Multi-bump type nodal solutions having a prescribed number of nodal domains : I

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Multi-bump type nodal solutions having a prescribed number of nodal domains: I

Solutions nodales de multi-bosse ayant un nombre de domaines nodales prescrites: I

Zhaoli Liu

, Zhi-Qiang Wang

aDepartment of Mathematics, Capital Normal University, Beijing 100037, PR China bDepartment of Mathematics and Statistics, Utah State University, Logan, Utah 84322, USA

Received 16 January 2004; accepted 6 October 2004 Available online 7 April 2005

Abstract

This paper is concerned with analyzing their nodal property of multi-bump type sign-changing solutions constructed by Coti Zelati and Rabinowitz [Comm. Pure Appl. Math. 45 (1992) 1217]. In some cases we provide both upper and lower bounds on the number of nodal domains for these solutions.

2005 Elsevier SAS. All rights reserved.

Résumé

Cet article concerne l’analyse de la propriété nodale des solutions de multi-bosse changeant de signe, construites par Coti Ze- lati et Rabinowitz [Comm. Pure Appl. Math. 45 (1992) 1217]. Dans certain cas, nous obtenons la borne supérieure et inférieure de nombre des domaines nodales de ces solutions.

2005 Elsevier SAS. All rights reserved.

MSC: 35B05; 35J60

Keywords: Semilinear elliptic equation; Multi-bump nodal solution; Number of nodal domains

* Corresponding author.

E-mail address: zliu@mail.cnu.edu.cdu (Z. Liu).

1 Supported by NSFC: 10441003.

0294-1449/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpc.2004.10.002

1. Introduction

This paper is concerned with constructing multi-bump type, nodal (sign-changing) solutions which have a pre- scribed number of nodal domains for nonlinear time-independent Schrödinger equations of the form

−u+V (x)u=f (x, u) inΩ, u=0 on∂Ω, (1)

which satisfyu(x)→0 as |x| → ∞, hereΩ is a smooth domain in R^N or the whole space R^N. This type of equations arise from study of steady state and standing wave solutions of time-dependent nonlinear Schrödinger equations. Ifu∈C(R^N,R)is a solution, each connected component of the set R^N\u⁻¹(0)is called a nodal do- main ofu. The potential function is assumed to be periodic in the unbounded directions ofΩ. Multi-bump type solutions for nonlinear elliptic PDEs with a periodic potential was first obtained by Coti Zelati and Rabinowitz [11]

by using a gluing method which was used initially for Hamiltonian ODEs in [10,22] (see the survey monographs of Rabinowitz [19–21] for more references therein). The gluing procedure is to use a variational argument to find a family of ground state solutions which constitute the basic ‘one-bump’ solutions. These one-bump solutions need to have a certain non-degeneracy property. Then further variational arguments are used to construct multi-bump so- lutions, i.e. solutions near sums of sufficiently separated translates of the basic solutions. In [11] the basic building blocks, one-bump solutions, are allowed to be either positive and negative ground state solutions. Thus multi-bump type nodal solutions potentially having many nodal domains are constructed. However, the question on the exact number of nodal domains for these multi-bump solutions was left open. A related question is to provide estimates on the number of nodal domains in terms of the minimax gluing procedure. To our knowledge this question has not been addressed either except in [1] nodal solutions having exactly two nodal domains were claimed. In this paper we shall partially address these questions. Another motivation for our study is that in recent years, nodal solutions of nonlinear elliptic BVPs have received much attention (e.g. [2–5,7,12,13,15–17]) in which many multiplicity re- sults of nodal solutions were given both for bounded domains and for entire space. However, without any symmetry of subgroups ofO(N )for the equations and the domains and without assuming the nonlinearity being odd inu, it is only known that there are nodal solutions having exactly two nodal domains. There seem no examples giving nodal solutions of more than two nodal domains, let alone giving a prescribed large number of nodal domains. We shall give results in this paper that claim the existence of nodal solutions having an arbitrary prescribed number of nodal domains. We shall build upon the ideas and approaches used by Coti Zelati–Rabinowitz [11] and exploit further finer estimates of the multi-bump type solutions so that their nodal property can be analyzed qualitatively.

