In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations
{√−Δ+m2u+λu=ϑ|u|p−2v+|u|2♯−2v,x∈RN, u>0, ∫RN|u|2dx=a2,
where N≥2, a,ϑ,m>0, λ is a real Lagrange parameter, 2<p<2♯=2NN−1 and 2♯ is the critical Sobolev exponent. The operator √−Δ+m2 is the fractional relativistic Schrödinger operator. Under appropriate assumptions, with the aid of truncation technique, concentration-compactness principle and genus theory, we show the existence and the multiplicity of normalized solutions for the above problem.
Citation: Xueqi Sun, Yongqiang Fu, Sihua Liang. Normalized solutions for pseudo-relativistic Schrödinger equations[J]. Communications in Analysis and Mechanics, 2024, 16(1): 217-236. doi: 10.3934/cam.2024010
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In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations
{√−Δ+m2u+λu=ϑ|u|p−2v+|u|2♯−2v,x∈RN, u>0, ∫RN|u|2dx=a2,
where N≥2, a,ϑ,m>0, λ is a real Lagrange parameter, 2<p<2♯=2NN−1 and 2♯ is the critical Sobolev exponent. The operator √−Δ+m2 is the fractional relativistic Schrödinger operator. Under appropriate assumptions, with the aid of truncation technique, concentration-compactness principle and genus theory, we show the existence and the multiplicity of normalized solutions for the above problem.
We consider the biharmonic nonlinear Schrödinger equation (NLS) with the mixed dispersion and a general nonlinear term
i∂tψ−γΔ2ψ+βΔψ+f(ψ)=0,ψ(0,x)=ψ0(x),(t,x)∈R×RN, | (1.1) |
where N≥1, i denotes the imaginary unit, γ>0,β∈R and the nonlinear term f satisfies the following conditions throughout this paper:
(F1) f∈C(C,C),f(0)=0.
(F2) f(s)∈R for s∈R,f(eiθz)=eiθf(z) for θ∈R,z∈C.
(F3) limz→0f(z)/|z|=0.
(F4) lim|z|→∞f(z)/|z|l−1=0, where l:=2+8/N.
(F5) There exists s0>0 such that F(s0)>0, where F(z)=∫|z|0f(τ)dτ for z∈C.
In nonlinear optics, the NLS is usually derived from the nonlinear Helmholtz equation for the electric field by separating the fast oscillations from the slowly varying amplitude. In the so-called paraxial approximation, the NLS appears in the limit as the equation solved by the dimensionless electric-field amplitude, see [1,Section 2]. The fact that its solutions may blow up in finite time suggests that some small terms neglected in the paraxial approximation play an important role in preventing this phenomenon. Therefore, a small fourth-order dispersion term was proposed in [1] (see also [2,3,4]) as a nonparaxial correction, which eventually gives rise to (1.1). For more background, see [5,6,7] and references therein.
Under these conditions (F1)-(F5), for a solution u of (1.1), it has been established that the following conservations laws:
|ψ(t,⋅)|2=|ψ(0,⋅)|2,I(ψ(t,⋅))=I(ψ(0,⋅))for any t∈R, |
where Lq(RN) is the usual Lebesgue space with norm |u|qq:=∫RN|u|qdx, 1≤q<∞, and I is the energy functional associated with (1.1) defined by
I(ψ)=γ2∫RN|Δψ|2dx+β2∫RN|∇ψ|2dx−∫RNF(ψ)dx |
for ψ∈H2(RN).
If ψ is a standing wave, i.e., ψ(t,x)=eiλtu(x), then u∈H2(RN) and λ∈R satisfy the following equation:
γΔ2u−βΔu+λu=f(u),x∈RN. | (1.2) |
In this paper, we look for solutions (u,λ) with a priori prescribed L2-norm. For a given a>0, we put
Sa:={u∈H2(RN):∫RN|u|2dx=a}. | (1.3) |
In this way, the parameter λ is unknown and appears as a Lagrange multiplier. We remark that this is natural, from a physical point view, to search for the normalized solutions which have prescribed mass.
When γ=0,β≠0, the problem (1.2)-(1.3) has attracted much attention in the last twenty years. The presence of the L2-constraint makes several methods developed to deal with unconstrained variational problems unavailable, and new phenomena arise. If we set f(u)=|u|p−2u, then a new critical exponent appears, i.e., the L2-critical exponent
ˉp=2+4N. |
This is the threshold exponent for many dynamical properties such as global existence vs. blow-up, and the stability or instability of ground states. From the variational point of view, if the problem is purely L2-subcritical, then I is bounded from below on Sa. Thus, for every a>0, a ground state can be found as a global minimizer of I|Sa, and moreover, the minimizer would be orbitally stable, see [8,9] for homogeneous nonlinear term and [10,11] for general nonlinear term. And multiplicity results of normalized solutions in the L2-subcritical case can be referred to [12,13] and the references therein. In the purely L2-supercritical case, on the contrary, I|Sa is unbounded from below; however, exploiting the mountain pass lemma and a smart compactness argument, L. Jeanjean [14] could show that a normalized ground state does exist for every a>0 also in this case. The associated standing wave is strongly unstable [15,16] for homogeneous nonlinear term, due to the supercritical character of the equation. We point out that, in [14,17,18,19], more general nonlinearities are considered. With regard to the combined nonlinearities, we refer the reader to [20,21] for the existence and stability results.
For the case γ≠0,β≠0, there is only a few papers about the normalized solutions. As we know, this kind of problem would give rise to a new L2-critical exponent, i.e.,
l=2+8/N. |
When γ>0,β>0, Bonheure et al. [5] have dealt with the L2-subcritical case and obtained the existence of normalized solutions as energy minimizers, while for the L2-critical and supercritical case, [6] is concerned with several questions including the existence of ground states and of positive solutions and the multiplicity of radial solutions, and the stability of the standing waves of the associated dispersive equation have also been discussed. Recently, in [22] the authors have improved some results to [5] and [6]. When γ>0,β<0, the problem is more involved, see [7,23] for the L2-subcritical case and [24] for the L2-supercritical case. We remark that all the aforementioned papers have only considered the homogeneous nonlinearity, i.e., f(u)=|u|p−2u. For the general nonlinear term, as far as we know, the results are not there yet. With regard to the point, we attempt to study this kind of problem in this paper. We point out that, when dealing with general nonlinearity, we will face some extra difficulties. Such as the loss of homogeneity, which often plays an important role to use the scaling transformations. Besides, some inequalities about energy would be more difficult to obtain. But these inequalities are the key to obtain the compactness of the minimizing sequences. Finally, other types of normalized solution problems can be referred to [25,26,27,28,29,30,31,32,33,34,35] and the references therein.
