Multiplicity of 2-nodal solutions for semilinear elliptic problems in R^{N}

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Multiplicity of 2–nodal solutions for

semilinear elliptic problems in R

N

Chi-hua Liu

Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

Hsiao-yun Wang and Tsung-fang Wu

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

Abstract

In this paper, we study the effect of the graph of weight functions on the number of 2–nodal solutions for semilinear elliptic problems in RN. Our results generalize and improve on the results of D. Cao and E. S. Noussair [Multiplicity of positive and nodal solutions for nonlinear elliptic problems in RN_{, Ann. Inst. H. Poincar´e Anal.}

Non Lineair´e 13 (1996) 567–588]. In the present paper, we give a new analytical method to study this issue.

Key words: 2–nodal solutions; Nehari manifold; Palais-Smale

1 Introduction

In this paper, we study the multiplicity of 2–nodal solutions for the following semilinear elliptic problem:

    

−∆u + λu = f (z) |u|p−2_u+_{+ g (z) |u|}p−2_u− _{in R}N_,

u ∈ H1³_RN´_, (E)

where λ > 0, 2 < p < 2∗₍₂∗ ₌ 2N

N −2 if N ≥ 3, 2∗ = ∞ if N = 2), u+ =

max {0, u} , u− _{= u − u}+ _{and f, g ∈ C}³_RN´ _{are are assumed to satisfy the} Email address: [email protected] (Tsung-fang Wu).

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following conditions: (D1) f, g ≥ 0 in RN_;

(D2) There exist some points a1_{, a}2_{, . . . , a}k _{in R}N _{such that f (a}i_{) are strict}

maxima and satisfy

f³ai´= fmax≡ maxnf (z) : z ∈ RNo for all i = 1, 2, . . . , k;

(D3) There exist some points b1_{, b}2_{, . . . , b}m _{in R}N _{such that g (y}i_{) are strict}

maxima and satisfy

g³bj´= gmax≡ maxng (z) : z ∈ RNo for all j = 1, 2, . . . , m.

It is known that if f and g are positive constant, equation (E) has a unique positive (or negative) solution for each λ > 0 (see Berestycki-Lions [5] and Kwong [11]). When f and g are not positive constants, several authors have made some progresses on the existence of positive solutions of equation (E) under various conditions (see Bahri-Lions [3], Lions [13], Li [12], Bahri-Li [2], Cao [7] and Cao-Noussair [8], etc.). Furthermore, if f, g are satisfying conditions (D1) − (D3) and addition f = g, k = m and ai _{= b}i_{, then}

Cao-Noussair [8] proved that equation (E) has at least m nodal solutions for λ sufficiently large.

The main goal of this paper is generalize and improve on the results of Cao-Noussair [8]. Our result is the following theorem.

Theorem 1.1 Assume conditions (D1)−(D3) hold. Then there exists λ0 > 0

such that for λ > λ0, equation (E) has at least k × m 2–nodal solutions. This paper is organized as follows. In section 2, we describe various prelimi-naries. In section 3, we construct the Palais–Smale (simply by (PS)) sequences. In section 4, we prove Theorem 1.1.

2 Preliminaries

To frame equation (E) in the abstract setting, we first make the change of variable ε = _√1

λ, v (z) = ε

2/(p−2)_{u (εz) and rewrite equation (E) as}

     −∆v + v = fε|v|p−2v++ gε|v|p−2v− in RN, v ∈ H1³_RN´_, ( e E)

where fε = f (εz) and gε = g (εz) . If vε(z) is a solution of equation

³ e

E´ then

uε(z) := ε2/(2−p)vε(z/ε) solves equation (E) . Associated with equation

³ e E´,

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we consider the energy functional Ifε,gε in H01 ³ RN´ Ifε,gε(u) = 1 2kuk 2₋ 1 p Z RNfε|u +_|p₋ 1 p Z RNgε|u −_|p

where kuk = ³R_RN|∇u|2+ u2

´_1/2

is a standard norm in H1³_RN´_{. It is well}

known that the functional Ifε,gε ∈ C

2³_H1³_RN´_{, R}´_{and the solutions of}

equa-tion (E) are the critical points of the energy functional Ie fε,gε in H1

³

RN´_{. (see}

Ambrosetti-Rabinowitz [1] and Willem [16]). In this section, we recall several known results which will be used for later sections. First, we consider the minimization problems α±_f_ε_,g_ε = inf u∈M± fε,gε Ifε,gε(u) ; θfε,gε = inf u∈NIfε,gε(u) , where M± fε,gε = n u ∈ H1³_RN´_{\{0} |} D_I0 fε,gε(u) , u E = 0, ±u ≥ 0o and Nfε,gε = n u ∈ H1³RN´ | u+ ∈ M+_f_ε_,g_ε, u− ∈ M−_f_ε_,g_εo.

Clearly, α±_f_max_,g_max ≤ α±_f_ε_,g_ε and α_f+_ε_,g_ε + α−_f_ε_,g_ε ≤ θfε,gε. Then we have the

fol-lowing results.

Lemma 2.1 If u ∈ H1³_RN´ _{is a nodal solution of equation} ³_Ee´ _and Ifε,gε(u) < α

+

fmax,gmax + αfmax,gmax + min

n α+ fmax,gmax, α − fmax,gmax o

then u is a 2–nodal solution of equation ³Ee´.

