# Semiclassical Solutions for a Kind of Coupled Schrödinger Equations

Authors: Jinmei Fan; Yi-rong Jiang; Qiongfen Zhang
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this paper, we are concerned with the following coupled Schrödinger equations−λ2Δu+a1xu=cxv+a2xup−2u+a3xu2∗−2u,x∈ℝN,−λ2Δv+b1xv=cxu+b2xvp−2v+b3xv2∗−2v,x∈ℝN, where 2<p<2∗,2<q<2∗,2∗=2N/N−2, andN≥3; λ>0 is a parameter; and a1,a2,a3,b1,b2,b3,c∈CℝN,ℝ and u,v∈H1ℝN. Under some suitable conditions that a10=infa1=0 or b10=infb1=0 and cx2≤ϑa1xb1x with ϑ∈0,1, the above coupled Schrödinger system possesses nontrivial solutions if λ∈0,λ0, where λ0 is related to a1,a2,a3,b1,b2,b3, and N.


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## Body

## 1. Introduction

We consider the following coupled Schrödinger equations in this paper:(1)−λ2Δu+a1xu=cxv+a2xup−2u+a3xu2∗−2u,x∈ℝN,−λ2Δv+b1xv=cxu+b2xvp−2v+b3xv2∗−2v,x∈ℝN,where 2<p<2∗, 2<q<2∗, N≥3, and 2∗=2N/N−2 are the Sobolev critical exponent;λ>0 is a parameter; and a1,a2,a3,b1,b2,b3,c∈CℝN,ℝ and u,v∈H1ℝN.As it is known in [1], this type of systems arises in nonlinear optics. In the past years, under different kinds of assumptions on the potential V and the nonlinearity f, many authors [2–8] focus on the following kind of Schrödinger equation:
(2)−λ2Δu+Vxu=fx,u,x∈ℝN.As one knows, single-mode optical fibers are not really “single mode” but actually bimodal because of the presence of birefringence. So recently, the coupled Schrödinger systems are investigated by the authors [9–12]. For more related results and physical background on Schrödinger systems, please see [13–23] and references therein.In [11], the authors investigated standing waves for the following kind of coupled Schrödinger equations:
(3)−λ2Δu+a1xu=cv+up−2u,x∈ℝN,−λ2Δv+b1xv=cu+v2∗−2v,x∈ℝN,where a1,b1∈CℝN,ℝ, N≥3, u,v>0, u,v∈H1ℝN, and ux,vx→0 as ∣x∣→∞. Under the following conditions,(A0) there exist positive constantsa10>0 and b10>0 such that a1x≥a10, b1x≥a10, and 0<c≤a10b10; they obtained the existence of a positive solution for (3) if λ is sufficiently small. But, if a10=infa1=0 or b10=infb1=0, then 0<c≤a10b10 cannot hold. So in the very recent paper [12], Peng et al. investigated the following coupled Schrödinger equations and generalize the result in [11]:
(4)−λ2Δu+a1xu=cxv+up−2u,x∈ℝN,−λ2Δv+b1xv=cxu+vq−2v,x∈ℝN,where a1,b1 are the same as in (3), N≥3. Under the following conditions,(A1)a1x≥a10=0 and b1x≥0, and there exist constants a10>0 and b10>0 such that the measure of the sets Aa10≔x:a1x<a10 and Bb10≔x:b1x<b10 are finite(A2) there exists a constantϑ∈0,1 such that cx2≤ϑa1xb1x for all x∈ℝN; Peng et al. proved that system (4) has at least one nontrivial solution. An interesting question is what will happen if the nonlinearity is also critical growth in system (4)? Motivated mainly by the above-mentioned results, we will answer this question and prove that system (1), under conditions (A1) and (A2), and(A3) there exist constantsa20, a21, a30, a31, b20, b21, b30, b31>0 such that
(5)a20≤a2x≤a21,a30≤a3x≤a31,b20≤b2x≤b21,b30≤b3x≤b31,∀x∈ℝN,possesses nontrivial solutions if λ∈0,λ0, where λ0 is related to a1,a2,a3,b1,b2,b3, and N. As far as we know, similar results for system (1) with a critical exponent have not been investigated by variational methods in the literature. The following condition is similar to condition (A1):(A1’)b1x≥b10=0 and a1x≥0, and there exist constants a10>0 and b10>0 such that the measure of the sets Aa10≔x∈ℝN:a1x<a10 and Bb10≔x∈ℝN:b1x<b10 are finite.Sinceq−2N−2q<0 and p−2N−2p<0, one can choose d0≥1 such that
(6)C1a21αp−2/p+C2b21βq−2/q+C3a31α2/N+C4b31β2/N≤121−ϑ,where
(7)α=ωNa20p−22NpN2+2N+2N+21−2−N2p/p−2d0p−2N−2p/p−2,β=ωNb20q−22NqN2+2N+2N+21−2−N2q/q−2d0q−2N−2q/q−2,C1=2pη0η2∗Np−2a20p−2/pa10pN−2−2N/2p,C2=2qη0η2∗Nq−2b20q−2/qb10qN−2−2N/2q,C3=22∗η0η2∗N2∗−2a302/N,C4=22∗η0η2∗N2∗−2b302/N,η0 and η2∗ are embedding constants and ωN is the volume of the unit ball in ℝN. From (A1’) and (A1), using b10=0 and a10=0, one can let μ0>1 such that
(8)supμ1/2∣x∣≤2d0b1x≤d0−2,supμ1/2∣x∣≤2d0a1x≤d0−2,∀μ≥μ0.Letw=u,v and λ−2=μ, then system (1) can be rewritten as
(9)−Δu+μa1xu=μcxv+μa2xup−2u+μa3xu2∗−2u,x∈ℝN,−Δv+μb1xv=μcxu+μb2xvp−2v+μb3xv2∗−2v,x∈ℝN,and the functional of (9) is given by
(10)Sμw=12∫ℝNΔu2+Δv2+μa1xu2+μb1xv2dx−μp∫ℝNa2xupdx−μq∫ℝNb2xvqdx−μ2∗∫ℝNa3xu2∗dx−μ2∗∫ℝNb3xv2∗dx−μ∫ℝNcxuvdx.As is known, the solutions of (1) are the critical points of Sλ−1/2w. The main results are the following.Theorem 1.
Suppose that (A1)–(A3) or (A1’)–(A3) hold. Then, (9) possesses at least one nontrivial solution wμ=uμ,vμ such that 0<Sμwμ≤βμ1−N/2 for μ≥μ0.Theorem 2.
Suppose that (A1)–(A3) or (A1’)–(A3) hold. Then, (1) possesses at least one nontrivial solution wλ=uλ,vλ such that 0<Sλ−1/2wλ≤βλN−2 for 0<λ<μ0−1/2.Remark 3.
Since the presence of the termsa2xup−2u, a3xu2∗−2u, b2xvp−2v, and b3xv2∗−2v, system (1) is more general than (4), and it is more difficult to deal with the nontrivial solutions. In order to prove that system (1) has nontrivial solutions, we need to find some conditions to restrict a2x, a3x, b2x, and b3x. It seems that there is no literature considering system (1).


