With the rapid development of the economy, the air quality is facing increasingly severe pollution challenges. The air quality is related to public health and the sustainable development of the environment of China. In this paper, we first investigate the changes in the monthly air quality index data of Hefei from 2014 to 2020. Second, we analyze whether the Spring Festival factors lead to the deterioration of the air quality index according to the time sequence. Third, we construct an improved model to predict the air quality index of Hefei. There are three primary discoveries: (1) The air quality index of Hefei has obvious periodicity and a trend of descent. (2) The influencing factors of Spring Festival have no significant effect on the air quality index series. (3) The air quality index of Hefei will maintain a fluctuating and descending trend for a period of time. Finally, some recommendations for the air quality management policy in Hefei are presented based on the obtained results.
Citation: Jia-Bao Liu, Xi-Yu Yuan. Prediction of the air quality index of Hefei based on an improved ARIMA model[J]. AIMS Mathematics, 2023, 8(8): 18717-18733. doi: 10.3934/math.2023953
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With the rapid development of the economy, the air quality is facing increasingly severe pollution challenges. The air quality is related to public health and the sustainable development of the environment of China. In this paper, we first investigate the changes in the monthly air quality index data of Hefei from 2014 to 2020. Second, we analyze whether the Spring Festival factors lead to the deterioration of the air quality index according to the time sequence. Third, we construct an improved model to predict the air quality index of Hefei. There are three primary discoveries: (1) The air quality index of Hefei has obvious periodicity and a trend of descent. (2) The influencing factors of Spring Festival have no significant effect on the air quality index series. (3) The air quality index of Hefei will maintain a fluctuating and descending trend for a period of time. Finally, some recommendations for the air quality management policy in Hefei are presented based on the obtained results.
In this paper, we consider the following Cauchy problem for the nonlinear Schrödinger equation with an attractive inverse-power potential
{i∂tψ+Δψ+γ|x|−σψ+|ψ|αψ=0,(t,x)∈R×RN,ψ(0,x)=ψ0(x),x∈RN, | (1.1) |
where N≥3, ψ:[0,T∗)×RN→C is an unknown complex valued function with 0<T∗≤∞, ψ0∈H1(RN), γ∈(0,+∞), σ∈(0,2) and 4N<α<4N−2.
In the case σ=1, i.e., the operator Δ+γ|x| with Coulomb potential, (1.1) describes the situation where the wave function of an electron, satisfying the Schrödinger evolution equation, is influenced by m nuclei, see [19] for a broader introduction. In the case 0<σ<2, i.e., the operator Δ+γ|x|σ with slowly decaying potentials, we refer the readers to [15] and references therein. It also attracted a great deal of attention from mathematicians in recent years, see, e.g. [4,13,14,23].
Recently, this type of equations has been studied widely in [1,4,5,6,7,8,9,10,11,13,17,20,22,23,24,25,26]. In particular, Eq (1.1) enjoys a class of special solutions, which are called standing waves, namely solutions of the form ψ(t,x)=eiωtu(x), where ω∈R is a frequency and u∈H1(RN) satisfies the elliptic equation
−Δu+ωu−γ|x|−σu−|u|αu=0. | (1.2) |
The Eq (1.2) is variational and its action functional is defined by
Sω(u):=Eγ(u)+ω2‖u‖2L2, | (1.3) |
where the corresponding energy functional Eγ(u) is defined by
Eγ(u):=12‖∇u‖2L2−γ2∫RN|u(x)|2|x|σdx−1α+2‖u‖α+2Lα+2. | (1.4) |
For the evolutional type Eq (1.1), one of the most interesting problems is to consider the stability of standing waves, which is defined as follows:
Definition 1.1. Assume u is a solution of (1.2). The standing wave eiωtu(x) is called orbitally stable in H1(RN), if for every ϵ>0, there exists δ>0 such that if ψ0∈H1(RN) satisfies
‖ψ0−u‖H1≤δ, |
then the solution ψ(t) to (1.1) with ψ|t=0=ψ0 satisfies
supt∈Rinfθ∈R,y∈RN‖ψ(t,⋅)−eiθu(⋅−y)‖H1≤ϵ. |
There are two main methods to study the stability of standing waves. One is the stability or instability criterion in [16] proposed by Grillakis, Shatah and Strauss. It says that the standing wave eiωtuω is stable if ∂∂ω‖uω‖2L2>0 and unstable if not, see [12] for more details. However, it is hard to estimate the sign of ∂∂ω‖uω‖2L2 for nonlinear Schrödinger equations without scaling invariance, e.g. (1.1). The other is the constrained minimization approach introduced by Cazenave and Lions in [3]. In this paper, we thus take into account the orbital stability of the set of minimizers by using the method from Cazenave and Lions in [3].
