Also we give a new characterization of the ground states, which provides a more convenient way for finding or checking ground states

Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 127, pp. 1–18.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

EXISTENCE, CHARACTERIZATION AND NUMBER OF GROUND STATES FOR COUPLED EQUATIONS

QIHAN HE, SHUANGJIE PENG

Abstract. This article concerns the existence, characterization and number of ground states for the system consisting ofmcoupled semilinear equations

−∆ui+λui=

m

X

j=1

kij

qij

p+ 1|uj|^pij|ui|^qij−2ui, x∈Ω, ui∈H₀¹(Ω), i= 1,2, . . . , m.

We extend the characterization results obtained by Correia [5, 6] to the above problem. Also we give a new characterization of the ground states, which provides a more convenient way for finding or checking ground states. This study may be the first result not only positive ground states but also for semi- trivial ground states, and it shows that the positive ground state is unique for some special cases.

1. Introduction and statement of main results

In this article, we study the existence, characterization and number of ground states of the system consisting ofmsemilinear equations

−∆ui+λu_i=

m

X

j=1

k_ij q_ij

p+ 1|uj|^pij|ui|^qij−2u_i, x∈Ω, u_i∈H₀¹(Ω), i= 1,2, . . . , m,

(1.1)

where Ω ⊂ R^N may be bounded or R^N and {pij},{qij},{kij} ⊂ R satisfy the following assumptions:

(1) 1 < qij =pji,pij+qij = 2p+ 2, where 0< p < 2/(N −2) ifN ≥3 and 0< p <+∞ifN = 1,2;

(2) kij = kji and for any fixed i ∈ {1,2, . . . , m}, there exists at least one j ∈ {1,2, . . . , m} such that kij >0, which implies that the problem (1.1) may contain attraction and repulsion.

2010Mathematics Subject Classification. 35B99, 35J47, 35J60.

Key words and phrases. Coupled equations; ground states.

c

2019 Texas State University.

Submitted July 18, 2018. Published November 26, 2019.

1

Problem (1.1) arises when one looks for standing waves Ψ(t, x) = e^−iλtU with U = (u₁, u₂, . . . , u_m)∈ H₀¹(Ω)^m

to the equations i(vi)t+ ∆vi+

m

X

j=1

kij

qij

p+ 1|vj|^pij|vi|^qij−2vi = 0, (t, x)∈R⁺×Ω, i= 1,2, . . . , m,

(1.2)

whereidenotes the imaginary unit. This coupled system of nonlinear Schr¨odinger equations with power-type nonlinearities comes from physical problems, such as nonlinear optics and Bose-Einstein condensates. It models a physical system in which the field has more than one component. According to the results in [1], one can see thatu_jdenotes thejth component of the beam in Kerr-like photo-refractive media and the coupling constantk_ij acts to the interaction between theith and the jth components of the beam. System (1.2) also stems from the Hartree-Fock theory for a m-component Bose-Einstein condensate. Readers can learn more about the derivation and applications of this system in[7, 18].

Because of both physical and mathematical reasons, the ground states are the most important solutions. At the same time, the existence, uniqueness and multi- plicity of solutions are important characteristic. Therefore, we pay attention to the existence, uniqueness or multiplicity of the ground states. Researchers studied the existence, non-existence, and uniqueness of ground states to the scalar equation in [2, 9, 14, 16, 17] and the references therein. Results about ground states for 2 and 3 coupled systems can be found in [3, 4, 10, 13, 15]. Recently, Correia [5, 6] studied the system ofmcoupled equations

−∆ui+λui =

m

X

j=1

kij|uj|^p+1|ui|^p−1ui, x∈Ω, i= 1,2, . . . , m, (1.3) and not only presented sufficient conditions for the existence of nontrivial ground states, but also gave a characterization of the ground states. We want to point out that the characterization given by Correia depends heavily on the maximum point of a function constraint on the unit spherical surface, which makes difficult to find or check a ground state. As far as we know, there are no results on the uniqueness or multiplicity of the ground states of (1.1) and (1.3). So we want to study the existence, the form, and the number of the ground states of (1.1). Also we give sufficient conditions for the existence of nontrivial ground states, a new characterization which is easy to find or to check a ground state, and an estimate on the number of the ground states.