In what follows, we give some notations. In H2(RN), when β>0, we define its norm by
‖u‖H2:=(∫RN[|Δu|2+|∇u|2+|u|2]dx)12, |
and for β≤0, by
‖u‖H2:=(∫RN[|Δu|2+|u|2]dx)12. |
Recalling that the following interpolation inequality
∫RN|∇u|2dx≤(∫RN|Δu|2dx)12(∫RN|u|2dx)12,∀u∈H2(RN), | (1.4) |
we easily see these two norms above are equivalent in H2(RN). We define the energy functional associated with (1.2) by
I(u)=γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−∫RNF(u)dx |
for u∈H2(RN), and we consider a constrained variational problem as follows:
ma=infu∈SaI(u). | (1.5) |
Denote the set of minimizers, called ground states for (1.1),
Ma={u∈Sa:I(u)=ma}. |
In this paper, we will study the orbital stability of standing waves for (1.1), in the following sense:
Definition 1.1. The set Ma is said to be orbitally stable if any given ε>0, there exists δ>0 such that for any initial data ψ0 satisfying
infu∈Ma‖ψ0−u‖H2<δ, |
the corresponding solution ψ(t,x) of the Cauchy problem (1.1) satisfies
infu∈Ma‖ψ(t,⋅)−u‖H2<ε for allt≥0. |
According to the sign of β, we consider the following two cases respectively: (Ⅰ) β≥0, (Ⅱ) β<0.
(Ⅰ): β≥0. In this case, we define
a0=inf{a>0;ma<0}, | (1.6) |
see Lemma 3.3 and (3.6) for more details about a0. For the existence and stability of the minimizer of ma, we have
Theorem 1.2. Under the case β≥0, suppose (F1)–(F5) and that a constant a0≥0 which satisfies (1.6) is uniquely determined. If a>a0,
(i) There exists a global minimizer with respect to ma, i.e., Ma≠∅.
(ii) Assume the local existence of the Cauchy problem (1.1), then Ma is orbitally stable, i.e., for any ε>0, there exists δ>0 such that for any solution u of (1.1) with dist(u(0,⋅),Ma)<δ, it holds that
dist(u(t,⋅),Ma)<εfor anyt∈R, |
where dist(ϕ,Ma)=infψ∈Ma‖ϕ−ψ‖H2.
If 0<a<a0, there is no global minimizer with respect to ma.
Remark 1.3. Note that under assumption (F1)–(F5) it is not known if (1.1) is locally well posed. Thus, we need to assume the local existence of the Cauchy problem (1.1) in Theorem 1.2, a similar assumption also appears in Theorem 1.7. However, when f(u)=|u|p−2u,2<p<2+8/N, the local (even global) existence of the Cauchy problem (1.1) has been known, see [1] for β≥0 and [40] for β<0.
We briefly outline the proof of Theorem 1.2. As the celebrated paper [8] to prove the orbital stability of ground state, the main method we use is the Concentration Compactness Principle. However, our situation is far more complex because we deal with the operator γΔ2−βΔ and the general nonlinearity. To rule out the vanishing case of the minimizing sequences, we need to know when the condition ma<0 holds for which mass a. This is the reason why we define the value of a0 in (1.6). Besides, the second difficulty we face is to exclude the dichotomy. And we prove a strict subadditivity (conditional) inequality for ma to overcome this obstacle, see Lemma 3.3. In addition, since we often use the scaling transformations of functions, the general nonlinearity also causes some extra difficulties.
Next, we give the characterization of ma. And it is a direct consequence of the definition of a0 and Lemma 3.3.
Corollary 1.4. (i) If a0=0, then ma<0 for any a>0.
(ii) If a0>0, then ma=0 for any a∈(0,a0], and ma<0 for any a>a0.
It is a natural question that "When a0>0 holds". To answer the question, the behavior of f near 0 is important. We can show that the following results:
Theorem 1.5. If β=0 and we assume f satisfies (F1)-(F5).
(i) If lim infs→0F(s)/|s|l=∞ holds, then a0=0 holds.
(ii) If lim sups→0F(s)/|s|l<∞ holds, then a0>0 holds.
Theorem 1.6. If β>0 and we assume f satisfies (F1)-(F5).
(i) If lim infs→0F(s)/|s|2+4/N=∞ holds, then a0=0 holds.
(ii) If lim sups→0F(s)/|s|2+4/N<∞ holds, then a0>0 holds.
Let us explain why the conditions in Theorems 1.5 and 1.6 are different. For β=0, the main effect for the integral ∫RNF(u)dx is the semi-norm |Δu|22. While for β>0, both |Δu|22 and |∇u|22 affect the integral ∫RNF(u)dx. In particular, whether ma is negative or not greatly depends on the "small" u. For u small, compared with |Δu|22, the gradient norm |∇u|22 (when it exists) dominates the effect.
(Ⅱ): β<0. Under this case, the problem (1.2)-(1.3) is more involved since the term β|∇u|22 in the energy functional I can't be a part of H2-norm but acts as an independent part which effects the behavior of the energy. At present, except for (F1)-(F5), we also assume f satisfies:
(F6) Assume that F(s)≥0 for every s≥0 and there exists a constant η>2 such that F(τs)≥τηF(s) for every τ≥1,s≥0.
We set
a1:=inf{a>0:ma<−β28γa}, | (1.7) |
see Lemma 4.4 and (4.2) for more details about a1. For the existence and stability of the minimizer of ma, we have
Theorem 1.7. Under the case β<0, suppose (F1)–(F6) and that a constant a1≥0 which satisfies (1.7) is uniquely determined. If a>a1, we have
(i) there exists a global minimizer with respect to ma, i.e., Ma≠∅.
(ii) assume the local existence of the Cauchy problem (1.1), then Ma is orbitally stable, i.e., for any ε>0, there exists δ>0 such that for any solution u of (1.1) with dist(u(0,⋅),Ma)<δ, it holds that
dist(u(t,⋅),Ma)<εfor anyt∈R, |
where dist(ϕ,Ma)=infψ∈Ma‖ϕ−ψ‖H2.
If 0<a≤a1, there is no global minimizer with respect to ma.
Remark 1.8. A typical example of the nonlinear term satisfying (F1)–(F6) is f(u)=|u|p−2u+μ|u|q−2u,2<q<p<2+8/N,μ≥0.
In the proof of Theorem 1.7, the situation is more involved compared with the case β≥0. To rule out the vanishing case of minimizing sequences, we need to analyse the spectrum of the operator γΔ2u−βΔu. Thanks to a result in [23] (see Lemma 4.2), we can infer the behavior of ma and overcome the difficulty by defining the value of a1 in (1.7). Besides, the dichotomy case is more hard to deal with. To this aim, we use suitable scaling transformations of functions to get the subadditivity condition for the minimizing energy and hence exclude this case. Once we get the precompactness of minimizing sequences, we can prove the existence and orbital stability of normalized solutions.
Next, we give the characterization of ma. And it is a direct consequence of the definition of a1 and Lemma 4.4.
Corollary 1.9. (i) If a1=0, then ma<−β28γa for any a>0.
(ii) If a1>0, then ma=−β28γa for any a∈(0,a1], and ma<−β28γa for any a>a1.
Finally, in the case β<0, we consider the nonlinearity f(u)=|u|p−2u+μ|u|q−2u,2<q<p<2+8/N,μ<0. It is easy to see that f satisfies the conditions (F1)–(F5), but does not satisfy (F6). However, with regard to the value of minimizing energy ma, we can still give its partial characterization as follows.
Theorem 1.10. Let β<0 and f(u)=|u|p−2u+μ|u|q−2u,2<q<p<2+8/N,μ<0. Then ma≤−β28γa for any a>0. Moreover, there exist two constants a∗,a∗∈[0,∞) with a∗≥a∗ such that
(i) if a∗=0, then ma<−β28γa for any a>0.