## 2. Preliminaries

Let(11)E=u,v:∫ℝNa1xu2+b1xv2dx<∞,u,v∈H1ℝN,(12)wμ†=∫ℝNΔu2+Δv2+μa1xu2+μb1xv2dx1/2,∀w=u,v∈E.From Lemma 1 of [17], by (A1) or (A1’) and the Sobolev inequality, there exists a positive constantη0>0 independent of μ such that
(13)wH1≔∫ℝNΔu2+Δv2+u2+v2dx1/2≤η0wμ†,∀w=u,v∈E,μ≥1,where H1≔H1ℝN. Then, E,·μ† is a Banach space for μ≥1 equipped with the norm given by (12). Moreover, for s∈2,2∗, one has
(14)ws≤ηswH1≤ηsη0wμ†,∀w∈E,μ≥1,where ws is the usual norm in space LsℝN. From (12), we rewrite Sμ as
(15)Sμw=12wμ†2−μp∫ℝNa2xupdx−μq∫ℝNb2xvqdx−μ2∗∫ℝNa3xu2∗dx−μ2∗∫ℝNb3xv2∗dx−μ∫ℝNcxuvdx,∀w∈E.It is not difficult to see thatSμ∈C1E,ℝ and
(16)S′μw,w¯=∫ℝNΔu·u¯+Δv·v¯+μa1xuu¯+μb1xvv¯dx−μ∫ℝNcxuv¯+vu¯dx−μ∫ℝNa2xup−2uu¯dx−μ∫ℝNb2xvq−2vv¯dx−μ∫ℝNa3xu2∗−2uu¯dx−μ∫ℝNb3xv2∗−2vv¯dx,∀w=u,v,w¯=u¯,v¯∈E.As in [12, 22], let
(17)θx=1d0,∣x∣≤d0,d0N−11−2−Nx−N−2d0−N,d0<∣x∣≤2d0,0,∣x∣>2d0.Then,θ∈H1ℝN; moreover,
(18)∇θ22=∫ℝN∇θx2dx≤NωNd0N−4N+21−2−N2,(19)θ22=∫ℝNθx2dx≤2ωNd0N−2N1−2−N2.In the next section, we will prove the main results.