Proof. Assume the contrary, without loss of generality, we may assume that RN_\u−1_{(0) has three connected components A}

1, A2 and A3 such that RN_\u−1_{(0) = A}

1 ∪ A2∪ A3, u (z) > 0 for all z ∈ A1 ∪ A2 and u (z) < 0 for all z ∈ A3. Define u+ _{= max {u, 0} and u}− _{= u}+_{− u. Let v}

i(z) = u+(z) for

z ∈ Ai for i = 1, 2 and v3(z) = u−(z) for z ∈ A3. We note that every solution

u of equation ³Ee´ is a C2_{–function on R}N_{. Hence, v}

i ∈ M+fε,gε for all i = 1, 2

and v3 ∈ M−fε,gε (see M¨uller–Pfeiffer [14, Lemma 1]). Moreover,

Ifε,gε(vi) ≥ α

+

fmax,gmax for i = 1, 2 and Ifε,gε(v3) ≥ α

−

fmax,gmax.

Then

Ifε,gε(u) = Ifε,gε(v1) + Ifε,gε(v2) + Ifε,gε(v3)

≥ α+

fmax,gmax+ αfmax,gmax+ min

n α+ fmax,gmax, α − fmax,gmax o

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which is a contradiction. ¤

3 Palais–Smale Sequences

First, we use the graph of the coefficient f to find some Palais–Smale se-quences which are used to prove Theorem 1.1. For η > 0, let Cη(z) denote

the hypercube ΠN

j=1(zj − η, zj + η) center at z = (z1, z2, . . . , zN). Let Cη(z)

and ∂Cη(z) denote the closure and the boundary of Cη(z) respectively. By

the conditions (D1) − (D3) , we can choose numbers l > 0 such that Cl(ai)

are disjoint, f (z) < f (ai_{) for x ∈ ∂C}

l(ai) for all i = 1, 2, . . . , k and Cl(bj)

are disjoint, g (z) < g (bj_{) for z ∈ ∂C}

l(bj) for all j = 1, 2, . . . , m. Moreover,

Cl(ai) ∩ Cl(bj) = ∅ if ai 6= bj and Cl(ai) = Cl(bj) if ai = bj.

Next, we need a generalized barycenter map. By this we mean a continuous map Φ : Lp³_RN´_{\ {0} → R}N _{which is equivalent with respect to the action}

of the group of Euclidian motions in RN_{, that is, for every ξ ∈ R}N_{, orthogonal}

N × N matrix A and u ∈ Lp³_RN´_{\ {0} , one has Φ (u) = Φ (|u|) ,}

Φ (u (x − ξ)) = ξ + Φ (u (x)) and Φ (u (Ax)) = Φ (u (x)) . (3.1) Such a map has been constructed in Bartsch-Weth [4, Theorem 2.1] and Cerami-Passaseo [9]. Let M+ i (ε) = ( u ∈ M+ fε,gε | Φ ³ u+´_{∈ C} l/ε Ã ai ε !) ; O+ i (ε) = ( u ∈ M+ fε,gε | Φ ³ u+´_{∈ ∂C} l/ε Ã ai ε !) ; M− j (ε) = ( u ∈ M− fε,gε | Φ ³ u−´_{∈ C} l/ε Ã bj ε !) ; O− j (ε) = ( u ∈ M− fε,gε | Φ ³ u−´_{∈ ∂C} l/ε Ã bj ε !) ; Ni,j(ε) = n u ∈ H1³RN´ | u+∈ M_i+(ε) and u−∈ M_j−(ε)o; Oi,j(ε) = n u ∈ H1³RN´ | u+∈ M_i+(ε) ∪ O+_i (ε) , u−∈ M_j−(ε) ∪ O_j−(ε) and u+_{∈ O}+ i (ε) or u−∈ O−j (ε) o ,

for i = 1, 2, . . . , k and j = 1, 2, . . . , m. It is easy to verify that Ni,j(ε) are

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and j = 1, 2, . . . , m, consider the minimization problems in Ni,j(ε) and Oi,j(ε)

for Ifε,gε,

γi,j(ε) = inf u∈Ni,j(ε)

Ifε,gε(u) ; γei,j(ε) = inf

u∈Oi,j(ε)

Ifε,gε(u) .

Let wfmax and wgmax be unique positive and negative radial solutions of

    

−∆u + u = fmax|u|p−2u++ gmax|u|p−2u− in RN,

u ∈ H1³_RN´_,

respectively such that Ifmax,gmax(wfmax) = α

+

fmax,gmax and Ifmax,gmax(wgmax) =

α−

fmax,gmax. For small ε > 0 satisfying

√ ε < 1, we define a function ψε ∈ C1³_RN_{, [0, 1]}´ _{such that} ψε(z) =      1 |z| < _√1 ε − 1, 0 |z| > _√1 ε, and |∇ψε| ≤ 2 in RN. Let e ∈ SN −1= n z ∈ RN _{| |z| = 1}o _and wε(z) = t+εwfmax Ã z −a i ε + e 2√ε ! ψε Ã z − a i ε + e 2√ε ! +t− εwgmax Ã z − bj ε − e 2√ε ! ψε Ã z −bj ε − e 2√ε ! , where t+

ε, t−ε > 0 are selected such that wε± ∈ M±fε,gε. Moreover, we have the

following results. Lemma 3.1 t±

ε → 1 as ε → 0.