(ii) if a∗>0 and a∗>0, then ma=−β28γa for any a∈(0,a∗], and ma<−β28γa for any a>a∗.
(iii) if a∗>0 and a∗=0, then ma<−β28γa for any a>a∗.
Compared with the results of Corollary 1.9, the present situation is more involved and the behavior of ma is more difficult to figure out. Based on this point, we think the condition (F6) may be crucial to determine the behavior of ma and hence the existence of minimizers for ma. Although we assume it holds that a∗>a∗>0, we can't infer the behavior of ma in (a∗,a∗] due to the combined effect of the nonlinear terms |u|p−2u and μ|u|q−2u. On the other hand, for a>a∗, we have ma<−β28γa, but we still don't know whether ma can be achieved. At this case, ruling out the vanishing case of the minimizing sequence is easy, however, the dichotomy case is difficult to deal with because the strict subadditivity condition is unclear. Thus, we can't deduce the precompactness of the minimizing sequence. All the facts above show that there is a sharp contrast between the conditions containing (F6) and these conditions lacking (F6).
This paper is organized as follows. In Section 2, we give some preliminaries which will be used later. Section 3 is devoted to studying the existence and orbital stability of normalized solutions which belong to the ground state set Ma under the case β≥0. The main method we use is the concentration compactness principle. And we rule out the vanishing case according to the negative of energy and exclude the dichotomy by proving a strict additivity inequality for ma, see Lemma 3.4. Also in this section, we give the proofs of Theorems 1.2, 1.5 and 1.6. In Section 4, we consider the case β<0. This situation is more involved. To obtain the existence and orbital stability of normalized solutions, we propose an extra condition on f, i.e., (F6). But it is more hard to rule out the vanishing case. We make use of the spectral analysis for the operator γΔ2u−βΔu which was given in [23] to rule out the vanishing case. The proof of Theorem 1.7 is finished in this section. Finally, to reveal the effect of the condition (F6), we consider a special nonlinearity which does not satisfy (F6) and investigate the behavior of the energy ma, i.e., Theorem 1.10.
We begin by recall two well-known Gagliardo-Nirenberg interpolation inequalities for functions u∈H2(RN), namely,
|u|p≤BN,p|Δu|(p−2)N4p2|u|1−(p−2)N4p2, | (2.1) |
where
{2≤pfor N≤4,2≤p≤2NN−4for N>4, |
and
|u|p≤CN,p|∇u|(p−2)N2p2|u|1−(p−2)N2p2, | (2.2) |
where
{2≤pforN≤2,2≤p≤2NN−2forN>2. |
In what follows, we give a concentration-compactness lemma for the sequence in H2(RN).
Lemma 2.1. Let {un}n∈N be a bounded sequence in H2(RN) which satisfies
supz∈RN∫B(z,1)|un|2dx→0asn→∞. |
Then, for p∈(2,4∗),
|un|p→0asn→∞ |
holds, where 4∗=2N/(N−2)+ is the critical Sobolev exponent.
Proof. For the proof, we can take a similar argument as that of the classical concentration-compactness lemma by Lions and omit the details. See, for instance, [38,Lemma 1.21].
Throughout this section, unless otherwise stated, we always assume f satisfies (F1)–(F5). First, we show that ma is bounded from blow for any a>0.
Lemma 3.1. (i) Let {un}n∈N be a bounded sequence in H2(RN). If either limn→∞|un|2=0 or limn→∞|un|l=0 holds, then it is true that limn→∞∫RNF(un)dx=0.
(ii) There exists a positive constant C=C(f,N,a,γ) depending f,N and a such that
I(u)≥γ4∫RN|Δu|2dx+β2∫RN|∇u|2dx−C | (3.1) |
holds for any u∈Sa. Specifically, ma≥−C>−∞.
Proof. (ⅰ): By the assumptions (F1)–(F4), for any ε>0, there exists a positive constant C(f,ε) which depends on ε and f such that
|F(u)|≤C(f,ε)|u|2+ε|u|l,|F(u)|≤ε|u|2+C(f,ε)|u|l, |
where l=2+8/N. For u∈H2(RN), we have
|∫RNF(u)dx|≤C(f,ε)|u|22+ε|u|ll, | (3.2) |
|∫RNF(u)dx|≤ε|u|22+C(f,ε)|u|ll. | (3.3) |
The Gagliardo-Nirenberg inequality implies that
|u|ll≤BN|Δu|22|u|8N2, |
where BN is a positive constant which depends on N. Thus, we obtain
|∫RNF(u)dx|≤C(f,ε)|u|22+εBN|Δu|22|u|8N2. | (3.4) |
We take the case where {un}n∈N is a bounded sequence in H2(RN) satisfying limn→∞|un|2=0. By (3.4), we have limn→∞∫RNF(un)dx=0. Alternatively, we can take the case where {un}n∈N is a bounded sequence in H2(RN) satisfying limn→∞|un|l=0. By (3.3), we have
lim supn→∞|∫RNF(u)dx|≤ε|u|22. |
Since we can choose ε>0 arbitrary, we obtain limn→∞∫RNF(un)dx=0.
(ⅱ): In (3.4), we choose ε>0 satisfying BNa4Nε=γ4. Then, for u∈Sa, we have
∫RNF(u)dx≤C+γ4∫RN|Δu|2dx, |
where C=C(f,N,a,γ) is a positive constant which depends on f,N,γ and a. This implies (3.1).
Lemma 3.2. Let {un}n∈N be a bounded sequence in H2(RN) satisfying limn→∞|un|22=a>0. Let αn=√a/|un|2 and ˜un=αnun. Then the following holds:
˜un∈Sa,limn→∞αn=1,limn→∞|I(˜un)−I(un)|=0. |
Proof. Clearly, ˜un∈Sa and limn→∞αn=1 hold. We can compute
I(˜un)−I(un)=γ(α2n−1)2∫RN|Δun|2dx+β(α2n−1)2∫RN|∇un|2dx−∫RN[F(αnun)−F(un)]dx=γ(α2n−1)2∫RN|Δun|2dx+β(α2n−1)2∫RN|∇un|2dx−∫RN[F(|αnun|)−F(|un|)]dx=γ(α2n−1)2∫RN|Δun|2dx+β(α2n−1)2∫RN|∇un|2dx−∫RN(∫10f(|un|+(|αn−1|)θ|un|)(|αn|−1)|un|dθ)dx=γ(α2n−1)2∫RN|Δun|2dx+β(α2n−1)2∫RN|∇un|2dx−(|αn|−1)∫RN(∫10f(|un|+(|αn−1|)θ|un|)|un|dθ)dx. |
We have 0≤|un|+(|αn−1|)θ|un|≤(|αn+2|)|un|. Under the assumptions (F1)–(F4), we have |f(s)|≤|s|+C|s|l−1. Hence, we obtain
|∫RN(∫10f(|un|+(|αn−1|)θ|un|)|un|dθ)dx|≤∫RN(∫10|(αn+2)|un|2+C(αn+2)l−1|un|ldθ)dx=∫RN(αn+2)|un|2+C(αn+2)l−1|un|ldx. |
Since {un}n∈N is bounded in H2(RN), we achieve our conclusion.
In what follows, we give some properties on ma.
Lemma 3.3. (i) ma≤0 for any a>0.
(ii) ma+b≤ma+mb for any a,b>0.
