Ann. I. H. Poincaré – AN 30 (2013) 1–22
www.elsevier.com/locate/anihpc
On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system
Yohei Sato
a, Zhi-Qiang Wang
b,c,∗aOsaka City University Advanced Mathematical Institute, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Smiyoshi-ku, Osaka 558-8585, Japan
bChern Institute Mathematics, Nankai University, Tianjin 300071, PR China cDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA Received 14 December 2011; received in revised form 5 May 2012; accepted 29 May 2012
Available online 15 June 2012
Abstract
The paper concerns multiplicity of vector solutions for nonlinear Schrödinger systems, in particular of semi-positive solutions.
New variational techniques are developed to study the existence of this type of solutions. Asymptotic behaviors are examined in various parameter regimes including both attractive and repulsive cases.
©2012 Elsevier Masson SAS. All rights reserved.
0. Introduction
In this paper, we consider the following nonlinear Schrödinger systems:
−u+λ1u=μ1u3+βuv2 inΩ,
−v+λ2v=μ2v3+βu2v inΩ, u, v∈H01(Ω).
(∗)
HereΩis a bounded domain inRn(n3) andλi, μi>0 fori=1,2. In this paper, we show the multiple existence of semi-positive solutions(uk, vk)for(∗). As there may be semi-trivial solutions (which are zero for some components) we call a solution non-trivial if every component is non-zero. Here we say a non-trivial solution(u, v) is a semi- positive solution for(∗)if and only if it satisfiesu >0 orv >0 inΩ.
For positive solutions (which meansu >0 andv >0 inΩ) of nonlinear Schrödinger systems, there has been ex- tensive work in recent years (cf.[1–7,11,13,15–22,24,27–30]and their references). In particular, we refer to results of [13]which partially inspire our work of the current paper. Dancer, Wei and Weth[13]showed that the a priori bounds of positive solutions and the multiplicity of positive solutions of nonlinear Schrödinger systems are complementary to each other depending on the parameter regimes. They showed the existence of a priori bounds of positive solutions
* Corresponding author at: Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA.
E-mail addresses:[email protected](Y. Sato),[email protected](Z.-Q. Wang).
0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.
http://dx.doi.org/10.1016/j.anihpc.2012.05.002
for some nonlinear Schrödinger systems which contain(∗). Applying their result to(∗), whenβ >−√μ1μ2, there exists a constantC=C(β, μ1, μ2, Ω)such thatuL∞(Ω),vL∞(Ω)C for any positive solutions(u, v). On the other hand, whenλ1=λ2=μ1=μ2=1 in(∗), they showed the multiple existence of positive solutions of(∗). More precisely, whenβ−1,(∗)has an unbounded sequence of positive solutions(uk)∞k=1such that
ukL∞(Ω)+ vkL∞(Ω)→ ∞ ask→ ∞.
These positive solutions were given by minimax method from making use of a symmetryσ (u, v)=(v, u). That is, the variational functionalIβ(u, v)associated with(∗)satisfiesIβ(σ (u, v))=Iβ(u, v)forσ (u, v)=(v, u). This multiplic- ity result was recovered and generalized to the non-symmetric case ofμ1=μ2by using a bifurcation method in[5]
in which an unbounded sequence of positive solutions was established forβ−√μ1μ2when the domain is radial.
For nonlinear Schrödinger systems(∗)withλ1=λ2=μ1=μ2=1, these results suggest thatβ= −√μ1μ2is the threshold that divides the existence of a priori bounds of positive solutions and the existence of an unbounded sequence of positive solutions. In this paper, we consider the existence and multiplicity of semi-positive solution of (∗). A natural question is to examine the coupling constant β and to find the coupling value that separates the a priori bounds and infinitely many semi-positive solutions. Our results suggest thatβ=0 is the threshold dividing the existence of a priori bounds of semi-positive solutions and the existence of an unbounded sequence of semi-positive solutions. This is the main motivation of the current work. We also study the asymptotic properties of semi-positive solutions whenβ→0 andβ→ ∞, and establish multiplicity results of semi-positive solutions in these regimes.
