Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth Jiaquan Liu a , Xiangqing Liu b , Zhi-Qiang Wang c,d,∗ a LMAM, School of Mathematics, Peking University, Beijing, 100871, PR China b Department of Mathematics, Yunnan Normal University, Kunming 650500, PR China c Center for Applied Mathematics, Tianjin University, Tianjin 300072, PR China d Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Received 18 March 2016

Abstract We consider the following nonlinear Schrödinger system with critical growth

−uj = λj uj +

k 

βij |ui |

2∗ 2

|uj |

2∗ 2 −2 uj ,

in , uj = 0, on ∂, j = 1, · · · , k,

i=1 2N , 0 < λ < λ (), j = 1, · · · , k, λ () is the where  is a bounded smooth domain in RN , 2∗ = N−2 j 1 1 first eigenvalue of − with zero Dirichlet boundary condition. We consider the repulsive case, namely βjj > 0, j = 1, · · · , k, βij = βj i ≤ 0, i = j , i, j = 1, · · · , k. The existence of infinitely many sign-changing solutions as bound states is proved, provided N ≥ 7, by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions. © 2016 Elsevier Inc. All rights reserved.

MSC: 35A15; 35J50; 58J70 Keywords: Systems of Schrödinger equations; Critical growth; Sign-changing solutions

* Corresponding author at: Center for Applied Mathematics, Tianjin University, Tianjin 300072, PR China, and Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA. E-mail address: [email protected] (Z.-Q. Wang).

http://dx.doi.org/10.1016/j.jde.2016.09.018 0022-0396/© 2016 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we consider the following coupled nonlinear Schrödinger equations with the critical exponent ⎧ ⎨ ⎩

−uj = λj uj +

k  i=1

2∗

βij |ui | 2 |uj |

2∗ 2 −2

uj , in 

(1.1)

uj = 0, on ∂, j = 1, · · · , k,

2N where  is a bounded smooth domain in RN , 2∗ = N−2 is the critical Sobolev exponent for the 1 p embedding of H0 () into L (), λj , βij are constants. This class of coupled equations have applications in some physical problems such as in nonlinear optics and in multispecies Bose Einstein condensates (e.g., [13,37]). In physics literatures the signs of the coupling constants βij being positive or negative determine the nature of the system being attractive or repulsive. Mathematical work has been done extensively in recent years and we refer [1,3–5,11,18–22,25,26,28,31–33,36,37] for more references for the existence theory and the studies of qualitative property of solutions depending upon the systems being attractive or repulsive. Compared with scalar equations, there are many new challenges for coupled equations in dealing with the existence of multiple solutions, in particular multiple sign-changing solutions. First, there are many semi-trivial solutions due to systems collapsing, i.e., solutions with one or more components being zeroes. In fact there are infinitely many such semi-trivial solutions. Secondly, due to repulsive effect when βij < 0 (i = j ), for coupled systems like (1.1) there can exist infinitely many positive solutions (solutions with each components being positive). This was established in [11,37] for the case of 2-systems (k = 2 in (1.1)) when β12 = β21 = β ≤ −1 and β11 = β22 = 1, and was extended in [33] to the case of k-equations when βjj = 1, j = 1, ..., k and 1 βij = β ≤ − k−1 for i = j . Infinitely many radially symmetric positive solutions for the 2-system in the repulsive case have been established in [3] by using a global bifurcation approach that works also for the general case β11 = β22 . We refer the multiplicity of positive solutions to [3,11, 24,32,33,37] and references therein. Recently semi-positive solutions (meaning one component positive and the other nonzero) was studied in [27] in which infinitely many such solutions was established (see also [8,9]). All these signed solutions are considered critical points of the variational formulation of the systems. Thus to obtain sign-changing critical points for the variational formulation one needs to distinguish them from the known existing semi-trivial critical points, the existing positive critical points and the existing semi-positive solutions. In [10] for a system with two equations with critical exponent (i.e., k = 2 in (1.1)) the authors constructed a solution with β12 < 0 such that the first component is sign-changing and the second component is positive. With these known results about existence of solutions it seems very difficult to develop a critical theory to construct multiple sign-changing solutions as there are many families of unbounded sequences of these known existing critical points to the variational problem. An attempt was made in [20] by the authors of the current paper in establishing an abstract framework to deal with sign-changing solutions for systems that share some of the above features. In [20] for the subcritical case (2∗ is replaced by 4 in (1.1) with N = 2, 3) infinitely many sign-changing solutions were established (see also [9] that treated k = 2 equations by a different approach). In this paper we consider the critical exponent case, we will prove that the critical case can be approximated by subcritical problems and infinitely many sign-changing solutions of (1.1) will be constructed through approximations of solutions for the corresponding subcritical problems.

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3

This idea was initially used by Devillanova and Solimini in [12] successfully for a scalar Brezis– Nirenberg problem to have infinitely many solutions constructed, and has been recently extended to the cases of quasilinear problems (e.g., [7,14]). More precisely, in this paper we consider the repulsive case and we assume the following conditions. () 0 < λj < λ1 (), j = 1, · · · , k, where λ1 () is the first eigenvalue of − with zero Dirichlet boundary condition. (B) βjj > 0, j = 1, · · · , k; βij = βj i ≤ 0, 1 ≤ i < j ≤ k. Under these conditions we will show there exists an unbounded sequence of sign-changing solutions to the system (1.1) provided N ≥ 7, Next let us outline our approach in more details. The system (1.1) is of critical growth. Replacing 2∗ in (1.1) by any p ∈ (2, 2∗ ), we obtain the system with subcritical growth ⎧ ⎨ ⎩

−uj = λj uj +

k  i=1

p

p

βij |ui | 2 |uj | 2 −2 uj , in 

(1.2)

uj = 0, on ∂, j = 1, · · · , k.

In [20] by making use of the method of invariant sets of decreasing flow the existence of infinitely many sign-changing solutions for (1.2) was proved for N = 2, 3, p = 4, λj < 0, j = 1, · · · , k and  = RN or  being a bounded smooth domain in RN . These solutions are sign-changing in the sense that each component of the solutions is sign-changing. This was done by using an abstract critical point theory combining with the method of invariant sets of decreasing flows. The arguments there can be easily extended to the general subcritical case (1.2) with p ∈ (2, 2∗ ). In order to establish the existence of multiple sign-changing solutions of (1.1), following the idea of [12] we use the solutions of (1.2) as approximating solutions and obtain infinitely many sign-changing solutions for (1.1) by passing the limit p → 2∗ . Here is our main results. We use H to denote H01 () × · · · × H01 (), the k-fold product of H01 (). We use  and → to denote weak and strong convergence in H respectively, and || · ||p to denote the Lp norm. Theorem 1.1. Assume λj > 0, j = 1, · · · , k. Let {pn } ⊂ (2, 2∗ ] be a sequence such that pn → 2∗ and Un = (un,1 , · · · , un,k ) be a solution of (1.2) with p = pn . Assume Un  U = (u1 , · · · , uk ) in H . Then Un → U in H and U is a solution of (1.1). Theorem 1.2. Assume (), (B) and N ≥ 7. Then (1.1) has infinitely many sign-changing solutions. Remark 1.1. Our method allows us to have more general results to give the existence of solutions with some of its components being sign-changing and the rest of its components being of one sign (i.e., being positive or negative). See Remark 6.1 in Section 6. The paper is organized as follows. In Section 2 we develop necessary concentration analysis and integral estimates which in conjunction with a local Pohozaev identity are used in Section 3 to prove Theorem 1.1 for convergence of approximating solutions from subcritical problems. Section 4 is devoted to an abstract framework of minimax schemes in critical point theory to construct multiple critical points outside a family of invariant sets of the descending flow. This

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treatment is an improvement of the abstract theory we developed in [20] and is especially useful in giving rise to Morse index estimates of the constructed critical points, which is an important ingredient used in showing multiplicity of solutions for the critical system in Section 5. In Section 5 we verify the abstract theory from Section 4 to give the construction of an infinite sequence of sign-changing solutions of the subcritical systems which are approximating solutions to the critical system. In Section 6 using the results from Sections 2 and 5 we finally give the proof of Theorem 1.2 for the existence of an unbounded sequence of sign-changing solutions of the original system (1.1). Appendices A and B contain the proofs of some technical results from Section 4 about an intersection property and about Morse index estimates, both of which may be of independent interests. In the remaining part of the paper, we often use a capital letter C to denote a constant that is independent of the sequences in the arguments but maybe different from line to line, and C(·) will be used to indicate the dependency of the constant C on the relevant quantity. 2. Concentration analysis and integral estimates Let D = D 1,2 (RN ) and D be the group of scaling operators over D D = {gλ,x |gλ,x u = λ

N−2 2

u(λ(· − x)), λ > 0, x ∈ RN }.

I.e., D contains all translations and dilations in RN . Let pn → 2∗ , Un be a solution for (1.2) with p = pn such that Un  U0 in H . Extended by zero outside , {Un } is a bounded sequence in E := H 1 (RN ) × · · · × H 1 (RN ). By Theorem 3.1 and Corollary 3.2 in [35] (see also [34]), we have the following profile decomposition for Un Un = U 0 +



gn(m) Vm + Rn

(2.1)

m (m)

where gn

:= gλn,m ,xn,m ∈ D, Vm ∈ D , such that

(1) Un  U0 in H , (gn(m) )−1 Un  Vm in D as n → ∞; l (2) gn(0) = I d, (gn(m) )−1 g n U  0, in D for any U ∈ D and any m = l as n → ∞; 2 2 (3) Un D = U0 D + Vm 2D + Rn 2D + o(1); m  (m) D (4) Rn = Un − U0 − gn Vm −→ 0, that is for any sequence {gn } ⊂ D, gn Rn  0 in D . m

∗

We first note that, by (4), Rn → 0 in L2 (RN ) (see [35, Lemma 5.3]). Recall N−2

2 gλn,m ,xn,m u = λn,m u(λn,m (· − xn,m )), xn,m ∈ .

Then the property (2) of (2.1) is equivalent to

(2.2)

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λn,m λn,l + + λn,m λn,l |xn,m − xn,l |2 → ∞, m = l, as n → ∞. λn,l λn,m

(2.3)

Since {Un } is bounded in L2 (RN ), we have also λn,m → ∞, m ≥ 1, as n → ∞.

(2.4)

Lemma 2.1. Let pn → 2∗ , Un be a solution for (1.2) with p = pn , Un  U0 in H . Assume the decomposition (2.1). Then (1) U0 = (u1 , · · · , uk ) satisfies the system (1.1). (2) V = Vm = (v1 , · · · , vk ) satisfies either the following system of equations in RN ⎧ k ∗ ∗ ⎪ ⎨ −v = α  β |v | 22 |v | 22 −2 v , x ∈ RN j ij i j j i=1 ⎪ ⎩ vj → 0, as |x| → ∞, j = 1, · · · , k,

(2.5)

or up to an orthogonal transformation of coordinates, the system of equations with the Dirichlet boundary condition ⎧ k  2∗ 2∗ ⎪ ⎪ βij |vi | 2 |vj | 2 −2 vj , x ∈ RN ⎪ −vj = α L ⎨ i=1

(2.6)

⎪ vj = 0 outside RN ⎪ L ⎪ ⎩ vj → 0, as |x| → ∞, j = 1, · · · , k, N−2

2 where α = αm = lim λn,m n→∞ xN > −L}.

pn −N

N ∈ (0, 1], L ≥ 0 and RN L = {x| x = (x1 , · · · , xN ) ∈ R ,

Proof. (1) Since Un  U0 in H , Un → U0 in Lp () for 1 ≤ p < 2∗ , we readily get that U0 satisfies the system (1.1) (in the weak form). (2) Denote L = lim λn,m dist(xn,m , ∂). We distinguish two cases, namely L = +∞ and n→∞ L < +∞. Consider the case L = +∞ firstly. Un = (un,1 , · · · , un,k ) satisfies the system (1.2) with N−2

2 p = pn . Let ϕ be a smooth function in C0∞ (RN ) and ψ = gn ϕ = λn,m ϕ(λn,m (· − xn,m )). ∞ For n large enough ψ ∈ C0 (). Take ψ as a test function in (1.2) with p = pn . We have for j = 1, ..., k,

(m)



 ∇un,j ∇ψ dx =



λj un,j ψ dx +

  k

βi,j |un,i |

pn 2

|un,j |

pn 2 −2

un,j ψ dx.