(iii) a↦ma is nonincreasing.
(iv) For sufficiently large a, ma<0 holds.
(v) a↦ma is continuous.
Proof. (ⅰ): Let u∈Sa. For τ>0, we set uτ(x)=τN/2u(τx), giving uτ∈Sa. Moreover, |uτ|ll=τ4|u|ll→0 as τ→0. By Lemma 3.1(ⅰ), we have
limτ→0∫RNF(uτ)dx=0. |
As
|Δuτ|22=τ4|Δu|22,|∇uτ|22=τ2|∇u|22, |
we see that limτ→0I(uτ)=0 holds. By the definition of ma, we have ma≤I(uτ). Thus, we obtain ma≤0.
(ⅱ): We fix ε>0. By the definition of ma and mb, there exist u∈Sa∩C∞0(RN) and v∈Sb∩C∞0(RN) such that
I(u)≤ma+ε,I(v)≤mb+ε. |
Since u and v have compact support, by using parallel translation, we can assume suppu∩suppv=∅. Therefore, we have u+v∈Sa+b. Thus, we find
ma+b≤I(u+v)=I(u)+I(v)≤ma+mb+2ε. |
Since ε is arbitrary, we have ma+b≤ma+mb.
(ⅲ): By (ⅰ) and (ⅱ), we have
ma+b≤ma+mb≤ma |
for any a,b>0. This gives (ⅲ).
(ⅳ): We set
M0:=sups∈[0,s0]|F(s)|, |
where s0 is a constant determined in (F5). By (F5), we know M0>0. And then we choose a constant α>1 such that
M0(αN−1)=F(s0)2. |
We take a cut-off function φ∈C∞0(RN) such that
φ(x)={s0,|x|≤1,0,|x|≥α, |
For R>0, we set φR(x)=φ(x/R), then there exist two constants C1,C2>0 such that
|∇φR|≤C1R,|ΔφR|≤C2R2. |
We write |SN−1| for the surface area of the unit sphere. If N=1, set |S0|=2. Now we estimate I(φR) as follows:
I(φR)=γ2∫RN|ΔφR|2dx+β2∫RN|∇φR|2dx−∫RNF(φR)dx=∫R≤|x|≤αR[γ2|ΔφR|2+β2|∇φR|2−F(φR)]dx−∫|x|≤RF(s0)dx≤∫R≤|x|≤αR[γC222R4+βC212R2+M0]dx−F(s0)|S|N−1RNN=(αN−1)|S|N−12N[γC22RN−4+βC21RN−2]+|S|N−1RNN[M0(αN−1)−F(s0)]=|S|N−1RNF(s0)2N[γC222M0R4+βC212M0R2−1]. |
Since
γC222M0R4+βC212M0R2−1→−1asR→∞, |
for a sufficiently large R, we have I(φR)<0. By choosing such a R and setting aR=|φR|22, we obtain maR≤I(φR)<0. By (ⅲ), we have mb≤maR<0 if b≥aR.
(ⅴ): We fix a>0. By (ⅲ), ma−h and ma+h are monotonic and bounded as h→0+, so therefore they has limits. Moreover, ma−h≥ma≥ma+h holds due to (ⅲ). Thus, we obtain
limh→0+ma−h≥ma≥limh→0+ma+h. |
Claim: limh→0+ma−h≤ma.
By (ⅰ), this is clear if ma=0. So we consider the case ma<0. Take u∈Sa and let uh(x)=√1−h/au(x) for 0<h<<1. Since |uh|22=(1−h/a)a=a−h, we have uh∈Sa. On the other hand, we have
‖uh−u‖H2=(1−√1−ha)‖u‖H2→0ash→0+. |
Thus, we obtain limh→0+I(uh)=I(u). By ma−h≤I(uh), we have
limh→0+ma−h≤limh→0+I(uh)=I(u). |
As we choose u∈Sa arbitrarily, we obtain limh→0+ma−h≤ma.
Claim: limh→0+ma+h≥ma.
Since the left hand side converges, it is sufficient to consider the case h=1/n, where n∈N. Choose a un∈Sa+1/n which satisfies I(un)≤ma+1/n+1/n for each n∈N. By (ⅰ), I(un)≤1/n. Lemma 3.1(ⅱ) asserts that {un}n∈N is a bounded sequence in H2(RN). By the definition of un, we have
limn→∞ma+1/n≤limn→∞I(un)≤limn→∞ma+1/n+1/n, |
which implies
limn→∞I(un)=limn→∞ma+1/n=limh→0+ma+h. | (3.5) |
Let vn=un/√1+1/(an) for n∈N. Then, {vn}n∈N is also a bounded sequence in H2(RN). Moreover, we have
|vn|22=|un|221+1/(an)=a+1/n1+1/(an)=a. |
Hence, vn∈Sa holds. By Lemma 3.2, we obtain
ma≤I(vn)=I(un)+o(1)asn→∞. |
By (3.5), the claim holds.
We define
a0=inf{a>0;ma<0}. | (3.6) |
By Lemma 3.3, a0 is well-defined. Moreover, if a0>0, by Lemma 3.3 (ⅴ), we know
ma0=0. | (3.7) |
Under certain conditions, we can further prove the strict subadditivity for ma.
Lemma 3.4. (i) Assume that there exists a global minimizer u∈Sa with respect to ma for some a>0. Then mb<ma for any b>a. In particular, we have mb<0 for any b>a.
(ii) Assume that there exist global minimizers u∈Sa and v∈Sb with respect to ma and mb respectively for some a,b>0. Then ma+b<ma+mb.
Proof. (ⅰ): By Lemma 3.3(ⅰ), we have I(u)≤0. Now setting τ=b/a>1 and ˜u(x)=u(τ−1/Nx), by the assumption, we have |˜u|22=b and
I(˜u)=τ(γτ−4/N2∫RN|Δu|2dx+βτ−2/N2∫RN|∇u|2dx−∫RNF(u)dx)<τI(u), |
Noticing that I(u)=ma and the definition of mb, we obtain mb≤I(˜u)<τma≤ma.
(ⅱ): By the assumption and the argument as above, we have
mηa<ηmafor anyη>1,mτa≤τmafor anyτ≥1. |
Noting that we can assume 0<b≤a without loss of generality, taking η=(a+b)/a and τ=a/b, we obtain
ma+b<a+bama=ma+bamab⋅b≤ma+mb. |
It completes the lemma.
With regard to the minimizing sequence for ma, we have
Theorem 3.5. Suppose (F1)–(F5) and that a>0. If {un}n∈N⊂Sa is a minimizing sequence with respect to ma, then one of the following holds:
(i)
lim supn→∞supz∈RN∫B(z,1)|un|2dx=0. | (3.8) |
(ii) Taking a subsequence if necessary, there exist u∈Sa and a family {yn}n∈N⊂RN such that un(⋅−yn)→u in H2(RN) as n→∞. Specifically, u is a global minimizer.