Before stating our results, we introduce some definitions and notation. We say a solution U of (1.1) is called positive if Ui >0 for any i ∈ {1,2, . . . , m} and a solutionU of (1.1) is nontrivial ifU ∈H := H₀¹(Ω)^m

\ {~0}. We denote byAmthe set consisting of all nontrivial solutions and callU ∈Ama ground state of (1.1) if

Sm(U) :=1

2Im(U)− 1

2p+ 2Jm(U)≤Sm(V), ∀ V ∈Am, where

I_m(v) :=

m

X

i=1

Z

Ω

(|Dv_i|²+λ|v_i|²),

J_m(v) :=

m

X

i,j=1

k_ij Z

Ω

|vj|^pij|vi|^qij. We denote byGm the set of ground states of (1.1). Let

G⁺_m:=

U ∈Gm:ui>0, i= 1,2, . . . , m , and forn∈ {1, . . . , m},

T_n:=

U ∈G_m:U has exactlynnontrivial components and

T_n⁺:=

U ∈T_n :U has exactlynpositive components . Forγ >0, we define

I_m^γ := inf

J_m(u)=γIm(u), S^γ_m:= inf

J_m(u)≥γIm(u), γG:= inf

J_m(u)=1Im(u)^p+1_p . It is easy to check thatI_m¹ =γ^−p+11 I_m^γ andI_m¹ =γ

p p+1

G . LetGbe the set of the ground states to the equation

−∆u+λu=u^2p+1, u >0, x∈Ω,

u= 0, x∈∂Ω, (1.4)

Let |G| be the number of the elements in G, and λ₁(Ω) be the first eigenvalue of −∆ in H₀¹(Ω), if Ω is bounded and 0 if Ω = R^N. According to the results of [2, 9, 11, 12, 17], we can see that if λ >−λ₁(Ω), then G6=∅. In particular, when Ω =R^N andλ >0,|G|= 1. Moreover, ones can check that ifw∈G, then

G=

u:I₁(u) = min

J(Q)=J(w)

I₁(Q), J(u) =J(w) , whereJ(v) =R

Ω|v|^2p+2.

The above facts are very important for our results since the existence of the ground states of (1.1) depends heavily on the existence of the ground states of the scalar equation (1.4). If (1.4) has no ground states, so does (1.1), which will be presented in Theorems 1.2 and 1.5. Our result about the existence of the ground states of (1.1) can be stated as follows.

Theorem 1.1. Assume that there exists u∈H such that

m

X

i,j=1

k_ij Z

Ω

|u_j|^pij|u_i|^qij >0.

If λ >−λ₁(Ω), thenG_m6=∅.

Set (R⁺0)^m:=

(x₁, . . . , x_m)∈R^m:x_i≥0, ∀i= 1, . . . , m , define f(x) :=

m

X

i,j=1

kij|xj|^pij|xi|^qij, x∈(R⁺0)^m (1.5) and letX⊂(R⁺0)^m be the set of solutions of the following maximization problem

fmax:= max

|x|=1f(x) (1.6)

andCbe the set of complex numbers.

Here are our results about the characterization of the ground states of (1.1).

Theorem 1.2. (1) If u ∈ Gm, then there exist ai ∈ C, i = 1,2, . . . , m, and w∈G such that u= (a1w, a2w, . . . , amw)and f

1 2p

max |a1|,|a2|, . . . ,|am|

∈ X;

(2) For any w∈G, ifb_i∈C, i= 1,2, . . . , m, satisfyf

1

max2p |b₁|,|b₂|, . . . ,|b_m|

∈ X, we have(b₁w, b₂w, . . . , b_mw)∈G_m;

(3) If w∈Gand c1w, c2w, . . . , cmw

is a ground state of (1.1), then

m

X

i=1

c²_i =f⁻

1

maxp, f(c) =f⁻

1

maxp, fmax= I1(w) γ_G

^p

and for any fixedi∈ {1,2, . . . , m},

m

X

j=1

k_ij q_ij

p+ 1|c_j|^pij|c_i|^qij−2=

(1, if |ci|>0, 0, if |ci|= 0,

where c = (c1, . . . , cm), f and fmax have been defined in (1.5) and (1.6) respectively, and we need a special definition: 0^q = 0, for any fixedq∈R. In particular, G⁺_m 6= ∅ if and only if there is an x ∈ X such that xi 6= 0, i = 1,2, . . . , m,andG_m=T_mif and only if all the elements ofX have no zero compo- nents.

From the above results, we can obtain the following corollary easily.

Corollary 1.3. λ >−λ1(Ω)is a necessary condition to the existence of the ground states of (1.1).

Using Theorem 1.2, we can show the following proposition.

Proposition 1.4. Suppose thatΩ = R^N and there exists a partition {Yk}_1≤k≤K of {1,2, . . . , m} such that for any giveni, j withi6=j,

k_ij ≥0 if and only if there existsk such thati, j∈Y_k.

Furthermore ifu= (u1, u2, . . . , um)∈Gm, then there existsk0∈ {1,2, . . . , K}such that ul6= 0for some l∈Yk0 and us= 0for any s6∈Yk0.

In Theorem 1.2,X is the set consisting of the solutions of a maximization prob- lem constraint on the unit spherical surface, which causes a big difficulty to find a point x∈X or check whether a point xbelongs to X. So we want to find a new characterization of the ground states of (1.1), which can give a more convenient way to find or check a ground state of (1.1). To get this goal, we consider the maximization problem

fˆmax:= max

|x|6=0

f(x)

|x|^2p+2 (1.7)

and letX⁰be the set of maximizers in (R⁺0)^m. By the homogeneity off(x)|x|^−2p−2, we can check that ifx∈X andt >0, thentx∈X⁰, which implies thatX⁰6=∅.

