Remarks on some fourth-order boundary value problems with non-monotone increasing

Remarks on some fourth-order boundary value problems with non-monotone increasing

eigenvalue sequences

Chengyue Li

and Yuhan Wu

Department of Mathematics, Minzu University of China, Zhongguancun South Avenue, Beijing, 100081, P.R. China

Received 25 November 2015, appeared 8 January 2016 Communicated by Dimitri Mugnai

Abstract. By Z2-index theory, the existence and multiplicity of solutions for some fourth-order boundary value problems

(u⁽⁴⁾+au⁰⁰=µu+Fu(t,u), 0<t<L, u(0) =u(L) =u⁰⁰(0) =u⁰⁰(L) =0

at resonance are studied, wherea>0 andµ∈_Ris an eigenvalue of the corresponding eigenvalue problem. The difficulty caused by the non-monotone eigenvalue sequence is handled concretely.

Keywords:fourth-order differential equation, boundary value problem, resonance, crit- ical point,Z2-index theory.

2010 Mathematics Subject Classification: 58E05, 34C37, 70H05.

1 Introduction

In order to study physical, chemical and biological systems, one has considered some fourth- order semilinear differential equation models, such as the extended Fisher–Kolmogorov equa- tion [4], the Swift–Hohenberg equation [10], suspended beam equations [2], etc. In the present paper, we are concerned with the existence and multiplicity of solutions for boundary value problems of fourth-order differential equations

(u⁽⁴⁾+au⁰⁰ =µu+Fu(t,u), 0<t< L,

u(0) =u(L) =u⁰⁰(0) =u⁰⁰(L) =0 (1.1) where a>0, µis a real parameter, F(t,u)∈ C¹([0,L]×_R,R), F_u(t,u)denotes the gradient of F(t,u)with respect to the variableu. If F(t,u)satisfies lim_|u|→∞F(t,u)/u²=_{0, we say} F(t,u) is sublinear at infinity.

BCorresponding author. Email: cunlcy@163.com

We first observe that the corresponding eigenvalue problem (u⁽⁴⁾+au⁰⁰ =λu,

u(₀) =u(L) =u⁰⁰(₀) =u⁰⁰(L) =₀ ^(1.2) has the eigenvalues

λ_k = kπ

L 4

−a kπ

L 2

, k=1, 2, 3, . . . (1.3)

and the eigenfunctions

u_k =sinkπt

L , k=1, 2, . . . (1.4)

One says (1.1) is resonant at infinity ifµ= λ_k andF(t,u)is sublinear at infinity.

On the other hand, for fixed a>0, we can find that from (1.2)

(i) ifLis extremely small such that(^π_L)² >a, then 0<λ₁<λ₂ <λ₃ <· · · →_∞;

(ii) ifLis suitable large such that ₂^a <(^π_L)² <a, thenλ₁<0<λ₂ <λ₃ <· · · →_∞;

(iii) if Lis sufficiently large , then there exists two integers ˆk ≥ 2 and ˆlsatisfying (^kπˆ_L)²< ₂^a and 0>λ₁ >λ₂> · · ·>λ_ˆk,λ_kˆ <λ_ˆk+1 <· · ·<λ_ˆk+lˆ<0<λ_kˆ+ˆl+1 <λ_kˆ+lˆ+2· · · →_∞.

We see easily that for the case (i) and (ii), the eigenvalue sequence{λ_k}increases strictly. In the last two decades, much of the research in critical point theory has examined the existence and multiplicity of solutions of (1.1), so, we shall focus on the most complicated case (iii) with non-monotone increasing eigenvalue sequence. For the sake of simplicity, we assume that λ_k1 6=λ_k2 ifk₁6= k₂ in case (iii) throughout this paper.

Up to now there have been a vast of literature about superlinear (1.1), namely, F(t,u) satisfies lim_|u|→∞F(t,u)/u² =∞; we refer the reader to [1,5,6,11,12], and references therein.