IfΩis the entire R^N, we rewrite Eq. (1) as

−u+V (x)u=f (x, u), in R^N. (2)

Depending on whetherΩ is a cylindrical domain or the entire R^N and on whetherV andf are periodic in allx variables, we shall consider three different cases.

Case (i). Our first result is for Eq. (2). We make the following assumptions onV andf: (V₁) V ∈C(R^N,R),V₀=inf_RNV (x) >0, is periodic inx₁, . . . , x_N.

(f₁) f ∈C¹(R^N×R,R)is periodic inx₁, . . . , xN. (f₂) f (x,0)=0=f_u(x,0).

(f₃) There isC >0 such that f_u(x, u)C

1+ |u|^p−2 for allx∈R^N, u∈R where 2< p <2^∗. (f₄) There isµ >2 such that

0< µF (x, u):=µ u

0

f (x, t )dtuf (x, u)

for allx∈R^N, u∈R\ {0}.

(f₅) f (x,−u)= −f (x, u), i.e.,f is odd inu.

The weak solutions of (2) correspond to critical points of I (u):=1

2

R^N

|∇u|²+V (x)u² dx−

R^N

F (x, u)dx,

inE=W^1,2(R^N). We shall use the following notations as in [10,11].I^b= {u∈E|I (u)b},I_a= {u∈E|I (u) a},I_a^b= {u∈E|aI (u)b},K= {u∈E|I(u)=0},K(c)= {u∈E|I(u)=0, I (u)=c},K^b=K∩I^b, K^ba=K∩I_a^b.

Due to the periodicity in thex₁, . . . , x_Ndirections the problem is invariant under translations in thex₁, . . . , x_N directions, i.e., being Z^N-invariant. With the superlinear nonlinearity f (x, u)we may define the mountain pass valuec >0.

c= inf

g∈Γ sup

t∈[0,1]I g(t ) where

Γ = g∈C

[0,1], E

|g(0)=0, g(1)∈I⁰\ {0} . As in [10,11], we assume

(∗) There isα >0 such thatK^c+α/Z^Nis finite.

According to [11], the above assumptions guarantee the existence of positive and negative solutions to Eq. (2) at the mountain-pass level. These solutions will be referred to as one-bump solutions. Letv₁, . . . , v_kbe one-bump solutions such that their barycenters are sufficiently separated. Solutions of (2) that are close tok

i=1v_i are called k-bump solutions. We understand this reference also applies to Eq. (1).

Theorem 1.1. Assume (V₁) and (f₁)–(f₅). Suppose (∗) holds. For multi-bump nodal solutions of Eq. (2), the number of nodal domains is bounded by the number of bumps. In particular, the two-bump nodal solutions have exactly two nodal domains. Moreover, there are infinitely many, geometrically different, two-bump, nodal solutions which have exactly two nodal domains.

Case (ii). Next we consider Eq. (1) on a cylinder domain Ω =ω×R and we write x =(x, xN)with x = (x1, . . . , xN−1), whereωis a bounded smooth domain in R^N−1. We assume that

(V₁) V ∈C(Ω,R),V0:=infΩV (x) >0, is periodic inxN. (f₁) f ∈C¹(Ω×R,R)is periodic inxN.

We understand(f2)–(f5)are satisfied now forx∈Ω. The weak solutions of (1) correspond to critical points of I (u):=1

2

Ω

|∇u|²+V (x)u² dx−

Ω

F (x, u)dx,

inE=H₀¹(Ω). Then we can still define the mountain pass valuec >0. The problem now is Z invariant.

(∗) There isα >0 such thatK^c+α/Z is finite.