Whenβ <0, we get infinite many semi-positive solutions of(∗)by the following theorem.
Theorem 0.1.Letβ <0. Then(∗)has a sequence of solutions(uk, vk)such that uk>0, ukL∞(Ω)+ vkL∞(Ω)→ ∞ask→ ∞.
Moreover, ifβ∈(−√μ1μ2,0), thenvkmust change sign for largek.
Whenβ >0 is small, we get multiplicity of semi-positive solutions of(∗)as follows.
Theorem 0.2.For givenk∈N, there existsβk>0such that, for anyβ∈(0, βk), we haveksemi-positive solutions (ui, vi)of(∗)withui>0inΩ (i=1,2, . . . , k).
Roughly speaking, our semi-positive solutions are given by making use of a symmetryσ (u, v)=(u,−v). That is, it is essential that the variational functional Iβ(u, v)satisfies Iβ(u, v)=Iβ(u,−v). More generally, we develop an abstract framework in Section2. We consider the following situation. LetH be a Hilbert space and suppose that σ :H→Hsatisfies
σ2=idH, (0.1)
σ=idH. (0.2)
Then, for C1-manifold M⊂H which does not contain fix points of σ andC1-functional J:M→R satisfying J (σ (u))=J (u) and some conditions, we can prove the multiple existence of the critical values ofJ. For details, see Section2. We point out that generalizations and variants of the genus theory have been established recently in [9,10,26]. Refs. [9,10] were for existence of multiple vector solutions of some elliptic systems. Ref.[26]was on existence of multiple sign-changing vector solutions with each component sign-changing for systems like(∗)in the defocussing case (i.e.,μj0). In the general perspective we use partial symmetry for variants of the genus theory in this paper.
Next, we consider the asymptotic behavior of semi-positive solutions as β →0. To state our result about the asymptotic behavior, we need the following notations: forJ2(v)=(4μ2v4L4(Ω))−1:Σ2= {v∈H01(Ω)|
Ω|∇v|2+ λ2|u|2dx=1} →R, we define symmetric mountain pass valuesbn2(n∈N∪ {0}) by
b2n= inf
γ2∈Γn2
maxθ∈SnJ2 γ2(θ )
, Γn2=
γ2(θ )∈C
Sn, Σ2 γ2(−θ )= −γ2(θ )for allθ∈Sn ,
whereSn= {θ=(θ1, . . . , θn+1)∈Rn+1| |θ| =1}. Now, we show the following theorem.
Theorem 0.3.For givenk∈N, there existsβk >0such that, for anyβ∈(−βk, βk), we haveksolutions(ui,β, vi,β) of(∗)withui,β>0inΩ (i=1,2, . . . , k)and(ui,β, vi,β)satisfy the following:extracting a subsequenceβj→0, we have
(ui,βj, vi,βj)→(ui,0, vi,0) inH01(Ω)×H01(Ω).
Hereui,0is a positive least energy solution of
−u+λ1u=μ1u3 inΩ,
u∈H01(Ω). (0.3)
vi,0is a solution of
−v+λ2v=μ2v3 inΩ,
v∈H01(Ω). (0.4)
In particular,vi,0corresponds to the critical valueb2i which is given by a symmetric mountain pass theorem.
Remark 0.4. The functional J2(v): Σ2 → R corresponds to (0.4). In fact, for a critical point v0 of J2, (√
μ2v02L4(Ω))−1v0is a non-trivial solution of (0.4).
Remark 0.5.The semi-positive solutions(ui,β, vi,β)in Theorem0.3may be different from the semi-positive solutions (ui, vi)in Theorem0.1or Theorem0.2.
Next, we consider the semi-positive solutions for the caseβis large. In[18], Liu and Wang showed that, for given k∈N, there existsβk>0 such that, for anyβ > βk,(∗)has at leastksolutions. In this paper, we get multiplicity of semi-positive solutions of(∗)as follows.
Theorem 0.6.For givenk∈N, there existsβk>0such that, for anyβ > βk,(∗)has at leastksemi-positive solutions (ui,β, vi,β)withui,β>0inΩ (i=1,2, . . . , k).