(2.7)

 i=1



Since (gn )−1 Un  V = Vm in D , (gn )−1 Un → V in Lloc (RN ) for 1 ≤ p < 2∗ , the limits of the terms in (2.7) as n → ∞ are as follows: (m)

(m)

p

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 ∇un,j ∇gn(m) ϕ dx

∇un,j ∇ψ dx = RN



∇(gn(m) )−1 un,j ∇ϕ dx

= RN

 ∇vj ∇ϕ dx,

→ RN

 |un,i | 

pn 2

|un,j |



=

|un,i |

pn 2

RN N−2

2 = λn,m

pn −N



|un,j | 

un,j ψ dx

pn 2 −2

un,j gn(m) ϕ dx

|(gn(m) )−1 un,i |

pn 2

|(gn(m) )−1 un,j |

pn 2 −2

(gn(m) )−1 un,j ϕ dx

RN

 →α

pn 2 −2

2∗ 2

|vi | |vj |

RN

2∗ 2 −2

vj ϕ dx,



un,j ψ dx =

un,j gn(m) ϕ dx

= λ−2 n,m

RN





(gn(m) )−1 un,j ϕ dx → 0.

RN

Take the limit n → ∞ in (2.7), we obtain (2.5) (in its weak form). In the case L < +∞, xn,m → x ∗ ∈ ∂ as n → ∞. Without loss of generality we assume ∗ x = 0 and the inner normal at x ∗ to ∂ is eN = (0, · · · , 0, 1). We choose functions ϕ in N−2

2 ∞ C0∞ (RN L ). Again for n large enough ψ = gn ϕ = λn,m ϕ(λn (· − xn,m )) ∈ C0 (). Taking ψ as test functions in (2.2) with p = pn and letting n → ∞, we obtain the system (2.6) (in its weak form). 2

(m)

Corollary 2.1. The summation



gn(m) Vm in the decomposition (2.1) contains only finitely many

m

terms.

Proof. Note that V = Vm satisfies either (2.5) or (2.6). In both cases we have   k RN

|∇vj | dx = α 2

j =1

  k

RN

2∗

βij |vi | 2 |vj |

2∗ 2

dx.

i,j =1

Notice that α ∈ (0, 1], we have   k RN

and

 RN

|∇vj | dx ≤ C 2

j =1

|∇V |2 dx =

  k

RN

 RN

k  j =1

2∗

|vj | dx ≤ C

j =1



  k

RN

|∇vj |2 dx

2∗ 2

j =1

|∇vj |2 dx ≥ m > 0 for some m > 0. By the property (3) of (2.1)

the terms of the summation are finitely many. 2

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Lemma 2.2. There exists C > 0 such that V = Vm satisfies |V (x)| ≤

C (1 + |x|2 )

(x) = Proof. Assume that V satisfies (2.5). Let V = ( Then V v1 , · · · , v k ) satisfies −v j = α

k 

2∗

βij | vi | 2 |v j |

2∗ 2 −2

N−2 2

, x ∈ RN .

1 V ( |x|x 2 ) |x|N−2

be the Kalvin transform of V .

v j , x ∈ RN \{0}, j = 1, · · · , k.

j =1

(x)| ≤ C in B1 , hence |V (x)| ≤ By Moser’s iteration, |V satisfies loss of generality, assume L = 0, then V

C (1+|x|2 )

N−2 2

. If V satisfies (2.6), without

⎧ k 2∗ 2∗ ⎪ ⎨ −v = α  β | j | 2 −2 v j , x ∈ RN j ij vi | 2 |v + i=1 ⎪ ⎩ v j = 0, on ∂RN + \{0}, j = 1, · · · , k. (x)| ≤ C in B + , hence |V (x)| ≤ Again by Moser’s iteration, we obtain |V 1

C (1+|x|2 )

N−2 2

.

2

Consider the decomposition (2.1). By the regularity theory for elliptic equations, as a solution of (1.1), U0 belongs to L∞ , and

U0 p1 ≤ C Suppose

N N−2

< p2 <

2N N−2 .

for

2N < p1 < +∞. N −2

(2.8)

By Lemma 2.2,

gn(m) Vm p2

≤C

 RN

N−2 p2  p12 λn,m 2 dx 1 + λ2n,m |x|2

N−2 N 2 − p2

= Cλn,m

 RN

N−2 N 2 − p2

= Cλn,m



dx (1 + |x|2 ) N−2 N 2 − p2

≤ Cλn

1 p2

(2.9)

N−2 2 p2

,

where λn = min{λn,m }. m

Inspired by the estimates (2.8), (2.9) we have the following definition from [12]. Definition 2.1. ([12, Definition 2.1]) Let p1 , p2 ∈ [1, +∞), p2 < 2∗ < p1 , α > 0, λ > 0. We consider the inequalities system

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u1 p1 ≤ α ,

u2 p2 ≤ αλ

N−2 N 2 − p2

(2.10) ,

and introduce a norm depending on p1 , p2 and λ by setting

u p1 ,p2 ,λ = inf{α > 0 : there exist u1 , u2 such that (2.10) holds and |u| ≤ u1 + u2 }. N 2N Proposition 2.1. Let N−2 < p2 < N−2 < p1 . Let pn → 2∗ , Un be a solution of (1.2) with p = pn and the decomposition (2.1) hold. Then

Un p1 ,p2 ,λn ≤ C. Proof. By (2.8), (2.9),

U0 +



gn(m) Vm p1 ,p2 ,λn ≤ C.

(2.11)

m

Define z, wm and y by 

−z = 1 in 

(2.12)

z = 0 on ∂ 

∗ −1

−wm = |gn(m) Vm |2 wm = 0 

in 

(2.13)

on ∂ ∗ −1

−y = |Rn |2

in 

(2.14)

y = 0 on ∂. For p1 >

2N N−2 ,

z p1 ≤ C z L∞ ≤ C. N For N−2 < p2 < and Lemma 2.2

2N N−2 , q

=

Np2 N+2p2

wm p2 = wm

(2.15)

∈ (1, N2 ). By the Lp -regularity theory for elliptic equations

Nq N−2q

∗ −1

≤ C |gn(m) Vm |2 ≤C

 RN

N−2 · N+2 ·q  q1 λn,m 2 N−2 dx 1 + λ2n,m |x|2

N+2 N 2 −q

= Cλn,m

 RN

N+2

2 = Cλn,m

q

− Nq

q

(1 + |x|2 ) N−2 N 2 − p2

= Cλn,m

(2.16)

1

dx N+2 2 q

N−2 N 2 − p2

≤ Cλn

.

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Let a = |Rn | N−2 , a N = 2 ten as



∗

2 RN |Rn | dx



2

N

9

→ 0 as n → ∞. The equation (2.14) can be writ-

−y = a|Rn | in  y = 0 on ∂.

By Lemma 2.1 in [12]

y p1 ,p2 ,λn ≤ C a N Rn p1 ,p2 ,λn = o(1) Rn p1 ,p2 ,λn .

(2.17)

2

For j = 1, ..., k, let fj = λj un,j +

k 

βij |un,i |

i=1 ∗ −1

|fj | ≤ C(1 + |Un |2

) ≤ C(1 +

pn 2



|un,j |

pn 2 −2

(2.18)

un,j ,

∗ −1

|gn(m) Vm |2

∗ −1

+ |Rn |2

).

(2.19)

m

We note that un,j , z, wm and y satisfy (1.2), (2.12), (2.13) and (2.14) respectively. By the maximum principle and (2.19) |un,j | ≤ C(|z| +



|wm | + |y|).

m

Hence by (2.15), (2.16), (2.17) and (2.11)

Un p1 ,p2 ,λn ≤ C

k 

||un,j ||p1 ,p2 ,λn

j =1

≤ C |z| +



|wm | p1 ,p2 ,λn + C y p1 ,p2 ,λn

m

≤ C(1 + o(1) Rn p1 ,p2 ,λn ) ≤ C + o(1)( Un p1 ,p2 ,λn + U0 +



gn(m) Vm p1 ,p2 ,λn )

m

≤ C + o(1) Un p1 ,p2 ,λn . Finally Un p1 ,p2 ,λn ≤ C.

2

Lemma 2.3. Let pn → 2∗ , Un be a solution of the system (1.2) with p = pn and Un  U0 in H . N ) there exists a constant c > 0 such that Then for γ ∈ (1, N−1 1 rN

 |Un |γ dy ≤ C, Dr (x)

where Dr (x) = {y|y ∈ , |y − x| < r}.

−1

for x ∈  and r ≥ λn 2

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∗

Proof. Let fj be defined as in (2.18), and f = (f1 , · · · , fk ). Then |f | ≤ C(1 + |Un |2 −1 ). By N Theorem 4.8 in [17] (see also [7, Lemma 3.2]) for γ ∈ (1, N−1 ) there exists a constant c > 0 such that ⎛ ⎜ 1 ⎝ N r



⎞1 γ

⎟ |Un | dy ⎠ ≤ C + C

r0



1

γ

t r

Dr (x)

|f | dy)dt

( N−1 Dt (x)

r0 ≤C +C

1 ( t N−1

r

(2.20)



∗ −1

|Un |2

dy)dt

for x ∈ 

Dt (x)

where Dt (x) = {y|y ∈ , |y − x| < t} and r0 is the diameter of . By Proposition 2.1 there exist N 2N u1 , u2 such that for N−2 < p2 < N−2 < p1 < +∞ 

|Un | ≤ u1 + u2 ,

N −N 2∗ p2

u1 p1 ≤ C, u2 p2 ≤ Cλn Choose p1 = N(2∗ − 1) = N N+2 N−2 > r0

2N N−2 ,



1

∗ −1

u21

t N−1 r

p2 = 2∗ − 1 =

N+2 N−2

(2.21) .

N 2N ∈ ( N−2 , N−2 ). Then

dy dt

Dt (x)

r0 ≤

1



t N−1 r

p

u1 1 dy

1

Dt (x)

r0 ≤C

1 t N−1



N

dy

1− 1

N

(2.22)

dt

Dt (x) 1

t N (1− N ) dt ≤ C.

r

r0

1



t N−1 r

Dt (x)

∗ u22 −1 dy dt

r0 ≤C

1 − N−2 λn 2 dt N−1 t

(2.23)

r − N−2 2

≤ Cr −N+2 λn The lemma follows from (2.20), (2.22) and (2.23).

≤C

−1

for r ≥ λn 2 .

2

Let λn = λn,1 = min{λn,m }, xn = xn,1 . Since the concentration points are finitely many, say s, m

there exists c > 0 such that An1 ∩ {xn,1 , · · · , xn,s } = ∅ where

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An1 = D

− 12

(xn )\D

(c+5)λn

11

(xn ).

− 12

cλn

Also denote An2 = D

− 12

(xn )\D

(c+4)λn

An3 = D

− 12

(xn ),

(c+1)λn

(xn )\D

− 12

(c+3)λn

− 12

(xn ).

(c+2)λn

As in [12] we call these sets safe regions, since we have the following estimates. Proposition 2.2. Let pn → 2∗ , Un be a solution of the system (1.2) with p = pn and Un  U0 in H . Then |Un (x)| ≤ C,

for x ∈ An2 .

Proof. The proof is divided into three steps.  ∗ Step 1. We prove D 1 (x) |Un |2 dx → 0 as n → ∞ uniformly in x ∈ An2 . 3 λ− 2 4 n

−1

−1

For x ∈ An2 and |y − x| ≤ 34 λn 2 , we have |y − xn,m | ≥ 14 λn 2 and by Lemma 2.2 |gn(m) Vm (y)| ≤ C. Hence 



2∗

∗

|Un | dx ≤ C D

1 3 λ− 2 4 n

(x)

D

(|U0 |2 + 1 3 λ− 2 4 n

− N2

≤ Cλn



∗

∗

|gn(m) Vm |2 + |Rn |2 ) dx

m

(x)



+C

∗

|Rn |2 dx = o(1),

as n → ∞.