Proof. Suppose that {un}n∈N⊂Sa is a minimizing sequence which does not satisfy (3.8). It is sufficient to show that (ⅱ) holds. Since (3.8) does not hold and {un}n∈N⊂Sa, we have
0<lim supn→∞supz∈RN∫B(z,1)|un|2dx≤α<∞. |
Taking a subsequence if necessary, there exists a family {yn}n∈N⊂RN, such that
0<limn→∞∫B(0,1)|un(x−yn)|2dx<∞. | (3.9) |
Since {un}n∈N⊂Sa is a minimizing sequence, Lemma 3.1(ⅱ) asserts that {un}n∈N is a bounded sequence in H2(RN). Hence {un(⋅−yn)}n∈N is a bounded sequence in H2(RN). Using the weak compactness of a Hilbert space and the Rellich compactness, for some subsequence, there exists u∈H2(RN) such that
un(⋅−yn)⇀uweakly inH2(RN), | (3.10) |
un(⋅−yn)→uinL2loc(RN), | (3.11) |
un(⋅−yn)→ua.e. inRN. | (3.12) |
Equations (3.9) and (3.11) assert that |u|2>0. We put vn=un(⋅−yn)−u. By (3.10), vn⇀0 weakly in H2(RN). Thus, we have
∫RN|Δu+Δvn|2dx=∫RN|Δu|2dx+∫RN|Δvn|2dx+2ℜ∫RNΔu¯Δvndx=∫RN|Δu|2dx+∫RN|Δvn|2dx+o(1)asn→∞, | (3.13) |
∫RN|∇u+∇vn|2dx=∫RN|∇u|2dx+∫RN|∇vn|2dx+2ℜ∫RN∇u⋅¯∇vndx=∫RN|∇u|2dx+∫RN|∇vn|2dx+o(1)asn→∞, | (3.14) |
∫RN|u+vn|2dx=∫RN|u|2dx+∫RN|vn|2dx+2ℜ∫RNu¯vndx=∫RN|u|2dx+∫RN|vn|2dx+o(1)asn→∞. | (3.15) |
Using (3.12), the Brezis-Lieb theorem(see [39] or [13,Lemma 3.2]) implies that
∫RNF(u+vn)dx=∫RNF(u)dx+∫RNF(vn)dx+o(1)asn→∞. |
Since I(un)=I(un(⋅−yn))=I(u+vn), we can obtain
I(un)=I(u)+I(vn)+o(1),|un|22=|u|22+|vn|22+o(1)asn→∞. | (3.16) |
We will show the following claim.
Claim.
lim supn→∞supz∈RN∫B(z,1)|vn|2dx=0. | (3.17) |
Suppose that (3.17) does not hold. Since {vn}n∈N is bounded in H2(RN), similarly as above, for some subsequence, there exist a family {zn}n∈N⊂RN and v∈H2(RN) satisfying |v|2>0 such that
vn(⋅−zn)⇀vweakly inH2(RN),vn(⋅−zn)→vinL2loc(RN),vn(⋅−zn)→va.e. inRN. |
We put wn=vn(⋅−zn)−v. Then, similarly as above, we can obtain
I(vn)=I(v+wn)=I(v)+I(wn)+o(1),|vn|22=|v|22+|wn|22+o(1)asn→∞. |
Consequently, we have
I(un)=I(u)+I(v)+I(wn)+o(1)asn→∞, | (3.18) |
|un|22=|u|22+|v|22+|wn|22+o(1)asn→∞. | (3.19) |
Here, we set η=|u|22,ζ=|v|22 and δ=a−η−ζ. Then, we have limn→∞|wn|22=δ≥0. We will consider cases δ>0 and δ=0.
In the case δ>0, we set ˜wn=αnwn and αn=√δ/|wn|2. By Lemma 3.2, we have ˜wn∈Sδ and I(wn)=I(˜wn)+o(1). Thus, by (3.18) and the definition of mδ, we have
I(un)=I(u)+I(v)+I(wn)+o(1)=I(u)+I(v)+I(˜wn)+o(1)≥I(u)+I(v)+mδ+o(1)asn→∞. |
As n→∞, Lemma 3.3 implies that
ma≥I(u)+I(v)+mδ≥mη+mζ+mδ≥mη+ζ+δ=ma. | (3.20) |
Hence u and v are global minimizers with respect to mη and mζ respectively. Here, we can apply Lemma 3.4 (ⅱ) to obtain
mη+ζ<mη+mζ. |
It contradicts to (3.20).
In the case δ=0, the equations a=η+ζ and limn→∞|wn|2=0 hold. By Lemma 3.1(ⅰ), we have
limn→∞∫RNF(wn)dx=0. |
Thus, we obtain
lim infn→∞I(wn)≥0. |
As n→ in (3.18), we have
ma≥I(u)+I(v)≥mη+mζ≥mη+ζ=ma. |
Hence u and v are global minimizers with respect to mη and mζ respectively. Here, we can apply Lemma 3.4 (ⅱ) to obtain
ma=mη+ζ<mη+mζ, |
which is a contradiction. It completes the proof of the claim.
By (3.17) and Lemma 2.1, we have limn→∞|vn|l=0. Lemma 3.1(ⅰ) asserts that
limn→∞∫RNF(vn)dx=0. | (3.21) |
Next, we estimate the L2 norm of vn.
Claim. limn→∞|vn|2=0. In particular, |u|22=a.
By (3.16) and η=|u|22, it is sufficient to show that η=a. Otherwise, η<a holds because η≤a. By (3.21), we have
lim infn→∞I(vn)≥lim infn→∞−∫RNF(vn)dx=0. |
Taking the limit in (3.16), we obtain ma≥I(u). Using Lemma 3.3(ⅲ) along with u∈Sη, we have
ma≥I(u)≥mη≥ma. | (3.22) |
This requires mη=ma. Moreover, u is a global minimizer with respect to mη. By Lemma 3.4(ⅰ), we obtain mη>ma because η<a. It contradicts to (3.22).
Finally, we estimate the H2-norm of vn. Using the above claim, u∈Sa. This gives I(u)≥ma. Therefore, we have
I(un)=I(u)+I(vn)+o(1)≥ma+I(vn)+o(1)asn→∞. |
As n→∞, we obtain
lim supn→∞I(vn)≤0, |
while (3.21) asserts that
lim supn→∞∫RN[γ2|Δvn|2+β2|∇vn|2]dx≤lim supn→∞I(vn)+lim supn→∞∫RNF(vn)dx≤0. |
Since limn→∞|vn|2=0, we have limn→∞‖vn‖H2=0. Hence limn→∞un(⋅−yn)=u in H2(RN).
For any minimizing sequence of ma, Theorem 3.5 shows that the dichotomy case can't occur. To rule out the vanishing case, we will use the condition ma<0. Thus, for a>a0 (a0 is given by (3.6)), we can obtain the compactness of the minimizing sequence for ma.
Proposition 3.6. Suppose that a>a0. If {un}n∈N⊂Sa is a minimizing sequence with respect to ma, i.e., limn→∞I(un)=ma. Then, taking a subsequence if necessary, there exist a family {yn}n∈N⊂RN and u∈Sa such that limn→∞un(⋅−yn)=u in H2(RN). In particular, u is a global minimizer, i.e., u∈Ma.
Proof. By the assumption of the proposition and (3.6), we have ma<0. Let {un}n∈N⊂Sa be a minimizing sequence with respect to ma. It is sufficient to show that {un}n∈N satisfies (ⅱ) in Theorem 3.5. Otherwise, by Theorem 3.5, {un}n∈N satisfies (3.8). By Lemma 3.1(ⅱ), {un}n∈N is bounded in H2(RN), so (3.8) and Lemma 2.1 imply that un→0 in Ll(RN). By Lemma 3.1(ⅰ), we have
limn→∞∫RNF(un)dx=0. |
Since I(un)≥−∫RNF(un)dx, we can obtain
ma=limn→∞I(un)≥lim infn→∞−∫RNF(un)dx=0, |
contradicting to ma<0.