Theorem 1.2. Assume(V₁),(f₁), and(f₂)–(f₅). Suppose (∗) holds. Then for any integerskm2, Eq. (1) has infinitely many, geometrically different,k-bump, nodal solutions inI_kc^kc₋⁺_α^α which have exactlymnodal domains.

More precisely, given any positive integers k₁, k₂, . . . , k_m such thatm

i=1k_i =k2, there are infinitely many, geometrically different,k-bump, nodal solutions inI_kc^kc₋⁺_α^α which have exactlymnodal domainsD_i,i=1, . . . , m, such thatu|Di is ak_i-bump positive or negative solution.

Case (iii). Finally, we consider Eq. (2) with differentx-dependence in different directions.

(V₁) V ∈C(R^N,R),V0:=inf_RNV (x) >0, is periodic inxNand radially symmetric in(x1, . . . , xN−1).

(f₁) f ∈C¹(R^N×R,R)is periodic inx_Nand radially symmetric in(x₁, . . . , x_N−1).

Then the problem is again Z-invariant. Withx=(x, x_N)andx=(x₁, . . . , x_N−1), we take E=

u∈W^1,2(R^N) u(x, x_N)=u

|x|, x_N ,

R^N

V (x)u²dx <∞

i.e., functions inEare radially symmetric in the firstN−1 variables. We can still define the mountain pass value cin the spaceE.

(∗) There isα >0 such thatK^c+α/Z is finite.

Theorem 1.3. Assume(V₁),(f₁)and(f₂)–(f₅). Suppose (∗) holds. For any integerk2, Eq. (2) has infinitely many, geometrically different,k-bump, nodal solutions inI_kc^kc₋⁺_α^α such that the numbers of their nodal domains are bounded between[k/2] +1 andk. In particular, there are nodal solutions such that the numbers of their nodal domains tend to infinity.

Remark 1.4. In the setting of Theorem 1.2, one can easily see thatΩcan be taken as more general smooth domain than a standard cylinder domain in R^N, which needs to be periodic in thex_N direction and whose projection on R^N−1is bounded. Furthermore, it follows from the proof that this(N−1)-dimensional projection may also be unbounded. In this case, we need to assume that there is a numbert such that{x∈R^N−1|(x, t )∈Ω}is bounded and impose additional assumptions on V andf in R^N−1direction, for example, V is coercive in the(N−1) space. Finally, in the case the equation is autonomous, namely, independent ofx, we may assume the domains are periodic in thexNdirection. By assuming an isolatedness condition of the critical points at the mountain pass level like(∗), a similar result to Theorem 1.2 can be stated.

Remark 1.5. The assumption thatf is odd inuis a technical condition but used in [11] in an essential way. In a sequel to this paper [18] we will provide techniques to remove this condition. This requires modifications of the gluing procedure of [11] by combining with invariant set method.

The paper is organized as follows. Section 2 contains a sketch of the original construction of multi-bump so- lutions in the setting of Theorem 1.1 due to Coti Zelati–Rabinowitz [11], and some variants of it for the settings of our Theorems 1.2 and 1.3. Section 3 is devoted to the proof of our main estimates aboutC¹-closeness of the multi-bump solutions to the sum of the basic one bump solutions. In Section 4, using the results from Section 3 we prove our main Theorems 1.1, 1.2, and 1.3.

2. Multi-bump type nodal solutions revisited

In this sectionEdenotes the Sobolev spaceW^1,2(R^N), and some times a subspace of it when the context is clear.

We consider the case (i) (Theorem 1.1) first which was considered in [11] in detail by Coti Zelati–Rabinowitz.

Without loss of generality we assume the periods in all directions are equal to 1.

Letj=(j₁, . . . , jN)∈Z^Nand the translations on the R^N will be defined by τju(x)=u(x1+j1, . . . , xN+jN).