Proof. The proofs of cases “ + ”, “ − ” are similar arguments. Therefore, we only need to prove the case “ + ”. Since w+

ε ∈ M+fε,gε, we have ³ t+ ε ´₂  Z RN ¯ ¯ ¯ ¯ ¯∇ Ã wfmax Ã z − a i ε + e 2√ε ! ψε Ã z − a i ε + e 2√ε !!¯ ¯ ¯ ¯ ¯ 2 + Ã wfmax Ã z − a i ε + e 2√ε ! ψε Ã z −a i ε + e 2√ε !!₂  =³t+ ε ´_pZ RNfεw p fmax Ã z − a i ε + e 2√ε ! ψp ε Ã z − a i ε + e 2√ε ! . Since kwfmaxk 2 H1 = R

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³ t+_ε´2kwfmaxk 2 H1 = ³ t+_ε´2 ° ° ° ° °wfmax Ã z − a i ε + e 2√ε ! ψε Ã z −a i ε + e 2√ε !°_° ° ° ° 2 H1 + o (ε) =³t+ ε ´_pZ RNfεw p fmax Ã z − ai ε + e 2√ε ! ψp ε Ã z − ai ε + e 2√ε ! + o (ε) =³t+ ε ´_pZ RNf Ã εz + ai₋ √ εe 2 ! w_fp_max + o (ε) ,

where o (ε) → 0 as ε → 0. Moreover, √εe → 0 as ε → 0 for all x ∈ SN −1_.

Thus, t+

ε → 1 as ε → 0. ¤

Lemma 3.2 For each positive number σ < minnα+

fmax,gmax, α

− fmax,gmax

o there exists εσ > 0 such that for ε < εσ

γi,j(ε) < αf+max,gmax + α

−

fmax,gmax + σ

for all i = 1, 2, . . . , k and j = 1, 2, . . . , m.

Proof. First, we show that Φ (w+

ε) ∈ Cl/ε ³ ai ε ´ and Φ (w− ε) ∈ Cl/ε ³ bj ε ´ .

Without loss of generality, we may assume that Φ (w+

ε) ∈ Cl/ε ³ ai ε ´ (Φ (w− ε) ∈ Cl/ε ³ bj ε ´

can be considered similarly). The proofs of two cases are similar arguments. Therefore, we only need to prove the case Φ (w+

ε) ∈ Cl/ε ³ ai ε ´ . Since Φ³w+ ε ´ = R RN zw_fp_max ³ z −ai ε +2√eε ´ ψp ε ³ z −ai ε +2√eε ´ R_p RNw p fmax ³ z − ai ε + 2√eε ´ ψpε ³ z − ai ε + 2√eε ´ and ψε Ã z − a i ε + e 2√ε ! = 0 if ¯ ¯ ¯ ¯ ¯z − ai ε ¯ ¯ ¯ ¯ ¯> 1 2√ε.

By the definition of ψε, we have

Φ³w+ ε ´ = R Cl/ε ³ ai ε ´_zwp fmax ³ z −ai ε + 2√eε ´ ψp ε ³ z −ai ε + 2√eε ´ R Cl/ε ³ ai ε ´_wp fmax ³ z − ai ε + e 2√ε ´ ψεp ³ z − ai ε + e 2√ε ´ provided 1

2√ε < εl. From the definition of ψε we conclude that Φ (w+ε) ∈

Cl/ε ³ ai ε ´ . Thus, w+

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Ifε,gε(wε) = (t+ ε) 2 2   Z RN ¯ ¯ ¯ ¯ ¯∇ Ã wfmax Ã z − ai ε + e 2√ε ! ψε Ã z − ai ε + e 2√ε !!¯ ¯ ¯ ¯ ¯ 2 (3.2) + Ã wfmax Ã z − a i ε + e 2√ε ! ψε Ã z −a i ε + e 2√ε !!₂  −(t+ε) p p Z RNfεw p fmax Ã z − ai ε + e 2√ε ! ψp ε Ã z − ai ε + e 2√ε ! +(t − ε) 2 2   Z RN ¯ ¯ ¯ ¯ ¯∇ Ã wgmax Ã z − b j ε − e 2√ε ! ψε Ã z − b j ε − e 2√ε !!¯_¯ ¯ ¯ ¯ 2 + Ã wgmax Ã z − bj ε − e 2√ε ! ψε Ã z −bj ε − e 2√ε !!₂  −(t − ε)p p Z RNgεw p gmax Ã z − b j ε − e 2√ε ! ψ_εp Ã z − b j ε − e 2√ε ! =1 2 Z RN|∇wfmax| 2_{+ w}2 fmax− 1 p Z RNf Ã εz + ai₋ √ εe 2 ! w_fp_max +1 2 Z RN|∇wgmax| 2_{+ w}2 gmax − 1 p Z RNg Ã εz + bj ₋ √ εe 2 ! w_fp_max + o (ε) . where o (ε) → 0 as ε → 0. Since√εe → 0 as ε → 0 and from (3.2) , we have

Ifε,gε(wε) = Ifmax,gmax(wfmax) + Ifmax,gmax(wgmax) + o (ε)

= α+

fmax,gmax + α

−

fmax,gmax+ o (ε) as ε → 0.

This completes the proof. ¤

Lemma 3.3 There are positive numbers δ and εe δ such that for ε < εδ

e

γi,j(ε) > α+fmax,gmax + α

−

fmax,gmax + δ

for all i = 1, 2, . . . , k and j = 1, 2, . . . , m.