After the above preparations have been done, we are now in position to prove our main results.
Proof of Theorem 1.2. First, we consider the case 0<a<a0 and suppose by contradiction that there exists a global minimizer with respect to ma. By the assumption, we have ma=0. Here, Lemma 3.4 (ⅰ) asserts that
0=ma>ma0. |
It contradicts to (3.7).
Next, we consider the case a>a0. Proposition 3.6 asserts Theorem 1.2 (ⅰ). For (ⅱ), we assume it does not hold by contradiction. Then there exists ε0>0 such that for a sequence of solutions un of (1.1) with dist(un(0,⋅),Ma)<1/n, it holds that
dist(un(tn,⋅),Ma)≥ε0, |
which implies that
|un(tn,⋅)|22=|un(0,⋅)|22→a,I(un(tn,⋅))=I(un(0,⋅))→ma. |
Let αn=√a/|un(tn,⋅)|2 and ˜un(x)=αnun(tn,x). Then by Lemma 3.2 the following holds:
˜un∈Sa,limn→∞αn=1,limn→∞I(˜un)→ma. |
By Proposition 3.6, there exist a family {yn}n∈N⊂RN and u∈Ma such that limn→∞˜un(⋅−yn)=u in H2(RN). Thus, we also get limn→∞‖un(tn,⋅−yn)−u‖H2=0. We can deduce a contradiction from the following inequalities:
dist(un(tn,⋅),Ma)≤‖un(tn,⋅)−u(⋅−yn)‖H2=‖un(tn,⋅−yn)−u‖H2→0asn→∞. |
In what follows, we prove Theorems 1.5 and 1.6, which answer the question that "When a0>0 holds".
Proof of Theorem 1.5. (ⅰ): We fix a>0 and take some function u∈Sa∩C∞0(RN)∖{0}. For τ>0, let uτ(x)=τN/2u(τx). Then, we see that uτ∈Sa. By the assumption of (ⅰ), there exists a positive constant δ such that
F(s)≥C|s|lif|s|<δ, |
where C is a constant determined by
C=γ∫RN|Δu|2dx/∫RN|u|ldx. |
Hence F(uτ)≥C|uτ|l holds for a sufficiently small τ. Thus we have
I(uτ)≤γ2∫RN|Δuτ|2dx−C∫RN|uτ|ldx=−γτ42∫RN|Δu|2dx. |
It concludes that ma≤I(uτ)<0 for any a>0.
(ⅱ): By the assumption of (ⅱ), there exists a positive constant C=C(f) such that F(s)≤C|s|l holds for any s≥0. For u∈Sa, using the Gagliardo-Nirenberg inequality, we have
∫RNF(u)dx≤C|u|ll≤CBN|Δu|22a4/N. |
For a sufficiently small a>0, it can be shown that CBNa4/N≤γ/2 holds. After choosing an appropriately small a, we have
I(u)≥γ2∫RN|Δu|2dx−γ2∫RN|Δu|2dx=0. |
This together with Lemma 3.3 (ⅰ) implies ma=0 for a small a>0. Hence, we obtain a0>0.
Proof of Theorem 1.6. (ⅰ): We fix a>0 and take some function u∈Sa∩C∞0(RN)∖{0}. For τ>0, let uτ(x)=τN/2u(τx). Then, we see that uτ∈Sa. By the assumption of (ⅰ), there exists a positive constant δ such that
F(s)≥C|s|2+4/Nif|s|<δ, |
where C is a constant determined by
C=β∫RN|∇u|2dx/∫RN|u|2+4/Ndx. |
Hence F(uτ)≥C|uτ|2+4/N holds for a sufficiently small τ. Thus we have
I(uτ)≤γ2∫RN|Δuτ|2dx+β2∫RN|∇uτ|2dx−C∫RN|uτ|2+4/Ndx=γτ42∫RN|Δu|2dx−βτ22∫RN|∇u|2dx. |
If necessary, we take a smaller τ, then we conclude that ma≤I(uτ)<0 for any a>0.
(ⅱ): By (F3)-(F4), there exist two positive constants C1=C1(f) and C2=C2(f) such that F(s)≤C1|s|2+4/N+C2|s|l holds for any s≥0. For u∈Sa, using the Gagliardo-Nirenberg inequality, we have
∫RNF(u)dx≤C1|u|2+4/N2+4/N+C2|u|ll≤C1CN|∇u|22a2/N+C2BN|Δu|22a4/N. |
For a sufficiently small a>0, it can be shown that C1CNa2/N≤β/2 and C2BNa4/N≤γ/2 hold. After choosing an appropriately small a, we have
I(u)=γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−∫RNF(u)dx≥0. |
This together with Lemma 3.3 (ⅰ) implies ma=0 for a small a>0. Hence, we obtain a0>0.
Throughout this section, unless otherwise stated, we always assume f satisfies (F1)-(F6).
Lemma 4.1. (i) Let {un}n∈N be a bounded sequence in H2(RN). If either limn→∞|un|2=0 or limn→∞|un|l=0 holds, then it is true that limn→∞∫RNF(un)dx=0.
(ii) There exist two positive constants C1=C1(a,β) and C2=C2(f,N,a,γ) such that
I(u)≥γ4|Δu|22−C1|Δu|2−C2 | (4.1) |
holds for any u∈Sa. Specifically, ma>−∞.
Proof. (ⅰ): the proof can be proceeded as that of Lemma 3.1 (ⅰ) and is omitted.
(ⅱ): First, notice that by (1.4), we have
∫RN|∇u|2dx≤√a|Δu|2,∀u∈Sa, |
and we set C1=−β√a/2. In addition, recalling that (3.4) in the proof of Lemma 3.1, we choose ε>0 satisfying BNa4Nε=γ4. Then, for u∈Sa, we have
∫RNF(u)dx≤C2+γ4∫RN|Δu|2dx, |
where C2=C2(f,N,a,γ) is a positive constant which depends on f,N,γ and a. Together with the two inequalities above, we get (4.1).
To character the properties of ma, we will use some results from [23].
Lemma 4.2. (see [23])
(i) For any γ>0,β∈R and u∈H2(RN), it follows that
γ|Δu|22−β|∇u|22+β24γ|u|22≥0. |
Thus
infu∈H2(RN)(γ|Δu|22−β|∇u|22|u|22)≥−β24γ. |
(ii) When β<0, we introduce
J(u):=γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx, |
and consider the constrained minimization problem
mJa:=infu∈SaJ(u). |
Then for all a>0, it follows that:
(J1) mJa=−β28γa.
(J2) mJa is never achieved.
(J3) All minimizing sequences present vanishing, i.e., if {un}n∈N⊂Sa is a minimizing sequence with respect to mJa, then {un}n∈N satisfies (3.8).
Lemma 4.3. Let {un}n∈N be a bounded sequence in H2(RN) satisfying limn→∞|un|22=a>0. Let αn=√a/|un|2 and ˜un=αnun. Then the following holds:
˜un∈Sa,limn→∞αn=1,limn→∞|I(˜un)−I(un)|=0. |
Proof. Since the proof is similar as that of Lemma 3.2, we omit it.