A new characterization of the ground states of (1.1) can be summarized as follows.

Theorem 1.5. (1) If u ∈ G_m, then there exist a_i ∈ C, i = 1,2, . . . , m, and w∈Gsuch that

u= a

|a|

fˆ_max^−1/(2p)w and (|a1|,|a2|, . . . ,|am|)∈X⁰,

wherea= (a1, . . . , am). In particular, ifu∈T_n⁺, then there exist x0∈X⁰ andw∈Gsuch that u=_|x^x0

0|fˆmax^−1/(2p)w.

(2) _|b|^b fˆmax^−1/(2p)w∈G_mfor any w∈Gand any b:= (b₁, b₂, . . . , b_m)with (|b1|,|b2|, . . . ,|bm|)∈X⁰.

Next we discuss the number of the ground states of (1.1).

Theorem 1.6. Assume that u¹, u²∈Gmwith u¹_i, u²_i ≥0,i∈ {1,2, . . . , m}. Then for any fixedi∈ {1,2, . . . , m}, one of the following two conclusions hold:

(1) there exists a positive constant c0 such that u¹_i =c0w1, u²_i = c0w2, where w1, w2∈G;

(2) u¹_i ≡0 oru²_i ≡0.

The following corollary is a direct consequence of Theorems 1.2 and 1.6.

Corollary 1.7. T_n⁺ has at most |G|C_mⁿ elements, whereC_mⁿ is the combinatorial number. In particular, ifG⁺_m6=∅, then|G⁺_m|=|T_m⁺|=|G|.

Before we end this section, we outline the main ideas and the approaches in the proofs of our main results. We will introduce a constraint minimization problem and show that if the constraint minimization problem can be obtained, then G_m exactly consists of all the reached function of the constraint minimization problem.

So to prove Theorem 1.1, we show that the constraint minimization problem can be obtained by the concentration-compactness lemma.

For Theorems 1.2 and 1.5, we firstly prove that ifu∈Gm, then X^m

i=1

|u_i|^21/2

=f_max^−1/(2p)w for somew∈G. Secondly, we show that

(|u1|,|u2|, . . . ,|um|) =x

m

X

i=1

u²_i1/2

for somex∈X. Finally, using complex analysis and the integration, we conclude that

u_i=x_ie^iθiX^m

i=1

u²_i^1/2

=x_ie^iθif_max^−1/(2p)w.

The proof of Theorem 1.6 is inspired by [3, 8]. But we encounter three main difficulties. Firstly, we can not consider a perturbation problem of (1.1) as [3, 8]

since k_ii may be zero; Secondly, when Ω is bounded, we have no results on the uniqueness of the ground states of (1.4). The last difficulty is that the extreme points, corresponding to semi-trivial ground state, cannot be interior points. Thus we can not determine that the first derivative at the extreme point is zero and the second derivative is not zero, which play a key role in using the Implicit Function Theorem. Therefore, we have to make some changes. Under our careful obser- vation, we find that the purpose of studying a perturbation problem is to obtain a perturbation least energy and get an equivalence by using the derivative of the perturbation least energy. So we introduce a new perturbation system which is different from that in [3, 8].

This article is organized as follows: We will show the existence of the ground states of (1.1) in part 2. The proofs of the characterization of the ground states

of (1.1) would be put into part 3. The last part contributes to the proof of the number of the ground states of (1.1).

2. Existence of ground states Lemma 2.1. Let

E₁:={U ∈H :I_m^γG is achieved byU}, E2:={U ∈H :S_m^γG is achieved byU}.

ThenE₁=E₂ andI_m^γG=S_m^γG.

A similar proof can be found in [5]. But for the readers’ convenience and the completeness, we would give a detailed proofs.

Proof. Step 1: We prove thatE1 ⊂E2. For any u⁰∈E1, we assume that there exists Q withJm(Q)≥γG, Im(Q)< Im(u⁰). Then we can choose some constant 0 < C ≤ 1 such that Jm(CQ) = γG. From the minimality of u⁰, Im(u0) ≤ Im(CQ)≤Im(Q), which is impossible. So for any Q∈ H with Jm(Q)≥γG, we haveIm(Q)≥Im(u⁰), which, combiningJm(u⁰) =γG, implies thatu⁰∈E2. Thus, E₁⊂E₂.

Step 2: We show that E2 ⊂ E1. For any v⁰ ∈ E2, we have Im(v⁰) = S_m^γG and Jm(v⁰) ≥ γG. If Jm(v⁰) > γG, then there is a constant 0 < c < 1 such that Jm(cv⁰) = γG and Im(cv⁰) = c²Im(v⁰) < Im(v⁰), which contradicts to the minimality ofv⁰. SoJ_m(v⁰) =γ_G. Hence

I_m(v⁰)≥ min

Jm(v)=γG

I_m(v) =I_m^γG. (2.1) By the minimality ofv⁰, we have

Im(v⁰) = min

J_m(v)≥γ_GIm(v)≤ min

J_m(v)=γ_GIm(v) =I_m^γG. (2.2) It follows from (2.1) and (2.2) that

Im(v⁰) =I_m^γG and Jm(v⁰) =γG. Thereforev⁰∈E1, and henceE2⊂E1.