Now we recall the following definition of the orbital stability of the set M.
Definition 1.2. The set M⊂H1(RN) is called orbitally stable if for every ϵ>0, there exists δ>0 such that for any initial data ψ0∈H1(RN) satisfying
infu∈M‖ψ0−u‖H1<δ, |
the corresponding solution u to (1.1) satisfies
infu∈M‖ψ(t)−u‖H1<ϵ,∀t>0. |
Based on the above definition, in order to study the stability, we require that the solution of (1.1) exists globally, at least for initial data ψ0 enough close to M. In the L2-subcritical case, all solutions for (1.1) exist globally. Therefore, the stability of standing waves has been studied widely in this case, see, e.g. [3,4,14,23]. However, in the L2-supercritical case, we know that the solution of (1.1) with small initial data exists globally, but for some large initial data, the solution of (1.1) may blow up in finite time by the local well-posedness theory of NLS, see [2] for further inference.
For Eq (1.1), in the mass subcritical and critical cases, i.e., 0<α≤4N, Dinh in [4] and Li-Zhao in [23] studied the stability of set of minimizers by using the concentration compactness principle. In the mass supercritical case, i.e., 4N<α<4N−2, Fukaya-Ohta in [14] proved the strong instability of standing wave eiωtu under the assumption ∂2λEγ(uλ)|λ=1≤0 with uλ(x):=λN2u(λx). Therefore, whether there are stable standing waves is an interesting problem in the mass supercritical case. In this paper, we will solve this problem by considering the following minimization problem
mγ(a):=infu∈S(a)Eγ(u), | (1.5) |
where
S(a)={u∈H1(RN),‖u‖2L2=a}. | (1.6) |
In the L2-supercritical case, the energy Eγ(u) is unbounded from below on S(a). Actually, when 4N<α<4N−2, taking u∈S(a) and setting uλ(x):=λN2u(λx), then ‖u‖2L2=‖uλ‖2L2=a, and we will find that
Eγ(uλ)=λ22‖∇u‖2L2−λσγ2∫RN|u(x)|2|x|σdx−λNα2α+2‖u‖α+2Lα+2→−∞, |
as λ→∞, where 2<Nα2<2NN−2. Thus, we can not discuss the global minimization problem (1.5) to study the existence and stability of standing waves for (1.1). However, inspired by the thought in [18], we consider the further constrained minimization problem:
mγ(a,r0):=infu∈V(a)Eγ(u). | (1.7) |
The sets V(a) and ∂V(a) are given by
V(a):={u∈S(a):‖∇u‖2L2<r0},∂V(a):={u∈S(a):‖∇u‖2L2=r0}, |
for an appropriate r0>0, depending only on a0 but not on a∈(0,a0). However, compared with the case (i.e., γ=0) considered in [18], the energy functional of (1.2) is not invariant under the scaling transform due to the inhomogeneous nonlinearity γ|x|−σu. Furthermore, we cannot prove the preconceived limiting problem of the energy functional under translation sequences. In fact, we can solve the minimization problem (1.7) by proving the boundedness of the translation sequences. We now denote all the energy minimizers of (1.7) by
M(a):={u∈V(a):Eγ(u)=mγ(a,r0)}. |
Our main results are as follows:
Theorem 1.3. Let N≥3, 0<σ<2 and 4N<α<4N−2. Then there exists a γ0>0 sufficiently small such that 0<γ<γ0 and for any a∈(0,a0), the following properties hold:
(i) ∅≠M(a)⊂V(a).
(ii) M(a) is orbitally stable.
This paper is organized as follows: In Section 2, we firstly give some variational problems and then prove that the solution ψ(t) of (1.1) with the initial data ψ0 exists globally. In Section 3, we will show that M(a) is orbitally stable.