We study the asymptotic behavior asβ→ ∞. For the solution(ui,β, vi,β)of Theorem0.6,(√
βui,β,√
βvi,β)is bounded inH01(Ω)×H01(Ω)asβ→ ∞. (See Section7.) Thus, extracting a subsequenceβj→ ∞, we expect that (
βjui,βj,
βjvi,βj)approaches to a solution of
−u+λ1u=uv2 inΩ,
−v+λ2v=u2v inΩ,
u, v∈H01(Ω). (0.5)
Here, we remark that (0.5) does not have semi-trivial solutions. In fact, letting(0, v)be a solution of (0.5), we also havev=0 from the second equation of (0.5). For the limiting equation (0.5), we have the following:
Theorem 0.7.Eq.(0.5)has infinitely many semi-positive solutions(uk, vk)such thatuk>0inΩ and
ukL∞(Ω)+ vkL∞(Ω)→ ∞ ask→ ∞. (0.6)
Moreover, whenλ1=λ2,vkmust change sign for largek∈N.
Remark 0.8. The solutions (uk, vk) of Theorem 0.7 are characterized by values ek,∞ which are defined as fol- lows. Let N= {(u, v)∈H01(Ω)×H01(Ω)|
RN|∇u|2+ |∇v|2+λ1|u|2+λ2|v|2dx=1, u+v≡0},J˜∞(u, v)= (8u+v2L2(Ω))−1. We defineek,∞(k∈N∪ {0}) by
ek,∞=inf
c∈Rγ
[ ˜J∞c]N
k .
Hereγ is a genus corresponding toσ (u, v)=(u,−v)which is defined in Section2.
Remark 0.9.Whenλ1=λ2=λ >0, all positive solutions(u, v)of (0.5) must satisfyu=v. In fact,u−vsatisfies
−(u−v)+λ(u−v)=uv(v−u).
Multiplyingu−vand integrating overΩthe above equation, we have
Ω
∇(u−v)2+λ(u−v)2dx= −
Ω
uv(u−v)2dx.
Thus we haveu=v. We also remark that there exist a priori bounds of−u+λu=u3inΩ andu=0 on∂Ω. Therefore, whenλ1=λ2=λ >0, (0.6) implies thatvkis a sign-changing solution for largek∈N. Whenλ1=λ2we do not know whethervkchanges sign.
Now, we get the following theorem about the asymptotic behavior asβ→ ∞.
Theorem 0.10.For givenk∈N, let(uk,β, vk,β)be a family of solutions of(∗)which are given in Theorem0.6. Then there exist a subsequenceβj→ ∞and(uk,∞, vk,∞)∈H01(Ω)×H01(Ω)such that
(
βjuk,βj,
βjvk,βj)→(uk,∞, vk,∞) inH01(Ω)×H01(Ω).
Here(uk,∞, vk,∞)is a solution of (0.5)and corresponds to critical valueek,∞.
We devote the next four sections to the proofs of our theorems. For the caseβ0 or the case β >0 small, we reduce the functionalIβ(u, v)to a functionalJβ(u, v)defined on a subset of a torusΣ1×Σ2in Section1. On the other hand, for the caseβ >0 is large, we reduce the functionalI˜β(u, v)to a functionalJ˜β(u, v)defined on a subset of the sphereΣ in Section6. In Section2, we give an abstract theory for the multiple existence of the critical values ofC1-functionalJ:M→RsatisfyingJ (σ (u))=J (u). We will get most of our multiple existence of semi-positive solutions by using these abstract results. In Section3, we will show Theorem0.1and Theorem0.2. In Sections4–5, we will prove Theorem0.3. To show this, we apply the method from[25]. In Sections6–7, we will show Theorems0.6, 0.7and0.10.
1. The functional setting for the caseβ0 or the caseβ >0 small
To prove the existence of semi-positive solutions(u, v)withu >0, we seek critical points of the following func- tional
Iβ(u, v)=1 2
|||u|||2λ1+ |||v|||2λ2
−1 4
μ1u+44+μ2v44
−β
2u+v22:H01(Ω)×H01(Ω)→R.