SoE1=E2, which also implies thatI_m^γG =S^γ_m^G. The next Lemma will give a relation ofG_mandE₁.

Lemma 2.2. If E16=∅, thenGm=E1, whereE1 is defined in Lemma 2.1.

Proof. Step 1: We prove thatE1⊂Gm. Ifu⁰∈E1, then we can find someµ∈R such that for anyh:= (h1, h2, . . . , hm)∈H and anyi∈ {1,2, . . . , m}, we have

Z

Ω

(Du⁰_iDh_i+λu⁰_ih_i) =µ(p+ 1)

m

X

j=1

k_ij qij

p+ 1 Z

Ω

|u⁰_j|^pij|u⁰_i|^qij−2u⁰_ih_i. (2.3)

Takingh=u⁰, we obtain γG=γ

1 p+1

G I_m¹ =I_m^γG =Im(u⁰)

=µ(p+ 1)

m

X

i=1 m

X

j=1

kij

qij

p+ 1 Z

Ω

|u⁰_j|^pij|u⁰_i|^qij

=µ(p+ 1)

m

X

i,j=1

kij

Z

Ω

|u⁰_j|^pij|u⁰_i|^qij

=µ(p+ 1)Jm(u⁰)

=µ(p+ 1)γG,

(2.4)

which implies

µ(p+ 1) = 1. (2.5)

From (2.3)–(2.5), we see that

Im(u⁰) =γG and u⁰∈Am. (2.6) Letv⁰ be a solution of (1.1). Then Im(v⁰) = Jm(v⁰)>0. From the definition of I_m^γ, we have

γ

p p+1

G =I_m¹ = I_m^γ γ^p+11

= Im^Jm(v0)

J

1

mp+1(v⁰)

≤ Im(v⁰) J

1

mp+1(v⁰)

=I

p

mp+1(v⁰), which implies thatIm(v⁰)≥γG. Therefore,u⁰∈GmandE1⊂Gm.

Step 2: We proof thatGm⊂E1. LetV¹∈Gmand V²∈E1. Then as above we haveI_m(V¹) =J_m(V¹)≥γ_G andV²∈A_m, I_m(V²) =J_m(V²) =γ_G. Thus

S_m(V¹) = p

2p+ 2I_m(V¹)≥ p

2p+ 2γ_G= p

2p+ 2I_m(V²) =S_m(V²).

SinceV¹ is a ground state, we haveI_m(V¹) =J_m(V¹) =I_m(V²) =J_m(V²) =γ_G, which means thatV¹∈E1. SoGm⊂E1.

ThereforeE1=Gm.

Proof of Theorem 1.1. According to Lemma 2.2, it suffices to prove thatE1 6=∅.

Let{Un}be a minimizing sequence ofI_m^γG. We divide the proof into two cases:

Case I:Ω is bounded. It is easy to check that{Un}is bounded inH. So using the compactness ofH₀¹(Ω),→L^q(Ω) (1≤q <2^∗), we can have, up to a subsequence, fori∈ {1,2, . . . , m},

U_ni* U_i weakly inH₀¹(Ω), Uni→Ui strongly inL^q(Ω), which implies that

Jm(U) = lim

n→+∞Jm(Un) =γG, Im(U)≤ lim

n→+∞Im(Un) =I_m^γG. (2.7) SinceJ_m(U) =γ_G,

I_m^γG= min

J_m(v)=γ_GIm(v)≤Im(U) (2.8) It follows from (2.7) and (2.8) thatU ∈E1. SoE16=∅.

Case II: Ω =R^N. For any fixed smallε > 0, we let δ(ε) :=C₀ε, where C₀ is a very large positive constant. By the concentration-compactness principle [11, 12],

up to a subsequence, it is possible to associate to each Uni,1 ≤ i ≤M, a set of functions{U_ni^l , Wni}1≤l≤L_i⊂H (a set of bubbles plus a remainder), such that

(1) each U_ni^l has support in a ball of radius R and the distance between the supports ofU_ni^l andU_ni^j,(j6=l), goes to +∞as n→+∞;