However, for the sublinear (1.1), only a few attempts have been done. In [9], Liu considered the existence of solutions with L = 1, µ = −a²/4. In [7], Han–Xu supposed L = 1, a = µ = 0, F(t,u) < γ|u|²+β (0 < γ < π²/2) and m⁴π⁴ < f_u(t, 0) < (m+1)⁴π⁴ with m ≥ 1 and proved the existence of three solutions. In [13], Yang–Zhang assumed L = 1, aπ⁴+ µ/π² < 1, u f(t,u)−2F(t,u) → _∞ (|u| → ∞), and discussed the existence of two solutions at resonance, basing on combining the minimax methods and the Morse theory. In [8], Li–

Wang–Xiao used the Clark theorem to prove the following result.

Theorem 1.1. Consider the problem

(u⁽⁴⁾(t)−V_u(t,u(t)) =0, 0< l<L,

u(0) =u(L) =u⁰⁰(0) =u⁰⁰(L) =0. (1.5) Let V(t,u)∈C¹([0,L]×_R,R)satisfy

(V1) V(t,u) =V(t,−u),∀t∈_R, u∈_R; (V2) there exist m>0, b>0such that L< ^√₄^π

m and

V(t,u)≤ ^m

2|u|²+b, t ∈[0,L], u∈_R; (1.6)

(V3) there exist p∈_Nand constants M>0,ρ>0such that M>mρ⁴, L>ρπ/√⁴ m and V(t,u)≥ ^M

2 |u|², ∀t∈ [0,L], |u| ≤ρ√

p. (1.7)

Then, for each L∈^√₄^πp

M, √4^π

m

,(1.5)has at least p distinct pairs(u(t),−u(t))of solutions.

Motivated mainly by the above papers [7–9,13], we obtain the existence and multiplicity results for (1.1) as stated in the following.

Theorem 1.2. Suppose that there exists some integer k such that 1 ≤ k ≤ k, and^ˆ µ = λ_k. Assume that F(t,u)∈C¹([0,L]×_R,R)and f(t,u) =F_u(t,u)satisfy that

(f1) F(t,u) =F(t,−u), ∀t∈[0,L], u∈_R;

(f2) there exist K₁,K₂>0andα∈[0, 1)such that

|f(t,u)| ≤K₁|u|^α+K₂, ∀t ∈[_0,L]_, u ∈_{R; (1.8)} (f3) there exist an integer pˆ >1, and M>0, p>0such thatλ_kˆ+lˆ+pˆ < M+λ_kˆ and

F(t,u)≥ ^M

2 |u|², ∀t∈ [0,L], |u| ≤ p; (1.9) (f4) lim inf

u=csin^jπ_L,|c|→_∞ Z _L

0 F(t,u(t))dt/|c|^α → −_∞, ∀j≥1.

Then(1.1)possesses at least(k^ˆ+l^ˆ+pˆ)−(k+1)distinct pairs(u(t),−u(t))of solutions, where, if λ_k >min{λ_j}_j≥1, then k satisfieskˆ+1≤k+k≤ ^ˆk+l, and^ˆ λ_j <λ_k if and only if k+1≤ j≤k+k (k is unique); ifλ_k =min{λ_j}_j≥1, then k=0.

Corollary 1.3. In Theorem1.2, if the condition µ= λ_k is replaced byλ_k < µ < λ_k+1, and (f4)is omitted, then the same conclusion still holds.

Remark 1.4. In Theorem1.2and Corollary 1.3, if the integer kis between ˆk+1 and ˆk+l^ˆ, or k > ^ˆk+l, then the similar results are still true. However, the details of their proofs must be^ˆ adapted. Evidently, one can also handle case (i) and (ii) by the same arguments used by us in (iii), moreover, the process seems easier than that of (iii). In addition, it is also not hard to see that Theorem1.1is equivalent to the special situation ofa =0, µ<λ₁ in Corollary1.3.

This paper is organized as follows. In Section 2, we will prove some lemmas. In Section 3, the proof of Theorem1.2shall be given by the following Z₂-type index theorem.

Theorem 1.5. Let Y be a Banach space and the functional ϕ ∈ C¹(Y,R) be even satisfying the Palais–Smale condition. Suppose that

(i) there exist a subspace V of Y with V=r andδ >0such thatsup_w∈V,kxk=δϕ(w)< ϕ(0); (ii) there exists a closed subspace W of Y with W =s<r such thatinfw∈Wϕ(w)>−_∞. Then f possesses at least r−s distinct pairs(u,−u)of critical points.