In what follows, we give some properties about ma.
Lemma 4.4. (i) ma≤−β28γa for any a>0.
(ii) ma+b≤ma+mb for any a,b>0.
(iii) a↦ma is decreasing.
(iv) maτ≤τma for any a>0 and τ≥1.
(v) For sufficiently large a, ma<−β28γa holds.
(vi) a↦ma is continuous.
Proof. (ⅰ): By (F5) and (F6), we know F(z)=F(|z|)≥0 for any z∈C. Thus, we get I(u)≤J(u) for any u∈Sa. By Lemma 4.2 (ⅱ), it holds that ma≤mJa=−β28γa.
(ⅱ): The proof is similar as that of Lemma 3.3 (ⅱ) and omitted.
(ⅲ): For any 0<a<b, we get from (ⅱ) that mb≤ma+mb−a. By (ⅰ), we have mb−a≤−β28γ(b−a)<0, which implies mb<ma.
(ⅳ): For any u∈Sa and τ≥1, we set uτ(x)=τ1/2u(x), then uτ∈Saτ. Moreover, by (F5)-(F6), we obtain
I(uτ)=γ2∫RN|Δuτ|2dx+β2∫RN|∇uτ|2dx−∫RNF(uτ)dx=τ(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−∫RNF(τ1/2|u(x)|)dx≤τ(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−τη/2∫RNF(|u(x)|)dx≤τI(u). |
Since u is arbitrary, we get (ⅳ).
(ⅴ): By (F5), we can choose a function u∈H2(RN) with |u|22=1 such that ∫RNF(u)dx>0. In fact, we take a cut-off function φ∈C∞0(RN) such that
φ(x)={s0,|x|≤1,0,|x|≥2, |
where s0>0 is a constant given by (F5). For R>0, we set φR(x)=φ(x/R), then |φR|22=RN|φ|22. Thus, we can choose a R0>0 such that |φR0|2=1. Now we take u(x):=φR0(x), then, by (F5)-(F6), we see
∫RNF(u)dx=∫RNF(φR0(x))dx≥∫|x|≤R0F(s0)dx>0. |
For the u above, we set ua(x)=a1/2u(x),a≥1, then ua∈Sa. Moreover, by (F5)-(F6), we obtain
I(ua)=γ2∫RN|Δua|2dx+β2∫RN|∇ua|2dx−∫RNF(ua)dx=a(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−∫RNF(a1/2u)dx≤a(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−aη2∫RNF(u)dx=a(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−aη2−1∫RNF(u)dx). |
Since η>2, we have
γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−aη2−1∫RNF(u)dx→−∞asa→∞. |
For a>0 large enough, we deduce that
ma≤I(ua)<−β28γa. |
(ⅵ): The proof is similar as that of Lemma 3.3 (ⅴ) and omitted.
Next we define
a1:=inf{a>0:ma<−β28γa}. | (4.2) |
By Lemma 4.4 (ⅴ), we see that a1<∞. And again by Lemma 4.4 (ⅳ) and (ⅵ), we know ma<−β28γa for a>a1. Moreover, if a1>0, then it concludes from Lemma 4.4 (ⅰ) and (ⅵ) that
ma=−β28γa,0<a≤a1. | (4.3) |
Under certain conditions, we can further prove the strict subadditivity for ma.
Lemma 4.5. (i) Assume Ma≠∅ for some a>0. Then mτa<τma for any τ>1.
(ii) Assume that there exists a global minimizer u∈Sa with respect to ma for some a>0 and let b>0. Then ma+b<ma+mb.
Proof. (ⅰ): First, if u∈Ma, then we claim that ∫RNF(u)dx>0. Otherwise, by Lemma 4.2(ⅱ), ma=I(u)=∫RN|Δu|2dx+β2∫RN|∇u|2dx≥−β28γa. Thus, we conclude together with Lemma 4.4 (ⅰ) that ma=−β28γa, which implies u is also a minimizer for mJa. This contradicts to (J2) of Lemma 4.2(ⅱ).
Next, for u∈Ma, we set uτ(x)=τ1/2u(x),τ>1, then uτ∈Saτ. Moreover, by (F5)-(F6), we obtain
I(uτ)=γ2∫RN|Δuτ|2dx+β2∫RN|∇uτ|2dx−∫RNF(uτ)dx=τ(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−∫RNF(τ1/2|u(x)|)dx≤τ(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx)−τη/2∫RNF(|u(x)|)dx<τI(u). |
Therefore, we get mτa≤I(uτ)<τI(u)=τma.
(ⅱ): Assume first that 0<b≤a. Then, by Lemma 4.4(ⅳ) and Lemma 4.5 (ⅰ), we have
ma+b<a+bama=ma+bama=ma+bamabb≤ma+baabmb=ma+mb. |
If 0<a<b, by again Lemma 4.4(ⅳ) and Lemma 4.5 (ⅰ), we obtain
ma+b≤a+bbmb=mb+abmb=mb+abmbaa<mb+abbama=ma+mb. |
With regard to the minimizing sequence for ma, we have
Theorem 4.6. Suppose (F1)–(F6) and that a>0. If {un}n∈N⊂Sa is a minimizing sequence with respect to ma, then one of the following holds:
(i)
lim supn→∞supz∈RN∫B(z,1)|un|2dx=0. | (4.4) |
(ii) Taking a subsequence if necessary, there exist u∈Sa and a family {yn}n∈N⊂RN such that un(⋅−yn)→u in H2(RN) as n→∞. Specifically, u is a global minimizer.
Proof. Suppose that {un}n∈N⊂Sa is a minimizing sequence which does not satisfy (4.4). It is sufficient to show that (ⅱ) holds. Since (4.4) does not hold and {un}n∈N⊂Sa, we have
0<lim supn→∞supz∈RN∫B(z,1)|un|2dx≤α<∞. |
Taking a subsequence if necessary, there exists a family {yn}n∈N⊂RN, such that
0<limn→∞∫B(0,1)|un(x−yn)|2dx<∞. | (4.5) |
Since {un}n∈N⊂Sa is a minimizing sequence, Lemma 4.1(ⅱ) asserts that {un}n∈N is a bounded sequence in H2(RN). Hence {un(⋅−yn)}n∈N is a bounded sequence in H2(RN). Using the weak compactness of a Hilbert space and the Rellich compactness, for some subsequence, there exists u∈H2(RN) such that
un(⋅−yn)⇀uweakly inH2(RN), | (4.6) |
un(⋅−yn)→uinL2loc(RN), | (4.7) |
un(⋅−yn)→ua.e. inRN. | (4.8) |
Equations (4.5) and (4.7) assert that |u|2>0. We put vn=un(⋅−yn)−u. By (4.6), vn⇀0 weakly in H2(RN). Thus, we have
∫RN|Δu+Δvn|2dx=∫RN|Δu|2dx+∫RN|Δvn|2dx+2ℜ∫RNΔu¯Δvndx=∫RN|Δu|2dx+∫RN|Δvn|2dx+o(1)asn→∞, | (4.9) |
∫RN|∇u+∇vn|2dx=∫RN|∇u|2dx+∫RN|∇vn|2dx+2ℜ∫RN∇u⋅¯∇vndx=∫RN|∇u|2dx+∫RN|∇vn|2dx+o(1)asn→∞, | (4.10) |
∫RN|u+vn|2dx=∫RN|u|2dx+∫RN|vn|2dx+2ℜ∫RNu¯vndx=∫RN|u|2dx+∫RN|vn|2dx+o(1)asn→∞. | (4.11) |
Using (4.8), the Brezis-Lieb theorem(see [39] or [13,Lemma 3.2]) implies that
∫RNF(u+vn)dx=∫RNF(u)dx+∫RNF(vn)dx+o(1)asn→∞. |
Since I(un)=I(un(⋅−yn))=I(u+vn), we can obtain
I(un)=I(u)+I(vn)+o(1),|un|22=|u|22+|vn|22+o(1)asn→∞. | (4.12) |
Claim: |vn|22→0 as n→∞.