Step 2. Assume q > 1, 0 < r < R ≤ 34 . Then there exist positive constants δq , CR , Cr such   ∗ −N that if D 1 (x) |Un |2 dx ≤ δq , D 1 (x) |Un |q dx ≤ CR λn 2 for x ∈ An2 , then 3 λ− 2 4 n 2N q· N−2

 D

−1 rλn 2

(x) |Un |

− Rλn 2

− N2

dx ≤ Cr λn .

Let vn = |Un |, ϕ ∈ H01 (), ϕ ≥ 0. Then we have

 Dvn Dϕ dx = 

  k



un,j Dun,j Dϕ

 dx ≤

k p p 1  βij |un,i | 2 |un,j | 2 ϕ dx |Un |

u2n,1 + · · · + u2n,k    ∗ ≤ C(vn2 −1 + vn )ϕ dx = an vn ϕ dx  j =1





i,j =1

(2.24)

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12



4

where an = C(vnN−2 + 1),

N 2

D

1 3 λ− 2 4 n

(x) an

dx → 0 as n → ∞ uniformly in x ∈ An2 . Take η ∈ −1

−1

C0∞ (RN ) such that η(y) = 1 if |y − x| ≤ rλn 2 ; η(y) = 0 if |y − x| ≥ Rλn 2 , |∇η| ≤ q−1

1

2 2 R−r λn .

If q ≥ 2, taking ϕ = η2 vn as a test function in (2.24), we have   N−2

 q 2N q N (vn2 η) N−2 dx ≤ C |∇(vn2 η)|2 dx 



 



q

an vn η2 dx + C

≤ Cq 

 q

vn |∇η|2 dx



N

L 2 (D

−1 Rλn 2

q

(vn η2 )

(x))

N

q





(vn η2 ) N−2 dx

N q· N−2

vn D

−1 rλn 2

N N−2



N−2 N



N

L 2 (D

vn dx

1 3 λ− 2 4 n

(x))

−1 Rλn 2

(x)

vn dx −1 Rλn 2

q 2N (vn2 η) N−2 dx ≤ Cλn

In the above we have used Cq an

q

+ Cλn

q

+ Cλn



(x)

N

(2.25)

 D

D

dx ≤

dx

N−2





and

vn |∇η|2 dx



≤ Cq an

1 2

q

η ) dx + C



≤



q−1 2

∇vn ∇(vn

≤ Cq

≤

(x)



q

vn dx

D

−1 Rλn 2

1 2

for x ∈ An2 .



N N−2

−N

≤ Cλn 2 .

(2.26)

(x)

If 1 < q < 2, we take ϕ = (vn + θ )q−1 − θ q−1 , θ > 0 as a test function in (2.24) and let θ → 0. Then again we have (2.25) and (2.26).  2N Step 3. Assume 0 < R < 34 and there exist γ > N−2 , C > 0, such that D 1 (x) |un |γ dx ≤ C. Then there exists CR > 0 such that |un | ≤ CR in D Denote

d

=

(N−2)γ 4

η ∈ C0∞ (RN ), η(y) = 1 q −1

− Rλn 2

1 2 2 Rλn N > d= < N−2 . Let qk > 2, 0 < Rk+1 < Rk ≤ R ≤ 34 . Choose 1 −1 −1 2 2 if |y − x| ≤ Rk+1 λn 2 ; η(x) = 0 if |y − x| ≥ Rk λn 2 , |∇η| ≤ Rk −R λ . n k+1

d d  −1

N 2,

Taking ϕ = η2 vnk as a test function in (2.24), we have  N−2

 qk 2N N (vn2 η) N−2 dx 

− 1 (x).



≤ Cqk

q



an vnk η2 dx + C 

q

vnk |∇η|2 dx 

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≤ Cqk





and dx

D

−1 Rλn 2

 1 

q

(vnk η2 )d dx



D

−1 R k λn 2

q d vn k

N (N−2)d



N 2

D

dx

1 dx D



N 1 1− N2 + 2d )λ n (Rk − Rk+1 )2

 dx

1 χd

−1 R k λn 2

−1 R k λn 2

 1 d

(x)

q d

vnk dx

1 d

.

(x)

N 1 ≤ C(qk + ) λn2 (Rk − Rk+1 )2

(x)

Let q0 =

2N (N−2)d

> 2, qk = q0 χ k , R0 = R, Rk =

N λn2



q d vnk+1

−1 Rk+1 λn 2



d

(x)

−1 Rk+1 λn 2

D

(2.27)

> 1, from (2.27) we have

q χd vn k

λn

d

1

D

Denoting χ =

1



(x)

Cλn + (RK − Rk+1 )2

≤ C(qk +

d

13

 dx

1 qk+1 d

(x)

R 2

+



q d

vnk dx D

−1 R k λn 2

1 d

(2.28)

.

(x)

R , 2k+1

1 N 1 qk ≤ C(qk + ) λn2 (Rk − Rk+1 )2



q d

vnk dx D

−1 R k λn 2



1 qk d

.

(x)

By iteration



N

q d

vnk dx

λn2 D

−1 R k λn 2



1 qk d

(x)

let k → ∞, |Un | = vn ≤ C in D

− 12 1 2 Rλn



N ≤ C λn2

2N

vnN−2 dx D

−1 Rλn 2

(x), ∀x ∈ An2 .

2

 N−2 N

≤ C,

(x)

Corollary 2.2. 

1− N2

|∇Un |2 dx ≤ Cλn

.

An3 −1

−1

Proof. Choose η ∈ C0∞ (RN ) such that η(y) = 1, (c + 2)λn 2 ≤ |y − x| ≤ (c + 3)λn 2 ; η(y) = 0, − 12

|y − x| ≥ (c + 4)λn p = pn , we have

−1

1

or |y − x| ≤ (c + 1)λn 2 ; |∇η| ≤ 2λn2 . Taking ϕ = η2 Un in (1.2) with

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14



 |∇Un | η dx ≤ C



 |Un | |∇η| dx + C

2 2

2

(|Un |pn + |Un |2 )η2 dx.

2





Hence 

 |∇Un | dx ≤ C

An3

1− N2

(λn Un2 + |Un |p ) dx ≤ Cλn

2

. 2

An2

3. Local Pohozaev identity and the proof of Theorem 1.1 In order to prove Theorem 1.1 we will follow the original idea of [12] to employ a local version of the Pohozaev identity. We have the following result. Lemma 3.1. Let Un = (u1 , . . . , uk ) satisfy the system (1.2) with p = pn . Then N N −2 ( − ) pn 2

  k

Dn j =1

=

1 2

  k

N N |∇uj | η dx + ( − ) 2 pn 2

+

N pn 1 2

λj u2j dx

Dn j =1

|∇uj |2 (x − x ∗ , ∇η) dx −

Dn j =1

−

  k

  k (∇uj , x − x ∗ )(∇uj , ∇η) dx

Dn j =1

   k k k pn pn 1 1  (∇uj , ∇η)uj dx − ( λj u2j + βij |ui | 2 |uj | 2 (x − x ∗ , ∇η) dx 2 pn

Dn j =1

  k ∂e D n

j =1

Dn

i,j =1

|∇uj |2 (x − x ∗ , n)η dσ,

(3.1)

j =1

where Dn = D

− 12

(c+3)λn

(xn ), ∂e Dn = ∂Dn ∩ ∂, n is the outward normal to ∂, x ∗ ∈ RN , η is −1

− 12

a cut-off function such that η = 0 if |x − x ∗ | ≥ (c + 3)λn 2 , η = 1 if |x − x ∗ | ≤ (c + 2)λn

and

1 2

|∇η| ≤ 2λn . Proof. Multiplying the equations in the system (1.2) with p = pn by (∇uj , x − x ∗ )η and integrating by parts, we have

−

N −2 2 +

  k Dn j =1

|∇uj |2 η dx −

1 2

  k

|∇uj |2 (x − x ∗ , ∇η) dx

Dn j =1

  k (∇uj , ∇η)(∇uj , x − x ∗ ) dx

Dn j =1

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−

    k k 1 (∇uj , n)(∇uj , x − x ∗ )η dσ + |∇uj |2 (x − x ∗ , n)η dσ 2

∂Dn j =1

 = −N

(

 −

(

k k pn pn 1 1  λj u2j + βij |ui | 2 |uj | 2 )η dx 2 pn i,j =1

k k pn pn 1 1  λj u2j + βij |ui | 2 |uj | 2 )(x − x ∗ , Dη) dx 2 pn j =1

Dn

 +

(3.2)

∂Dn j =1

j =1

Dn

15

(

i,j =1

k k pn pn 1 1  λj u2j + βij |ui | 2 |uj | 2 )(x − x ∗ , n)η dx. 2 pn j =1

∂Dn

i,j =1

Note that η = 0 on ∂Dn ∩  and uj = 0, ∇uj =

∂uj ∂n

n on ∂e Dn = ∂Dn ∩ ∂. We have

    k k 1 ∗ (∇uj , n)(∇uj , x − x )η dσ − |∇uj |2 (x − x ∗ , n)η dx 2

∂Dn j =1

1 = 2

∂Dn j =1

  k

|∇uj |2 (x − x ∗ , n)η dσ

∂e Dn j =1

and  ( ∂Dn

k k pn pn 1 1  λj u2j + βij |ui | 2 |uj | 2 )(x − x ∗ , n)η dσ = 0. 2 pn j =1

(3.3)

i,j =1

Taking  = U η as test function in (1.2) with p = pn , we have   k

    k k k  pn pn |∇uj | η dx + (∇uj , ∇η)uj dx = ( λj u2j + βij |ui | 2 |uj | 2 )η dx. 2

Dn j =1

Dn j =1

Dn j =1

i,j =1

(3.4) By eliminating the term and (3.4).

2



k  Dn

i,j =1

βij |ui |

pn 2

|uj |

pn 2

η dx, the lemma follows from (3.2), (3.3)

−1

Proof of Theorem 1.1. Choose x ∗ such that |x ∗ − xn | ≤ (c + 8)λn 2 and (x − x ∗ , n) ≤ 0 for all x ∈ ∂e Dn . If ∂e Dn = ∅, we simply choose x ∗ = xn . With this choice of x ∗ and noticing that pn < 2∗ , we obtain by (3.1) that

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16

N N ) − 2 pn

  k

λj u2j η dx

Dn j =1

1 ≤ 2

  k

  k |∇uj | (x − x , ∇η) dx − (∇uj , x − x ∗ )(∇uj , ∇η) dx ∗

2

Dn j =1

−

N pn

Dn j =1

   k k k pn pn 1 1  (∇uj , ∇η)uj dx − ( λj u2j + βij |ui | 2 |uj | 2 )(x − x ∗ , ∇η) dx. 2 pn

Dn j =1

j =1

Dn

i,j =1

(3.5)

The integrals of the right hand of (3.5) are taken over the domain An3 , due to the fact that ∇η = 0 1

−1

outside An3 . Since |∇η| ≤ 2λn2 and |x − x ∗ | ≤ Cλn 2 in An3 , we have 

 |Un | η dx ≤ C

(|∇Un |2 + |Un |2 |∇η|2 + |Un |pn + |Un |2 ) dx.

2

(3.6)

An3

Dn

By Proposition 2.2 and Corollary 2.2, the right hand side of (3.6) can be estimated as 1− N2

RHS of (3.6) ≤ cλn

(3.7)

.

To estimate the left hand side of (3.6) we make use of the decomposition (2.1). Let Dn = DLλ−1 (xn ) where L > 0 is so large that for some α > 0 n

 |V1 |2 dx ≥ α > 0 .

(3.8)

BL (0)

For n large enough Dn ⊆ D have

− 12

(c+2)λn

(xn ) ⊆ Dn . Doing a change of variable y = λn (x − xn ) we



 |Un | η dx ≥ 2

|Un |2 dx

Dn

Dn

= λ−2 n



(3.9) |(gn(1) )−1 Un |2 dy.