Here we use notationsu+=max{u,0},u−=min{u,0}and
|||u|||2λ=
Ω
|∇u|2+λu2dx, upp=
Ω
|u|pdx.
For a critical point(u, v)ofIβ(u, v), the positivity ofucomes from the following proposition.
Proposition 1.1.Let(u, v)be a critical point ofIβ(u, v)withu=0. Then we haveu >0inΩ.
Proof. Let(u, v)be a critical point ofIβ(u, v). Then∇Iβ(u, v)(u−,0)= |||u−|||2λ1 =0. Thus we haveu+≡u0.
Now, forβ0,usatisfies
−u+
λ1−βv2
u=μ1u30.
Forβ >0,usatisfies
−u+λ1u=
μ1u2+βv2 u0.
Since the maximum principle works foruin both cases, we haveu >0 inΩ. 2
We setΣi= {u∈H01(Ω)| |||u|||λi=1}fori=1,2. We remark that there existsC1>0 such that
u4, v4< C1 for all(u, v)∈Σ1×Σ2. (1.1)
To seek non-trivial critical points ofIβ(u, v), sometimes one may reduce Iβ(u, v) to a functional defined on a Nehari manifold with co-dimension 2. In this paper, we reduceIβ(u, v) to a functional defined on an open subset of torusΣ1×Σ2. Since we also consider a perturbation problem forβ (Theorem0.3), it is easy to treat a domain which does not depend onβ. This is the main reason to reduce the functional to one on the torus but not on a Nehari manifold.
1.1. The reduction to a functional on a torus
Whenβ∈R, we set
Nβ=
⎧⎪
⎨
⎪⎩(u, v)∈Σ1×Σ2
g1(u, v):=μ1μ2u+44v44−β2u+v42>0, g2(u, v):=μ1u+44−βu+v22>0,
g3(u, v):=μ2v44−βu+v22>0
⎫⎪
⎬
⎪⎭. From the Hölder inequality, we see that
Nβ=
⎧⎨
⎩
{(u, v)∈Σ1×Σ2|g1(u, v) >0}, β∈(−∞,−√μ1μ2], {(u, v)∈Σ1×Σ2|u+≡0}, β∈(−√μ1μ2,0], {(u, v)∈Σ1×Σ2|g2(u, v) >0, g3(u, v) >0}, β∈(0,∞).
We remark that, for allβ ∈R,(u, v)∈Nβ impliesg1(u, v) >0 and u+≡0. We can define a functionalJβ(u, v) onNβ by the following proposition.
Proposition 1.2.For any(u, v)∈Nβ, a function (s, t)→Iβ(su, t v):R2+→R
has a unique maximum point(sβ(u, v), tβ(u, v)). Moreover, setting Jβ(u, v)= sup
s,t>0
Iβ(su, t v), we have
Jβ(u, v)=1 4
sβ(u, v)2+tβ(u, v)2
(1.2)
=1 4
μ1sβ(u, v)4u+44+μ2tβ(u, v)4v44+2βsβ(u, v)2tβ(u, v)2u+v22
(1.3)
=1
4·μ1u+44+μ2v44−2βu+v22
μ1μ2u+44v44−β2u+v42 (1.4)
and
(i) sβ(u, v),tβ(u, v):N→R+areC1-functions.
(ii) Jβ(u, v):Nβ→Ris aC1-function.
(iii) If(u, v)∈Nβ is a critical point ofJβ(u, v), then(sβ(u, v)u, tβ(u, v)v)is a non-trivial critical point ofIβ(u, v).
(iv) Jβ(u, v)satisfies(PS)-condition.
Proof. For any(u, v)∈Nβ, we set f (s, t )=Iβ(su, t v):R2+→R.
Differentiatingf (s, t ), we have
∂f
∂s(s, t)=s−s3μ1u+44−st2βu+v22,
∂f
∂t(s, t)=t−t3μ2v44−s2tβu+v22. Thus critical points(s, t)off (s, t )satisfy
μ1u+44, βu+v22
βu+v22, μ2v44 s2 t2
= 1
1
.
