Existence of infinitely many nodal solutions for a superlinear Neumann boundary value problem

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EXISTENCE OF INFINITELY MANY NODAL SOLUTIONS FOR

A SUPERLINEAR NEUMANN BOUNDARY VALUE PROBLEM

AIXIA QIAN

Received 12 January 2005

We study the existence of a class of nonlinear elliptic equation with Neumann boundary condition, and obtain infinitely many nodal solutions. The study of such a problem is based on the variational methods and critical point theory. We prove the conclusion by using the symmetric mountain-pass theorem under the Cerami condition.

1. Introduction

Consider the Neumann boundary value problem:

−u+αu=f(x,u), x∈Ω, ∂u

∂ν=0, x∈∂Ω,

(1.1)

whereΩ⊂RN₍_N_≥_{1) is a bounded domain with smooth boundary}_∂_Ω_and_{α >}_{0 is a}

constant. Denote byσ(−) := {λi|0=λ1< λ2≤ ··· ≤λk≤...}the eigenvalues of the

eigenvalue problem:

−u=λu, x∈Ω, ∂u

∂ν=0, x∈∂Ω.

(1.2)

LetF(x,s)=s

0 f(x,t)dt,G(x,s)=f(x,s)s−2F(x,s). Assume

(f1) f ∈C(Ω×R), f(0)=0, and for some 2< p <2∗=2N/(N−2) (forN=1, 2,

take 2∗= ∞),c >0 such that

_f₍_x_,_u₎_≤_c

1 +|u|p−1

, (x,u)∈Ω×R. (1.3)

(f2) There existsL≥0, such thatf(x,s) +Lsis increasing ins.

(f3) lim|s|→∞(f(x,s)s)/|s|2=+∞uniformly for a.e.x∈Ω.

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(f4) There existθ≥1,s∈[0, 1] such that

θG(x,t)≥G(x,st), (x,u)∈Ω×R. (1.4)

(f5) f(x,−t)= −f(x,t), (x,u)∈Ω×R.

Because of (f3), (1.1) is called a superlinear problem. In [6, Theorem 9.38], the author

obtained infinitely many solutions of (1.1) under (f1)–(f5) and

(AR)∃µ >2,R >0 such that

x∈Ω, |s| ≥R=⇒0< µF(x,s)≤f(x,s)s. (1.5)

Obviously, (f3) can be deduced from (AR). Under (AR), the (PS) sequence of

corre-sponding energy functional is bounded, which plays an important role for the applica-tion of variaapplica-tional methods. However, there are indeed many superlinear funcapplica-tions not satisfying (AR), for example, takeθ=1, the function

f(x,t)=2tlog1 +|t| (1.6)

while it is easy to see that the above function satisfies (f1)–(f5). Condition (f4) is from

[2] and (1.6) is from [4].

In view of the variational point, solutions of (1.1) are critical points of corresponding functional defined on the Hilbert spaceE:=W1,2₍_Ω_{). Let}_X_:_{= {u}_∈_C1₍_Ω₎_|_∂u/∂_ν₌

0,x∈∂Ω}a Banach space. We consider the functional

J(u)=1 2

Ω

|∇u|2₊_αu2_dx₋

ΩF(x,u)dx, (1.7)

whereEis equipped with the norm

u =

Ω|∇u|

2₊_α

Ωu

2

1/2

. (1.8)

By (f1),Jis ofC1and

J(u),v =

Ω(∇u∇v+αuv)dx−

Ωf(x,u)vdx, u,v∈E. (1.9)

Now, we can state our main result.

Theorem1.1. Under assumptions (f1)–(f5), (1.1) has infinitely many nodal solutions.

Remark 1.2. [8, Theorem 3.2] obtained infinitely many solutions under (f1)–(f5) and

(f3)lim|u|→∞inf(f(x,u)u)/|u|µ≥c >0 uniformly forx∈Ω, whereµ >2. (f4) f(x,u)/uis increasing in|u|.