For the convenience of the reader, let us recall that the functional ϕis said to satisfy the Palais–Smale condition if any sequence {u_j}in Y is such that ϕ(u_j) is bounded, ϕ⁰(u_j)→ 0, possesses a convergent subsequence.

2 Preliminary

Let X= H²(0,L;R)∩H₀¹(0,L;R)be a Hilbert space with the inner product (u,w) =

Z _L

0

[u⁰⁰(t)w⁰⁰(t) +u⁰(t)w⁰(t) +u(t)w(t)]dt, ∀u,w∈ X, (2.1) and the corresponding norm

kuk_X = (u,u)¹² = Z _L

0

|u⁰⁰|²+a|u⁰|²+|u|²dt ¹₂

. (2.2)

Setting

kuk= _{Z L}

0

|u⁰⁰|²dt ¹₂

, (2.3)

we infer from the Poincaré inequality Z _L

0

u²dt≤ ^L

4

π⁴ Z _L

0

(u⁰⁰)²dt, (2.4)

Z _L

0 u²dt=

Z _L

0 u⁰du=−

Z _L

0 uu⁰⁰dt

≤ ¹ 2

Z _L

0

(u²+ (u⁰⁰)²)dt

≤ ¹ 2

L⁴ π⁴ +1

_{Z L}

0

(u⁰⁰)²dt, (2.5)

sok · k_Xis equivalent with k · k.

It is well known that solutions of (1.1) are exactly the critical points of the corresponding functional given by

I(u) = ¹ 2

Z _L

0

|u⁰⁰(t)|²−a|u⁰(t)|²−µ|u|²dt−

Z _L

0 F(t,u)dt (2.6) onX. Direct computation shows, for∀u,w∈ X,

I⁰(u)w=

Z _L

0

u⁰⁰(t)w⁰⁰(t)−au⁰(t)w⁰(t)−µu(t)w(t)dt−

Z _L

0 f(t,u)vdt. (2.7) Obviously, there is an orthogonal basis on(X, k · k)as follows

sinπ

Lt, sin2π

L t, sin3π L t, . . .

. (2.8)

Lete_j =sin^jπ_L,s_j = (^jπ_L)⁴, j≥1, then Z _L

0

|e⁰⁰_j(t)|²dt= ^L 2

jπ L

4

=s_j Z _L

0

|e_j(t)|²dt. (2.9) Definev_j =^q_Ls²

je_j, we havekv_jk=1, and Z _L

0

h|v⁰⁰_j(t)|²−a|v⁰_j(t)|²ⁱdt= ² Ls_j

Z _L

0

h|e⁰⁰_J(t)|²−a|e⁰_j(t)|²ⁱdt= ^λj

s_j , (2.10)

Z _L

0

|v_j(t)|²dt= ² Ls_j

Z _L

0

|e_j(t)|²dt= ² Ls_j · ^L

2 = ¹

s_j . (2.11)

From (2.10) and (2.11), we obtain Z _L

0

h|v⁰⁰_j(t)|²−a|v⁰_j(t)|²ⁱdt=λ_j Z _L

0

|v_j|²dt. (2.12)

Lemma 2.1. Under the assumptions of Theorem1.2, there exists a norm k · k_∗ equivalent with k · k on X; X has an orthogonal decomposition X = X⁺⊕X⁻⊕X⁰, and the functional I(u)in(2.6)is of the form

I(u) = ¹

2 ku⁺k²_∗− ku⁻k²_∗−

Z _L

0 F(t,u)dt, ∀u∈X, (2.13) where u=u⁺⊕u⁻⊕u⁰, u⁺∈ X⁺, u⁻∈ X⁻, u⁰∈ X⁰.

Proof. Consider two cases. Case (i): µ = λ_k > _min{λ_j}_j≥1. For this, there exists a unique integerksuch that ˆk+1≤k+k≤k^ˆ+l, and^ˆ λ_j < λ_k if and only ifk+1≤ j≤ k+k. Define

I₁(u) =

Z _L

0

|u⁰⁰(t)|²−a|u⁰(t)|²−µ|u|²dt. (2.14) For∀u∈ X,u=_∑^∞_j=1α_jv_j, we getkuk²=_∑^∞_j=1α²_j, and