## 3. Proof of the Main Results

Proof of Theorem 1.
The proof of Theorem1 is divided into four steps.Step 1.
We first prove that for anyμ≥μ0>1, one has
(20)supSμ0,teμ:t≥0≤βμ1−N/2,supSμteμ,0:t≥0≤αμ1−N/2,where eμx=θμ1/2x. From (8), (9), (17), (18), (19), and (A3), we have
(21)Sμ0,teμ=t22∫ℝN∇eμ2+μb1xeμ2dx−μq∫ℝNb2xteμqdx−μ2∗∫ℝNb3xteμ2∗dx=μ1−N/2t22∫ℝN∇θ2+b1μ−1/2xθ2dx−1q∫ℝNb2μ−1/2xtθqdx−12∗∫ℝNb3μ−1/2xtθ2∗dx≤μ1−N/2t22∇θ22+θ22sup∣x∣≤2d0b1μ−1/2x−1q∫∣x∣≤d0b2μ−1/2xtd0qdx−12∗∫∣x∣≤d0b3μ−1/2xtd02∗dx≤μ1−N/2t22∇θ22+d0−2θ22−ωNb20qNtqd0N−q−ωNb302∗Nt2∗d0N−2∗≤μ1−N/2t22∇θ22+d0−2θ22−ωNb20qNtqd0N−q≤μ1−N/2q−2∇θ22+d0−2θ22q/q−22q−1ωNb20d0N−qN−2/q−2≤μ1−N/2ωNb20q−22NqN2+2N+2N+21−2−N2q/q−2d0q−2N−2q/q−2≔βμ1−N/2.
Similarly, from (8), (9), (17), (18), (19), and (A3), we have
(22)Sμteμ,0≤μ1−N/2ωNa20p−22NpN2+2N+2N+21−2−N2p/p−2d0p−2N−2p/p−2≔αμ1−N/2,which together with (21) implies that (20) holds.Step 2.
Letcμ∗=minSμteμ,0,Sμ0,teμ, we should prove that there exists a constant cμ∈0,cμ∗ and a sequence wn⊂E satisfying
(23)Sμwn→cμ,S′μwnE∗1+wnμ†,asn→∞.
By a standard argument, one can obtain (23) by employing the mountain-pass lemma without the (PS) condition, so we omit the details here.Step 3.
We prove that any sequencewn⊂E satisfying (23) is bounded in E. From (A2) and Young’s inequality, we have
(24)μ∫ℝ3cxunvndx≤μϑ∫ℝ3a1xb1xunvndx≤ϑ2∫ℝ3μa1xun2+μb1xvn2dx≤ϑ2wnμ†2.
For2<p≤q<2∗, from (15), (16), (23), and (24), we have
(25)cμ+o1=Sμwn−1pS′μwn,wn=12−1pwnμ†2+1p−1qμ∫ℝNb2xvnqdx+1p−12∗μ∫ℝNa3xun2∗dx−1−2pμ∫ℝNcxunvndx+1p−12∗μ∫ℝNb3xvn2∗dx≥12−1p1−ϑwnμ†2.
For2<q≤p<2∗, from (15), (16), (23), and (24), we obtain
(26)cμ+o1=Sμwn−1qS′μwn,wn=12−1qwnμ†2+1q−1pμ∫ℝNa2xvnpdx+1q−12∗μ∫ℝNa3xun2∗dx−1−2qμ∫ℝNcxunvndx+1q−12∗μ∫ℝNb3xvn2∗dx≥12−1q1−ϑwnμ†2.
It follows from (25) and (26) that wn is bounded in E.Step 4.
We show that there exists a nontrivial solution. By Steps1–3, we know that there exists a bounded sequence wn⊂E satisfying (23) with
(27)cμ≤cμ∗,∀μ≥μ0.
Passing to a subsequence, one can suppose thatwn=un,vn⇀wμ=uμ,vμ in E,·μ† and S′μwn→0, as n→∞. Now, we verify that wμ≠0,0. Arguing by contradiction, assume that wμ=0,0, that is, wn⇀0,0 in E, so by [24], we have wn→0,0 in LlocsℝN, s∈2,2∗, and wn→0,0 a.e. on ℝN. Since Aa10 and Bb10 are sets with finite measure, we have