(2) one has

Z

R^N

|Uni|^2p+2−

Li

X

l=1

|U_ni^l |^2p+2

< δ(ε), (2.9)

kDUnik²₂≥

Li

X

l=1

kDU_ni^l k²₂−δ(ε), kUnik²₂≥

Li

X

l=1

kU_ni^l k²₂−δ(ε). (2.10) Essentially, one applies successively the concentration-compactness principle toUni

to obtain the various bubbles. This process will end in finite steps since the total L² norm is finite and one always picks up the bubble whose L² norm is larger than a positive constant uniformly, which implies that, afterLi steps, the remain- der Wni has L^2p+2 norm smaller than ε. So it is easy to see that the bubbles {U_ni^l }_1≤l≤Li satisfy the above condition (1) and inequality (2.10). We give a proof of the inequality (2.9) as below. Since the supports ofU_ni^l andU_ni^j (j6=l) have no intersection,

Z

R^N

|Uni|^2p+2−

L_i

X

l=1

|U_ni^l |^2p+2

= Z

R^N

|

L_i

X

l=1

U_ni^l +Wni|^2p+2− |

L_i

X

l=1

U_ni^l |^2p+2

≤C Z

R^N

|

Li

X

l=1

U_ni^l |^2p+1|Wni|+|Wni|^2p+2

≤CZ

R^N

|Wni|^2p+2_2p+2¹ +C

Z

R^N

|Wni|^2p+2

< Cε < δ(ε).

Setting L = max_1≤i≤M{Li}, we define, for each i, U_ni^l = 0 if Li < l≤L. Up to a subsequence, it is possible to group the bubbles into several clusters in such a way that:

(3) each cluster has one and only one bubble fromUni(1≤i≤M);

(4) the supports of two bubbles U_ni^l , U_nj^s (1 ≤ i 6= j ≤ M,1 ≤ l, s ≤ L) have a nonempty intersection if and only if U_ni^l and U_nj^s belong to the same cluster.

Obviously, we shall end up with L clusters. Define U_n^l as the vector of bubbles from the clusterl. Then, by the definition ofU_n^l and the fact that

|a|^p− |b|^p

q ≤ C

|a|^pq− |b|^pq

for anyq >1 and some positive constantC, dependent ofq, we have U_n^l ·U_n^j = 0, x∈R^N for anyl6=j

and

Jm(Un)−

L

X

l=1

Jm(U_n^l)

=

m

X

i,j=1

kij

Z

R^N

|Unj|^pij|Uni|^qij−

L

X

l=1 m

X

i,j=1

kij

Z

R^N

|U_nj^l |^pij|U_ni^l |^qij

≤

m

X

i,j=1

|kij| Z

R^N

|Unj|^pij|Uni|^qij− Z

R^N

(

L

X

l=1

|U_nj^l |)^pij(

L

X

l=1

|U_ni^l |)^qij

≤C Z

R^N

|Unj|^pijh

|Uni|^qij−(

L

X

l=1

|U_ni^l |)^qiji

+C Z

R^N

h|Unj|^pij −(

L

X

l=1

|U_nj^l |)^pijiX^L

l=1

|U_ni^l |^qij

≤CZ

R^N

|Uni|^qij −X^L

l=1

|U_ni^l |^qij

2p+2 qij _2p+2^qij

+CZ

R^N

U_nj|^pij −(

L

X

l=1

|U_nj^l |)^pij

2p+2 pij _2p+2^pij

≤CZ

R^N

|Uni|^2p+2−X^L

l=1

|U_ni^l |2p+2

_2p+2^qij

+CZ

R^N

Unj|^2p+2−X^L

l=1

|U_nj^l |^2p+2

_2p+2^pij

=CZ

R^N

Uni|^2p+2−

L

X

l=1

|U_ni^l |^2p+2

_2p+2^qij

+CZ

R^N

U_nj|^2p+2−

L

X

l=1

|U_nj^l |^2p+2

_2p+2^pij

≤Cδ(ε)^2p+2qij +Cδ(ε)^2p+2pij . It follows form (2.10) that

m

X

i=1

kUnik²₂≥

m

X

i=1 L

X

l=1

kU_ni^l k²₂−M δ(ε),

m

X

i=1

kDU_nik²₂≥

m

X

i=1 L

X

l=1

kDU_ni^l k²₂−M δ(ε).

Up to a subsequence, we can define γ_l := lim_n→+∞J_m(U_n^l), 1 ≤ l ≤ L. Using a diagonalization process, we obtain, for each n, a decomposition of {U_n} in L_n bubbles (whereLn→Lˆ ∈N∪ {∞}) such that

m

X

i=1

kUnik²₂≥

m

X

i=1 L_n

X

l=1

kU_ni^l k²₂−M δ(1

n), (2.11)

m

X

i=1

kDUnik²₂≥

m

X

i=1 L_n

X

l=1

kDU_ni^l k²₂−M δ(1

n), (2.12)

|J_m(U_n)−

L_n

X

l=1

J_m(U_n^l)| ≤δ^σ(1

n), (2.13)

γG=

Lˆ

X

l=1

γl, (2.14)

whereσ:= min{_2p+2^qij ,_2p+2^pij }.