I₁(u) =

Z _L

0





∑

α_jv_j

!00

2

−a

∑

α_jv_j

!0

2

−λ_k

∑

α_jv_j

!2

dt

=

∑

α²_j

Z _L

0

|v⁰⁰_j|²dt−aα²_j Z _L

0

|v⁰_j|²dt−λ_kα²_j Z _L

0

|v_j|²dt

=

∑

λ_j−λ_k s_j

α²_j

=

k−1

∑

λ_j−λ_k s_j

α²_j +

k+k j=

∑

λ_j−λ_k S_j

α²_j +

∑

λ_j−λ_k s_j

α²_j. (2.15) We define

u⁺=

k−1 j

∑

α_jv_j+

∑

α_jv_j, u⁻=

k+k j=

∑

a_jv_j, u⁰= a_kv_k, (2.16) and

X⁺=Span{v_j |1≤ j≤ k−1 or j≥k+k+1}, X⁻=Span{v_j |k+1≤ j≤k+k},

X⁰ =Span{v_k}_,

(2.17)

then we derive

u=u⁺+u⁻+u⁰, u⁺∈ X⁺, u⁻ ∈X⁻, u⁰ ∈X⁰, X=X⁺⊕X⁻⊕X⁰. (2.18)

Moreover, one can estimate the terms in (2.15) as follows:

k−1

∑

λ_j−λ_k s_j

α²_j +

∑

λ_j−λ_k s_j

α²_j

≥

λ_k−1−λ_k s_j

_k−1 j

∑

α²_j + ^λk+k+1−λ_k s_k+k+1

! _∞

∑

j=k+k+1

α²_j, (2.19)

k−1

∑

λ_j−λ_k s_j

α²_j +

∑

λ_j−λ_k s_j

α²_j ≤

λ₁−λ_k s₁

_k−1

∑

α²_j +

∑

α²_j, (2.20) and

k+k j=

∑

λ_k−λ_j s_j

α²_j ≥ ^µ

1 k

s_k+k

k+k j=

∑

α²_j, (2.21)

k+k j=

∑

λ_k−λ_j s_j

α²_j ≤ ^µ

2 k

s_k+1

k+k j=

∑

α²_j, (2.22)

with

µ¹_k = min

k+1≤j≤k+k

{λ_k−λ_j}>0, µ²_k = max

k+1≤j≤k+k

{λ_k−λ_j}>0. (2.23) Combining (2.19)–(2.20) with (2.21)–(2.22), we can define a new normk · k_∗ on Xby

kuk²_∗ =

k−1 j

∑

λ_j−λ_k s_j

α²_j +

k+k j=

∑

λ_j−λ_k S_j

α²_j +

∑

λ_j−λ_k s_j

α²_j +α²_k, (2.24) equivalent withk · k. Ifu= _∑^∞_j=1α_jv_j,w =_∑^∞_j=1β_jv_j ∈X, then the inner product correspond- ing tok · k_∗ is

hu, wi_∗ =

k−1 j

∑

λ_j−λ_k s_j

α_jβ_j+

∑

λ_j−λ_k s_j

α_jβ_j

+

k+k j=

∑

λ_k−λ_j s_j

α_jβ_j+α_kβ_k. (2.25)

From (2.24) one gets ku⁺k²_∗=

k−1 j

∑

λ_j−λ_k s_j

α²_j +

∑

λ_j−λ_k s_j

α²_j,

ku⁻k²_∗=

k+k j=

∑

λ_k−λ_j s_j

α²_j, (2.26)

thus

I(u) = ¹

2 ku⁺k²_∗− ku⁻k²_∗−

Z _L

0 F(t,u)dt, (2.27)

I⁰(u)w=hu⁺, wi_∗− hu⁻, wi_∗−

Z _L

0 f(t,u)wdt, ∀u, w∈X. (2.28)

Case (ii): µ = λ_k = min{λ_j}_j≥1. Under this assumption, for ∀u ∈ X,u = _∑^∞_j=1α_jv_j, we also have

