2 Statement of the Problem and Main Result

www.i-csrs.org

Available free online at http://www.geman.in

Resonance Problem of a Class of Quasilinear Parabolic Equations

Wang Zhong-Xiang¹, Jia Gao² and Zhang Xiao-Juan³

1,2,3College of Science, University of Shanghai for Science and Technology Shanghai, China

1E-mail: [email protected]

2E-mail: [email protected]

3E-mail: [email protected] (Received: 4-9-14 / Accepted: 21-10-14)

Abstract

In this paper, we study the resonance problem of a class of singular quasi- linear parabolic equations with respect to its higher near-eigenvalues. Under a generalized Landesman-Lazer condition, it is proved that the resonance prob- lem admits at least one nontrivial solution in weighted Sobolev spaces. The proof is based upon applying the Galerkin-type technique, the Brouwer’s fixed- point theorem and a compact embedding theorem of weighted Sobolev spaces by Shapiro.

Keywords: Weighted Sobolev Space, Quasilinear Parabolic Equation, Res- onance.

1 Introduction

Resonance problems of quasilinear elliptic (or parabolic) partial differential equations have been studied extensively in the usual Sobolev spaces. Since the celebrated paper by Landesman and Lazer [8], many existence results were obtained under various nonlinearity growth conditions and the Landesman- Lazer conditions (see [1–4, 6, 7, 9, 11–15] and references therein). However, there has been very limited existence results for the case of singular quasilinear elliptic(or parabolic) equations in the existing literature.

In 2001, Shapiro published a paper [12] on the resonance problems of singu- lar quasilinear equations. An important element of that paper is the existence of a complete orthonormal basis in the weighted Sobolev space associated with singular coefficients of the differential operator. In that paper, a new concept of near-eigenvalues for singular quasilinear elliptic operators was introduced, a new compact embedding theorem in the weighted Sobolev spaces was es- tablished, and some new existence results for the resonance problems were obtained.

In 2002, Chung-Cheng Kuo [7] applied Galerkin-type techniques and Brouwer’s fixed point theorem to obtain existence theorems of time-periodic solutions for quasilinear parabolic partial differential equations with respect to its first eigenvalue in which the Landesman-Lazer condition may be excluded.

In 2005, Rumbos and Shapiro [11] introduced a generalized Landesman- Lazer condition and studied the resonance problem of the semilinear elliptic equations with respect to its first eigenvalue by using the linking argument and a deformation theorem in weighted Sobolev spaces.

Inspired by papers [9, 10, 12, 14], we have studied the resonance problem of quasilinear or singular quasilinear elliptic(or parabolic) equations in weighted Sobolev spaces with respect to their first eigenvalues by using the Galerkin- type technique and the Brouwer’s fixed-point theorem [2–4].

Motivated by [10–12], in this paper, we show the existence of solutions for a class of singular quasilinear parabolic equations with respect to its higher near-eigenvalue in the Hilbert space H(e Ω,e Γ):

(ρD_tu+Mu= (λ_j0u+b(x, t, u)u⁻+f(x, t, u))ρ−G, (x, t)∈Ω,e

u∈H(e Ω,e Γ), (P)

where

Mu=−

N

X

i,j=1

D_i[p

1 2

i (x)p

1 2

j(x)s

1 2

i (u)s

1 2

j(u)a_ij(x)D_ju] +a₀(x)s₀(u)qu, (1.1) and λ_j0 is an eigenvalue of L.

As in paper [3], we assume the existence of a linear uniformly elliptic op- erator which is close to the original singular quasilinear operator in a certain sense, and hence the existence of a complete orthonormal basis in the weighted Sobolev space associated with singular coefficients of the differential operator.

However, unlike the case of the first near-eigenvalue which is simple and whose eigenfunction is of one-sign, the case of higher near-eigenvalue is challenging to study due to the fact that the multiplicity of higher near-eigenvalue is greater than 1 and their corresponding eigenfunctions are sign-changing. By using a space decomposition technique, we are able to prove that the resonance prob- lem has at least one solution under a generalized Landesman-Lazer condition.

The proof method is similar to [12] and [4], which is based also upon apply- ing the Galerkin-type technique, the Brouwer’s fixed-point theorem and the compact embedding theorem of weighted Sobolev spaces by Shapiro [12].