Here, notingμ1μ2u+44v44−β2u+v42>0, we have s2
t2
= 1
μ1μ2u+44v44−β2u+v42
μ2v44, −βu+v22
−βuv22, μ1u+44 1 1
= 1
μ1μ2u+44v44−β2u+v42
μ2v44−βu+v22 μ1u+44−βu+v22
. (1.5)
Since(u, v)∈Nβ,f (s, t )has a unique critical point(s0, t0)=(sβ(u, v), tβ(u, v)). Next, to show(s0, t0)is a maximum point, we calculate the second derivatives off (s, t ).
∂2f
∂s2(s, t)=1−3s2μ1u+44−t2βu+v22=1 s
∂f
∂s(s, t)−2s2μ1u+44,
∂2f
∂t ∂s(s, t)= −2stβu+v22,
∂2f
∂t2(s, t)=1−3t2μ2v44−s2βu+v22=1 t
∂f
∂t(s, t)−2t2μ2v44. Therefore, we have
A=∂2f
∂s2(s0, t0)= −2s20μ1u+44, B= ∂2f
∂t ∂s(s0, t0)= −2βs0t0u+v22. C=∂2f
∂t2(s0, t0)= −2t02μ2v44.
SinceA <0 andAC−B2=4s02t02(μ1μ2u+44v44−β2u+v42) >0,(s0, t0)is a maximum point off (s, t ). Thus, by direct calculations, we get (1.2)–(1.4).
Next we show (i). To show (i), we use the implicit function theorem. We consider the following function:
F(s, t, u, v)=
F (s, t, u, v) G(s, t, u, v)
= ∂f
∂s(s, t)
∂f
∂t(s, t)
:R2+×Nβ→R2. Now, for any(u, v)∈Nβ, we have
F(s0, t0, u, v)=0, ∂F
∂s(s0, t0, u, v) ∂F∂t(s0, t0, u, v)
∂G
∂s(s0, t0, u, v) ∂F∂t(s0, t0, u, v)
= A B
B C
.
Thus from the implicit function theorem, we can easily see theC1-property of(s0, t0)=(sβ(u, v), tβ(u, v)).
We show (ii). Noting Jβ(u, v)=Iβ
sβ(u, v)u, tβ(u, v)v ,
we can easily find thatJβ(u, v)is aC1-function. Moreover we have
∇uJβ(u, v)ϕ= ∇uIβ
sβ(u, v)u, tβ(u, v)v
∇usβ(u, v)ϕu+sβ(u, v)ϕ + ∇vIβ
sβ(u, v)u, tβ(u, v)v
∇utβ(u, v)ϕv
= ∇uIβ
sβ(u, v)u, tβ(u, v)v
sβ(u, v)ϕ, (1.6)
∇vJβ(u, v)ψ= ∇vIβ
sβ(u, v)u, tβ(u, v)v
tβ(u, v)ψ. (1.7)
Thus, if(u, v)∈Nβ is a critical point ofJβ(u, v), then(sβ(u, v)u, tβ(u, v)v)is a non-trivial critical point ofIβ(u, v) and we get (iii).
Finally, we show (iv). If (un, vn)∈Nβ is a (PS)-sequence forJβ, thenJβ(un, vn)are bounded and this means the boundedness of(sβ(un, vn), tβ(un, vn))from (1.2). Thus from (1.6)–(1.7),(sβ(un, vn)un, tβ(un, vn)vn)is also a (PS)-sequence forIβ. SinceIβ(u, v)satisfies (PS)-condition,Jβ(u, v)also satisfies (PS)-condition. 2
From (1.2), for all β∈R, it is obvious that Jβ(u, v)is bounded from below. Moreover, we have the following proposition.