(28)∥un∥22=∫ℝN\Aa10un2dx+∫Aa10un2dx=∫ℝN\Aa10un2dx+∫Aa10un2dx≤∫ℝN\Aa101μa10μa1xun2dx+∫Aa10un2dx≤1μa10wnμ†2+o1,(29)vn22=∫ℝNBb10vn2dx+∫Bb10vn2dx=∫ℝN\Bb10vn2dx+∫Bb10vn2dx≤∫ℝN\Bb101μb10μb1xvn2dx+∫Bb10vn2dx≤1μb10wnμ†2+o1.
Similar to [12], from (14), (28), (29), and the Hölder inequality, we obtain
(30)unss=∫ℝNunsdx≤∫ℝNun22∗−s/2∗−2dx∫ℝNun2∗s−2/2∗−2dx≤η0η2∗2∗s−2/2∗−2μa10−2∗−s/2∗−2wnμ†s+o1,s∈2,2∗(31)vnss=∫ℝNvnsdx≤∫ℝNvn22∗−s/2∗−2dx∫ℝNvn2∗s−2/2∗−2dx≤η0η2∗2∗s−2/2∗−2μb10−2∗−s/2∗−2wnμ†s+o1,s∈2,2∗.
It follows from (15), (16), (23), and (A3) that
(32)cμ+o1=Sμwn−12S′μwn,wn=12−1pμ∫ℝNa2xunpdx+12−1qμ∫ℝNb2xvnqdx+12−12∗μ∫ℝNa3xun2∗dx+12−12∗μ∫ℝNb3xvn2∗dx≥p−2μa202punpp+q−2μb202qvnqq+2∗−2μa3022∗un2∗2∗+2∗−2μb3022∗vn2∗2∗.
From (14), (30), (31), and (32), we have
(33)μunpp=μunpp−2unp2≤μμa10−22∗−p/p2∗−2η0η2∗22∗p−2/p2∗−22pcμμa20p−2p−2/pwnμ†2+o1≔C1μN−2/2cμp−2/pwnμ†2+o1,(34)μvnqq=μunq2vnqq−2≤μμb10−22∗−q/q2∗−2η0η2∗22∗q−2/q2∗−22qcμμb20q−2q−2/qwnμ†2+o1≔C2μN−2/2cμq−2/qwnμ†2+o1.
From (14) and (32), we have
(35)μun2∗2∗=μun2∗2un2∗2∗−2≤μη0η2222∗cμ2∗−2μa302∗−2/2∗wnμ†2+o1=η0η2222∗cμ2∗−2a302∗−2/2∗μN−2/Ncμ2/Nwnμ†2+o1≔C3μN−2/Ncμ2/Nwnμ†2+o1,(36)μvn2∗2∗=μvn2∗2vn2∗2∗−2≤μη0η2222∗cμ2∗−2μb302∗−22∗wnμ†2+o1=η0η2222∗cμ2∗−2b302∗−22∗μN−2Ncμ2Nwnμ†2+o1≔C4μN−2Ncμ2Nwnμ†2+o1.
It follows from (6), (16), (20), (33), (34), (35), and (36) that
(37)o1=S′μwn,wn=wnμ†2−μ∫ℝNa2xunpdx−μ∫ℝNb2xvnqdx−μ∫ℝNa3xun2∗dx−μ∫ℝNb3xvn2∗dx−2μ∫ℝNcxunvndx≥1−ϑwnμ†2−C1a21μN−2/2cμp−2/p+C2b21μN−2/2cμq−2/q+C3a31μN−2/Ncμ2/N+C4b31μN−2/Ncμ2/Nwnμ†2+o1≥1−ϑwnμ†2−C1a21αp−2/p+C2b21βq−2/q+C3a31α2/N+C4b31β2/Nwnμ†2+o1≥1−ϑ2wnμ†2+o1.
Hence, we obtain(38)limn→∞wnμ†2=0.
From (15), (23), and (38), we have
(39)0<cμ=limn→∞Sμwn≤12wnμ†2=0,a contradiction, which implies that wμ≠0,0. We can easily check that S′μwn=0 and Sμwn≤cμ by a standard argument. Hence, wμ is a nontrivial solution for (9).It is easy to see that Theorem2 is a direct consequence of Theorem 1.



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*Source: 4378691-2020-08-17.xml*