It turns out that (f3)and (f4)are stronger than (f3) and (f4), respectively, furthermore,

the function (1.6) does not satisfy (f3), thenTheorem 1.1applied to Dirichlet boundary

value problem improves [8, Theorem 3.2].

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2. Preliminaries

Let Ebe a Hilbert space and X⊂E, a Banach space densely embedded inE. Assume thatEhas a closed convex conePEand thatP=:PE

Xhas interior points inX, that is, P=P◦ ∂P, withP◦ the interior and∂Pthe boundary ofP inX. LetJ∈C1₍_E_{,R), denote}

K= {u∈E: J(u)=0},Jc_{= {u}_∈_E_: _J₍_u₎_≤_c}_,_K

c= {u∈K: J(u)=c},c∈R.

Definition 2.1. We say thatJsatisfies Cerami condition (C), if for allc∈R

(i) Any bounded sequence{un} ⊂EsatisfyingJ(un)→c,J(un)→0 possesses a

con-vergent subsequence.

(ii) There exist σ,R,β >0 such that for any u∈J−1_([_c₋_σ_,_c₊_σ_{]) with} _{u ≥}_R_,

J₍_u₎_{u ≥}_β_.

Definition 2.2(see [3]). LetM⊂Xbe an invariant set underσ. We sayMis an admissible invariant set forJ, if (a)Mis the closure of an open set inX, that is,M=M◦ ∂M; (b) ifun=σ(tn,v) for somev∈M andun→uinEastn→ ∞for someu∈K, thenun→u

inX; (c) ifun∈K

Msuch thatun→uinE, thenun→uinX; (d) for anyu∈∂M\K, σ(t,u)∈M◦ fort >0.

In [5], we provedJ∈C1₍_E_{,R) satisfier the deformation}_{Lemma 2.3}_{under (}_PS₎

condi-tion and assumpcondi-tion (Φ):K(J)⊂X,J(u)=u−A(u) foru∈E,A:X→Xis continuous. It turns out that the same lemma still holds ifJsatisfies (C), that is.

Lemma2.3. AssumeJ∈C1₍_E_,_R)_satisfies₍_Φ₎_and₍_C₎_{condition. Let}_M_⊂_X_{be an}

admissi-ble invariant set to the pseudo-gradient flowσ ofJ. DefineK1

c =Kc ◦

M,K2

c =Kc

(X\M) for somec. Assume Kc∂M= ∅, there exitsδ >0such that(Kc1)4δ(Kc2)4δ= ∅, where

(Ki

c)4δ= {u∈E: dE(u,Kci)<4δ}fori=1, 2. Then there isε0>0, such that for any0< ε <

ε0and any compact subsetA⊂(Jc+ε

X)M, there isη∈C([0, 1]×X,X)such that (i)η(t,u)=u,ift=0oru∈J−1_([_c₋₃_ε

0,c+ 3ε0])\(K_c2)_δ;

(ii)η(1,A\(K2

c)3δ)⊂Jc−ε

M, andη(1,A)⊂Jc−ε_M_if_K2

c = ∅;

(iii)η(t,·)is a homeomorphism ofXfort∈[0, 1]; (iv)J(η(·,u))is nonincreasing for anyu∈X;

(v)η(t,M)⊂Mfor anyt∈[0, 1];

(vi)η(t,·)is odd, ifJis even andMis symmetric about the origin.

Indeed,σ > ε0>0can be chosen small, whereσis from(ii)of(C), such thatJ(u)2/(1 +

2J₍_u₎₎_≥₆_ε₀_/δ_,_∀u_∈_J−1_([_c₋₃_ε

0,c+ 3ε0])\(Kc)δ.

In [3,5], a version of symmetric mountain-pass theorem holds under (PS). (C) is weaker than (PS), but by above deformation Lemma 2.3, a version of “symmetric mountain-pass theorem” still follows.