Following [11], one can prove that there isν >0 such that for allv∈K\ {0},vν. Under (∗), it is proved in [11] that the mountain pass valuecis a critical value. LetΛ= {+1,−1}andλ=(λ₁, . . . , λ_k)∈Λ^k. Then one chooses a finite setA⊂K(c)which contains only positive solutions (see [11] for the details) to define for any fixed integerk2,λ=(λ₁, . . . , λ_k)∈Λ^k,

M=M(j₁, . . . , j_k, A, λ)= _k

i=1

λ_iτ_jiv_i v_i∈A

,

wherej_i∈Z^N fori=1, . . . , kare fixed such that_k

i=1τ_jiv_i^kν₂.

Theorem 2.1 [11]. Assume(V₁)and(f₁)–(f₅). Suppose (∗) holds. There isr₀>0 such that for anyr∈(0, r₀) Nr

M(lj1, . . . , ljk, A, λ)

∩(K^kc_kc⁺_−α^α/Z^N)= ∅

for all but finitely manyl∈N, whereN_r(·)is ther-neighborhood inE.

Next consider case (ii). In the setting of Theorem 1.1 which was sketched in [11] we state the following result.

Here the problem is only Z-invariant and we understandj ∈Z here and the translationsτ_j are only in thex_N direction.

Theorem 2.2. Assume(V₁),(f₁), and(f₂)–(f₅). Suppose (∗) holds. There isr₀>0 such that for anyr∈(0, r₀) Nr

M(lj1, . . . , ljk, A, λ)

∩(K^kc_kc⁺_−α^α/Z)= ∅

for all but finitely manyl∈N, whereNr(·)is ther-neighborhood inE=W₀^1,2(Ω).

Though this result was not stated in [11], the multi-bump solutions without nodal information were stated there.

With the modifications for the case of Theorem 2.1 little needs to be added for getting Theorem 2.2.

Finally we consider the case of Theorem 1.3 corresponding to case (iii). We note again the spaceEwe employ here is a subspace of W^1,2(R^N)which contains functions radially symmetric with respect to the first (N−1) variables.

Theorem 2.3. Assume(V₁),(f₁), and(f₂)–(f₅). Suppose (∗) holds. There isr₀>0 such that for anyr∈(0, r₀) Nr

M(lj₁, . . . , ljk, A, λ)

∩(K^kc_kc⁺_−α^α/Z)= ∅

for all but finitely manyl∈N, whereN_r(·)is ther-neighborhood inE.

To establish this result we note that checking through the arguments in [11] for proving Theorem 2.1 all the procedures used in [11] can be confined in the subspace E in our case by taking care of the symmetry in the first(N−1)variables. We omit the proofs here. Theorem 2.3 is also valid ifV andf are radially symmetric in x₁, . . . , x_nand periodic inx_n+1, . . . , x_N for some 1< n < N.

By using the arguments in Proposition 7.32 of [11], one can prove that ifλ=(+1, . . . ,+1)or(−1, . . . ,−1), and for r sufficiently small and l sufficiently large, the solutions given in the above three theorems are nodal (sign-changing) solutions.

Next we state a result which is one of the main ingredients in establishing more detailed nodal property of these solutions.

Theorem 2.4. In the above three theorems, ifris sufficiently small andlsufficiently large, we may replace ther- neighborhood inEwithr-neighborhood inXwhereX=C¹(Ω) in Theorem 2.2 andX=C¹(R^N)in Theorems 2.1 and 2.3.

Remark 2.5. Theorem 2.4 says the multi-bump solutions constructed in [11] are close to the sum of translates of the one bump basic solutions not only in the Sobolev norm but also in the strongerC¹norm. This in some sense resembles Brezis and Nirenberg’s remarks on minimizers [6] as well as the remarks by Chang on minimax critical points [8,9] and suggests that the minimax procedure of Coti Zelati and Rabinowitz [11] can be posed in theC¹ topology where the multi-bump feature is more apparent.

The proof of this result is given in Section 3. Using it and some maximum principle arguments we prove Theorems 1.1, 1.2 and 1.3 in Section 4.