Case 1: Ifγ_l≥0 for anyl, one has J_m

( γ_l

Jm(U_n^l))^2p+21 U_n^l

=γ_l, which, combining (2.11) and (2.12), implies that

I_m^γG = lim

n→∞Im(Un)

≥lim sup

n→∞

L_n

X

l=1

Jm(U_n^l) γ_l

_p+1¹ Im

( γl

J_m(U_n^l))^2p+21 U_n^l

≥lim sup

n→∞

Ln

X

l=1

I_m^γl=

Lˆ

X

l=1

I_m^γl.

(2.15)

However, the function

γ→I_m^γ =γ^p+11 I_m¹

is strictly concave inR⁺, which implies that there existsl₀such that, for anyl6=l₀, γ_l= 0. By (2.14), we see thatγ_l0=γ_G. Therefore, defining

V_n:= γG

J_m(Un^l0) _2p+2¹

U_n^l0,

it follows from (2.15) that lim inf_n→+∞I_m(V_n) =I_m^γG,J_m(V_n) =γ_G and so{V_n} is a minimizing sequence forI_m^γG, for which the compactness alternative from the concentration-compactness principle is verified (recall that Vn is, up to a multi- plicative factor, the vector of a group of bubbles ofUn). Since{Vn}is bounded in H, there existsW ∈H such thatVn * W weakly in H andVn →W strongly in (L²(R^N)∩L^2p+2(R^N))^m. In particular, it holds that

I_m(W)≤limI_m(V_n) =I_m^γG, J_m(W) = limJ_m(V_n) =γ_G. ThereforeW is a minimizer ofI_m^γG.

Case 2: Now suppose that

L⁻_n :=

l:γl<0 6=∅.

LetL⁺_n be the complementary set ofL⁻_n, set L⁺:= lim_n→+∞L⁺_n and ηl:=

PL^ˆ j=1γj

P

i∈L⁺nγ_iγl. Notice that (2.14) impliesL⁺6=∅. Furthermore,

γG= X

l∈L⁺

ηl. (2.16)

Since

J_m ( η_l

Jm(U_n^l))^2p+21 U_n^l

=η_l>0, l∈L⁺_n,

using (2.11) and (2.12), one has I_m^γG= lim

n→+∞Im(Un)

≥lim sup

n→+∞

X

l∈L⁺n

J_m(U_n^l) ηl

_p+1¹ Im

η_l Jm(U_n^l)

_2p+2¹ U_n^l

≥lim sup

n→+∞

X

l∈L⁺n

I_m^ηl

= X

l∈L⁺

I_m^ηl.

(2.17)

Similarly, since

γ→I_m^γ =γ^p+11 I_m¹

is strictly concave inR⁺, which, combining (2.16), implies that there existsl₀such thatγ_l0 =γ_G. Proceeding as the previous case, we can complete the proof.

3. Form of the ground states

Proof of Theorem 1.2. Step 1: We show that if u∈G_m, then (Pm

i=1|ui|²)^1/2 = fmax^−1/(2p)wfor somew∈G. Letu∈Gm,|u|:= (|u1|,|u2|, . . . ,|um|) andku(x)k²

R¹ :=

Pm

i=1|ui(x)|². Since J_m(u) =J_m(|u|) andI_m(u)≥ I_m(|u|), we have, by Lemma 2.2, that|u| ∈G_m. Fixingx₀∈X, we can conclude that

Jm(|u|) =Jm(u)

= Z

Ω

f(u) = Z

Ω

f u

ku(x)k_R¹

ku(x)k^2p+2

R¹

≤ Z

Ω

f(x0)ku(x)k^2p+2

R¹

= Z

Ω

f x₀ku(x)k_R¹

=Jm x0ku(x)k_R1

(3.1)

and

I_m x₀ku(x)k_R1

= Z

Ω

λku(x)k²_R1|x₀|²+

Dku(x)k_R1

2|x₀|²

= Z

Ω

X^m

i=1

λ|ui|²+

Pm i=1|ui|

D|ui| (Pm

i=1|ui|²)¹²

2

≤

m

X

i=1

Z

Ω

λ|u_i|²+ D|u_i|

2

=Im(|u|),

(3.2)

where we have used the Cauchy-Schwarz inequality. Let 0< c ≤1 be such that J_m cx₀ku(x)k_R1

=J_m(|u|). Then by the minimality of |u|and (3.2), we can see that

Im(|u|)≤Im cx0ku(x)k_R1

=c²Im x0ku(x)k_R1

≤Im x0ku(x)k_R1

≤Im(|u|), which implies that c = 1. So J_m x₀ku(x)k_R¹

= J_m(|u|) and I_m x₀ku(x)k_R¹

= I_m(|u|). Therefore,x₀ku(x)k_R¹ is also a ground state of (1.1).