Proposition 1.3.Whenβ <0, we have lim inf
(u,v)∈Nβ,dist{(u,v),∂Nβ}→0Jβ(u, v)= ∞. (1.8)
Proof. For any sequence ((un, vn))∞n=1⊂Nβ with g1(un, vn)→0 (n→ ∞), we need to show Jβ(un, vn)→ ∞ (n→ ∞). Since|||un|||λ1= |||vn|||λ2=1, for someu0, v0∈H01(Ω), we may assume
un→u0, vn→v0 strongly inL4(Ω).
Here ifg2(u0, v0)+g3(u0, v0) >0, then it is obvious that (1.8) holds. Thus we assumeg2(u0, v0)+g3(u0, v0)=0.
Sinceβ <0, we haveu0=v0=0 and we findun44→0,vn44→0 asn→ ∞. SinceJβ(u, v)is written by (1.4), we get (1.8). 2
Remark 1.4.From Proposition1.3, whenβ <0, the behavior ofJβ(u, v)in the neighborhood of∂Nβdoes not disturb deformation arguments. Whenβ >0, it is complicated by the behavior ofJβ(u, v)in the neighborhood of∂Nβ and we cannot expect the property like (1.8). But for β >0 small,Jβ(u, v)satisfies the property like (1.8) on a proper subsetMδ⊂Nβ. (See Proposition1.9.)
1.2. The caseβ >0small Forδ >0, we set
Mδ=
(u, v)∈Σ1×Σ2μ1u+44> δ, μ2v44> δ . We remark thatMδ= ∅ifδ <4b1
0 whereb0is given by b0=min
b10, b20
>0, b01= inf
u∈Σ1
1
4μ1u44>0, b20= inf
v∈Σ2
1
4μ2v44>0. (1.9)
Herebi0(i=1,2) is a least energy level of (1.15) and (1.17) respectively. (See Remark1.8.) We also remark thatMδ is independent ofβ.
Lemma 1.5.For any givenδ∈(0,4b1
0), there existsβδ∈(0,√μ1μ2)such that Mδ⊂Nβ for allβ∈(−√
μ1μ2, βδ).
Proof. Whenβ∈(−√μ1μ2,0),Mδ⊂Nβ is obvious. Forδ∈(0,4b1
0), we chooseβδ>0 satisfyingδ > βδC14. Here C1is a constant given in (1.1). Then it holds
μ1u+44> δ > βδC14βu+v22 for all(u, v)∈Mδ, β∈ [0, βδ).
By a similar way, we haveμ2v44> βu+v22. Thus we getMδ⊂Nβ for allβ∈(−√μ1μ2, βδ). 2
From Lemma1.5,Jβ(u, v)is defined onMδ. Lemma 1.6.For any givenδ∈(0,4b1
0), there exists a constantCδ>0which does not depend onβ such that sβ(u, v)Cδ, tβ(u, v)Cδ for all(u, v)∈Mδ, β∈(−βδ, βδ). (1.10) Hereβδwas given in Lemma1.5. Moreover it holds
sβ(u, v), tβ(u, v)
→
1
√μ1u+24, 1
√μ2v24
uniformly for(u, v)∈Mδasβ→0. (1.11)
Proof. Suppose(u, v)∈Mδ,β∈(−βδ, βδ). Sincesβ(u, v)was written by (1.5), we have sβ(u, v)2= μ2v44−βu+v22
μ1μ2u+44v44−β2u+v22 (μ2+βδ)C14 (μ1μ2−βδ2)μδ2
1μ2
.
HereC1is a constant given in (1.1) and we have used the fact thatμ1u+44, μ2v44δfor all(u, v)∈Mδ. And we also have
sβ(u, v)2→ 1
μ1u+44 uniformly for(u, v)∈Mδasβ→0.
Sincetβ(u, v)also was similarly written by (1.5), we obtain (1.10) and (1.11). 2 Proposition 1.7. For any given δ∈(0,4b1
0), there exists a constant cδ(β) withcδ(β)→0 (as β →0) such that Jβ(u, v)satisfies
Jβ(u, v)−J1(u)−J2(v)cδ(β) for all(u, v)∈Mδ, β∈(−βδ, βδ), (1.12) ∇uJβ(u, v)− ∇J1(u)
λ1∗cδ(β) for all(u, v)∈Mδ, β∈(−βδ, βδ), (1.13) ∇vJβ(u, v)− ∇J2(v)
λ2∗cδ(β) for all(u, v)∈Mδ, β∈(−βδ, βδ), (1.14) where, fori=1,2,Ji(u)=4μ1
iu44
,TuΣi= {v∈H01(Ω)| u, vλi=0}and ∇Ji(u)
λi∗= sup
v∈TuΣi,|||v|||λi=1
∇Ji(u)v.