I₁(u) = ¹ 2

Z _L

0





∑

α_jv_j

!00

2

−a

∑

α_jv_j

!0

2

−λ_k

∑

α_j|v_j|

!2

dt

=

∑

α²_j

Z _L

0

|v⁰⁰_j|²dt−aα²_j Z _L

0

|v⁰_j|²dt−λ_kα²_j Z _L

0

|v_j|²dt

=

∑

λ_j−λ_k s_j

α²_j. (2.29)

Let

u⁺=

∑

α_jv_j, u⁰ =α_kv_k, (2.30)

and

X⁺ =Span{v_j | j6= k}, X⁰=Span{v_k}, (2.31) then we conclude

u=u⁺+u⁰, u⁺∈ X⁺, u⁰∈ X⁰, X=X⁺⊕X⁰, (2.32) and

λ₁−λ_k s₁

_k−1

j

∑

α²_j +

∑

α²_j ≥

∑

j6=k

λ_j−λ_k s_j

α²_j

≥

λ_k−1−λ_k s_k−1

_k−1 j

∑

α²_j +

λ_k+1−λ_k s_k+1

_∞

j=

∑

α²_j, (2.33) which implies a new normk · k_∗ as follows:

kuk²_∗=

∑

j6=k

λ_j−λ_k s_j

α²_j +α²_k. (2.34)

equivalent withk · k. Obviously, one obtains ku⁺k²_∗ =

∑

j6=k

λ_j−λ_k s_j

α²_j. (2.35)

Consequently, we also have the same results as (2.27)–(2.28) withu⁻=0 .

Lemma 2.2. Under the assumptions of Theorem1.2, the functional I(u)defined in(2.6)satisfies the Palais–Smale condition.

Proof. Assume that{u_m} ⊂Xsatisfy

|I(u_m)| ≤C₀, I⁰(u_m)→_{0. (2.36)} Writing u_m = u⁺_m+u⁻_m+u⁰_m,u⁺_m ∈ X⁺, u⁻_m ∈ X⁻,u⁰_m ∈ X⁰, then, formsufficiently large, one has

|I⁰(u_m)u⁺_m|=hu⁺_m,u⁺_mi_∗− hu_m , u⁺_mi_∗−

Z _L

0 f(t,u⁺_m)u⁺_mdx

=ku⁺_mk²_∗−

Z _L

0

f(t,u⁺_m)u⁺_mdt≤ ku⁺_mk_{∗, (2.37)}

Using(f2)and the Sobolev embedding inequality, we obtain

Z _L

0 f(t,u⁺_m)u⁺_mdt

≤K₁ Z _L

0

|u⁺_m|^α+1dt+K₂ Z _L

0

|u⁺_m|dt

≤C₁(ku⁺_mk_∗+ku⁺_mk^α_∗⁺¹), (2.38) with some constantC₁ >0. Combining (2.37) with (2.38) derives

ku⁺_mk²_∗ ≤(₁+C₁)ku⁺_mk_∗+C₁ku⁺_mk^α_∗⁺¹_{, (2.39)} thus we reduce that {u⁺_m} is bounded since 1 ≤ α+1 < 2. In the same way, {u⁻_m} is also bounded. Therefore there is aC₂>0 such that

ku_m−u⁰_mk_∗ =ku⁺_m+u⁻_mk_∗ ≤C₂, ∀m≥1. (2.40) Next, with the aid of(f2)and the mean value theorem, we easily show

Z _L

0

[F(t,um)−F(t,u⁰_m)]dx

≤

Z _L

0

f(t,u⁰_m+ξ(u_m−u⁰_m))(u_m−u⁰_m)dt(0<ξ <1)

≤C_{3 Z L}

0

(u_m−u⁰_m) u⁰_m

αdt+

Z _L

0

(u_m−u⁰_m)

α+1

dt+

Z _L

0

(u_m−u⁰_m)dt

≤C₃

ku⁰_mk^α_∞

Z _L

0

|(u_m−u⁰_m)|dt+

Z _L

0

|(u_m−u⁰_m)|^α+1dt+

Z _L

0

|(u_m−u⁰_m)|dt

≤C₄(ku⁰_mk^α_∞kum−u⁰_mk_∗+kum−u⁰_mk^α_∗⁺¹+kum−u⁰_mk_∗) (2.41) for some constantsC₃,C₄ >0. Since dimX⁰ =1, u⁰_m ∈ X⁰, we know thatku⁰_mk_∞is equivalent withku⁰_mk, it is easy to conclude by (2.40) and (2.41)