Sincex0ku(x)k_R1 is a ground state of (1.1), it is easy to check that

−∆ku(x)k_R¹+λku(x)k_R¹ =f_maxku(x)k^2p+1

R¹ . (3.3)

Letc0:=f

1 2p

max. Thenc0ku(x)k_R1 is a solution of (1.4). By the maximum principle and the fact thatJm(|u|)>0, we obtainku(x)k_R1 >0 in Ω. In fact,x0ku(x)k_R1 is a ground state of (1.1) implies that c0ku(x)k_R¹ is a ground state of (1.4). Hence ku(x)k_R1 =c⁻¹₀ (c0ku(x)k_R1) =:c⁻¹₀ w, wherewis a ground state of (1.4).

Step 2: We show that (|u1|,|u2|, . . . ,|um|) =x(Pm

i=1u²_i)^1/2for somex∈X. Since

|u|

ku(x)k

R1

= 1, by the definition ofX, we havef(x0)≥f _ku(x)k^|u|

R1

. It follows from J_m(|u|) =J_m x₀ku(x)k_R¹

that Z

Ω

f |u|

ku(x)k_R¹

ku(x)k^2p+2

R¹ = Z

Ω

f(x0)ku(x)k^2p+2

R¹ . Combiningku(x)k_R¹ >0 in Ω andf(x₀)≥f _ku(x)k^|u|

R1

, implies thatf _ku(x)k^|u|

R1

= f(x0) a.e x ∈ Ω. That is, _ku(x)k^|u|

R1 = X(x) a.e x ∈ Ω, where X(x) satisfies f(X(x)) =f(x0) i.eX(x)∈X. So|u|=X(x)ku(x)k_R1 a.e x∈Ω.

Since |u| = X(x)ku(x)k_R1 is a solution of (1.1), inserting this expression into (1.1) and using (3.3), we can have

2Dku(x)k_R1DX_i(x) +ku(x)k_R1∆X_i(x) = 0, i= 1,2, . . . , m. (3.4) Using integration by parts and (3.4), we obtain

− Z

Ω

ku(x)k²_R1X_i(x)∆X_i(x)

= Z

Ω

DXi(x)D ku(x)k²_R1Xi(x)

= Z

Ω

DXi(x)

ku(x)k²_R1DXi(x) + 2Xi(x)ku(x)k_R1Dku(x)k_R1

= Z

Ω

|DXi(x)|²ku(x)k²_R1+ 2Xi(x)DXi(x)ku(x)k_R1Dku(x)k_R1

= Z

Ω

|DXi(x)|²ku(x)k²_R1− Z

Ω

ku(x)k²_R1Xi(x)∆Xi(x),

(3.5)

which implies that Z

Ω

|DXi(x)|²ku(x)k²_R1 = 0, i= 1,2, . . . , m.

So we know thatXi(x) is a constant. Therefore, there exists an ˆX ∈X such that

|u|= ˆXku(x)k_R1.

Step 3: We show that ui = xie^iθi Pm

i=1u²_i1/2

. Since ku(x)k_R1 > 0 in Ω, one may assume that ui(x) = |ui|e^iθi(x) = Xiku(x)k_R1e^iθi(x). Then it follows from Im(u) =Im(|u|) that

Z

Ω

Dku(x)k_R1

2+λku(x)k²_R1

=

m

X

i=1

Z

Ω

|X_i|²

Dku(x)k_R1

2+λ|X_i|²ku(x)k²

R¹

=

m

X

i=1

Z

Ω

D|ui|

2+λ u_i

2

=I_m(|u|) =I_m(u)

=

m

X

i=1

Z

Ω

|Du_i|²+λ|u_i|²

=

m

X

i=1

Z

Ω

|Xi|²

Dku(x)k_R1

2+λ|Xi|²ku(x)k²_R1+|Xi|²ku(x)k²_R1

Dθi(x)

2

= Z

Ω

Dku(x)k_R1

2+λku(x)k²_R1+

m

X

i=1

|Xi|²ku(x)k²_R1

Dθi(x)

2 , where the first, second and sixth equalities have used Pm

i=1|Xi|² = 1,|ui| = Xiku(x)k_R1, ui=Xiku(x)k_R1e^iθi(x)respectively. The above equality implies that

Z

Ω

|Xi|²ku(x)k²_R1

Dθi(x)

2= 0, i= 1,2, . . . , m.

So we can conclude that for anyi∈ {1,2, . . . , m}, X_i = 0 orθ_i is a constant. So u_i= 0 oru_i=X_iku(x)k_R1e^iθi. In a word,u_i =X_iku(x)k_R1e^iθi.

From to Steps 1, 2 and 3, we can get the conclusion (1) directly.