Remark 1.8. For anyu∈Σ1 withu+=0, a function s→I1(su)=s22 −s44μ1u+44has a maximum value at a unique maximum points=√μ11u+24
and we can write as follows J1(u)=sup
s>0
I1(su)= 1
4μ1u+44, (1.15)
∇J1(u)ϕ= − 1 μ1u+84
Ω
u3+ϕ dx for allϕ∈TuΣ1. (1.16)
By a similar way, for anyu∈Σ2, a functiont→I2(t u)=t22 −t44μ2v44has a unique maximum point and we have J2(u)=sup
t >0
I2(tv)= 1
4μ2v44, (1.17)
∇J2(v)ψ= − 1 μ2v84
Ω
v3ψ dx for allψ∈TvΣ2. (1.18)
Proof of Proposition1.7. From (1.4), (1.15) and (1.17), we can directly calculateJβ(u, v)−J1(u)−J2(v)as follows:
Jβ(u, v)−J1(u)−J2(v)=1
4· βu+v22
μ1μ2u+44v44−β2u+v42
βu+v22
μ1u+44+βu+v22 μ2v44 −2
. For(u, v)∈Mδ,β∈(−βδ, βδ), we have
Jβ(u, v)−J1(u)−J2(v) C14|β| 4(μ1μ2−β2)μδ2
1μ2
C14|β|
δ +C14|β| δ +2
. (1.19)
Here C1 is a constant given in (1.1) and we have used the fact that μ1u+44, μ2v44δ for all (u, v)∈Mδ. From (1.19), we get (1.12). Next we calculate∇uJβ(u, v)ϕ− ∇J1(u)ϕfor anyϕ∈TuΣ1. From (1.6),
∇uJβ(u, v)ϕ= −sβ(u, v)4μ1
Ω
u3+ϕ dx−βsβ(u, v)2tβ(u, v)2
Ω
u+v2ϕ dx.
Combining (1.16), we have
∇uJβ(u, v)ϕ− ∇J1(u)ϕ
sβ(u, v)4− 1 μ21u+84
μ1 Ω
u3+|ϕ|dx+ |β|sβ(u, v)2tβ(u, v)2
Ω
u+v2|ϕ|dx
sβ(u, v)4− 1 μ21u+84
μ1C14|||ϕ|||λ1+ |β|Cδ4C14|||ϕ|||λ1.
We obtain (1.13) from the above inequality and Lemma1.6. (1.14) also holds from a similar calculation. 2 For smallβ >0, the following proposition plays a role similar to Proposition1.3.
Proposition 1.9.For anyβ∈(−βδ, βδ), we have sup
(u,v)∈Mδ
Jβ(u, v) 1
2δ +cδ(β), (1.20)
(u,v)inf∈∂Mδ
Jβ(u, v) 1
4δ +b0−cδ(β). (1.21)
Hereb0was given in(1.9).
Proof. From Proposition1.7, for(u, v)∈Mδ,β∈(−βδ, βδ), we have J1(u)+J2(v)−cδ(β)Jβ(u, v)J1(u)+J2(v)+cδ(β).
We remark that
u∈Σinf1,u+≡0J1(u)b10b0, inf
v∈Σ2J2(v)b02b0.
Here(u, v)∈∂Mδ impliesJ1(u)=4δ1 orJ2(v)=4δ1 and(u, v)∈Mδ impliesJ1(u)4δ1 orJ2(v)4δ1. Therefore we get (1.20) and (1.21). 2
2. The multiplicity of critical values forσ-invariant functionals
In this section, we construct abstract theories to get the multiple existence of critical points of functionals having symmetryJ (σ (u))=J (u)whereuis in a Hilbert space andσ satisfies (0.1)–(0.2). To do so, we construct a genus type index for the symmetryσ. In[23]or[13], the authors constructed the genus type index forσ (−u)=uin the scaler case orσ (u, v)=(v, u)in the vector case respectively.