Theorem2.4. AssumeJ∈C1₍_E_,_R)_{is even,}_J₍₀₎₌₀_satisfies₍_Φ₎_and₍_C₎

ccondition forc >

0. Assume thatPis an admissible invariant set forJ,Kc∂P= ∅for allc >0,E=∞j=1Ej,

whereEjare finite dimensional subspaces ofX, and for eachk, letYk= _k

j=1Ej andZk= _∞

j=kEj. Assume for eachk there existρk> γk>0, such thatlimk→∞ak<∞, whereak=

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pointsun∈X\(P

(−P))such thatJ(un)→ ∞asn→ ∞, providedZk

∂Bγk(0)

P= ∅ for largek.

3. Proof ofTheorem 1.1

Proposition3.1. Under (f1)–(f3) and (f4),Jsatisfies the(C)condition.

Proof. For allc∈R, since Sobolev embeddingH1₍_Ω₎_→_L2₍_Ω_{) is compact, the proof of}

(i) in (C) is trivial.

About (ii) of (C). If not, there existc∈Rand{un} ⊂H1(Ω) satisfying, asn→ ∞

Jun

−→c, un−→ ∞, J

unun−→0, (3.1)

then we have

lim

n→∞

Ω

1 2f

x,unun−Fx,undx=lim n→∞

Jun−1₂Jun,un =c. (3.2)

Denotevn=un/un, thenvn =1, that is,{vn}is bounded inH1(Ω), thus for some v∈H1₍_Ω_{), we get}

vnv inH1(Ω), vn−→v inL2(Ω), vn−→v a.e. inΩ.

(3.3)

Ifv=0, as the similar proof in [2], define a sequence{tn} ∈R:

Jtnun= max t∈[0,1]J

tun. (3.4)

If for somen∈N, there is a number oftnsatisfying (3.4), we choose one of them. For all m >0, let ¯vn=2√mvn, it follows that

lim

n→∞

ΩF

x, ¯vn

dx=lim

n→∞

ΩF

x, 2√mvn

dx=0. (3.5)

Then fornlarge enough

Jtnun

≥Jv¯n

=2m−

ΩF

x, ¯vn

dx≥m, (3.6)

that is, limn→∞J(tnun)=+∞. SinceJ(0)=0 andJ(un)→c, then 0< tn<1. Thus

Ω

∇tnun

2

+αtnun 2

−

Ωf

x,tnun

tnun

=Jtnun

,tnun =tn_dtd

t=tn Jtun

=0.

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We see that Ω ₁ 2f

s,tnun

tnun−F

x,tnun dx =1 2 Ω

∇tnun2+αtnun2

−

ΩF

x,tnun

=Jtnun−→+∞, n−→ ∞.

(3.8)

From above, we infer that

Ω

₁

2f

s,un

un−F

x,un dx =1 2 ΩG

x,un

dx≥ 1 2θ

ΩG

x,tnun dx =1 θ Ω ₁ 2f

s,tnun

tnun−F

x,tnun

dx−→+∞, n−→ ∞,

(3.9)

which contradicts (3.2). Ifv≡0, by (3.1)

Ω

∇un2+αu2n

−

Ωf

x,unun=Jun,un =o(1), (3.10)

that is,

1−o(1)=

Ω fx,un

un _u_n2 dx=

v=0+

v=0

_f_x_,_u n

un _u_n2 vn

2

dx. (3.11)

Forx∈Ω_:_{= {x}_∈_Ω_: _v₍_x₎₌₀_}_{, we get}_|u

n(x)| →+∞. Then by (f3)

fx,un(x)un(x) _u_n₍_x₎2 vn(x)

2

dx−→+∞, n−→ ∞. (3.12)

By using Fatou lemma, since|Ω|>0 (| · |is the Lebesgue measure inRN_),

v=0

fx,unun _u

n

2 vn

2

dx−→+∞, n−→ ∞. (3.13)

On the other hand, by (f3), there existsγ >−∞, such that f(x,s)s/|s|2≥γfor (x,s)∈

Ω×R. Moreover,

v=0

_v_n2

dx−→0, n−→ ∞. (3.14)

Now, there existsΛ>−∞such that

v=0

fx,un

un _u_n2 vn

2

dx≥γ

v=0

_v_n2

dx≥Λ>−∞, (3.15)

together with (3.11) and (3.13), it is a contradiction.