Proof. Fix i ∈ {1, 2, . . . , k} and j ∈ {1, 2, . . . , m} . Assume the contrary, there exists a sequence {εn} with εn → 0 as n → ∞ such that

e

γi,j(εn) → c ≤ α+fmax,gmax + α

− fmax,gmax. Since e γi,j(εn) ≥ θfεn,gεn ≥ α + fεn,gεn + α − fεn,gεn ≥ α + fmax,gmax+ α − fmax,gmax,

we haveγei,j(εn) → α+fmax,gmax+α

−

fmax,gmax. Consequently, there exists a sequence

{un} ⊂ Oi,j(εn) such that

Ifεn,gεn(un) → α

+

fmax,gmax + α

−

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and Φ (u+ n) ∈ ∂Cl/εn ³ ai εn ´ or Φ (u− n) ∈ ∂Cl/εn ³ bj εn ´ . Moreover, Z RN ¯ ¯ ¯∇u+_n ¯ ¯ ¯2 + ³ u+ n ´₂ = Z RNf (εnz) ¯ ¯ ¯u+_n ¯ ¯ ¯p ≤ Z RNfmax ¯ ¯ ¯u+_n ¯ ¯ ¯p (3.4) and _Z RN ¯ ¯ ¯∇u−_n ¯ ¯ ¯2+ ³ u− n ´₂ = Z RNg (εnz) ¯ ¯ ¯u−_n ¯ ¯ ¯p ≤ Z RNgmax ¯ ¯ ¯u−_n ¯ ¯ ¯p. (3.5) Since Ifεn,gεn(u±n) ≥ α±fεn,gεn ≥ α ±

fmax,gmax, this implies

Ifεn,gεn ³ u± n ´ = α± fmax,gmax + o (1) . (3.6)

Next we will show that

Z RN[fmax− f (εnz)] ¯ ¯ ¯u+_n ¯ ¯ ¯p = o (1) and Z RN[gmax− g (εnz)] ¯ ¯ ¯u−_n ¯ ¯ ¯p = o (1) . (3.7) Suppose otherwise, we may assume that there exists a positive constant C0 such that for large n

Z RN[fmax− f (εnz)] ¯ ¯ ¯u+_n ¯ ¯ ¯p > C0. (3.8) By (3.4) , (3.5) and (3.8), there exist s±

n > 0 such that s±nu±n ∈ M±fmax,gmax and

for large n ³ s+ n ´_p−2 = R RN|∇u+_n|2+ (u+_n)2 R RNf_max|u+_n|p < Ã 1 + R C0 RN|∇u+_n|2+ (u+_n)2 !₋₁ (3.9) and ³ s− n ´_p−2 = R RN|∇u−_n|2+ (u−_n)2 R RNg_max|u−_n|p ≤ R RN|∇u−_n|2+ (u−_n)2 R RNg (ε_nz) |u−_n|p = 1. (3.10)

Moreover, by (3.3) − (3.5) and (3.9), there exist d1, d2 > 0 such that

d1 ≤ Z RN ¯ ¯ ¯∇u+_n ¯ ¯ ¯2+ ³ u+ n ´₂ ≤ Z RN|∇un| 2_{+ u}2 n≤ d2 (3.11) and ³ s+ n ´₂ < µ 1 + C0 d2 ¶−2 p−2 < 1 − c0 for some c0 > 0 (3.12) Thus by (3.8) − (3.10) , (3.11) − (3.12) and the Sobolev inequality,

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Ifεn,gεn(un) = Ã 1 2 − 1 p ! ·Z RN ¯ ¯ ¯∇u+_n¯¯¯2+³u+_n´2+ Z RN ¯ ¯ ¯∇u−_n¯¯¯2+³u−_n´2 ¸ > Ã 1 2 − 1 p ! ·³ s+ n ´₂Z RN ¯ ¯ ¯∇u+_n ¯ ¯ ¯2+ ³ u+ n ´₂ +³s− n ´₂Z RN ¯ ¯ ¯∇u−_n ¯ ¯ ¯2 + ³ u− n ´₂¸ + Ã 1 2− 1 p ! c0 ·Z RN ¯ ¯ ¯∇u+_n ¯ ¯ ¯2+ ³ u+ n ´₂¸ ≥ α+ fmax,gmax + α − fmax,gmax + Ã 1 2 − 1 p ! c0d1,

for n sufficiently large, this contradicts (3.3) . Then by (3.3) − (3.7)

Z RN ¯ ¯ ¯∇u+_n ¯ ¯ ¯2+ ³ u+ n ´₂ = Z RN f (εnz) ¯ ¯ ¯u+_n ¯ ¯ ¯p = Z RNfmax ¯ ¯ ¯u+_n ¯ ¯ ¯p + o (1) , Z RN ¯ ¯ ¯∇u−_n ¯ ¯ ¯2+ ³ u− n ´₂ = Z RN g (εnz) ¯ ¯ ¯u−_n ¯ ¯ ¯p = Z RNgmax ¯ ¯ ¯u−_n ¯ ¯ ¯p+ o (1) . and Ifmax,gmax ³ u± n ´ = α± fmax,gmax+ o (1) (3.13) By Wang-Wu [15, Lemma 7] {u±

n} are (PS)α±_fmax,gmax–sequences in H1

³

RN´_for

Ifmax,gmax. It follows that {u±n} is uniformly bounded in H1

³

RN´_{. Since u}± n ∈

M±_f_εn_,g_εn, we deduce from the Sobolev imbedding theorem that ku±

nkH1 > ν >

0 for some constant ν and for all n. Applying the concentration-compactness principle of P. L. Lions [13] to |u±

n|

p_{, there are positive constants R, θ and}

{z± n} ⊂ RN such that Z BN₍_z± n;R) ¯ ¯ ¯u±_n ¯ ¯ ¯p ≥ θ for all n, (3.14) where BN_(z± n; R) = n z ∈ RN _{| |z − z}± n| < R o . Suppose that Φ (u+ n) ∈ ∂Cl/εn ³ ai εn ´ (Φ (u− n) ∈ ∂Cl/εn ³ ai εn ´

can be considered similarly). Let uen = u+n (z + z+n) ,

from the translation invariance of the functional, we get that also {uen} is a

(PS)_α+

fmax,gmax–sequences in H

1³_RN´ _{for I}

fmax,gmax. Then by (3.14) there exist

a subsequence {uen} and a nonzero u0 ∈ H1

³ RN´ _{such that} e un* u0 weakly in H1 ³ RN´_, e un→ u0 a.e. in RN, Z BN_(0;R)|uen| p_→Z BN_(0;R)|u0| p _{≥ θ.}