BL (0)

Using the property (1) of the decomposition (2.1) we have (gn )−1 Un → V1 in L2 (BL (0)) as n → ∞. Thus by (3.9) we have an estimate of the left hand side of (3.6) (1)

LHS of (3.6) ≥ cλ−2 n . 1− N

(3.10)

2 From (3.7), (3.10), λ−2 , which is a contradiction if N ≥ 7. Therefore, in the decomn ≤ Cλn ∗ position s = 0 and Un = U0 + Rn . By (4) of (2.1), Rn → 0 in L2 (RN ), that is Un → U0 in

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17

∗

L2 (RN ). U = (u1 , · · · , uk ) and Un = (un,1 , · · · , un,k ) satisfy the systems (1.1) and (1.2) with p = pn , respectively. We have Un  U in H and

Un 2 =

  k (|∇un,j |2 − λj u2n,j ) dx  j =1

=

  N

βij |un,i |

pn 2

|un,j |

pn 2

dx

 i,j =1

→

  N

2∗

βij |ui | 2 |uj |

2∗ 2

dx

 i,j =1

=

  k (|∇uj |2 − λj u2j ) dx = U 2 .  j =1

Hence Un → U in H .

2

4. Minimax method in the presence of multiple invariant sets of descending flows With the approximation approach established in Sections 2 and 3, we need to construct multiple solutions for the subcritical problems, in particular, multiple sign-changing solutions. We adopt the abstract framework established in [20] by the authors and make necessary generalizations of the abstract theory to fit our needs here, especially about the Morse index estimates of the critical points constructed. The abstract theory developed in this section will be used in Section 5 to construct multiple sign-changing solutions for the subcritical problems. Though the variational formulations involved here are smooth, the abstract theory can be set up in a much more general setting without much additional work. We refer to [6,15,16] for the background of critical point theory of continuous functionals on metric spaces. Let X be a complete metric space with the metric d and f : X → R be a continuous functional on X. For c ∈ R we denote f c = {x ∈ X | f (x) ≤ c} and Kc = {x ∈ X | f (x) = c, f  (x) = 0}. We say G : X → X is an isometric involution if G satisfies G2 = I d and d(Gx, Gy) = d(x, y) for x, y ∈ X. Definition 4.1. ([20]) Let X be a complete metric space with the metric d, G be an isometric involution on X. Let f be a G-invariant continuous functional on X. Let {Pj }kj =1 be a family k  of open subsets of X, Qj = GPj , j = 1, · · · , k. Set W = (Pj ∩ Qj ). We say {Pj , Qj }kj =1 j =1

is a G-admissible family of invariant sets with respect to f at the level c if the following deformation property holds: There exist ε0 > 0 and a G-invariant open neighborhood N of Kc \W with γ (N) < +∞ (in case Kc \W = ∅, we ask N = ∅) such that for 0 < ε < ε0 there exists a continuous map η : X → X satisfying (1) (2) (3) (4)

η(Pj ) ⊂ Pj , η(Qj ) ⊂ Qj , j = 1, · · · , k, η ◦ G = G ◦ η, η|f c−2ε = I d, η(f c+ε \(N ∪ W )) ⊂ f c−ε .

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18

Theorem 4.1. Let X be a complete metric space, G be an isometric involution, f be a G-invariant continuous functional on X. Let Pj , Qj = GPj , j = 1, · · · , k be open subsets of X. k k   (Pj ∪ Qj ),  = (∂Pj ∩ ∂Qj ), FG = {x| x ∈ X, Gx = x}. Assume Denote W = j =1

j =1

c0 = sup f (x) < c∗ = inf f (x).

(m)

x∈

x∈FG

(DF ) {Pj , Qj }kj =1 is a G-admissible family of invariant sets with respect to f at every level c with c ≥ c∗ . Define  = {σ | σ : X → X continuous, σ ◦ G = G ◦ σ, σ (Pj ) ⊂ Pj , σ (Qj ) ⊂ Qj , j = 1, · · · , k, σ (x) = x if f (x) ≤ c0 }, and for j = 1, 2, ..., j = {A| A ⊂ X, compact, G-invariant , γ (A ∩ σ −1 ()) ≥ j, for all σ ∈ }. Assume () j is nonempty, j = 1, 2, · · · . Define for j = 1, 2, ..., cj = inf

sup f (x).

A∈j x∈A\W

Then for j = 1, 2, ..., cj ≥ c∗ are critical values of f , Kc∗j := Kcj \W = ∅, and cj → +∞ as j → ∞. Proof. Firstly we show Kc∗j = ∅. Since the identity I d ∈  and W ∩  = ∅, we have γ ((A\W ) ∩ ) = γ (A ∩ ) ≥ j, for A ∈ j , hence cj = inf

sup f (x) ≥ inf f (x) = c∗ .

A∈j x∈A\W

x∈

If Kc∗j = ∅, by the deformation property there exist ε > 0 and a continuous map η : X → X such that (1) η(Pj ) ⊂ Pj , η(Qj ) ⊂ Qj , j = 1, · · · , k, (2) η ◦ G = G ◦ η,

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19

(3) η|f cj −2ε = I d, (4) η(f cj +ε \W ) ⊂ f cj −ε . Choose ε small such that cj − 2ε > c0 , then η ∈ . By the definition of cj there exists B ∈ j such that B\W ⊂ f cj +ε \W . Let A = η(B), then A ∈ j , since γ (A ∩ σ −1 ()) = γ (η(B) ∩ σ −1 ()) ≥ γ (B ∩ (σ η)−1 ()) ≥ j, ∀σ ∈ . We have A\W =η(B)\W ⊂ (η(B\W ) ∪ η(W ))\W ⊂ η(B\W ) ⊂ η(f cj +ε \W ) ⊂ f cj −ε which is a contradiction. Next we show cj → +∞ as j → ∞. Obviously cj is increasing in j and cj → c ≥ c∗ > c0 as j → ∞. Assume c < +∞. By the deformation property there exist a constant ε > 0, a G-invariant open neighborhood of N of Kc∗ with m = γ (N) < +∞, and a continuous map η : X → X such that (1) (2) (3) (4)

η(Pj ) ⊂ Pj , η(Qj ) ⊂ Qj , j = 1, · · · , k, η ◦ G = G ◦ η, η|f c−2ε = I d, η(f c+ε \(N ∪ W )) ⊂ f c−ε .

Choose ε small c − 2ε > c0 , then η ∈ . Choose j large enough such that c − 12 ε < cj ≤ cm+j (≤ c). There exists a set B ∈ j +m such that B\W ⊂ f cj +m +ε ⊂ f c+ε . Let A = η(B\N ). For σ ∈  B ∩ (σ η)−1 () ⊂ ((B\N ) ∩ (σ η)−1 ()) ∪ N , γ (A ∩ σ −1 ()) = γ (η(B\N ) ∩ σ −1 ()) ≥ γ ((B\N ) ∩ (σ η)−1 ()) ≥ γ (B ∩ (σ η)−1 ()) − γ (N) ≥j +m−m=j. Hence A ∈ j , but A\W = η(B\N )\W ⊂ (η(B\(N ∪ W )) ∪ η(W ))\W ε

⊂ η(B\(N ∪ W )) ⊂ η(f c+ε \(N ∪ W )) ⊂ f c−ε ⊂ f cj − 2 which is a contradiction.

2

Now we give a sufficient condition for j to be nonempty. First we need the following intersection lemma which closely relates to and improves slightly Lemma 2.2 in [20]. In Appendix A we give a sketch of the proof.

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20

Lemma 4.1. Let X be a complete metric space with the metric d, G be an isometric involution. Let Pj , Qj = GPj , j = 1, · · · , k be open subsets of X. Set Mj = Pj ∩ Qj , j = 1, · · · , k, k k   Mj ,  = (∂Pj ∩ ∂Qj ). Denote by B nk the closed unit ball in R nk and t = M = j =1

j =1

(t1 , · · · , tk ) ∈ B nk with t1 , · · · , tk ∈ R n . Assume a continuous map ϕ : B nk → X satisfies (1) ϕ(−t) = Gϕ(t), t ∈ B nk , (2) ϕ(t) ∈ Mj , if tj = 0, j = 1, · · · , k, (3) ϕ(t) ∈ / M, if t ∈ ∂B nk . Then γ (B nk ∩ ϕ −1 ()) ≥ n − k. Lemma 4.2. Assume ϕ : B nk → X, n = j + k satisfies (1) ϕ(−t) = Gϕ(t), t ∈ B nk , (2) ϕ(t) ∈ Mj , if tj = 0, j = 1, · · · , k, (3) ϕ(t) ∈ / M, if t ∈ ∂B nk . Moreover assume (4)

sup f (ϕ(t)) ≤ c0 = sup f (x). x∈FG

t∈∂B nk

Then A = ϕ(B nk ) ∈ j , that is γ (A ∩ σ −1 ()) ≥ j , ∀σ ∈ . Proof. Let σ ∈  and ψ = σ ϕ : B nk → X. We have (1) ψ(−t) = σ ϕ(−t) = σ Gϕ(t) = Gσ ϕ(t) = Gψ(t), t ∈ B nk , (2) ψ(t) = σ ϕ(t) ∈ σ (Mj ) ⊂ Mj , if tj = 0, j = 1, · · · , k, / M. (3) if t ∈ ∂B nk , f (ϕ(t)) ≤ c0 , ψ(t) = σ ϕ(t) = ϕ(t) ∈ By Lemma 4.1, γ (B nk ∩ ψ −1 ()) ≥ n − k = j , hence γ (A ∩ σ −1 ()) = γ (ϕ(B nk ) ∩ σ −1 ()) ≥ γ (B nk ∩ (σ ϕ)−1 ()) ≥ j . 2 5. Multiple sign-changing solutions for the systems with subcritical growth In this section we consider the system (1.2) with subcritical growth. In [20] by the method of invariant sets of decreasing flow the existence of infinitely many sign-changing solutions of the system (1.2) was proved for N = 2, 3, p = 4, λj < 0, j = 1, · · · , k and  = RN . The arguments there can be applied to the general cases. We will apply the abstract theory from Section 4 to obtain multiple sign-changing solutions. Let H = H01 () × · · · × H01 () be the k-fold product space, endowed the inner product (U, V ) =

  k (∇uj ∇vj − λj uj vj ) dx, U = (u1 , · · · , uk ), V = (v1 , · · · , vk ),  j =1

which gives rise to a norm on H . In the following we use || · ||p to denote the Lp () norm.

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21

Solutions of (1.2) correspond to critical points of the functional Ip : 1 1 Ip (U ) = U 2 − 2 p

  k

p

p

βij |ui | 2 |uj | 2 dx, U ∈ H.

(5.1)

 i,j =1

Ip is a C 1 -functional on H and satisfies the (PS) condition. Let P be the positive cone in H01 () P = {u ∈ H01 (), u(x) ≥ 0 a.e. x ∈ }. For δ > 0, j = 1, · · · , k, define open convex subsets of H : Pj = {U ∈ H, U = (u1 , · · · , uk ), dist(uj , P ) < δ}, Qj = −Pj = {U ∈ H, U = (u1 , · · · , uk ), dist(uj , −P ) < δ}, where dist(u, P ) = distH 1 () (u, P ). Denote also 0

W=

k  j =1

(Pj ∪ Qj ), M =

k 

k 

(Pj ∩ Qj ),  =

j =1

(∂Pj ∩ ∂Qj ).

j =1

At this point we assume, in addition to () and (B), (B+ ): the matrix B = (βij ) is positively definite. We define an operator A : H → H as follows, which will be used as a pseudo-gradient vector field to construct a decreasing flow. Given the fact that N ≥ 7 is assumed in Theorem 1.2, we assume p < 2∗ ≤ 4 here. Given U = (u1 , · · · , uk ) ∈ H , define W = (w1 , · · · , wk ) = AU ∈ H by the following system ⎧ k ⎪ ⎨ −wj − λj wj −  βij |ui | p2 |wj | p2 −2 wj = βjj |uj |p−2 uj , in  i=1,i=j ⎪ ⎩ wj = 0, on ∂, j = 1, · · · , k.

(5.2)

Alternatively in the weak form we have for j = 1, · · · , k 

 (∇wj ∇ϕ − λj wj ϕ) dx −





k 

p

 i=1,i=j

βjj |uj |p−2 uj ϕ dx, ∀ϕ ∈ H01 ().

=

p

βij |ui | 2 |wj | 2 −2 wj ϕ dx



Lemma 5.1. A is well-defined and locally Lipschitz continuous.

(5.3)

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22

Proof. In the following we only consider p < 2∗ ≤ 4. Note that W can be obtained by solving the following minimization problem ⎧ ⎨ Inf {G(W )|W ∈ H } 1 ⎩ G(W ) = 2 W 2 −



2 p 

k 

p

i,j =1,i=j

p

βij |ui | 2 |wj | 2 dx −

k   

j =1

βjj |uj |p−2 uj wj dx.