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Proposition3.2. Under(f4), then for|t| ≥ |s|andts≥0,G(x,t)≥G(x,s), that is, (f4)

holds forθ=1. Proof. for 0≤s≤t,

G(x,t)−G(x,s)=2

₁

2

f(x,t)t−f(x,s)s−F(x,t)−F(x,s)

=2

_t

0

f(x,t)

t τdτ−

_s

0

f(x,s)

s τdτ−

_t

s f(x,τ)

τ τdτ

=2

_t

s

f(x,t)

t −

f(x,τ) τ

τdτ+

_s

0

f(x,t)

t −

f(x,s) s

τdτ

≥0. (3.16)

In like manner, fort≤s≤0,G(x,t)−G(x,s)≥0.

OnE:=H1₍_Ω_{), let us define}_P

E= {u∈E: u(x)≥0, a.e. inΩ}, which is a closed

convex cone. LetX=C1

ν(Ω), which is a Banach space and embedded densely inE. Set P=PEX, then P is a closed convex cone inX. Furthermore, P=

◦

P∂P under the topology of X, that is, there exist interior points inX. We may define a partial order relation:u,v∈X,u > v⇔u−v∈P\{0},uv⇔u−v∈P◦.

As the proof of those propositions in [5, Section 5], it turns out that conditionΦis satisfied andPis an admissible invariant set forJunder (f1), (f2), and (C) condition.

Proof ofTheorem 1.1. Let Ei=ker(− −λi),Yk=ik=1Ei and Zk=∞i=kEi. It shows

thatJis continuously diﬀerentiable by (f1) and satisfies the (C)ccondition for everyc∈R

byProposition 3.1.

(1) As the proof of [7, Theorem 3.7(3)], there existsγk>0 such that foru∈Zk,u = γk, we have

bk:= inf Zk∂B_γk(0)

J(u)−→ ∞, k−→ ∞. (3.17)

(2) Since dimYk<+∞and all norms are equivalent on the finite dimensional space,

there existsCk>0, for allu∈Yk, we get

1 2

Ω

|∇u|2₊_αu2₌1

2u

2_≤_C

k|u|22≡Ck

Ω|u|

2_dx. _(3.18)

Next, by (f3), there existsRk>0 such thatF(x,s)≥2Ck|s|2for|s| ≥Rk. TakeMk:=

max{0, inf|s|≤RkF(x,s)}, then for all (x,s)∈Ω×R, we obtain

F(x,s)≥2Ck|s|2−Mk. (3.19)

It follows from (3.18) and (3.19) that, for allu∈Yk

J(u)=1 2

Ω

|∇u|2₊_αu2₋

ΩF(x,u) ≤ −Ck|u|22+Mk|Ω| ≤ −1₂u2+Mk|Ω|,

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which implies that forρklarge enough (ρk> γk),

ak:= max Yk∂B_ρk(0)

J(u)≤0. (3.21)

Moreover, fork≥2,Zk

P= {0}. This can be seen by noting that for allu∈P\{0},

Ωuφ1(x)dx >0, while for u∈Zk,

Ωuφ1(x)dx=0, whereφ1 is the first eigenfunction

corresponding toλ1, which impliesZk∂Bγk(0)

P= ∅.

By Theorem 2.4, J has a sequence of critical points un∈X\(P(−P)) such that J(un)→ ∞asn→ ∞, that is, (1.1) has infinitely many nodal solutions.

Example 3.3. ByTheorem 1.1, the following equation withα >0

−u+αu=2ulog1 +|u|, x∈Ω, ∂u

∂ν=0, x∈∂Ω

(3.22)

has infinitely many nodal solutions, while the result cannot be obtained by either [6, Theorem 9.12] or [8, Theorem 3.2].

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Aixia Qian: Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, China