Z _L

0

F(t,u_m)−F(t,u⁰_m)dx

≤C₅ku⁰_mk^α+C₆ (2.42) for some constantsC₅,C₆ >0. By Lemma2.1, I(u_m)can be written as

c₀ ≥ I(u_m) = ¹

2 ku⁺_mk²_∗− ku⁻_mk²_∗−

Z _L

0

F(t,u_m)−F(t,u⁰_m)dt−

Z _L

0 F(t,u⁰_m)dt. (2.43) From (2.42) and (2.43), we obtain

C5ku⁰_mk^α+

Z _L

0 F(t,u⁰_m)dt≥C7 (2.44) with some constantC₇. If{u⁰_m}is unbounded, then

lim inf

m→_∞ Z _L

0 F(t,u⁰_m)dt/ku⁰_mk^α ≥ −C₅. (2.45) We note that for ∀m ≥ _1, u⁰_m can be expressed by u⁰_m = c⁰_msin^kπt_L ,c⁰_m ∈ _{R, so} ku⁰_mk^α =

|c⁰_m|^α(^L₂s_k)^α/2. Here, k is fixed, consequently, (2.45) contradicts (f4). Thus, we conclude that {u_m}is bounded onX, and, using the standard method,{u_m}has a convergent subsequence.

Lemma 2.3. Under the assumptions of Theorem1.2, the functional I(u)defined in (2.6)is bounded from below on X⁺.

Proof. According to(f2), for anyu∈X, we have

Z _L

0 F(t,u)dt

≤ ^K1 α+1

Z _L

0

|u|^α+1dt+K₂ Z _L

0

|u|dt≤C₈(kuk^α_∗⁺¹+kuk_∗) (2.46) with some constantC₈>0. In particular, ifu∈ X⁺, then forkuk_∗ →_{∞, we get}

I(u) = ¹

2kuk²_∗−

Z _L

0 F(t,u)dt≥ ¹

2kuk²_∗−C₈(kuk^α_∗⁺¹+kuk_∗)→_∞. (2.47) So I is bounded from below on X⁺.

3 Proof of the theorems

3.1 Proof of Theorem1.2

Proof. In this section, we shall prove Theorem1.2 by Theorem1.5. Define

V=Span{v₁,v₂, . . . ,vkˆ+l^ˆ+pˆ}, (3.1)

Z= S_ρ˜∩V= (

u=

kˆ+l^ˆ+pˆ j

∑

α_jv_j,α₁,α₂, . . . ,α_kˆ+lˆ+pˆ ∈_R,

ˆk+l^ˆ+pˆ j

∑

α²_j =ρ˜² )

(3.2)

with ˜ρ=^{q Ls1}

2(k^ˆ+l^ˆ+pˆ)ρ. For eachu∈ Z, according to (3.1), we have

u(t) =

kˆ+^ˆl+pˆ

∑

α_jv_j =

kˆ+l^ˆ+pˆ j

∑

α_j s 2

Ls_j sinjπt

L . (3.3)

By the Cauchy–Schwarz inequality, one shows

|u(t)|²≤ ² L





ˆk+l^ˆ+pˆ j

∑

α²_j s_j









kˆ+l^ˆ+pˆ j

∑

sin² jπt L





≤ ² L

2(k^ˆ+l^ˆ+pˆ) Ls₁

kˆ+_l^ˆ+pˆ j

∑

α²_j

= ²(^ˆk+l^ˆ+pˆ)

Ls₁ ρ˜² =ρ². (3.4)

Therefore|u(t)| ≤_ρ, ∀t∈ [_0,L]_.