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This implies u0 is a nonzero nonnegative solution of equation      −∆v + v = fmax|v|p−2v++ gmax|v|p−2v− in RN, v ∈ H1³_RN´_. (Efmax,gmax)

By the strong maximum principle, u0is a positive solution of equation (Efmax,gmax) .

Moreover,by the Fatou lemma

α+

fmax,gmax ≤ Ifmax,gmax(u0) ≤ lim inf Ifmax,gmax(uen) = α

+

fmax,gmax,

and so Ifmax,gmax(u0) = α

+

fmax,gmax and kuenk = ku0k + o (1) . Since uen * u0

weakly in H1³_RN´_{, this implies}_u_e

n→ u0strongly in H1 ³ RN´_{. From Φ (u}+ n) ∈ ∂Cl/εn ³ ai εn ´ and uen(z) = u+n(z + zn+) , we have εnz+n = εnΦ ³ u+_n´− εnΦ (uen) = εnΦ ³ u+_n´− εnΦ (u0) + o (1) ,

and so dist (εnzn+, ∂Cl(ai)) → 0 as n → ∞. Without loss of generality, we

may assume εnzn+ → z0 ∈ ∂Cl(ai) . By condition (D2) , f (z0) < fmax. Then by (3.4) , (3.7) anduen → u0 strongly in H1 ³ RN´ _{we can conclude} Z RN|∇u0| 2_{+ u}2 0 = Z RNf (z0) |u0| p _<Z RNfmax|u0| p_,

this contradicts to u0 is a positive solution of equation (Efmax,gmax) . This

com-pletes the proof. ¤

By the conditions (D1) − (D3) , we can choose number σ0 > 0 such that

Cl ε+σ0 ³ ai ε ´ ∩ Cl ε+σ0 ³ bj ε ´ = ∅ if ai _{6= b}j _{and C}_l ε+σ0 ³ ai ε ´ = Cl ε+σ0 ³ bj ε ´ if ai _{= b}j_.

Here we will use the idea of Bartsch-Weth [4] and Clapp-Weth [10] to get the following results.

Lemma 3.4 For each positive number σ ≤ σ0 there exist positive numbers µ (σ) , δ (σ) and ε =e ε (δ (σ)) such that for every 0 < ε <e ε and everye v ∈ Ni,j(ε) with Ifmax,gmax(v) ≤ α

+

fmax,gmax + α

−

fmax,gmax + δ (σ) and every

u ∈ H1³_RN´ _{such that kv − uk < µ (σ) , there holds Φ (u}+_{) ∈ C}_l

ε+σ ³ ai ε ´ and Φ (u−_{) ∈ C}_l ε+σ ³ bj ε ´ .

Proof. By Lemmas 3.2 and 3.3, we let σn > 0, σn → 0 as n → ∞ and

{vn} ⊂ Ni,j(εn) be such that

J (vn) = α+fmax,gmax + α

−

fmax,gmax + o (σn)

and

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where εn > 0, εn → 0 and o (σn) → 0 as n → ∞. In order to prove the result

via an indirect argument it suffices to show that

¯ ¯ ¯Φ ³ u± n ´ − Φ³v± n ´¯ ¯ ¯→ 0 as n → ∞.

As the argument of proof in Lemma 3.3 we have {v±

n} are (PS)α±_fmax,gmax–

sequences in H1³_RN´ _{for I}

fmax,gmax. Applying the concentration-compactness

principle of P. L. Lions [13], there are positive constants R, d and two sequences

{zn} ⊂ RN such that _Z BN_(z_n_;R) ¯ ¯ ¯v_n+ ¯ ¯ ¯p ≥ d. (3.15)

Let vbn(z) = vn+(z + zn) , from the translation invariance of the functional,

we get that also {vbn} is a (PS)_α+

fmax,gmax–sequences in H

1³_RN´ _{for I}

fmax,gmax.

Similar to the argument of proof in Lemma 3.3, there exist a subsequence {vbn}

and v0 is a positive solution of equation (Efmax,gmax) such that Ifmax,gmax(v0) =

α+

fmax,gmax and vbn → v0 strongly in H

1³_RN´_{. Let} _u_b n(z) = u+n (z + zn) . By translation invariance, kbvn−ubnk = ° ° °v_n+− u+_n ° ° °≤ kvn− unk → 0. Thusubn → v0 strongly in H1 ³ RN´_{. This implies} ¯ ¯ ¯Φ ³ u+ n ´ − Φ³v+ n ´¯ ¯ ¯= |Φ (ubn) − Φ (vbn)| → 0.