Let W = AU , W = AU , W = (w1 , · · · , wk ), U = (u1 , · · · , uk ). By (5.3) we have

W − W 2   k = (|∇(wj − wj )|2 − λj (wj − wj )2 ) dx  j =1



k 

=

p

p

p

p

βij (|ui | 2 |wj | 2 −2 wj − |ui | 2 |wj | 2 −2 wj )(wj − wj ) dx

 i,j =1,i=j

+

  k

βjj (|uj |p−2 uj − |uj |p−2 uj )(wj − wj ) dx

 j =1

 ≤C

k 

 p p p |ui | 2 − |ui | 2  |wj | 2 −1 |wj − wj | dx

 i,j =1,i=j

+C

  k   |uj |p−2 uj − |uj |p−2 uj  |wj − wj | dx  j =1 p

≤ C(||U ||p2

−1

p

||W ||p2

−1

p

+ ||U ||p2

−1

p

||W ||p2

−1

p−2

+ ||U ||p

p−2

+ ||U ||p

)||U − U ||p ||W − W ||p

≤ C(||U ||p , ||U ||p ) U − U · W − W , hence

AU − AU ≤ C(||U ||p , ||U ||p ) U − U . 2 Lemma 5.2. For sufficient small δ > 0, A(Pj ) ⊂ Pj , A(Qj ) ⊂ Qj , j = 1, · · · , k. Proof. Taking ϕ = wj− in (5.3), we obtain 

(|∇wj− |2



Then we have

− λj (wj− )2 ) dx

 −

k 

 i=1,i=j

βij |ui |

p 2

p (wj− ) 2

 dx = 

βjj |uj |p−2 uj wj− dx .

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 dist (wj , P ) ≤ C 2

23

(|∇wj− |2 − λj (wj− )2 ) dx





≤C

βjj |uj |p−2 uj wj− dx ≤ C



p−1 − (u− wj dx j )



 p−1 − p−1 − ≤ C||uj ||p ||wj ||p ≤ CdistLp (uj , P ) · distLp (wj , P ) ≤ Cdistp−1 (uj , P ) · dist(wj , P ). p−2

Choosing Cδ0

=

1 2

and δ < δ0 , for U ∈ Pj we have 1 dist(wj , P ) ≤ dist(uj , P ). 2

Hence A(Pj ) ⊂ Pj . Similarly A(Qj ) ⊂ Qj , j = 1, · · · , k.

2

Lemma 5.3. (DIp (U ), U − AU )



= U − AU − 2

k 

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )(uj − wj ) dx.

(5.4)

 i,j =1,i=j

Consequently (DIp (U ), U − AU ) ≥ U − AU 2 .

(5.5)

Moreover if we assume (B)+ , then there exists a constant C > 0 such that p

1

DIp (U ) ≤ C( U − AU + U − AU p−1 + U − AU 2 −1 |Ip (U )| 2 ).

(5.6)

Proof. Let  = (ϕ1 , · · · , ϕk ) ∈ H , W = AU . By (5.3) (DIp (U ), )     k k p p (∇uj ∇ϕj − λj uj ϕj ) dx − βij |ui | 2 |uj | 2 −2 uj ϕj dx =

=

 j =1   k

 i,j =1

(∇(uj − wj )∇ϕj − λj (uj − wj )ϕj ) dx

(5.7)

 j =1



−

k 

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )ϕj dx

 i,j =1,i=j

= (U − AU, ) −



k 

 i,j =1,i=j

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )ϕj dx .

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24

Taking  = U − AU in (5.7), we obtain (DIp (U ), U − AU ) = U − AU 2 

k 

−

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )(uj − wj ) dx . (5.8)

 i,j =1,i=j

Now (5.5) follows from (5.8), since βij ≤ 0 for i = j . Also it follows from (5.8) that   

 p p p  βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )ϕj dx 

k 

 i,j =1,i=j

≤C



−

i,j =1,i=j



≤C

 

 p−2  p p p p p βij |ui | 2 |uj | 2 −2 uj − |wj | 2 −2 wj  p−2 dx ||U ||p ||||p

k 

−

k 

(5.9) p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )(uj − wj ) dx

 p−2 p

||U ||p ||||p

i,j =1,i=j

≤ C(DIp (U ), U − AU )

p−2 p

||U ||p ||||p .

In the above we have used the elementary inequality that for 2 ≤ p ≤ 4 there exists C(p) > 0 such that   p −2 p p |s| 2 s − |t| 2 −2 t  ≤ C(p)|s − t| 2 −1 ,

for s, t ∈ R.

From (5.7) and (5.9)

DIp (U ) ≤ C U − AU + C(DIp (U ), U − AU )

p−2 p

||U ||p .

(5.10)

We estimate ||U ||p . Choose 2 < s < p. Then 1 Ip (U ) − (U, U − AU ) s 1 1 1 1 = ( − ) U 2 + ( − ) 2 s s p −

1 s



k 

p

  k

p

p

βij |ui | 2 |uj | 2 dx

 i,j =1 p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )uj dx .

 i,j =1,i=j

By (5.11), (5.9) and the assumption (B+ ), we have

(5.11)

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25

p

U 2 + ||U ||p ≤ C(|Ip (U )| + |(U, U − AU )|  k    p p p + βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )uj dx )  i,j =1,i=j

≤ C(|Ip (U )| + U U − AU + (DIp (U ), U − AU )

p−2 p

||U ||2p ) .

(5.12)

Hence p

U 2 + ||U ||p ≤ C(|Ip (U )| + U − AU 2 + (DIp (U ), U − AU )) and 1

2

1

||U ||p ≤ C(|Ip (U )| p + U − AU p + (DIp (U ), U − AU ) p ).

(5.13)

Substitute (5.13) into (5.10), we obtain

DIp (U ) ≤ C U − AU + C(DIp (U ), U − AU ) 2

p−2 p

1

(|Ip (U )| p

1

+ U − AU p + (DIp (U ), U − AU ) p ) ≤ C U − AU + ε DIp (U ) + Cε U − AU

p−2 2

1

|Ip (U )| 2

p

+ Cε U − AU 2 + Cε U − AU p−1 . Finally, we obtain (5.6).

2

Lemma 5.4. Let Kc = {U ∈ H | DIp (U ) = 0, Ip (U ) = c} and N be a symmetric open neighborhood of Kc . Then there exists a positive constant ε0 such that for 0 < ε < ε  < ε0 there exists a continuous map σ : [0, 1] × H → H satisfying (1) (2) (3) (4) (5)

σt : U → σ (t, U ) is a diffeomorphism of H , σ0 = I d. σt (U ) = U for t ∈ [0, 1], |Ip (U ) − c| ≥ ε  . σt (−U ) = −σt (U ) for (t, U ) ∈ [0, 1] × H . σ1 (Ipc+ε \N) ⊂ Ipc−ε . σt (Pj ) ⊂ Pj , σt (Qj ) ⊂ Qj , j = 1, · · · , k, t ∈ [0, 1].

Proof. This is similar to the proof of [20, Lemma 3.5] and we sketch it here. For δ > 0 sufficiently small N (δ) = {U ∈ H |dist(U, Kc ) < δ} ⊂ N . Since Ip satisfies the P.S. condition, there exist constants ε0 , b0 > 0 such that 1

DIp (U ) ≥ b0 , for U ∈ Ip−1 [c − ε0 , c + ε0 ]\N ( δ). 2 By Lemma 5.3, (5.6), there exists a constant b > 0 such that 1

U − AU ≥ b, for U ∈ Ip−1 [c − ε0 , c + ε0 ]\N ( δ). 2 Decreasing ε0 if necessary, we assume ε0 ≤ 14 bδ.

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26

Define an even functional g : H → [0, 1]:  0, U ∈ N ( 14 δ) or |Ip (U ) − c| ≥ ε  , g(U ) = 1, U ∈ / N ( 12 δ) and |Ip (U ) − c| ≤ ε. Let V (U ) = −

2ε0 U − AU g(U ) . p

U − AU

Consider the initial value problem 

dσ dt

= V (σ ),

(5.14)

σ (0, U ) = U.

Then we can verify (1)–(4) as usual. To verify (5), we need only to notice that A(Pj ) ⊂ Pj , A(Qj ) ⊂ Qj , j = 1, · · · , k. 2 Corollary 5.1. Let N be an open symmetric neighborhood of Kc∗ = Kc \W . Then there exists a constant ε0 such that for 0 < 2ε < ε0 there exists a diffeomorphism η of H satisfying (1) η(−U ) = −η(U ), U ∈ H . (2) η|I c−2ε = I d. p

(3) η(Ipc+ε \(N ∪ W )) ⊂ Ipc−ε . (4) η(Pj ) ⊂ Pj , η(Qj ) ⊂ Qj , j = 1, · · · , k. Proof. N ∪ W is an open neighborhood of Kc . According to Lemma 5.4, we can choose η = σ (1, ·). 2 Lemma 5.5. For sufficiently small δ, it holds for some c0 > 0 Ip (U ) ≥ c0 δ 2 , for U ∈  = Ip (U ) ≥ 0, for U ∈ M =

k  j =1

k 

j =1

(∂Pj ∩ ∂Qj ) (5.15)

(Pj ∩ Qj ).

− p Proof. For U = (u1 , · · · , uk ) ∈ ∂Pj , u− j ≥ dist(uj , P ) = δ and ||uj ||p = distL (uj , P ) ≤ + Cdist(uj , P ) = Cδ. Similar estimates hold for uj . Hence we have for U ∈ 

1 1 Ip (U ) = U 2 − 2 p

  k

p

p

βij |ui | 2 |uj | 2 dx

 i,j =1

1 ≥C1 δ 2 − C2 δ p ≥ C1 δ 2 2

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27

if 12 C1 δ 2 ≥ C2 δ p . For U ∈ M, ||U ||p ≤ C3 δ, hence p

Ip (U ) ≥ C4 ||U ||2p − C5 ||U ||p ≥ 0 p−2

if C4 − C5 ||U ||p

≥ C4 − C5 (C3 δ)p−2 ≥ 0.

2

Now we want to apply the abstract theory in Section 4, in particular Theorem 4.1. Define cj (p) = inf

sup Ip (U )

(5.16)

A∈j U ∈A\W

j = {A| A ⊂ H, −A = A, A compact, γ (A ∩ σ −1 ()) ≥ j, ∀σ ∈ }

(5.17)

 = {σ |σ : H → H continuous, σ (−U ) = −σ (U ), ∀ U ∈ H, σ (Pj ) ⊂ Pj , σ (Qj ) ⊂ Qj , j = 1, · · · , k, σ (U ) = U if Ip (U ) ≤ 0}.

(5.18)

Lemma 5.6. j is nonempty. Proof. We construct a map ϕ (j ) : B nk → H , n = j + k satisfying the assumptions of Lemma 4.2. Denote t ∈ R nk by t = (t1 , · · · , tk ) with tj = (t1j , · · · , tnj ) ∈ RN for j = 1, · · · , k. Choose nk functions vij ∈ C0∞ (), 1 ≤ i ≤ n, 1 ≤ j ≤ k, with disjoint supports. Define ϕ (j ) : B nk → H by n n n    ϕ (j ) (t) = R( ti1 vi1 , ti2 vi2 , · · · , tij vij ) i=1

i=1

(5.19)

i=1

where R is sufficiently large such that Ip (ϕ (j ) (t)) < 0 for t ∈ ∂B nk . Then (1) ϕ (j ) (−t) = −ϕ (j ) (t), t ∈ B nk . (2) For l = 1, · · · , k, if tl = 0, the l-th component of ϕ (j ) is zero, hence ϕ (j ) (t) ∈ Pl ∩ Ql = Ml . / M. (3) For t ∈ ∂B nk , Ip (ϕ (j ) (t)) < 0 = inf Ip (U ), hence ϕ (j ) (t) ∈ U ∈M

By Lemma 4.2, A = ϕ (j ) (B nk ) ∈ j .