In this section, we first establish the following classical inequalities: If N≥2 and α∈[2,2NN−2), then the following Gagliardo-Nirenberg inequality holds that
‖u‖α+2Lα+2≤C(α)‖u‖α+2−Nα2L2‖∇u‖Nα2L2,∀u∈H1(RN), | (2.1) |
where C(α) is the sharp constant. Let 1≤p<∞, if σ<N is such that 0≤σ≤p, then |u(⋅)|p|⋅|σ∈L1(RN) for any u∈H1(RN). Moreover,
∫RN|u(x)|p|x|σdx≤C1‖u‖p−σLp‖∇u‖σLp,∀u∈H1(RN), | (2.2) |
where C1=(pN−σ)σ.
Secondly, by (2.1) and (2.2), we have
Eγ(u)≥‖∇u‖2L2(12−γC12‖u‖2−σL2‖∇u‖σ−2L2−C(α)α+2‖u‖α+2−Nα2L2‖∇u‖Nα2−2L2)=‖∇u‖2L2f(‖u‖2L2,‖∇u‖2L2). | (2.3) |
Next, setting
η0=2−σ,η1=σ−2,η2=α+2−Nα2,η3=Nα2−2, |
we now consider the function f(a,r) defined on (0,∞)×(0,∞) by
f(a,r)=12−γC12aη02rη12−C(α)α+2aη22rη32, | (2.4) |
and, for each a∈(0,∞), its restriction ga(r) defined on (0,∞) by r↦ga(r)=f(a,r). For further reference, note that for any N≥3, η0∈(0,2), η1∈(−2,0), η2∈(0,4N) and η3∈(0,4N−2).
Lemma 2.1. [2,Theorem 9.2.6] Let N≥3, 0<σ<2 and 4N<α<4N−2. Then for any initial data ψ0∈H1(RN), there exists T=T(‖ψ0‖H1) such that (1.1) admits a unique solution with ψ(0)=ψ0. Let [0,T∗) be the maximal time interval on which the solution ψ is well-defined, if T∗<∞, then ‖ψ(t)‖H1→∞ as t↑T∗.Moreover, for all 0<t<T∗, the solution ψ(t) satisfies the following conservations of mass and energy
‖ψ(t)‖2L2=‖ψ0‖2L2,Eγ(ψ(t))=Eγ(ψ0), |
where the energy Eγ(u) is defined by (1.4).
Lemma 2.2. For every a>0, the function ga(r)=f(a,r) has a unique global maximum and the maximum value satisfies
{maxr>0ga(r)>0,ifa<a0,maxr>0ga(r)=0,ifa=a0,maxr>0ga(r)<0,ifa>a0, |
where
a0:=[12K]2(η3−η1)η0η3−η1η2>0, | (2.5) |
with
K:=γC12[−C1η1(α+2)2C(α)η3]η1η3−η1+C(α)α+2[−C1η1(α+2)2C(α)η3]η3η3−η1. | (2.6) |
Proof. According to the definition of ga(r), we have
g′a(r)=−γC1η14aη02rη12−1−C(α)η32(α+2)aη22rη32−1. |
Hence, the equation g′a(r)=0 has a unique solution given by
ra=[−γC1η1(α+2)2C(α)η3]2η3−η1aη0−η2η3−η1. | (2.7) |
Noticing that ga(r)→−∞ as r→0 and ga(r)→−∞ as r→∞, we obtain that ra is the unique global maximum point of ga(r). Actually, the maximum value is
maxr>0ga(r)=12−γC12aη02rη12a−C(α)α+2aη22rη32a=12−γC12aη02[−C1η1(α+2)2C(α)η3]η1η3−η1aη1(η0−η2)2(η3−η1)−C(α)α+2aη22[−C1η1(α+2)2C(α)η3]η3η3−η1aη3(η0−η2)2(η3−η1)=12−Kaη0η3−η1η22(η3−η1). |
According to the definition of a0, we have maxr>0ga0(r)=0. Hence, the lemma follows.
Now let a0>0 be given by (2.5) and r0:=ra0>0 being determined by (2.7). Note that by the proof of Lemma 2.2, we have f(a0,r0)=0 and f(a,r0)≥0 for all a∈(0,a0). We denote
B(r0):={u∈H1(RN):‖∇u‖2L2<r0}andV(a):=S(a)∩B(r0). |
We now consider the following local minimization problem:
mγ(a,r0):=infu∈V(a)Eγ(u),∀a∈(0,a0). | (2.8) |
And if (a1,r1)∈(0,∞)×(0,∞) be such that f(a1,r1)≥0, then for any a2∈(0,a1], by Lemma 2.2 and direct calculations, we have
f(a2,r2)≥0ifr2∈(a2a1r1,r1). | (2.9) |
Theorem 2.3. Let N≥3, 0<σ<2, 4N<α<4N−2 and a∈(0,a0), there exists a γ0>0 sufficiently small such that 0<γ<γ0. Then, there exists u∈H1(RN) such that Eγ(u)=mγ(a,r0).