3. C¹-estimates

In this section, we will prove Theorem 2.4. It suffices to prove the following lemma.

Lemma 3.1. Letl_n→ ∞asn→ ∞andu_n∈K^kc_kc⁺_−α^αsuch that

nlim→∞dist_E

u_n,M(l_nj₁, . . . , l_nj_k, A, λ)

=0.

Then

nlim→∞dist_X

u_n,M(l_nj₁, . . . , l_nj_k, A, λ)

=0.

Proof. We will only prove the result in the case of R^N. The other case can be treated similarly. SinceAis a finite set, without loss of generality we may assume that asn→ ∞

u_n−

k

i=1

λ_iτ_lnjiv_i →0

for some{v₁, . . . , v_k} ⊂A. Denote forn=1,2, . . . , w_n=u_n−_k

i=1λ_iτ_lnjiv_i. Since

−(λ_iτ_lnjiv_i)+V (x)(λ_iτ_lnjiv_i)=f (x, λ_iτ_lnjiv_i) and

−u_n+V (x)u_n=f (x, u_n), wnsatisfies

−wn+V (x)wn=f (x, un)− k

i=1

f (x, λiτl_nj_ivi)=z⁽¹⁾_n +z_n⁽²⁾, (3)

where

z_n⁽¹⁾:=f (x, u_n)−f

x, k

i=1

λ_iτ_lnjiv_i

= 1 0

f_u

x, k

i=1

λ_iτ_lnjiv_i+sw_n

ds·w_n (4)

and

z_n⁽²⁾:=f

x, k

i=1

λ_iτ_lnjiv_i

− k

i=1

f (x, λ_iτ_lnjiv_i). (5)

The rest of the proof will be divided into four steps.

Step 1: For anyr >2,

R^N|w_n|^rdx→0 asn→ ∞. WithV₀=infV (x) >0, by(f₂)and(f₃), for allx∈R^N andu∈R,

f_u(x, u)V0

2 +C₁|u|^p−2. (6)

Here and in the sequel,Ci stands for positive constants independent ofnandx. For anyα >0, multiplying (3) with|wn|^αwnand integrating over R^N, we have

R^N

|∇w_n|^α/2w_n²+V (x)|w_n|^α+2

dxC₂

R^N

|z⁽¹⁾_n | + |z⁽²⁾_n |

|w_n|^α+1dx. (7)

Since (4) and (6) imply

|z⁽¹⁾_n |V₀

2 |w_n| +C₃

|u_n|^p−2+ k

i=1

|τ_lnjiv_i|^p−2

|w_n|, it follows from (7) that

R^N

∇

|w_n|^α/2w_n²+

V (x)−V₀ 2

|w_n|^α+2

dx

C₄(α)

R^N

|u_n|^2∗dx ^p_2∗⁻²

+ k

i=1

R^N

|v_i|^2∗dx ^p_2∗⁻²

R^N

|w_n|^{(α+2)22∗−p+2∗}dx ^2∗−_2∗^p+2

+

R^N

|z⁽²⁾_n |2∗+(α+1)(p−2)^(α+2)2∗ dx

^2∗+_(α+2)2∗^{(α+1)(p−2)}

R^N

|w_n|^{(α+2)22∗−p+2∗} dx

^{(α+1)(2∗−}_(α+2)2∗^p+2) .

Noting that{un}is bounded and{z⁽²⁾_n L^r(R^N)}has a bound independent ofnfor anyr >2 and using Sobolev inequality, we see that

R^N

|w_n|^(α+22)2∗dx _2∗²

C₅(α)

R^N

|w_n|^{(α2∗−p+2+2)2∗}dx ^2∗−p+2_2∗

+

R^N

|w_n|^{(α2∗−p+2+2)2∗} dx

(α+1)(2∗−p+2) (α+2)2∗

. (8)

