N|w*|4fs′(x,|w*|2)∫ℝN‍)dx+σ4∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx=σ2Q(w*)+(σ4-σ2)×∫ℝN‍{(x·∇W(x))⋆|w*|2}|w*|2dx>0
for 0<σ<1. By continuity, we can choose such σ closing to 1 enough such that Sω(wσ)<0 and Iω(wσ)<dII. That is to say, wσ∈𝒦+ and 𝒦+ is not empty.
(ii) In order to prove that𝒦 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 thatu(x,t) is a solution of (1) with u0∈𝒦. Then we can obtain
(111)Iω(u(·,t))=E(u(·,t))+ω∥u(·,t)∥22=E(u0)+ω∥u0∥22=Iω(u0)<dII
for t∈[0,T). Next we prove that Sω(u(·,t))<0 for t∈[0,T). Otherwise, by continuity, there exists a t0∈(0,T) such that Sω(u(·,t0))=0 because of Sω(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 d𝒩 and dII, we know that Iω(u(·,t0))≥d𝒩≥dII, which is a contradiction to Iω(u(·,t))<dII for t∈[0,T). Hence Sω(u(·,t))<0 for all t∈[0,T).
Now we only need to prove thatQ(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. Sω(u(·,t1))<0 means that u(·,t1)∈𝒞ℳ. By the definitions of dℳ and dII, we obtain Iω(u(·,t1))≥dℳ≥dII, which is a contradiction to Iω(u(·,t))<dII for t∈[0,T). Hence Q(u(·,t))<0 for all t∈[0,T).
By the discussions above, we know thatu(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 that𝒦+, and ℛ+ are also invariant manifolds of (1).Remark 27.
By the definitions ofdII, d𝒩, dℳ, 𝒦, 𝒦+, and ℛ+, it is easy to see that
(112){u∈Σ∖{0}:Iω(u)<dII}=𝒦∪𝒦+∪ℛ+.


## 5.2. Proof of Theorem7

Proof of Theorem7 depends on the following two lemmas.Lemma 28.
Assume that the conditions of Theorem7 hold. Then the solutions of (1) with u0∈𝒦 will blow up in finite time.Proof.
Sinceu0∈𝒦 and 𝒦 is the invariant manifold of (1), we have Q(u(x,t))<0, Sω(u(x,t))<0, and Iω(u(x,t))<dII.
Under the conditions of Theorem7, 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 Theorem7 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 Iω(u(·,t))<dII and Q(u(·,t))>0. Q(u(·,t))>0 and (23) imply that
(113)2∫ℝN‍|∇u|2dx-∫ℝN‍(x·∇V(x))|u|2dx≥Nl∫ℝN‍F(x,|u|2)dx-12∫ℝN‍{(x·∇W(x))⋆|u|2}|u|2dx.
By the definition of Iω(u) and using (113), we have
(114)dII>Iω(u(·,t))=ω∫ℝN‍|u|2dx+12∫ℝN‍[|∇u|2+V(x)|u|2]dx-12∫ℝN‍F(x,|u|2)dx-14∫ℝN‍(W(x)⋆|u|2)|u|2dx≥ω∫ℝN‍|u|2dx+Nl-22Nl∫ℝN‍|∇u|2dx+∫ℝN‍NlV(x)+(x·∇V(x))2Nl|u|2dx-14Nl∫ℝN‍{∫ℝN‍[NlW(x)+(x·∇W(x))]⋆|u|2∫ℝN‍[NlW(x)+(x·∇W(x))]}|u|2dx≥C(∫ℝN‍|u|2dx+∫ℝN‍|∇u|2dx+∫ℝN‍V(x)|u|2dx).
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 Iω(u(·,t))<dII and Sω(u(·,t))>0. Since Sω(u)>0, we can get
(115)ω∥u∥22+12∫ℝN‍(|∇u|2+V(x)|u|2)dx>12∫ℝN‍f(x,|u|2)|u|2dx+12∫ℝN‍(W(x)⋆|u|2)|u|2dx≥min(l+1,2)(12∫ℝN‍F(x,|u|2)dx+14∫ℝN‍(W(x)⋆|u|2)|u|2dx).
From (115), we can obtain
(116)Iω(u)=ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2-F(x,|u|2)]dx-G(|u|2)≥min(l(l+1),12)×(ω∥u∥22+12∫ℝN‍[|∇u|2+V(x)|u|2]dx).
Equation (116) implies that the solution u(x,t) exists globally.Proof of Theorem7.
By the results of Lemmas28 and 29, we know that Theorem 7 is right.As a corollary of Theorem7, we obtain a sharp threshold for the blowup in finite time and global existence of the solution of (9) as follows.Corollary 30.
Assume thatf(x,|u|2)≡0, V(x)≡0, W(x)>0 for all x∈ℝN, W(x) is even, and W(x)∈L∞(ℝN)+Lq(ℝN) with some q>N/4. Suppose further that there exists l satisfying 2<Nl and
(117)NlW(x)+(x·∇W(x))≤0.
If u0∈H1(ℝN), |x|u0∈L2(ℝN), and Iω(u0)=ω∥u0∥22+E(u0)<dII, then the solution of (9) blows up in finite time if and only if u0∈𝒦.Remark 31.
A typical example is(118)-iut=Δu+(|x|-K⋆|u|2)u,x∈ℝN,t>0,u(x,0)=u0(x),x∈ℝN,
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
(119)W1(x)={0,|x|≤1,|x|-K,|x|>1,(120)W2(x)={|x|-K,|x|≤1,0,|x|>1,
we can see that W1(x)∈L∞(ℝN) and W2(x)∈Lq(ℝN) with some N/4<q<N/2. Corollary 30 gives the sharp threshold for blowup and global existence of the solution to (118).



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*Source: 238410-2013-11-30.xml*