Lemma 2.4. For any a∈(0,a0), the following property holds,
infu∈V(a)Eγ(u)<0≤infu∈∂V(a)Eγ(u). | (2.10) |
Proof. For any u∈∂V(a), we have ‖∇u‖2L2=r0. Thus, by (2.3) and Lemma 2.2, we have
Eγ(u)≥‖∇u‖2L2f(a,‖∇u‖2L2)=r0f(a,r0)≥r0f(a0,r0)=0. | (2.11) |
Now let u∈S(a) be arbitrary but fixed. We denote uλ(x):=λN2u(λx) for λ∈(0,∞). It is obvious that uλ∈S(a) for any λ∈(0,∞). We set the map on (0,∞) by
Eγ(uλ)=λ22‖∇u‖2L2−γλσ2∫RN|u|2|x|σdx−λNα2α+2‖u‖α+2Lα+2. | (2.12) |
Noticing that 0<σ<2,2<Nα2<2NN−2, thus Eγ(uλ)<0 and ‖∇uλ‖2L2=λ2‖∇u‖2L2<r0 for sufficiently small λ>0. The proof follows.
Lemma 2.5. Let 4N<α<4N−2, r0>0 be determined as Lemma 2.2 and 0<a<a0 be as in Lemma 2.4. Then, there exists δ>0 such that for any initial data ψ0∈H1(RN) and infu∈M(a)‖ψ0−u‖H1<δ, the solution ψ(t) of (1.1) with the initial data ψ0 exists globally.
Proof. Firstly, we denote the right hand of (2.10) by A. According to the continuity of energy functional Eγ(u) with respect to u∈M(a), we deduce Eγ(u)=mγ(a,r0)<A and ‖∇u‖2L2<r0. Moreover, there exists δ>0 such that for any ψ0∈H1(RN) and ‖ψ0−u‖H1<δ, we have
Eγ(ψ0)<Aand‖∇ψ0‖2L2<r0. | (2.13) |
Secondly, let us prove this result by contradiction. If not, there exists ψ0∈H1(RN) such that ‖ψ0−u‖H1<δ and the solution ψ(t) with an initial value of ψ0 blows up in finite time. By continuity, there exists T1>0 such that ‖∇ψ(T1)‖2L2>r0. We now assume the initial value ~ψ0=√a‖ψ0‖L2ψ0. When δ>0 is sufficiently small, we have
Eγ(~ψ0)<Aand~ψ0∈S(a). |
When √a<‖ψ0‖L2, ‖∇~ψ0‖2L2<‖∇ψ0‖2L2<r0. When √a>‖ψ0‖L2, considering 0<a<a0, we have ‖∇~ψ0‖2L2<r0. This implies that ˜ψ0∈V(a). Since the solution of (1.1) is continuously dependent on the initial data and ‖∇ψ(T1)‖2L2>r0, there exists T2>0 such that ‖∇˜ψ(T2)‖2L2>r0, where ˜ψ(t) is the solution of (1.1) that satisfies ˜ψ(t)|t=0=˜ψ0. We consequently deduce from the continuity that there exists T3>0 such that ‖∇˜ψ(T3)‖2L2=r0. This indicates that ˜ψ(T3)∈∂V(a). Then we infer from Lemma 2.4 that A>Eγ(˜ψ0)>Eγ(˜ψ(T3))≥infu∈∂V(a)Eγ(u), which is a contradiction. The proof follows.
Lemma 2.6. It holds that
(i) a∈(0,a0), a↦mγ(a,r0) is a continuous mapping.