Choosingα=α₁:=2^∗−pin (8) and using Sobolev inequality again, we have

R^N

|wn|^{(2∗−p2+2)2∗}dx _2∗²

C6

wn^2∗−p+2+ wn^(α1+1)(2∗−p+2) α1+2

, (9)

while choosingα=α_m:=2(^2∗−₂^p+2)^m−2 in (8) yields for any positive integerm,

R^N

|w_n|^{(2∗−p2+m2)m2∗}dx _2∗²

C₇

R^N

|w_n|^{(2∗−p2+m2)m−12∗−1} dx ^2∗−_2∗^p+2

+

R^N

|w_n|^{(2∗−p2+m−12)m−12∗}dx

^{(αm+1)(2∗−p+2)}

(αm+2)2∗

. (10)

Sincewn →0 asn→ ∞and(2^∗−p+2)^m/2^m→ ∞asm→ ∞, an iterative process based on (9) and (10) shows that

R^N|wn|^rdx→0 asn→ ∞for anyr >2.

Step 2: For anyr >2,

R^N|z⁽¹⁾n |^rdx→0 asn→ ∞. Indeed, by (4) and(f3),

|z⁽¹⁾_n |C₈

|w_n| + |w_n|^p−1 . Thus the result in step 1 implies that

R^N|zn⁽¹⁾|^rdx→0 for anyr >2.

Step 3: For anyr >2,

R^N|z⁽²⁾_n |^rdx→0 asn→ ∞. For any >0, chooseR >0 such that for alli∈ {1, . . . , k},

R^N\BR(0)

|vi|^rdx < . (11)

EnlargingRif necessary, we can also assume that|v_i(x)|<1 for alli∈ {1, . . . , k}andx∈R^N\B_R(0). Thus for x∈G(R, n):=R^N\_k

i=1B_R(l_nj_i),

k

i=1

λ_iτ_lnjiv_i < k.

Since|f (x, u)|C9|u|for|u|k, from (5) we see that

G(R, n)

|z⁽²⁾_n |^rdx=

G(R, n)

f

x,

k

i=1

λ_iτ_lnjiv_i

− k

i=1

f (x, λ_iτ_lnjiv_i)

r

dx

C₁₀ k

i=1

G(R, n)

|τ_lnjiv_i|^rdx. (12)

Combining (11) and (12) yields

G(R, n)

|z⁽²⁾_n |^rdxC₁₁. (13)

ChooseN₀such that ifnN₀thenl_n|j_i −j_m|>2Rfor anyi=m. On each ballB_R(l_nj_m)withm∈ {1, . . . , k} andnN₀, we have from (5) again

BR(lnjm)

|z⁽²⁾_n |^rdx2^r

BR(lnjm)

f

x,

k

i=1

λ_iτ_lnjiv_i

−f (x, λ_mτ_lnjmv_m)

r

dx

+2^r

BR(lnjm)

i=m

f (x, λiτl_nj_ivi) ^rdx

2^r

BR(lnjm)

1

0

f_u

x, λ_mτ_lnjmv_m+s

i=m

λ_iτ_lnjiv_i

ds

r

i=m

λ_iτ_lnjiv_i ^rdx +2^r

BR(lnjm)

i=m

f (x, λ_iτ_lnjiv_i) ^rdx.

Since foruin a bounded interval,f_u(x, u)is bounded and|f (x, u)|C₁₂|u|, it follows that

BR(lnjm)

|z⁽²⁾_n |^rdxC₁₃

i=m

BR(ln(jm−ji))

|v_i|^rdx.

Then the fact thatB_R(l_n(j_m−j_i))∩B_R(0)= ∅and (11) imply that k

m=1

BR(lnjm)

|z⁽²⁾_n |^rdxC₁₄. (14)

That

R^N|z⁽²⁾_n |^rdx→0 asn→ ∞follows from (13) and (14).