Thus, condition(f3)implies that I(u) = ¹

2 Z _L

0

|u⁰⁰(t)|²−a|u⁰(t)|²−λ_k|u|²dt−

Z _L

0 F(t,u)dt

≤ ¹ 2

Z _L

0

|u⁰⁰(t)|²−a|u⁰(t)|²−λ_k|u|²dt−

Z _L

0

M 2 |u|²dt

= ¹ 2

ˆk+l^ˆ+pˆ j

∑

α²_j · ² Ls_j

"

jπ L

_{4Z L}

0

sinjπt

L ₂

dt−a jπ

L _{2Z L}

0

cos jπt

L ₂

dt

#

− ^λk+M 2

Z _L

0 ˆk+l^ˆ+pˆ

j

∑

α²_j 2 Ls_j

sin jπt

L 2

dt

= ¹ 2

ˆk+l^ˆ+pˆ j

∑

α²_j s_j

"

jπ L

4

−a jπ

L 2

−(λ_k+M)

#

. (3.5)

Again, in view of (f3), λ_kˆ+ˆl+pˆ < λ_k+ M, and the property of the eigenvalue sequence {λ_k}, we have, for 1≤j≤k^ˆ+l^ˆ+pˆ ,

jπ L

4

−a jπ

L 2

<(λ_k+M), (3.6)

hence

I(u)≤ ¹ 2

kˆ+^ˆl+pˆ

∑

α²_j s_j

"

jπ L

4

−a jπ

L 2

−(λ_k+M)

#

<0. (3.7)

Therefore, we get sup{l(u):u∈Z}<0.

The functional (2.6) satisfies all hypotheses of Theorem 1.5, therefore it has at least (k^ˆ+l^ˆ+pˆ)−(k+1) distinct pairs (u_j,−u_j) of critical points. Since I(u_j) < 0 for 1 ≤ j ≤ (k^ˆ+l^ˆ+pˆ)−(k+1), we getu_j 6=0 for 1≤ j≤(k^ˆ+l^ˆ+pˆ)−(k+1).

3.2 Proof of Corollary1.3

Proof. We first demonstrate the following result similar to Lemma 2.1. Forλ_k < µ < λ_k+1, u=_∑^∞_j=1α_jv_j, one has

I₁(u) =

Z _L

0

|u⁰⁰(t)|²−a|u⁰(t)|²−µ|u|²dt

=

Z _L

0





∑

α_jv_j

!00

2

−a

∑

α_jv_j

!0

2

−µ

∑

α_j|v_j|

!2

dt

=

∑

α²_j

Z _L

0

|v⁰⁰_j|²dt−aα²_j Z _L

0

|v⁰_j|²dt−µα²_j Z _L

0

|v_j|²dt

=

∑

λ_j

s_jα²_j − ^µ s_jα²_j

=

∑

λ_j−µ s_j

α²_j +

∑

λ_j−µ S_j

α²_j −

k+k j=

∑

µ−λ_j s_j

α²_j. (3.8)

Let

u⁺=

∑

α_jv_j+

∑

α_jv_j, u⁻=

k+k j=

∑

α_jv_j, (3.9)

X⁺ =Span{v_j |1≤ j≤k, orj≥k+k+1},

X⁻ =Span{v_j |k+₁≤j≤k+k}_, ^(3.10) then u = u⁺+u⁻, u⁺ ∈ X⁺, u⁻ ∈ X⁻, X = X⁺⊕X⁻, and I(u) = ¹₂(ku⁺k²_∗ − ku⁻k²_∗)− RL

0 F(t,u)dt.

We can continue to follow the same ideas as in Lemma2.1, Lemma2.2, and Lemma2.3to complete the proof of Corollary1.3. In addition, it is not hard to see that, for the present case of λ_k < µ<λ_k+1, condition(f4)appearing in Theorem1.2is not indispensable sinceX⁰ =0 in the orthogonal decomposition ofX. The details should be left to the reader.

Remark 3.1. For ∀β∈(0,¹₂),γ∈ [0, 1), we can take a functionH(s)∈_C¹([0,∞),R)such that s^1+2β ≤ H(s)≤s^1+β, ∀s ∈[0, 1], (3.11)

−¹

2s^γ ≤ H⁰(s)≤ −¹

4s^γ, ∀s∈[2,∞). (3.12) Let F(_t,u) = _H(|u|)[(sint)^2j+₂]_, ∀j ≥ 1. Since limu→0F(_t,u)_/|u|² = _∞ uniformly in t ∈ R, straightforward estimates show that F(t,u) satisfies (f1)–(f4) in Theorem 1.2 with α∈[γ, 1).

Acknowledgements

We would like to thank the referee.

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