(ii) Leta∈(0,a0). For all μ∈(0,a), we have
mγ(a,r0)<mγ(μ,r0)+mγ(a−μ,r0). |
Proof. (ⅰ) Let a∈(0,a0) be arbitrary and {an}n≥1⊂(0,a0) be such that an→a. According to the definition of mγ(an,r0), we know that there exists un∈V(a) such that
Eγ(un)≤mγ(an,r0)+εandEγ(un)<0foranyε>0sufficientlysmall. | (2.14) |
We set vn:=√aanun and hence vn∈S(a). We find that vn∈V(a). Indeed, if an≥a, then
‖∇vn‖2L2=aan‖∇un‖2L2≤‖∇un‖2L2<r0. |
If an<a, by Lemma 2.2, (2.4) and (2.9) we have f(an,r)≥f(an,r0)≥f(a0,r0)=0 for any r∈[anar0,r0]. Indeed, since f(a,r) is a non-increasing function, then we have
f(an,r0)≥f(a0,r0)=0. | (2.15) |
And then by direct calculations we have
f(an,r)≥f(an,r0). | (2.16) |
Hence, we deduce from (2.3) and (2.14) that f(an,‖∇un‖2L2)<0, thus ‖∇un‖2L2<anar0 and
‖∇vn‖2L2=aan‖∇un‖2L2<aananar0=r0. |
Since vn∈V(a) we can write
mγ(a,r0)≤Eγ(vn)=Eγ(un)+[Eγ(vn)−Eγ(un)], |
where
Eγ(vn)−Eγ(un)=12(aan−1)‖∇un‖2L2−γ2(aan−1)∫RN|un|2|x|σdx−1α+2[(aan)α+22−1]‖un‖α+2Lα+2. |
Since ‖∇un‖2L2<r0, also ‖un‖α+2Lα+2 and ∫RN|un|2|x|σdx are uniformly bounded. Thus, we have
mγ(a,r0)≤Eγ(vn)=Eγ(un)+on(1)asn→∞. | (2.17) |
Combining (2.14) and (2.17), we get
mγ(a,r0)≤mγ(an,r0)+ε+on(1). |
Now, let u∈V(a) be such that
Eγ(u)≤mγ(a,r0)+εandEγ(u)<0. |
Set un:=√anau and hence un∈S(an). Clearly, ‖∇u‖2L2<r0 and an→a imply ‖∇un‖2L2<r0 for n large enough, so that un∈V(a). Also, Eγ(un)→Eγ(u). We thus have
mγ(an,r0)≤Eγ(un)=Eγ(u)+[Eγ(un)−Eγ(u)]≤mγ(a,r0)+ε+on(1). | (2.18) |
Therefore, since ε>0 is arbitrary, we deduce that mγ(an,r0)→mγ(a,r0). The point (ⅰ) follows.
(ⅱ) Note that, fixed μ∈(0,a), it is sufficient to prove that the following holds
mγ(θμ,r0)<θmγ(μ,r0),∀θ∈(1,aμ]. | (2.19) |
Indeed, if (2.19) holds, then we have
mγ(a,r0)<a−μa⋅aa−μmγ(a−μ,r0)+μa⋅aμmγ(μ,r0)=mγ(a−μ,r0)+mγ(μ,r0). |
Next, we prove that (2.19) holds. According to the definition of mγ(μ,r0), we have that there exists u∈V(μ) such that
Eγ(u)≤mγ(μ,r0)+εandEγ(u)<0foranyε>0sufficientlysmall. | (2.20) |
By (2.9), f(a,r)≥0 for any r∈[μar0,r0]. Hence, we can deduce from (2.3) and (2.20) that
‖∇u‖2L2<μar0. | (2.21) |
Now we set v(x):=u(θ−1Nx). On the one hand, we note that ‖v‖2L2=θ‖u‖2L2=θμ and also, because of (2.21), ‖∇v‖2L2=θ‖∇u‖2L2<r0. Thus v∈V(θμ). On the other hand, we obtain from (2.2) that
γ∫RN|u(x)|2|x|σdx≤γC1μ2−σ2‖∇u‖σL2<γ0C1a2−σ2‖∇u‖σL2, |
and it follows easily that
‖∇u‖2L2−γ∫RN|u(x)|2|x|σdx>‖∇u‖2L2−γ0C1a2−σ2‖∇u‖σL2>0 |
for 0<γ<γ0 and γ0 small sufficiently. We can obtain that
mγ(θμ,r0)≤lim infn→∞Eγ(v)=lim infn→∞(θEγ(u)+‖∇u‖2L2(θ1−σN2−θ2)−γ(θ1−σN2−θ2)∫RN|u(x)|2|x|σdx)=θmγ(μ,r0)+(θ1−σN2−θ2)lim infn→∞(‖∇u‖2L2−γ∫RN|u(x)|2|x|σdx)<θmγ(μ,r0). |
Lemma 2.7. Let {vn}n≥1⊂B(r0) be such that ∫RN|vn|2|x|σdx→0. Then, there exist a β0>0 such that
Eγ(vn)≥β0‖∇vn‖2L2+on(1). |
Proof. Indeed, using the Sobolev inequality, we obtain that