2

Now we have the following existence theorem for sub-critical system (1.2). Theorem 5.1. Assume N ≥ 2, 2 < p < 2∗ . Assume () and (B) hold. Then the system (1.2) has infinitely many sign-changing solutions corresponding to the critical values cj (p) of the functional Ip . Proof. Firstly we assume the additional condition (B+ ) holds. By Lemma 5.6, j is nonempty. By Lemma 5.5, cj (p) ≥ c∗ = inf Ip (U ) > 0. By Lemma 5.4 (and Corollary 5.1) Ip satisfies the U ∈

deformation property and {Pj , Qj }kj =1 is a G-admissible family (G = −I d). By Theorem 4.1,

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cj (p), j = 1, 2, · · · are critical values of Ip , Kc∗j (p) = Kcj (p) \W = ∅ and cj (p) → +∞ as j → +∞. Next we consider the general case. Some of the previous estimates do not hold anymore, in particular, (5.12) in Lemma 5.3. We introduce the perturbed systems and the corresponding functionals ⎧ k ⎪ ⎨ −u = λ u +  β |u | p2 |u | p2 −2 u + μ|u |q−2 u , in  j j j ij i j j j j i=1 ⎪ ⎩ uj = 0 on ∂, j = 1, · · · , k,

(5.20)

where q = 12 (p + 2∗ ) and μ ∈ (0, 1] is a parameter. μ Ip,μ (U ) = Ip (U ) − q

  k

|uj |q dx

 j =1

1 1 = U 2 − 2 p

  k  i,j =1

μ βij |ui | |uj | dx − q p 2

p 2

  k

(5.21) |uj |q dx, U ∈ H.

 j =1

Ip,μ is a C 1 -functional on H and satisfies the (PS) condition. We define the Pj , Qj , j = 1, · · · , k as before and define the operator Aμ , W = Aμ U , W = (w1 , · · · , wk ) by ⎧ k ⎪ ⎨ −wj + λj wj −  βij |ui | p2 |wj | p2 −2 wj = βjj |uj |p−2 uj + μ|uj |q−2 uj , in  i=1,i=j ⎪ ⎩ wj = 0 on ∂, j = 1, · · · , k .

(5.22)

In the weak form W satisfies for j = 1, · · · , k, 

 (∇wj ∇ϕ − λj wj ϕ) dx −



p

p

βij |ui | 2 |wj | 2 −2 wj ϕ dx

 i=1,i=j

 (βjj |uj |

=

k 

p−2

uj + μ|uj |

q−2

uj )ϕ dx, ∀ϕ

(5.23) ∈ H01 ().



Parallel to Lemmas 5.1, 5.2, 5.3 and 5.5 we have the following results (Lemmas 5.7, 5.8, 5.9 and 5.10), and we omit their proofs. Lemma 5.7. Aμ is well-defined and locally Lipschitz continuous. Lemma 5.8. For sufficiently small δ > 0 independent of μ, Aμ (Pj ) ⊂ Pj , Aμ (Qj ) ⊂ Qj , j = 1, · · · , k.

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Lemma 5.9. It holds (DIp,μ (U ), U − Aμ U )  = U − Aμ U 2 −

k 

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )(uj − wj ) dx .

(5.24)

 i,j =1,i=j

Consequently (DIp,μ (U ), U − Aμ U ) ≥ U − Aμ U 2 . Moreover there exists a constant C = C(μ) such that p

1

DIp,μ (U ) ≤ C(μ)( U − Aμ U + U − Aμ U p−1 + U − Aμ U 2 −1 |Ip,μ (U )| 2 ).

(5.25)

Lemma 5.10. For δ > 0 sufficiently small, independent of μ, for some C0 > 0 it holds that Ip,μ (U ) ≥ C0 δ 2 for U ∈  and Ip,μ (U ) ≥ 0 for U ∈ M. The deformation property in Lemma 5.4 and Corollary 5.1 also holds for the functionals Ip,μ . Here we verify (5.25) only. Instead of (5.11), we have 1 Ip,μ (U ) − (U, U − Aμ U ) 4   k 1 1 1 = U 2 + ( − )μ |uj |q dx 4 4 q  j =1

−

1 p



k 

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )uj dx .

(5.26)

 i,j =1,i=j

By Hölder inequality we have for some c > 0 p

U 2 + μ||U ||p ≤ C( U + μ 2

  k

|uj |q dx)

 j =1

 = C(|Ip,μ (U )| + |(U, U − Aμ U )| + |

k 

p

p

p

βij |ui | 2 (|uj | 2 −2 uj − |wj | 2 −2 wj )uj dx|)

 i,j =1,i=j

≤ C(|Ip (U )| + U U − Aμ U + (DIp (U ), U − Aμ U ) From (5.27), parallel to (5.13), we have

p−2 p

||U ||2p ) .

(5.27)

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1

2

1

||U ||p ≤ C(μ)(|Ip (U )| p + U − Aμ U p + (DIp (U ), U − Aμ U ) p ) and finally p

1

DIp,μ (U ) ≤ C(μ)( U − Aμ U + U − Aμ U p−1 + U − Aμ U 2 −1 |Ip,μ (U )| 2 ) . Define cj (p, μ) = inf

sup Ip,μ (U )

(5.28)

A∈j U ∈A\W

where j is defined as in (5.17). By Theorem 4.1, cj (p, μ), j = 1, 2, · · · are critical values of the functional Ip,μ and there exist Uj (p, μ), j = 1, 2, · · · such that Uj (p, μ) ∈ H \W , Ip,μ (Uj (p, μ)) = cj (p, μ), DIp,μ (Uj (p, μ)) = 0. Finally, cj (p, μ) → ∞ as j → ∞. It is clear that cj (p, μ) ≤ cj (p) and is increasing in μ. Denote cj (p) = lim cj (p, μ), μ→0

cj (p) → ∞ as j → ∞. The following Lemma 5.11 shows that cj (p), j = 1, 2, · · · are critical values of the functional Ip , and there exist Uj (p), j = 1, 2, · · · such that Uj (p) ∈ H \W , Ip (Uj (p)) = cj (p) and DIp (Uj (p)) = 0. We have completed the proof of Theorem 5.1. 2 Lemma 5.11. Let μn → 0 and Un satisfy DIp,μn (Un ) = 0, Ip,μn (Un ) ≤ c. Then up to a subsequence, Un converges to U in H , DIp (U ) = 0, Ip (U ) = lim Ip,μn (Un ). Moreover, if Un , n→∞

n = 1, 2, · · · are sign-changing, then so is U . Proof. Ip,μn (Un ) −

1 1 1 1 1 (DIp,μn (Un ), Un ) = ( − ) Un 2 + ( − )μn p 2 p p q

  k

|un,j |q dx,

 j =1

{Un } is bounded in H . By standard argument a subsequence of {Un } converges to U in H and DIp (U ) = 0, Ip (U ) = lim Ip,μn (Un ). n→∞

Un = (un,1 , · · · , un,k ) satisfies the system (5.22) with μ = μn . Suppose Un is sign-changing, + + then u± n,j = 0, j = 1, · · · , k. Multiply (5.22) with μ = μn by un,j we have for un,j = +

un,j H 1 () 0

2

u+ n,j ≤ C



+ 2 2 (|∇u+ n,j | − λj (un,j ) ) dx



=C

  k

βij |un,i |

p 2

p 2 (u+ n,j )

 i=1



≤C

p βjj (u+ n,j ) dx + Cμ



 dx + Cμ



q (u+ n,j ) dx

 q (u+ n,j ) dx

 2∗

+ p0 ∗ ≤ C( u+ n,j + un,j ), for p0 ≤ p < 2 .

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+ Hence u+ n,j ≥ α > 0 for some α > 0. Since Un → U = (u1 , · · · , uk ), uj ≥ α > 0. Similarly

u± j ≥ α > 0, j = 1, · · · , k. 2

6. The proof of Theorem 1.2 Using the multiple sign-changing solutions of the subcritical systems constructed in Section 5 as approximating solutions to the critical system, and applying the convergence approach from Section 2 and 3, we are ready to give the proof of Theorem 1.2. Proof of Theorem 1.2. First we assume B = (βij ) is positively definite. Let cj (p), j = 1, 2, · · · be defined as in (5.16). By Theorem 5.1, cj (p), j = 1, 2, · · · are critical values of Ip , and there exists Uj (p) ∈ H \W such that Ip (Uj (p)) = cj (p), DIp (Uj (p)) = 0. We fix a p0 ∈ (2, 2∗ ) and let p0 ≤ p < 2∗ . We choose positive constants C0 , C1 , C2 such that for all (ξ1 , ..., ξk ) ∈ Rk and all p ∈ [p0 , 2∗ ] k p p 1  |βij ||ξi | 2 |ξj | 2 ≤ C1 p i,j =1

k 

2∗ 2

|ξi | |ξj |

2∗ 2

− C2

i,j =1,i=j

k 

|ξj |p0 + C0 .

j =1

Define 1 H (u) = U 2 + C1 2





2∗ 2

|ui | |uj |

2∗ 2

dx − C2

 i,j =1,i=j

  k

|uj |p0 dx + C0 .

 j =1

Then we have 1 1 Ip (U ) = U 2 − 2 p

  k

p

p

βij |ui | 2 |uj | 2 dx ≤ H (u).

(6.1)

 i,j =1

Let ϕ (j ) be defined as in (5.19), choose R so large such that H (ϕ (j ) (t)) < 0 for t ∈ ∂B nk . Then we have the upper bound for cj (p) as p → 2∗ : cj (p) ≤ βj := sup H (ϕ (j ) (t)).

(6.2)

t∈B nk

Let pn → 2∗ , {Uj (pn )}∞ n=1 is bounded in H . By Theorem 1.1, a subsequence of {Uj (pn )} converges to Uj in H , and DI2∗ (Uj ) = 0, I2∗ (Uj ) = cj = lim cj (pn ) and Uj ∈ H \W . n→∞ We show that cj → ∞ as j → ∞, hence the system (1.1) has infinitely many sign-changing solutions. In order to prove cj → ∞, we will make use of the estimates for Morse indices of the critical points. However for 2 < p < 4 the functional Ip is only a C 1 -functional. We will compare the critical values of Ip with that of some C 2 -functional for which Morse index type arguments can be applied. Define 1 1 Jp (U ) = U 2 − 2 p

  k  j =1

βjj |uj |p dx .

(6.3)

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Notice that Jp = Ip if βij = 0 for i = j . Define also dj (p) = inf

sup Jp (U )

(6.4)

A∈j U ∈A\W

where j is defined as in (5.17). We have dj = lim ∗ dj (pn ) ≤ lim ∗ cj (pn ) = cj . pn →2

(6.5)

pn →2

We note that Jpn is a C 2 -functional. Applying the abstract theory in Section 4 to Jp we have the following assertions with an additional claim about the Morse indices: there exists Uj (pn ) ∈ H \W such that Jpn (Uj (pn )) = dj (pn ), DJpn (Uj (pn )) = 0 and m∗ (Uj (pn )) ≥ j , where m∗ (·) is the augmented Morse index. A proof of the latter fact about the estimates on the augmented Morse index will be given in the Appendix B. Now suppose dj → d < +∞. By Theorem 1.1, we can choose a sequence {n(j )}∞ j =1 , n(j ) → ∞ as j → ∞ such that dj (pn(j ) ) → d, DIpn(j ) (Uj (pn(j ) )) = 0, Ipn(j ) (Uj (pn(j ) )) = dj (pn(j ) ) → d, m∗ (Uj (pn(j ) )) ≥ j and Uj (pn(j ) ) → U in H . U satisfies I2∗ (U ) = d, DI2∗ (U ) = 0 and m∗ (U ) = m < +∞, which contradicts the fact Uj (pn(j ) ) → U in H and m∗ (Uj (pn(j ) )) ≥ j → +∞. Next we consider the general case without assuming (B)+ . Let Ip,μ and cj (p, μ) be defined as in (5.21) and (5.28), respectively, and cj (p) = lim cj (p, μ). The estimate (6.2) holds for cj (p, μ), too. Hence we define cj = lim∗ cj (p).