Step 4: We complete the proof here. By the regularity theory of elliptic equations [14], for anyx∈R^N, w_nW^2,2N(B₁(x))C₁₅

w_nL^2N(B₂(x))+ z⁽¹⁾_n L^2N(B₂(x))+ z_n⁽²⁾L^2N(B₂(x))

C₁₅

wnL^2N(R^N)+ z_n⁽¹⁾L^2N(R^N)+ z⁽²⁾_n L^2N(R^N)

. By Sobolev inequality,

wn_C1(B₁(x))C16wnW^2,2N(B₁(x)). Thus, sincex∈R^Nis arbitrary,

w_nC¹(R^N)C₁₇

w_nL^2N(R^N)+ z⁽¹⁾_n L^2N(R^N)+ z⁽²⁾_n L^2N(R^N)

. Then the results from steps 1–3 imply

nlim→∞w_nC¹(R^N)=0, finishing the proof. 2

We remark that the proof above does not use the condition off being odd.

4. Proofs of the main results

In the following we referkas the number of bumps of the solutions constructed in Section 2. The following lemma is true in all three cases we consider.

Lemma 4.1. Ifris small enough andlis large enough then, for the solutionsugiven in Section 2, the number of nodal domains is bounded above by the number of bumps of the solutions.

Proof. Let infV (x)V₀>0. Then there isδ >0 such that for|t|δ,|f (x, t )|V₀|t|/2. SinceAis finite there isR₀>0 such thatv(x)δ/2kfor allv∈Aand|x|R₀. In cases (i) and (iii), we have inf_BR

0(0)v(x)a₀>0 for allv∈Aand for somea₀>0. LetF_l=_k

i=1B_R0(lj_i). Supposeu∈N_r(M(lj₁, . . . , lj_k, A, λ))∩K_kc^kc₋⁺_α^α be

ak-bump solution given in Section 2. By Theorem 2.4, ifr is small enough andl is large enough then for any i∈ {1, . . . , k}eitheru|BR0(lji)>0 oru|BR0(lji)<0, and max|u|F_l^c|< δwhereF_l^c=R^N\F_l. Thenu|Fl hasknodal domains. We claim any pointxinF_l^cfor whichu(x)=0 has to be connected in{x∈R^N: u(x)=0}to some nodal domains ofu|Fl. If this is not the case,u|F_l^chas a nodal domainDwhich is not connected in{x∈R^N:u(x)=0}to any of the nodal domains ofu|Fl, thereforeDitself becomes a nodal domain ofu. Sinceudecays to 0 at infinityu has a maximum or a minimum inD, for example, a maximum atx₀. But sinceu(x₀) < δ,|f (x₀, u(x₀))|^V₂⁰u(x₀), andu(x₀)0, we get a contradiction with the equation.

In case (ii), we have ^∂v_∂ν(x)=0 on∂Ω and|^∂v_∂ν(x)|has a positive lower bound onB_R0(0)∩∂Ω for anyv∈A, whereνis the outer unit normal to∂Ω. Again by Theorem 2.4, ifris small enough andlis large enough then for anyi∈ {1, . . . , k}eitheru|BR0(lji)∩Ω>0 oru|BR0(lji)∩Ω<0, and max|u|F_l^c∩Ω|< δ. The rest is the same. 2

Theorem 1.1 follows from Lemma 4.1 right away.

Proof of Theorem 1.2. Let km2 and k₁, k₂, . . . , k_m be fixed positive integers such that_m

s=1k_s =k. In this case the problem is Z-invariant. Denote i₀=0 and i_n =_n

s=1k_s for n∈ {1, . . . , m}. We may assume in Theorem 2.2,j₁< j₂<· · ·< j_k and we takeλ=(λ₁, . . . , λ_k)∈Λ^k such that the firstk₁ components are+1, the nextk₂ components are−1, and keep taking alternating signs forλ_i in this way. That is,λ_i =(−1)ⁿ⁺¹for i∈ {i_n−1+1, . . . , i_n}andn∈ {1, . . . , m}. We claim that with these choices the solutions given in Theorem 2.2 for sufficiently smallr and sufficiently largelhave exactlymnodal domainsD₁, . . . , D_mwithu|Di being a positive (negative, resp.)k_i-bump solution foriodd (even, resp.). First, there isδ >0 such that|f (x, t )|^V0₂^|t| for|t|δ.