Eγ(vn)=12‖∇vn‖2L2−1α+2‖vn‖α+2Lα+2+on(1)≥‖∇vn‖2L2(12−C(α)α+2aη22raη32)+on(1). |
Now, since f(a0,r0)=0, we have
β0:=12−C(α)α+2aη220r0η32=γC12aη020rη120>0. |
Lemma 2.8. For any a∈(0,a0), let {un}n≥1⊂B(r0) be such that ‖un‖2L2→a and Eγ(un)→mγ(a,r0). Then, there exists a β1>0 and a sequence {yn}n≥1⊂RN such that
∫B(yn,R)|un|2dx≥β1>0,forsomeR>0. | (2.22) |
Proof. We assume by contradiction that (2.22) does not hold. Since {un}n≥1⊂B(r0) and ‖un‖2L2→a, the sequence {un}n≥1 is bounded in H1(RN). From Lemma I.1 in [21] and since 4N+2<α+2<2NN−2, we deduce that ‖un‖Lα+2→0 as n→∞. At this point, Lemma 2.7 implies Eγ(un)≥on(1). This contradicts the fact that mγ(a,r0)<0 and the lemma follows.
Theorem 2.9. For any a∈(0,a0), if {un}n≥1⊂B(r0) is such that ‖un‖2L2→a and Eγ(un)→mγ(a,r0) then, up to translation, un→u in H1(RN).
Proof. Let {un}n≥1⊂B(r0) be a minimizing sequence of the energy functional Eγ(u), that is,
‖un‖2L2=aandlimn→∞Eγ(un)=mγ(a,r0). |
By similar proof as [23], we known that {un}n≥1 is bounded in H1(RN). By Lemma 2.6, Lemma 2.8 and Rellich compactness theorem, there exist sequences {unk}k≥1⊂H1(RN) and {ynk}k≥1⊂RN such that for any ϵ>0, there exists R(ϵ)>0 such that for all k≥1,
∫B(ynk,R(ϵ))|unk|2dx≥β2−ϵ>0. | (2.23) |
Denote ˜unk(⋅)=unk(⋅+ynk), then there exists ˜u such that ˜unk⇀˜u weakly in H1(RN), ˜unk→˜u strongly in Lrloc(RN) with r∈[2,2NN−2), combining with (2.23) we have
∫B(0,R(ϵ))|˜u|2dx≥β2−ϵ>0. |
Thus ∫RN|˜u|2dx=β2. Indeed, we assume by contradiction that ∫RN|˜u|2dx=¯β2<β2. According to Brézis-Lieb Lemma and ˜unk=(˜unk−˜u)+˜u, we have
‖˜unk‖2H1=‖˜unk−˜u‖2H1+‖˜u‖2H1+on(1), |
‖˜unk‖qLq=‖˜unk−˜u‖qLq+‖˜u‖qLq+on(1),1≤q<∞, |
‖˜unk‖p+2Lp+2=‖˜unk−˜u‖p+2Lp+2+‖˜u‖p+2Lp+2+on(1), |
∫RN|˜unk|2|x|σdx=∫RN|˜unk−˜u|2|x|σdx+∫RN|˜u|2|x|σdx+on(1). |
Then,
E(˜unk)=E(˜unk−˜u)+E(˜u)+on(1). |
Hence,
mγ(β2,r0)≥mγ(β2−¯β2,r0)+mγ(¯β2,r0), |
which contradicts Lemma 2.6. Thus ∫RN|˜u|2dx=β2, i.e., ˜unk→˜u strongly in L2(RN). By the Gagliardo-Nirenberg inequality, ˜unk→˜u strongly in Ls(RN), where s∈[2,2NN−2). Next, our aim is to show that {ynk}k≥1⊂RN is bounded. If it was not the case, we deduce that
∫RN|unk|2|x|σdx=∫RN|˜unk|2|x+ynk|σdx→0ask→∞. |
Hence limn→∞Eγ(unk)=limn→∞E∞γ(unk), where E∞γ(unk) is the corresponding energy functional of the sequence {unk}k≥1⊂H1(RN) via translation transformation. On the other hand, we know that limn→∞E∞γ(un) is attained by a nontrivial function ur0∈B(r0). Hence,
limk→∞E∞γ(unk)=limn→∞E∞γ(˜unk)=limk→∞12∫RN|∇˜unk|2dx−limk→∞1α+2∫RN|˜unk|α+2dx≥12∫RN|∇˜u|2dx−1α+2∫RN|˜u|α+2dx. |
According to the definition of E∞γ(unk), we see that ˜u is a minimizer of limk→∞E∞γ(unk). Consequently,
limk→∞Eγ(unk)<limk→∞E∞γ(unk)−∫RN|ur0|2|x|σdx<limk→∞E∞γ(unk), |