In order to prove this, we set ζ=|u|22>0. By (4.12), if we show that ζ=a, the claim follows. We assume by contradiction that ζ<a and we define
˜vn=√a−ζ|vn|2vn. |
By Lemma 4.3 and (4.12), it follows that
I(un)=I(u)+I(vn)+o(1)=I(u)+I(˜vn)+o(1)≥I(u)+ma−ζ+o(1). |
Let n→∞, and by Lemma 4.4 (ⅱ), we have
ma≥I(u)+ma−ζ≥mζ+ma−ζ≥ma, | (4.13) |
and so, I(u)=mζ, i.e., u∈Sζ is a global minimizer with respect to mζ. Thus, by Lemma 4.5 (ⅱ), we get
ma<mζ+ma−ζ, |
which contradicts (4.13). Hence, the claim follows and |u|22=a.
At this point, since {vn} is a bounded sequence in H2(RN), it follows from (1.4) that |∇vn|2→0 as n→∞. By Lemma 4.1 (ⅰ), we obtain that
lim infn→∞I(vn)=lim infn→∞γ2|Δvn|22≥0. | (4.14) |
On the other hand, since |u|22=a, we deduce from (4.12) that
I(un)=I(u)+I(vn)+o(1)≥ma+I(vn)+o(1), |
and so, that
lim supn→∞I(vn)≤0 | (4.15) |
From (4.14) and (4.15) we deduce that |Δvn|22→0 as n→∞ and so, that un(⋅−yn)→u in H2(RN).
For any minimizing sequence of ma, Theorem 4.6 shows that the dichotomy case can't occur. To rule out the vanishing case, we will use the condition ma<−β28γa. Thus, for a>a1 (a1 is given by (4.2)), we can obtain the compactness of the minimizing sequence for ma.
Proposition 4.7. Suppose that a>a1. If {un}n∈N⊂Sa is a minimizing sequence with respect to ma, i.e., limn→∞I(un)=ma. Then, taking a subsequence if necessary, there exist a family {yn}n∈N⊂RN and u∈Sa such that limn→∞un(⋅−yn)=u in H2(RN). In particular, u is a global minimizer, i.e., u∈Ma.
Proof. By the assumption of the proposition and (4.2), we have ma<−β28γa. Let {un}n∈N⊂Sa be a minimizing sequence with respect to ma. It is sufficient to show that {un}n∈N satisfies (ⅱ) in Theorem 4.6. Otherwise, by Theorem 4.6, {un}n∈N satisfies (4.4). By Lemma 4.1(ⅱ), {un}n∈N is bounded in H2(RN), so (4.4) and Lemma 2.1 imply that un→0 in Ll(RN). By Lemma 4.1(ⅰ), we have
limn→∞∫RNF(un)dx=0. |
Since un∈Sa, by Lemma 4.2(ⅱ), we can obtain
ma=limn→∞I(un)≥lim infn→∞γ2∫RN|Δun|2dx+β2∫RN|∇un|2dx≥mJa=−β28γa, |
contradicting to ma<−β28γa for a>a1.
With the above preparations at hand, we prove our Theorem 1.7.
Proof of Theorem 1.7. First, we consider the case 0<a≤a1 and suppose by contradiction that there exists a global minimizer u with respect to ma. By (4.3), we have
I(u)=ma=−β28γa, | (4.16) |
From (4.16), we deduce that u≠0. We choose a sequence {un} in H2(RN) such that un→u in H2(RN). Then |un|2→|u|2 as n→∞. We define
˜un(x)=|u|2|un|2un(x), |
then we easily see that
˜un∈Sa,˜un→uinH2(RN)andI(˜un)→I(u) |
as n→∞. Thus, ˜un is a minimizing sequence of ma. By (J1) of Lemma 4.2 (ⅱ), we know ma=mJa for 0<a≤a1. So ˜un is also a minimizing sequence of mJa. By (J3) of Lemma 4.2 (ⅱ), it must be vanishing, i.e., it satisfies (4.4). Combining with Lemma 2.1, we infer that u=0 a.e. in RN. This contradicts to (4.16).
Next, we consider the case a>a1. Proposition 4.7 asserts Theorem 1.7 (ⅰ).
(ⅱ): The proof is similar as that of Theorem 1.2 (ⅱ) and omitted.
Finally, in the case β<0, we consider the nonlinearity f(u)=|u|p−2u+μ|u|q−2u,2<q<p<2+8/N,μ<0. We give the partial characterization of the value of minimizing energy ma.
Proof of Theorem 1.10. Let {un}∈Sa be a minimizing sequence for mJa, then by (J3) of Lemma 4.2 (ⅱ), it must be vanishing. Thus, we have
lim supn→∞I(un)≤lim infn→∞γ2∫RN|Δun|2dx+β2∫RN|∇un|2dx=mJa=−β28γa, |
which implies ma≤−β28γa for any a>0. On the other hand, let u∈S1 and ua(x)=√au(x), then we see
I(ua)=γa2∫RN|Δu|2dx+βa2∫RN|∇u|2dx−ap/2p∫RN|u|p−aq/2μq∫RN|u|q=a(γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−ap/2−1p∫RN|u|p−aq/2−1μq∫RN|u|q) |
Since p>q>2, we have
γ2∫RN|Δu|2dx+β2∫RN|∇u|2dx−ap/2−1p∫RN|u|p−aq/2−1μq∫RN|u|q→−∞ |
as a→∞, and we conclude that
I(ua)<−β28γa | (4.17) |
for a large enough.
In what follows, we set
a∗=sup{a>0:mτ=−β28γτ,0<τ≤a}, | (4.18) |
if the a above does not exist, we set a∗=0. Besides, we define
a∗=inf{a>0:mτ<−β28γτ,τ≥a}, | (4.19) |
then we know a∗<∞ from (4.17). Noticing that ma is continuous (the continuity of ma can be proved as that of Lemma 3.3 (ⅴ)), together with the definitions of a∗ and a∗, we get the conclusion.
The authors would like to thank the reviewers for the valuable suggestions and comments to improve the manuscript.
H. Luo is supported by National Natural Science Foundation of China, No. 11901182, by Natural Science Foundation of Hunan Province, No. 2021JJ40033, and by the Fundamental Research Funds for the Central Universities, No. 531118010205. Z. Zhang is supported by National Natural Science Foundation of China, No. 12031015, 11771428, 12026217.
The authors declare there is no conflicts of interest.
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