μ→0

p→2

In Section 5 we proved that cj (p, μ), j = 1, 2, · · · are critical values of Ip,μ , and cj (p), j = 1, 2, · · · are critical values of Ip . Moreover there exists Uj (p) ∈ H \W such that DIp (Uj (p)) = 0, Ip (Uj (p)) = cj (p). By Theorem 1.1, for a sequence pn → 2∗ , Uj (pn ) → Uj in H , U ∈ H \W , DI2∗ (Uj ) = 0, I2∗ (Uj ) = cj = lim cj (pn ). Again we show that cj → ∞ as n→∞

j → ∞. We have 1 1 Ip,μ (U ) = U 2 − 2 p 1 1 ≥ U 2 − 2 p

  k  i,j =1

  k

μ βij |ui | |uj | dx − q p 2

p 2

βjj |uj |p dx −

 j =1

cε 1 ≥ ( − ε) U 2 − 2 q

  k

μ q

  k  j =1

|uj |q dx

 j =1

=: Jq (U ) . Define dj (q) = inf

sup Jq (U ) ,

A∈j U ∈A\W

  k

|uj |q dx

 j =1

|uj |q dx

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dj = lim ∗ dj (qn ) . qn →2

Then cj (p, μ) ≥ dj (q), cj ≥ dj . It is clear that dj → ∞ as j → +∞ as dj does.

2

Remark 6.1. Our method may give more general existence results of vector solutions. Given 1 ≤ m ≤ k, we may obtain infinitely many solutions whose first m components are sign-changing and whose last k − m components are positive. For the general case 1 ≤ m < k, this can be done as follows. Instead of working in H we work in a closed subset of H given by X = ∩kj =k+1 P¯j , which is a complete metric space. Then we may apply Theorem 4.1 to get the result. We omit the details here. Acknowledgments The first author is supported by NSFC 11271331. The second author is supported by NSFC 11361077 and Yunnan Province, Young Academic and Technical Leaders Program (2015HB028). The third author is supported by NSFC 11271201 and a Simons Grant (348496). Appendix A. On the intersection lemma We sketch the proof of Lemma 4.1, see [20] for the details of the proof. Denote O = {t| t ∈ B nk , ϕ(t) ∈ M} . Since ϕ(0) ∈

k  j =1

Mj = M and ϕ(t) ∈ / M for t ∈ ∂B nk . O is a symmetric open neighborhood of 0

in R nk and ϕ(∂O) ⊂ ∂M. By Borsuk’s theorem γ (∂O) = nk. Let S, T be open sets. Then ∂(S ∩ T ) ⊂ (∂S ∩ T ) ∪ (S ∩ ∂T ) ∪ (∂S ∩ ∂T ) .

(a1 )

Repeatedly using (a1 ), we decompose ∂M as follows ∂M = ∂(

k 

M j ) ⊆ C 1 ∪ C2 ∪ · · · ∪ C k

(a2 )

j =1

where Cp =

   ( ∂Mi ∩ Mj ), s∈Sp i∈s

j ∈s c

Sp is the index set: Sp = {s = (i1 , · · · , ip )| 1 ≤ i1 < i2 < · · · < ip ≤ k} and s c is the complement of s ∈ Sp in the set {1, 2, · · · , k}. For Ck we decompose further.

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Ck =

k 

k 

∂Mj =

j =1

⊂

k 

∂(Pj ∩ Qj )

j =1

(a3 )

((∂Pj ∩ Qj ) ∪ (Pj ∩ ∂Qj )) ∪  .

j =1

Define Ap , p = 1, · · · , k − 1 and Bj , j = 1, · · · , k, Ap = {t ∈ ∂O| ϕ(t) ∈ Cp }, Bj = {t ∈ ∂O| ϕ(t) ∈ (∂Pj ∩ Qj ) ∪ (Pj ∩ ∂Qj )} and D = ∂O ∩ ϕ −1 () = {t ∈ ∂O| ϕ(t) ∈ } . Then ∂O ⊂

k−1 

Ap ∪

p=1

k 

Bj ∪ D .

(a4 )

j =1

If we can define odd maps fp , p = 1, · · · , k − 1, B nk → R n and gj , j = 1, · · · , k, B nk → R such that fp (t) = 0 for t ∈ Ap and gj (t) = 0 for t ∈ Bj , then by (a4 ) γ (B nk ∩ ϕ −1 ()) ≥ γ (∂O ∩ ϕ −1 ()) ≥ γ (∂O) − (k − 1)n − k · 1 = nk − (k − 1)n − k =n−k=j. The map fp : B nk → R n is defined by fp (t) =



ti(s) dist(ϕ(t), ∂(

s∈Sp



Mj ))

j ∈s c

where i(s) = i1 , for s = (i1 , · · · , ip ) ∈ Sp . The map gj is defined by gj (t) = dist(ϕ(t), ∂Pj ) − dist(ϕ(−t), ∂Qj ) . Then we can verify that (1) fp , p = 1, · · · , k − 1; gj , j = 1, · · · , k are odd maps, (2) fp (t) = 0 for t ∈ Ap , p = 1, · · · , k − 1, (3) gj (t) = 0 for t ∈ Bj , j = 1, · · · , k .

(a5 )

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Here we check the property 5 ). Suppose t ∈ Ap , that is ϕ(t) ∈ Cp . There exists  (2) in (a ∂Mi ∩ Mj . Then ϕ(t) ∈ / Mi for i ∈ s0 . By the assumps0 ∈ Sp such that ϕ(t) ∈ i∈s0 j ∈s0c  Mj , tion (2) of Lemma 4.1, ti = 0 for i ∈ s0 . In particular ti(s0 ) = 0. Moreover ϕ(t) ∈ j ∈s0c  Mj ) > 0. If s ∈ Sp and s = s0 , then there exists an index i0 ∈ s0 ∩ s c . On one d(ϕ(t), ∂ j ∈s0c

hand ϕ(t) ∈



∂Mi ∩



Mj ⊂ ∂Mi0 .

(a6 )

j ∈s0c

i∈s0

On the other hand ϕ(t) ∈ ∂M, by (a1 ) ϕ(t) ∈ ∂M ⊂



Mj ∪ ∂

j ∈s c





Mj ⊂ M i 0 ∪ ∂

j ∈s c

(a7 )

Mj .

j ∈s c

From (a6 ), (a7 ) ϕ(t) ∈ ∂Mi0 ∩ {Mi0 ∪ ∂



Mj } ⊂ ∂

j ∈s c

and dist(ϕ(t), ∂

 j ∈s c

f (t) =



Mj

j ∈s c

Mj ) = 0. Altogether 

ti(s) dist(ϕ(t), ∂

s∈Sp



Mj ) = ti(s0 ) dist(ϕ(t), ∂

j ∈s c



Mj ) = 0 .

j ∈s0c

Notice that the inclusion in the formula (a1 ), hence (a2 ) may be strict as shown by the following example. Example A. Let S, T be subsets of R2 : S = (−1, 1) × (−1, 1)\[0, 1) × [0, 1), T = (−1, 1) × (−1, 1)\[0, 1) × (−1, 0] . Then the segment L = {(x, 0)| x ∈ [0, 1)} ⊂ (∂S ∩ ∂T )\∂(S ∩ T ). Appendix B. Morse index estimates Let H be a Hilbert space, I be an even C 2 -functional on H , Pj , j = 1, · · · , k be open k k   (Pj ∪ Qj ), M = (Pj ∩ Qj ) and convex subsets of H . Qj = −Pj j = 1, · · · , k, W = =

k 

j =1

(∂Pj ∩ ∂Qj ). Assume

j =1

(I1 ) I satisfies the (PS) condition. (I2 ) For every critical point x of I , D 2 I (x) is a Fredholm operator.

j =1

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(AI ) There exists an odd, Lipschitz continuous map A : H → H such that (a1 ) If x − Ax = 0, then DI (x) = 0. (a2 ) Given c0 , b0 > 0, there exists b = b(c0 , b0 ) such that if |I (x)| ≤ c0 , DI (x) ≥ b0 , then DI (x), x − Ax ≥ b x − Ax . (AP ) A(P j ) ⊂ Pj , A(Qj ) ⊂ Qj , j = 1, · · · , k. (m) c∗ = inf I (x) > 0. x∈

Define j = {A| A ⊂ H, A compact, −A = A, γ (A ∩ σ −1 ()) ≥ j, ∀σ ∈ }  = {σ | σ : H → H continuous, σ (−x) = −σ (x), ∀x ∈ H, σ (Pj ) ⊂ Pj , σ (Qj ) ⊂ Qj , j = 1, · · · , k. σ (x) = x if I (x) ≤ 0} . Assume () j is nonempty, j = 1, 2, · · · . Define cj = inf

sup I (x), j = 1, 2, · · · .

A∈j x∈A\W

Denote for c ∈ R, Kc = {x| DI (x) = 0, I (x) = c} . Kc∗ = Kc \W . Theorem B1 . Assume (I1,2 ) (AI ), (AP ), (m) and () hold. Then (1) cj ≥ c∗ , Kc∗j = ∅. (2) There exists x ∈ Kc∗j with m∗ (x) ≥ j , where m∗ (·) is the augmented Morse index. Lemma B2 . Assume (I1 ) (AI ) and (AP ). Then the family {Pj , Qj }kj =1 is a G-admissible (G = −I d) family of invariant sets with respect to I at every level c > 0, that is, the following deformation property holds: there exist a symmetric open neighborhood N of Kc∗ with γ (N) = γ (Kc∗ ) < +∞ and ε0 > 0 such that for 0 < ε < ε0 there exists a continuous map η : H → H satisfying (1) (2) (3) (4)

η(−U ) = −η(U ), U ∈ H . η|I c−2ε = I d. η(I c+ε \(N ∪ W )) ⊆ I c−ε . η(Pj ) ⊂ Pj , η(Qj ) ⊂ Qj , j = 1, · · · , k.

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Proof. The proof is the same as that of Lemma 5.4 and Corollary 5.1, by using the flow defined by the ordinary differential equation (5.14). 2 We need the following result of Marino–Prodi type [23]. Proposition B3 . Let H be a Hilbert space, I be an even C 2 -functional. Assume (I1,2 ) hold. Define K = {x| DI (x) = 0, |I (x) − c| ≤ ε} . Then for given d, δ > 0, 0 <  ε < ε there exist a symmetric open neighborhood N of K and an even C 2 -functional I on H satisfying (1) N is contained in the d-neighborhood of K. (2) I − I C 2 ≤ δ, I(x) = I (x), if x ∈ / N . In the neighborhood N , D I is proper (hence I 2  satisfies the (PS) condition), D I is a Fredholm operator. (3) Denote  = {x|D I(x) = 0, |I(x) − c| ≤ K ε}.  ⊂ N and K  consists of finitely many pairs of nondegenerate critical points of I. Then K Proof. Since I satisfies the (PS) condition, the set K is compact. According to [30, Lemma 3.9], we can find a (symmetric open) neighborhood N0 of K and a positive constant δ0 such that if a functional I has distance from I less than δ0 in the C 2 -norm, then D I is a proper map in N0 . Since D 2 I (x) is a Fredholm operator for x ∈ K, we can assume D 2 I(x) is a Fredholm operator for x ∈ N0 . In the following we always perturb the functional I in such a way that Iis C 2 -closed to I and  I = I out of N0 . / K, m = We assume 0 ∈ / K, and deal with the case 0 ∈ K later. Since K is compact, 0 ∈ γ (K) < +∞. We prove the proposition by induction on m. 1. γ (K) = 1. In this case K is divided into two disjoint compact sets K+ and K− . For a given set A we denote by Nμ (A) the (open) μ-neighborhood of A: Nμ (A) = {x|dist (x, A) < μ}. Choose μ > 0 such that N3μ (K+ ) ∩ N3μ (K− ) = ∅. Let ϕ be an odd smooth function such that ϕ(x) = ±1 if / N2μ (K+ ) ∪ N2μ (K− ). Denote x ∈ Nμ (K± ), ϕ(x) = 0 if x ∈ I(x) = I (x) − ϕ(x)(x, y), y ∈ X  = {x|D I(x) = 0, |I(x) − c| ≤ K ε}

(b1 )

 for x ∈ Out of N2μ (K+ ) ∪ N2μ (K− ), I(x) = I (x), hence x ∈ /K / N2μ (K+ ) ∪ N2μ (K− ). In N 2μ (K+ )\Nμ (K+ ) (similarly in N 2μ (K− )\Nμ (K− )), there exists α > 0 such that if |I (x) − c| ≤ ε then DI (x) ≥ α. Choose I so closed to I (that is choose y sufficiently small) that

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|I(x) − I (x)| ≤ 12 (ε − ε)

D I(x) − DI (x) ≤ 12 α, ∀x ∈ H .