Then there isR₀>0 such thatv(x)δ/2kfor allv∈Aand forx∈Ω\ω× [−R₀, R₀]. For anyv∈A,v(x) >0 forx∈ω× [−R₀, R₀]and ^∂v_∂ν(x) <0 forx∈∂Ω∩(R^N−1× [−R₀, R₀]). By Theorem 2.4, ifr is small enough andl is large enough then |u(x)|< δ for x ∈Ω \(ω×_k

i=1[−R₀+lj_i, R₀+lj_i])and(−1)ⁿ⁺¹u(x) >0 for x ∈ω× [−R₀+lj_i, R₀+lj_i] and fori∈ {i_n−1+1, . . . , i_n} andn∈ {1, . . . , m}. Thus u has at least mnodal domains. To see that u has exactlym nodal domains, it suffices to prove that (−1)ⁿ⁺¹u(x) >0 for x ∈ω× [−R₀+lj_in−1+1, R₀+lj_in]for all n∈ {1, . . . , m}and that any nodal domain ofu contains one of the msets ω× [−R₀+lj_in−1+1, R₀+lj_in]. And it is in turn sufficient to prove that there is no nodal domain ofucontained completely inΩ\(ω×k

i=1[−R0+lji, R0+lji]). Arguing indirectly, we assume thatΩ^∗is a nodal domain ofu which is contained completely inΩ\(ω×_k

i=1[−R₀+lj_i, R₀+lj_i])and assumeu|Ω^∗>0. Assumeu|Ω^∗attains its maximum atx^∗. Sinceu|Ω^∗< δ, we have−u(x^∗)0,V (x^∗)u(x^∗)V₀u(x^∗), andf (x^∗, u(x^∗)) <^V₂⁰u(x^∗), but this is a contradiction with the equation. 2

Proof of Theorem 1.3. Let k2 be fixed. In this case the problem again is Z-invariant, we may assume in Theorem 2.3,j1< j2<· · ·< jk and we takeλ=(λ1, . . . , λk)∈Λ^ksuch thatλi takes alternating+1 and−1 with λ₁= +1, that is,λ_i = +1 fori odd andλ_i = −1 fori even. By Lemma 4.1, a solutionugiven in Theorem 2.3 has at mostk nodal domains. We argue in the following thatu has at least[^k₂] +1 nodal domains. Again, there is R₀>0 such that for r small andl large and for some a₀>0, u(x) > a₀ for x∈B_R0(lj_i) when i is odd, u(x) <−a₀for x∈B_R0(lj_i)wheniis even, and any pointx in R^N for whichu(x)=0 has to be connected in {x∈R^N: u(x)=0}to one of the ballsB_R0(lj_i). Herelj_iis understood as(0, . . . ,0, lj_i). A useful observation here is that if 1i < nkandB_R0(lj_i)andB_R0(lj_n)are subsets of one nodal domain in which uis positive, then BR₀(ljp)withi < p < nandBR₀(ljq)withq < iorq > ncannot be subsets of one nodal domain in whichuis negative. This is a consequence of the fact thatuis radially symmetric in the first(N−1)variables. Similarly, if 1i < nkandBR₀(lji)andBR₀(ljn)are subsets of one nodal domain in whichuis negative, then BR₀(ljp) withi < p < nandB_R0(lj_q)withq < i orq > ncannot be subsets of one nodal domain in whichuis positive.

Now, we use an induction procedure to prove the result. With our selected configurations of positive and negative bumps, we may identify the bump atB_R0(lj_i)to the integeriso that positive bumps correspond to odd numbers and negative bumps correspond to even numbers. With this identification, two integersi, n∈ {1, . . . , k}are said