In this section, letHbe a Hilbert space andσ:H→Hbe a bounded linear operator satisfying (0.1)–(0.2). Setting H0= {u∈H|σ (u)=u},H0 is a subspace composed of fixed points ofσ. HereH0=H from (0.2). We also set H1=H0⊥= {0}. For anyu∈H, we uniquely writeu=u0+u1,(u0, u1)∈H0⊕H1. Then, from (0.1)–(0.2), we have
σ (u0+u1)=u0−u1 for allu=u0+u1∈H0+H1.
For thisσ:H→H, we define a genus as follows:
Definition 2.1.For anyσ-invariant closed setA⊂H\H0,γ (A)is the least integernsuch that there exists a function g∈C(A,Rn\ {0})with
g σ (u)
= −g(u) for allu∈A. (2.1)
If there is no suchg, we defineγ (A)= ∞. We also defineγ (∅)=0.
Here, whengsatisfies (2.1), we saygis aσ-odd function. WhenJ∈C(A,R)satisfies J
σ (u)
=J (u) for allu∈A,
we sayJ is aσ-invariant functional or aσ-even functional. Whenh∈C(A, H )satisfies h
σ (u)
=σ h(u)
for allu∈A, we sayhisσ-equivariant.
The following theorem is the main theorem in this section:
Theorem 2.2.LetM⊂H\H0be aσ-invariantC1-manifold andJ:M→Rbe aσ-evenC1-functional satisfying (PS)-condition. Moreover, we assume that
uinf∈MJ (u) >−∞, (2.2)
lim inf
u∈M,dist{u,∂M}→0J (u)= ∞, (2.3)
and, for anyk∈N, there existsψ∈C(Sk, M)withψ (−x)=σ (ψ (x)). ThenJ has an unbounded nondecreasing sequence of critical values(ck)∞k=1. Hereckis defined by
ck=inf
c∈Rγ
[Jc]M
k , [Jc]M=
u∈MJ (u)c
. (2.4)
Firstly we state the properties of our genus. These are similar to the properties of the genus type index constructed in[23]or[13].
Lemma 2.3.LetA, B⊂H\H0beσ-invariant closed sets. Then we have:
(i) IfA⊂B, thenγ (A)γ (B).
(ii) γ (A∪B)γ (A)+γ (B).
(iii) Ifh∈C(A, H\H0)satisfiesh(σ (u))=σ (h(u)), thenγ (A)γ (h(A)).
(iv) γ (A\B)γ (A)−γ (B).
(v) Ifγ (A) >1, thenAis an infinite set.
(vi) IfAis a compact set, thenγ (A) <∞. Moreover there existsσ-invariant neighborhood ofN ofAinM such thatγ (A)=γ (N ).
(vii) Ifψ∈C(Sn, H\H0)satisfiesψ (−u)=σ (ψ (u)), thenγ (ψ (Sn))n+1.
Proof. First of all, we show (iii). If γ (h(A))= ∞, (iii) is trivial. Supposing γ (h(A))=m <∞, there exists σ-odd function g∈C(h(A),Rm\ {0}). Then (g ◦h)∈C(A,Rm\ {0}) satisfies (g◦h)(σ (u))=g(σ (h(u)))=
−(g◦h)(u). Thus we haveγ (A)m=γ (h(A))and (iii) holds. We get (i), taking an inclusion mapidA∈C(A, B) in (iii). Next, we show (v). When A is a finite set, A is written by A= {u1, . . . , uk, σ (u1), . . . , σ (uk)} where u1, . . . , uk, σ (u1), . . . , σ (uk)are different from each other. Then we have g∈C(A,R\ {0})such that g(xi)=1, g(σ (xi))= −1 fori=1, . . . , k. Thus we findγ (A)=1. This implies (v).