(b2 )

Then if x ∈ N 2μ (K+ )\Nμ (K+ ) and |I(x) − c| ≤ ε, we have 1 |I (x) − c| ≤ |I (x) − I(x)| + |I(x) − c| ≤ (ε + ε) < ε, 2 hence DI (x) ≥ α and 1

D I(x) ≥ DI (x) − D I(x) − DI (x) ≥ α . 2 In N μ (K+ ), ϕ(x) = 1 and I(x) = I (x) − (x, y), D I(x) = DI (x) − y. By Sard–Smale theorem [29] we choose as y a small regular value for DI , then all critical points of I in N μ (K+ ) are nondegenerate, hence finitely many. This complete the proof for m = γ (K) = 1. 2. γ (K) = m ≥ 2. There is an odd map h = (h1 , · · · , hm ) : K → S m−1 ⊂ R m . Extend h : H → R m . Denote 1 K± = {x|x ∈ K, ±hm (x) ≥ }, 2

1 K0 = {x|x ∈ K, |hm (x)| ≤ } . 2

K+ , K− are disjoint compact sets and K− = −K+ . Choose μ > 0 such that N3μ (K+ ) ∩ N3μ (K− ) = ∅. Since h21 + · · · + h2m = 1, ±hm ≥ 12 for x ∈ K+ ∪ K− , we can assume that h21 + · · · + h2m ≥ 34 and ±hm ≥ 14 for x ∈ N3μ (K+ ) ∩ N3μ (K− ). Choose an odd smooth function ϕ  be defined such that ϕ(x) = ±1 if x ∈ Nμ (K± ), ϕ(x) = 0 if x ∈ / N2μ (K+ ) ∪ N2μ (K− ). Let I, K as in (b1 ). Denote  ±hm (x) ≥ 1 }, ± = {x| x ∈ K, K 2

 |hm (x)| ≤ 1 } . 0 = {x| x ∈ K, K 2

We discuss case by case (a) Firstly assume hm (x) ≥ 12 . Three subcases occur.  (a1 ) If hm (x) ≥ 12 and x ∈ / N2μ (K+ ), then I= I and x ∈ / K. 1 (a2 ) If hm (x) ≥ 2 and x ∈ N2μ (K+ )\Nμ (K+ ), we still find a positive constant α > 0 such that 1

DI (x) ≥ α, if hm (x) ≥ , x ∈ N 2μ (K+ )\Nμ (K+ ) and |I (x) − c| ≤ ε. 2 Assume I and I are closed in the C 2 -distance and (b2 ) holds. If |I(x) − c| ≤ ε then 1 |I (x) − c| ≤ |I (x) − I(x)| + |I(x) − c| ≤ (ε + ε) < ε 2 and DI (x) ≥ α, hence 1

D I(x) ≥ DI (x) − D I(x) − DI (x) ≥ α. 2

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(a3 ) If hm (x) ≥ 12 and x ∈ Nμ (K+ ), then I(x) = I (x) − (x, y), D I(x) = DI (x) − y. Assume y is a regular value of DI , then all the critical points of Iin this domain are nondegenerate,  hm (x) ≥ 1 } = {P1 , · · · , Pk }, + = {x| x ∈ K, K 2 P1 , · · · , Pk are nondegenerate critical points of I. (b) Consequently, since I is even we have  hm (x) ≤ − 1 } = {−P1 , · · · , −Pk } . − = {x| x ∈ K, K 2 (c) Finally consider the domain {x| |hm (x)| ≤ 12 }. Two subcases occur. (c1 ) By the choice of μ, in N2μ (K+ ) ∪ N2μ (K− ), h21 + · · · + h2m ≥ 34 . Since |hm (x)| ≤ 12 we have h21 + · · · + h2m−1 ≥ 12 .  then x ∈ K, hence h2 + · · · + (c2 ) Out of N2μ (K+ ) ∪ N2μ (K− ), I(x) = I (x). If x ∈ K 1 3 2 2 2 hm = 1 and h1 + · · · + hm−1 ≥ 4 . 0 = {x| x ∈ K,  |hm (x)| ≤ 1 } → R m−1 \{0}, γ (K 0 ) ≤ Altogether we have (h1 , · · · , hm−1 ) : K 2        m − 1. Since K = K+ ∪ K− ∪ K0 = {±P1 , · · · , ±Pk } ∪ K0 , we have γ (K) ≤ max{1, γ (K0 )} ≤  ≤ m − 1. This complete the induction step. m − 1. In this way we reduce I into I with γ (K) Now we consider the case 0 ∈ K. If 0 is a nondegenerate (hence isolated) critical point of I , K\{0} is an isolated component of K. Argue for K\{0} instead of K, we are done. If 0 is degenerate, replace I by I: 1 I(x) = I (x) + εϕ(x) x 2 , 2 where ε is a small constant, ϕ is a smooth cut-off function such that ϕ(x) = 1, x ≤ μ; ϕ(x) = 0,

x ≥ 2μ. Then D I(0) = 0, D 2 I(0) = D 2 I (0) + εI d is invertible, 0 is a nondegenerate critical point of I. 2 The conclusion of Proposition B3 may be localized to closed sets. More precisely we have the following result. Corollary B4 . Let I be as in Proposition B3 . Let Q be a closed subset of H . Denote KQ = {x| DI (x) = 0, |I (x) − c| ≤ ε, x ∈ Q} . Then for given d, δ > 0, 0 <  ε < ε, there exists a symmetric open neighborhood N of KQ and 2  an even C -functional I on H satisfying (1) N is contained in the d-neighborhood of KQ . (2) I − I C 2 ≤ δ. I(x) = I (x), if x ∈ / N . In the neighborhood N , D I is proper (hence I 2  satisfies the (PS) condition). D I is a Fredholm operator.

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(3) Denote Q = {x| D I(x) = 0, |I(x) − c| ≤ ε, x ∈ Q} . K Q ⊂ N and K Q consists of finitely many pairs of nondegenerate of I. Then K Proof of Theorem B1 . Denote c = cj . By Theorem 4.1, c is a critical value of I and Kc∗ = ∅. To prove that there exists x ∈ Kc∗ with m∗ (x) ≥ j , we use an indirect argument. (1) Firstly assume that Kc∗ = {±x0 }, ±x0 are nondegenerate critical points of I and m(x0 ) = m(−x0 ) = m < j , where m(·) is the Morse index. We follow [2]. In a coordinate chart ϕ around x0 described by (Y, Z) the functional I reads as 1 I (x) = c + ( Y 2 − Z 2 ) 2 1

where ϕ(x) = (Y, Z), ϕ(x0 ) = (0, 0), c = I (x0 ) and ( Y 2 + Z 2 ) 2 is the norm of H , dimZ = m < j . Choose δ small. Let N1+ = {x|ϕ(x) = (Y, Z), Z ≤ δ, Y ≤ 2δ} δ N + = {x|ϕ(x) = (Y, Z), Z < , Y ≤ 2δ} 2 δ N0+ = {x| ϕ(x) = (Y, Z), Z < , Y < δ} 2 δ C + = {X| ϕ(X) = (0, Z), Z < } 2 N1− = −N1+ , N − = −N + , N0− = −N0+ and C − = −C + . By the deformation property for small ε > 0 there exists a map η : H → H such that η ∈  and η(I c+ε \(N + ∪ N − ∪ W )) ⊂ I c−ε . By the definition of c, there exists A ∈ j and A\W ⊂ I c+ε . We define a map ψ ∈  as follows. Out of N1+ ∪ N1− , ψ(x) = x. In N + (and N − ) by using ψ , Z) is defined by the coordinate ϕ, (Y, Z) − → (Y

⎧ Z=Z ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (2 Y − 2δ) YY

⎪ ⎪ ⎨ = Y ⎪ ⎪Y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (4 Z − 2δ) YY

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ (2 Y − 2δ) Y

Y

if Z ≤ 2δ , Y ≤ δ if Z ≤ 2δ , δ ≤ Y ≤ 2δ if 2δ ≤ Z ≤ δ, Y ≤ 4 Z − 2δ

(b3 )

if 2δ ≤ Z ≤ δ, 4 Z − 2δ ≤ Y ≤ 2 Z

if 2δ ≤ Z ≤ δ, 2 Z ≤ Y ≤ 2δ .

Choose ε < 38 δ 2 . Then I (x) ≥ c + 38 δ 2 > c + ε for x ∈ N ± \N0± hence A ∩ N ± ⊂ N0± . Notice that

ψ(N0± ) = C ± . Replacing A by ψ(A)(ψ(A) ∈ j ), if necessary, we can assume A ∩ N ± ⊆ C ± . We have

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η(A\(N + ∪ N − ))\W ⊂ (η(A\(N + ∪ N − ∪ W )) ∪ η(W ))\W ⊂ η(A\(N + ∪ N − ∪ W )) ⊂ I c−ε , hence η(A\(N + ∪ N − )) ∈ / j and A\(N + ∪ N − ) ∈ / j . There exist σ ∈  and an odd map h such that γ ((A\(N + ∪ N − )) ∩ σ −1 ()) ≤ j − 1, h : A\(N + ∪ N − ) ∩ σ −1 () → S j −1 . Consider the retraction h on (A ∩ ∂N ± ) ∩ σ −1 (). Since (A ∩ ∂N ± ) ∩ σ −1 () ⊂ ∂C ± ∼ S m−1 , m < j , we can extend h to ∂C ± , then to C ± , and obtain an odd map h : A ∩ σ −1 () → S j −1 , which contradicts the fact that A ∈ j and γ (A ∩ σ −1 ()) ≥ j . (2) Secondly assume Kc∗ = {±x1 , · · · , ±xk }, consists of nondegenerate critical points with the Morse index less than j . Since the deformation in the previous case of one pair of nondegenerate critical points is localized in nature, namely inside a suitable neighborhood of the critical points, the argument can be easily generalized to the case of multiple pairs of nondegenerate critical points. (3) Finally, we consider the general case for Kc∗ . Assume that every point of Kc∗ has the augmented Morse index less than j . By the (PS) condition and Proposition B3 , there exist  ε > 0, an open neighborhood N of Kc∗ and a C 2 -functional I such that (1) I = I out of N , D I is proper in N , D 2 I is Fredholm in N . I is arbitrarily closed to I in C 2 -distance. (2) Denote ∗ = {x| x ∈ H \W, D I(x) = 0, |I(x) − c| ≤ ε} . K ∗ ⊂ N , and K ∗ consists of finitely many pairs of nondegenerate critical points of I. Then K We verify that I satisfies the assumptions (I1 ), (AI ), (AP ) and (m). Let N  be a symmetric open neighborhood of Kc∗ such that N ⊂ N  ⊂ N  ⊂ H \W . Let ϕ be a smooth cut-off function such that ϕ(x) = 0 out of N  and ϕ(x) = 1 in N . Define  = (1 − ϕ(x))A(x) + ϕ(x)(x − D I(x)) . A(x)  = A in a neighborhood of W . (AP ) and (m) hold obviously, since I= I , A  Let {xn } be a (PS) sequence of I . If {xn } ⊂ H \N , then I = I . Since I satisfies the P.S. condition, {xn } possesses a convergent subsequence. D I is proper in N . If {xn } ⊂ N , {xn } again possesses a convergent subsequence. We verify the condition (AI ). For two given positive constants c0 , b0 , suppose |I(x)| ≤ c0 ,

D I(x) ≥ b0 . For x ∈ H \N , I(x) = I (x), D I(x) = DI (x), we have DI (x), x − Ax ≥ b x − Ax . And

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 = DI (x), (1 − ϕ(x))(x − Ax) + ϕ(x)DI (x) D I(x), x − Ax ≥ (1 − ϕ(x))b x − Ax + ϕ(x)b0 DI (x)

 ≥ b (1 − ϕ(x))(x − Ax) + ϕ(x)DI (x) =  b x − Ax

 = D I(x), where  b = min(b, b0 ). For x ∈ N , x − Ax  = D I(x) 2 ≥   . D I(x), x − Ax b x − Ax

By Lemma B2 the deformation property holds for the functional I. Define  c = inf

sup I(x) .

A∈j x∈A\W

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