A similar argument gives |Φ (u−

n) − Φ (vn−)| → 0. ¤

By Lemmas 3.2, 3.3 and 3.4 there exist δ0 ≤ min

n e δ, δ (σ) , α+ fmax,gmax, α − fmax,gmax o

and ε0 = ε (δ0) > 0 such that for every ε < ε0

γi,j(ε) < min

n α+

fmax,gmax + α

−

fmax,gmax + δ0,γei,j(ε)

o

(3.16) for all i = 1, 2, . . . , k and j = 1, 2, . . . , m, and the result is holds in Lemma 3.4. Furthermore, we have the following results.

Lemma 3.5 For each v0 ∈ Ni,j(ε) there exists a map φ : H1

³ RN´ _{→ R}2 such that (i) φ³s1v0++ s2v−0 ´ = (s1, s2) for s1, s2 ≥ 0; (ii) φ (u) = (1, 1) if and only if u ∈ Nfε,gε.

Proof. Similar to the method used in Clapp-Weth [10, Lemma 13]. ¤ Proposition 3.6 Let σ, µ (σ) , δ (σ) > 0 be as in Lemma 3.4 and λ0 = α+fmax,gmax+

α−

fmax,gmax + δ0 − γi,j(ε). Then for every λ ∈ (0, λ0) and every µ ∈ (0, µ (σ))

there exists u0 ∈ H1

³

RN´ _{such that}

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(ii) Ifε,gε(u0) ∈ [γi,j(ε) , γi,j(ε) + λ);

(iii) k∇Ifε,gε(u0)k ≤ max

n_√ λ,λ µ o ; (iv) Φ³u+₀´∈ Cl ε+σ ³ ai ε ´ and Φ³u−₀´∈ Cl ε+σ ³ bj ε ´ .

Proof. Fix v0 ∈ Ni,j(ε) such that Ifε,gε(v0) < γi,j(ε) + λ, and fix l0 > 1

such that Ifε,gε

³ l0v0±

´

≤ 0. Let φ : H1³_RN´_{→ R}2 _{as in Lemma 3.5. We put}

K = [0, l0] × [0, l0] and define

η : K → H1³_RN´_{, η (s}

1, s2) = s1v0++ s2v0−. Then φ ◦ η = id : K → K, in particular

deg (φ ◦ η, K, (1, 1)) = 1. (3.17)

Notice also that

Ifε,gε(η (s1, s2)) ≤ Ifε,gε(v0) < γi,j(ε) + λ for all (s1, s2) ∈ K. (3.18)

Now we choose a Lipschitz continuous function χ : R → R such that 0 ≤

χ ≤ 1, χ (s) = 1 for s ≥ 0 and χ (s) = 0 for s ≤ −1. Since Ifε,gε ∈

C2³_H1³_RN´_{, R}´_{, there is a semiflow ϕ : [0, ∞) × H}1³_RN´ _{→ H}1³_RN´ satisfying      ∂

∂tϕ (t, u) = −χ (Ifε,gε(ϕ (t, u))) ∇Ifε,gε(ϕ (t, u)) ,

ϕ (0, u) = u.

We will frequently write ϕt _{in place of ϕ (t, ·) . Since}

Ifε,gε

³

v₀±´< γi,j(ε)+λ−α±fmax,gmax < α

+

fmax,gmax+α

−

fmax,gmax and Ifε,gε

³ l0v0± ´ ≤ 0, it follows that sup Ifε,gε(η (∂K)) < α + fmax,gmax + α − fmax,gmax. Hence _³ ϕt◦ η´(∂K) ∩ Nfε,gε = ∅ for all t ≥ 0.

By Lemma 3.5, this implies

³

φ ◦ ϕt_{◦ η}´_{(y) 6= (1, 1) for all y ∈ ∂K, t ≥ 0.}

Equality (3.17) and the global continuation principle of Leray-Schauder (see e.g. Zeider [17, p.629]) imply that there exists a connected subset Z ⊂ K×[0, 1]

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such that (1, 1, 0) ∈ Z; ϕt_{(η (s} 1, s2)) ∈ Nfε,gε for all (s1, s2, t) ∈ Z; Z ∩ (K × {1}) 6= ∅. We put e Z =nϕt_{(η (s} 1, s2)) ∈ Nfε,gε : (s1, s2, t) ∈ Z o . By inequality (3.18) , sup u∈Ze Ifε,gε(u) < γi,j(ε) + λ < b.

Therefore, since Z is connected, we obtain that Z ⊂ Ne i,j(ε) . Now we pick

(¯s1, ¯s2, 1) ∈ Z ∩ (K × {1}) and write

v1 := ε (¯s1, ¯s2) , v2 := ϕ1(v1) . Then v2 ∈Z ⊂ Ne i,j(ε) . We distinguish two cases.

Case 1. kϕt_(v

1) − v2k ≤ µ for all t ∈ [0, 1] . Then by Lemma 3.4, we have

¯ ¯ ¯ ¯Φ µ³ ϕt_(v 1) ´₊¶ − Φ³v+ 2 ´¯¯ ¯ ¯< σ and ¯ ¯ ¯ ¯Φ µ³ ϕt_(v 1) ´₋¶ − Φ³v− 2 ´¯¯ ¯ ¯< σ

for all t ∈ [0, 1]. We choose t0 ∈ [0, 1] with

° ° °∇Ifε,gε ³ ϕt0_(v 1) ´° ° °= min 0≤t≤1 ° ° °∇Ifε,gε ³ ϕt_(v 1) ´° ° °

and put u0 = ϕt0(v1) . Thus,

λ ≥ Ifε,gε(v1) − Ifε,gε(v2) = − Z ₁ 0 ∂ ∂tIfε,gε ³ ϕt(v1)´dt = Z ₁ 0 ° ° °∇Ifε,gε ³ ϕt(v1)´°°°2dt ≥ k∇Ifε,gε(u0)k 2_. We obtain u0 has the desired properties.