which contradicts limk→∞Eγ(unk)≥limk→∞E∞γ(unk). Thus, {ynk}k≥1⊂RN is bounded. We can deduce, up to subsequences, limk→∞ynk=y0 for some y0∈RN. Consequently,
‖unk(x)−˜u(x−y0)‖Ls≤‖unk(x)−˜u(x−ynk)‖Ls+‖˜u(x−ynk)−˜u(x−y0)‖Ls=‖unk(x+ynk)−˜u(x)‖Ls+‖˜u(x+y0−ynk)−˜u(x)‖Ls→0 | (2.24) |
as k→∞ for any s∈[2,2NN−2), that is, unk converges strongly to ˜u(x−y0) in Ls(RN) for s∈[2,2NN−2). We denote u(x)=˜u(x−y0). Hence,
limk→∞Eγ(unk)≥12∫RN|∇u|2dx−γ2∫RN|u|2|x|σdx−1α+2∫RN|u|α+2dx. |
We see that Eγ(u)=mγ(a,r0) and hence unk→u in H1(RN).
Proof of Theorem 2.3. The existence of γ0 and a minimizer for Eγ(u) on V(a) were proved in Lemma 2.6, Lemma 2.7 and Theorem 2.9.
Proof of Theorem 1.3. (ⅰ) The property that M(a)⊂V(a) is non-empty follows from Theorem 2.3.
(ⅱ) We will show that M(a) is orbitally stable. Firstly, we note that the solution ψ of (1.1) exists globally by Lemma 2.5. Now we suppose by contradiction that there exist ϵ0>0, a sequence of initial data {ψ0,n}n≥1⊂H1(RN) and a sequence {tn}n≥1⊂R such that for all n≥1,
infu∈M(a)‖ψ0,n−u‖H1<1n,infu∈M(a)‖ψn(tn)−u‖H1≥ϵ0, | (3.1) |
where ψn(t) is the solution to (1.1) with initial data ψ0,n. Next, we claim that there exists v∈M(a) such that
limn→∞‖ψ0,n−v‖H1=0. |
Indeed, by (3.1), we see that for each n≥1, there exists vn∈V(a) such that
‖ψ0,n−vn‖H1<2n. | (3.2) |
Since {vn}n≥1⊂M(a) is a minimizing sequence of (1.7) and by the same argument as in Lemma 2.3, there exists v∈M(a) such that
limn→∞‖vn−v‖H1=0. | (3.3) |
By (3.2) and (3.3), the claim follows. And then, due to v∈V(a), we have ~ψn=√aψn(tn)‖ψn(tn)‖L2∈V(a) and
limn→∞Eγ(~ψn)=limn→∞Eγ(ψn(tn))=limn→∞Eγ(ψ0,n)=Eγ(v)=mγ(a,r0), |
which implies that {~ψn}n≥1 is a minimizing sequence for (1.7). Thanks to the compactness of all minimizing sequence of (1.7), there is a ˜u∈V(a) satisfies
~ψn→˜uinH1(RN). |
Moreover, by the definition of ~ψn, it follows easily that
~ψn−ψn(tn)→0inH1(RN). |
Consequently, we have
ψn(tn)→˜uinH1(RN), |
which contradicts (3.1). The proof is now complete.
In this work, we study the stability of the set of energy minimizers in the mass supercritical case. In the mass supercritical case, the energy functional is unbounded from below on S(a). Thus, we consider the further constrained minimization problem to study the existence and stability of standing waves for (1.1). And, the energy functional is not invariant under the scaling transform due to inverse-power potentials γ|x|−σu. It is intrinsically difficult for us to prove the compactness of minimizing sequences. By a rather delicate analysis, we can overcome this difficulty by proving the boundedness of any translation sequence.
This work is supported by the Outstanding Youth Science Fund of Gansu Province (No. 20JR10RA111).
The author declares no conflicts of interest.
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