Step 4: We show that for any w∈G, ifbi∈C, i= 1,2, . . . , m, satisfy f

1

max2p |b1|,|b2|, . . . ,|bm|

∈ X, then we have (b1w, b2w, . . . , bmw) ∈ Gm. For any w ∈G, letb_i ∈ C,1 ≤i ≤M, be such that f

1

max2p (|b₁|,|b₂|, . . . ,|b_m|)∈X and set U := (b₁w, b₂w, . . . , b_mw). Then

f(b) =f⁻

1

maxp,

m

X

i=1

b²_i =f⁻

1

maxp, (3.6)

Jm(U) =f(b) Z

Ω

|w|^2p+2=f⁻

1 p

max

Z

Ω

|w|^2p+2=f⁻

1 p

max

Z

Ω

|Dw|²+λ|w|²

, (3.7) Im(U) =

m

X

i=1

b²_i Z

Ω

|Dw|²+λ|w|²) =f⁻

1

maxp

Z

Ω

(|Dw|²+λ|w|²

. (3.8) Sincex0ku(x)k_R1 is a ground state of (1.1), it follows that

γ_G=I_m x₀ku(x)k_R1

=I₁ ku(x)k_R1

=I1(c⁻¹₀ w) =f⁻

1 p

max

Z

Ω

(|Dw|²+λ|w|²

. (3.9)

From (3.7)–(3.9), we can see thatU is a minimizer ofI_m^γG. By Lemma 2.2,U ∈G_m, which implies that our conclusion (2) is true.

Following from the process of the proof of our conclusion (2), it is easy to get

our conclusion (3).

Proof of Proposition 1.4. We define an equivalence relation in{Yk}1≤k≤K, ij if and only if existsksuch thati, j∈Yk. (3.10) Let

k_ij^−∞:=

(kij, ifij,

−∞, ifi6j,

B:=

x∈(R⁺0)^m:f^−∞(x) =fmax,|x|= 1 , where

f^−∞(x) :=

m

X

i,j=1

k^−∞_ij |xj|^pij|xi|^qij. We firstly prove thatX =B.

(1) For all x⁰ ∈ B, we have |x⁰| = 1, f^−∞(x⁰) = fmax, which implies that x⁰_i ·x⁰_j = 0 if i 6 j. Sof(x⁰) = f^−∞(x⁰) = fmax. Therefore, x⁰ ∈X and then B⊂X.

(2) If x¹ ∈ X and x¹_i ·x¹_j = 0,∀ i 6 j, then x¹ ∈ B and X ⊂ B. We may assume that ∃ i0 6 j0 such that x¹_i

0 ·x¹_j

0 6= 0. Sincex¹ ∈ X, by Theorem 1.2, Q:=fmax^−1/(2p)x¹wis a ground state of (1.1), wherew∈G. That is,

I_m(Q) =J_m(Q) =γ_G.

Note that Qi₀·Qj₀ 6≡ 0, sincex¹_i0 ·x¹_j0 6= 0. Let Q^R_i = Qi ifi 6j0, and Q^R_i = Q_i(·+Re₁) ifij₀. Then for largeR,

Z

Ω

|Q^R_j|^pij|Q^R_i |^qij ≤ Z

Ω

|Qj|^pij|Qi|^qij, ifi6j, Z

Ω

|Q^R_j0|^pi0j0|Q^R_i0|^qi0j0 <

Z

Ω

|Qj0|^pi0j0|Qi0|^qi0j0 Z

Ω

|Q^R_j|^pij|Q^R_i |^qij = Z

Ω

|Q_j|^pij|Q_i|^qij, ifij.

Combining the fact thatk_ij ≥0 if and only ifij, implies thatJ_m(Q^R)> J_m(Q).

From

J_m J_m(Q) Jm(Q^R)

_2p+2¹ Q^R

=J_m(Q) =γ_G and the minimality ofQ,

Im(Q)≤Im

J_m(Q) Jm(Q^R)

_2p+2¹ Q^R

= J_m(Q) Jm(Q^R)

_p+1¹

Im(Q^R)< Im(Q^R) =Im(Q), which is impossible. Sox¹_i ·x¹_j = 0,∀ i6j. Therefore,X⊂B.

From the above discussions, we can get that X = B. If u ∈ G_m, then|u| :=

|u₁|,|u₂|, . . . ,|u_m|

∈ G_m and so, by Theorem 1.2, there exists anx₀ ∈ X = B such that|u|=fmax^−1/(2p)x0w. Since x0∈B, it is easy to check that there exists a k0 ∈ {1,2, . . . , K} such thatul6= 0 for some l ∈Yk0 and us = 0 for anys6∈Yk0.

This completes the proof.

Proof of Theorem 1.5. Let ˜X⁰:=n

x

|x|:x∈X⁰o

. We would prove thatX = ˜X⁰. (1) For anyx¹∈X, we have|x¹|= 1 and

f(x¹)

|x¹|^2p+2 =f(x¹) = max

|y|=1f(y) = max

|y|6=0

f(y)

|y|^2p+2. Sox¹∈X⁰ andx¹= _|x^x11|∈X˜⁰, which implies thatX⊂X˜⁰.

(2) For anyy⁰∈X˜⁰, there exists an x¹∈X⁰such thaty⁰= _|x^x11|. Therefore, f(y⁰) =f x¹

|x¹|

= f(x¹)

|x¹|^2p+2 = max

|y|6=0

f(y)

|y|^2p+2 = max

|y|=1f(y),