Case 2. There exists ¯t ∈ [0, 1] such that °°°ϕt¯_(v

1) − v2 ° ° ° > µ. Then let t1 = sup n t ≥ ¯t | °°°ϕt_(v 1) − v2 ° ° ° > µo. By Lemma 3.4, ¯ ¯ ¯ ¯Φ µ³ ϕt_(v 1) ´₊¶ − Φ³v+ 2 ´¯¯_¯ ¯< σ and ¯ ¯ ¯ ¯Φ µ³ ϕt_(v 1) ´₋¶ − Φ³v− 2 ´¯¯_¯ ¯< σ

for all t ∈ [t1, 1] . We choose t0 ∈ [t1, 1] with

° ° °∇Ifε,gε ³ ϕt0_(v 1) ´° ° °= min t1≤t≤1 ° ° °∇Ifε,gε ³ ϕt_(v 1) ´° ° °

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and put u0 = ϕt0_{(v1) . Then} µ ≤ Z ₁ t1 ° ° ° ° ° ∂ ∂tϕ t_(v1₎ ° ° ° ° °dt ≤ Z ₁ t1 ° ° °∇Ifε,gε ³ ϕt(v1)´°°°dt and λ ≥ Ifε,gε ³ ϕt1_(v1) ´ − J (v2) = Z ₁ t1 ° ° °∇Ifε,gε ³ ϕt(v1)´°°°2dt ≥ k∇Ifε,gε(u0)k Z ₁ t1 ° ° °∇Ifε,gε ³ ϕt_(v 1) ´° ° °dt.

We conclude that k∇Ifε,gε(u0)k ≤

λ

µ. Thus, u0 has the desired properties. ¤

Corollary 3.7 For each ε < ε0 there exists a sequence {ui,j

n } ⊂ H1

³

RN´

such that

(i) dist (ui,j

n , Ni,j(ε)) → 0;

(ii) Ifε,gε(ui,jn ) → γi,j(ε) ;

(iii) I0 fε,gε(u i,j n ) = o(1) strongly in H−1 ³ RN´_. 4 Proof of Theorem 1.1

We need the following proposition to provide the precise description of the Palais–Smale sequences for Ifε,gε.

Proposition 4.1 Assume that {ui,j

n } ⊂ H1

³

RN´ _{is a sequence satisfying}

(i) dist (ui,j

n , Ni,j(ε)) → 0 as n → ∞;

(ii) Ifε,gε(ui,jn ) → γi,j(ε) as n → ∞;

(iii) I0 fε,gε(u i,j n ) → 0 strongly in H−1 ³ RN´ _{as n → ∞.}

Then there exist a subsequence {ui,j

n } and ui,j0 ∈ Ni,j(ε) such that ui,jn → ui,j0

strongly in H1³_RN´_.

Proof. Since {ui,j

n } is bounded in H1

³

RN´_{, we can assume that there exists}

ui,j₀ ∈ H1³_RN´ _{such that}

ui,j n * ui,j0 and ³ ui,j n ´_± *³ui,j₀ ´± weakly in H1³_RN´_; _(4.1)

ui,j_n → ui,j₀ and ³ui,j_n ´± →³ui,j₀ ´± strongly in Lp_loc³RN´; (4.2)

ui,j n → ui,j0 and ³ ui,j n ´_± →³ui,j₀ ´± a.e. in RN_. _(4.3)

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First, we claim that ³ui,j₀ ´±6≡ 0. Suppose otherwise, that is ³

ui,j₀ ´+≡ 0 or ³ui,j₀ ´−≡ 0.

Without loss of generality, we can assume³ui,j₀ ´+≡ 0. Since dist (ui,j

n , Ni,j(ε)) →

0 as n → ∞ and γi,j(ε) > 0, we deduce from the Sobolev imbedding

theo-rem that °°°(ui,j n )

+°°

° > ν > 0 for some constant ν and for all n. Applying

the concentration-compactness principle of P. L. Lions [13], there are positive constants R, θ and a sequence {zn} ⊂ RN such that

Z BN_(0;R) ¯ ¯ ¯ ¯ ³ ui,j_n ´+(z + zn) ¯ ¯ ¯ ¯ p

≥ θ for n sufficiently large. (4.4) We will show that {zn} is a unbounded sequence in RN. Suppose otherwise,

we can assume that zn→ z0 for some z0 ∈ RN. By (4.2) and (4.4) ,

Z BN_(z₀_;R) ¯ ¯ ¯ ¯ ³ ui,j₀ ´+ ¯ ¯ ¯ ¯ p ≥ θ,

this contradicts ³ui,j₀ ´+ ≡ 0. Thus, {zn} is a unbounded sequence in RN. Set

e ui,j

n (z) = (ui,jn )

+_{(z + z}

n) . Since {uei,jn } is bounded in H1

³

RN´_{, we may assume}

that there existsuei,j₀ ∈ H1³_RN´_{such that}

e ui,j

n *uei,j0 weakly in H1

³

RN´ _(4.5)

From (4.4) we have uei,j₀ ≥ 0 and uei,j₀ 6≡ 0 in RN_{. Set v}

n = uei,jn −uei,j0 . We distinguish the cases:

Case I : kvnk → 0 as n → ∞;

Case II : kvnk ≥ θ for large n and for some constant θ > 0.

Assume Case I, we employ the argument in Lemma 3.3 to obtain

zn= Φ µ³ ui,j_n ´+ ¶ − Φ³uei,j₀ ´+ o (1) , and so¯¯¯Φ ³ (ui,j n ) +´¯¯

¯→ ∞ as n → ∞, this contradicts dist ³ Φ³(ui,j n ) +´ , Cl/ε ³ ai ε ´´ → 0.

In Case II, we notice first that I0 fε,gε(u i,j n ) → 0 strongly in H−1 ³ RN´ _and dist (ui,j n , Ni,j(ε)) → 0 as n → ∞ implies Z RN ¯ ¯ ¯∇uei,j₀ ¯¯¯2+ ³ e ui,j₀ ´2− Z RNf (εz + εzn) ¯ ¯ ¯eui,j₀ ¯ ¯ ¯p = o (1) (4.6) and Z RN ¯ ¯ ¯∇uei,j n ¯ ¯ ¯2+ ³ e ui,j n ´₂ − Z RNf (εz + εzn) ¯ ¯ ¯eui,j_n ¯ ¯ ¯p = o (1) . (4.7)

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By (4.6) , (4.7) and Brezis-Lieb lemma [6] we obtain Z RN |∇vn| 2 _{+ v}2 n+ Z RNf (εz + εzn) |vn| p _{= o (1) .}

Since kvnk ≥ θ for large n, is is easy to find a sequence {sn} ⊂ R+ with sn→ 1

as n → ∞ such that snvn ∈ M+_{f (εz+εz}_n_),g_ε, and so

1 2 Z RN|∇vn| 2_{+ v}2 n− 1 p Z RN f (εz + εzn) |vn| p _{≥ α}+ fmax,gmax + o (1) . Similarly 1 2 Z RN ¯ ¯ ¯∇uei,j₀ ¯¯¯2+ ³ e ui,j₀ ´2− 1 p Z RNf (εz + εzn) ¯ ¯ ¯eui,j₀ ¯ ¯ ¯p ≥ α+_f_max_,g_max + o (1) and 1 2 Z RN ¯ ¯ ¯ ¯∇ ³ ui,j_n ´− ¯ ¯ ¯ ¯ 2 +µ³ui,j_n ´− ¶₂ − 1 q Z RNg (εz) ¯ ¯ ¯ ¯ ³ ui,j_n ´− ¯ ¯ ¯ ¯ q ≥ α−_f_max_,g_max + o (1) . Thus by Brezis-Lieb lemma [6] we have

Ifε,gε ³ ui,j n ´ =1 2 Z RN ¯ ¯ ¯ ¯∇ ³ e ui,j n ´₊¯_¯ ¯ ¯ 2 +µ³uei,j n ´₊¶2 − 1 p Z RNf (εz + εzn) ¯ ¯ ¯ ¯ ³ e ui,j n ´₋¯_¯ ¯ ¯ p +1 2 Z RN ¯ ¯ ¯ ¯∇ ³ ui,j n ´₋¯_¯ ¯ ¯ 2 +µ³ui,j n ´₋¶2 − 1 q Z RNg (εz) ¯ ¯ ¯ ¯ ³ ui,j n ´₋¯_¯ ¯ ¯ q =1 2 Z RN|∇vn| 2_{+ v}2 n− 1 p Z RNf (εz + εzn) |vn| p +1 2 Z RN ¯ ¯ ¯∇uei,j₀ ¯¯¯2+ ³ e ui,j₀ ´2− 1 p Z RNf (εz + εzn) ¯ ¯ ¯eui,j₀ ¯ ¯ ¯p +1 2 Z RN ¯ ¯ ¯ ¯∇ ³ ui,j_n ´− ¯ ¯ ¯ ¯ 2 +µ³ui,j_n ´− ¶₂ − 1 q Z RNg (εz) ¯ ¯ ¯ ¯ ³ ui,j_n ´− ¯ ¯ ¯ ¯ q + o (1) ≥ 2α+ fmax,gmax + α − fmax,gmax + o (1)

which implies that lim n→∞Ifε,gε ³ ui,j n ´

= γi,j(ε) ≥ 2α+fmax,gmax + α

−

fmax,gmax (4.8)

this contradicts (3.16) . Next we will show that ui,j

n → ui,j0 strongly in H1

³

RN´_.

Using Case II we can prove that we result, otherwise, we may use a sim-ilar argument as above to reach the contradiction (4.8) . Finally, we will show that ui,j₀ ∈ Ni,j(ε) . Since dist (ui,jn , Ni,j(ε)) → 0 as n → ∞, we have

ui,j₀ ∈ Ni,j(ε) ∪ Oi,j(ε) . Moreover, Ifε,gε

³

ui,j₀ ´ = γi,j(ε) < γei,j(ε) and so

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Now, we begin to show the proof of Theorem 1.1: By Corollary 3.7 and Proposition 4.1 for each ε ∈ (0, ε0) , there exist ui,j0 ∈ Ni,j(ε) are nodal

solutions of equation ³Ee´ such that Ifε,gε

³

ui,j₀ ´= γi,j(ε) . By Lemma 2.1 ui,j0 is a 2–nodal solution of equation ³Ee´. Since ui,j₀ ∈ Ni,j(ε), this implies ui,j0 are distinct. Letting U₀i,j(z) = ε2/(2−p)_ui,j

0 (z/ε) . We obtain U0i,j are 2–nodal

solutions of equation (E) . ¤

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