This paper is organized as follows. In Section 2, we describe the resonance problem of a class of singular quasilinear parabolic equations to be studied, and state the main result. In Section 3, we prove the main theorem.

2 Statement of the Problem and Main Result

Let Ω⊂R^N(N ≥1), be an open set(possibly unbounded) and letρ(x), p_i(x)∈ C⁰(Ω) be positive functions with the property that

Z

Ω

ρ(x)dx <∞, Z

Ω

q(x)dx <∞, Z

Ω

pi(x)dx <∞, i= 1,2,· · · , N. (2.1) Letq(x)∈C⁰(Ω) be a nonnegative function and Γ⊂∂Ω be a fixed closed set.

Note that Γ may be an empty set and q(x) may be zero. On the other hand, q(x) will satisfy: there exists K >0, such that

0≤q(x)≤Kρ(x), for all x∈Ω. (2.2) HereA is a set of real-valued functions defined as

A={u:u∈C⁰( ¯Ω×R), u(x, t+ 2π) =u(x, t), for all (x, t)∈Ω¯×R}.

SettingΩ = Ω×T, Te = (−π, π), p= (p₁,· · ·, p_N) andD_i = _∂x^∂u

i(i= 1,2,· · · , N), we consider the following pre-Hilbert spaces (see [12]):

Ce_ρ⁰(eΩ) =

u∈C⁰(eΩ) : Z

Ωe

|u(x, t)|²ρ(x)dxdt < ∞

, with inner producthu, vi^∼_ρ =R

Ωeu(x, t)v(x, t)ρ(x)dxdt, and the space Ce_p,ρ¹ (eΩ,Γ) = {u∈ A ∩C¹(Ω×R)

u(x, t) = 0, for all (x, t)∈Γ×R;

Z

Ωe

[

N

X

i=1

|D_iu|²p_i+ (u²+|D_tu|²)ρ]<∞}

with inner product hu, vi_He =

Z

Ωe

" _N X

i=1

p_iD_iuD_iv+ (uv+D_tuD_tv)ρ

# dxdt.

LetLe²_ρ,L²_ρ(Ω) denote the Hilbert space obtained from the completion ofe Ce_ρ⁰ with the norm ||u||_ρ = (hu, ui^∼_ρ)¹², and He , H(ee Ω,Γ) denote the completion of the space Ce_p,ρ¹ with the norm ||u||_He = hu, ui

1 2

He. Similarly, we have Le²_pi(i = 1,2,· · ·, N) andLe²_q.

It is assumed throughout the paper that s_i(u)(i= 0,1,· · · , N) meets:

(S1) s_i(u): He →R is weakly sequentially continuous;

(S2) there exist η₀, η₁ > 0 such that η₀ ≤ s_i(u)≤ η₁, and s_i(u) is measur- able, foru∈H.e

The functionsa_ij(i, j = 1,2,· · · , N) and a₀(x) satisfy(also b_ij(x) and b₀(x)):

(A1)a0(x), aij(x)∈C⁰(Ω)T

L^∞(Ω), aij(x) =aji(x),∀x∈Ω;

(A2)a₀(x)≥β₀ >0, ∀x∈Ω;

(A3) there existsc₀ >0, for x∈Ω andξ ∈R^N, such that

N

P

i,j=1

a_ij(x)ξ_iξ_j ≥ c₀ |ξ |² .

Furthermore, we assume both Caratheodory functionsb(x, t, s) andf(x, t, s) satisfy the following conditions.

(B1) There exist constants δ >0 andk > 1 such that

|b(x, t, s)| ≤

δ|s|, |s| ≤γ₁,

δγ1

(|s|+1−γ1)^m, |s|> γ₁, (2.3) and 0< γ1 <1, where γ1 = ^λj0+j_k^1−λj0 and m≥1.

Conditions on f(x, t, s):

(f1) There exists a nonnegative function f₀(x, t)∈Le²_ρ such that

|f(x, t, s)| ≤f₀(x, t), fora.e. x∈Ω and ∀s∈R;

(f2) lim sup_s→+∞f(x, t, s) = f⁺(x, t) ∈ L^∞(Ω), lim infs→−∞f(x, t, s) = f⁻(x, t)∈L^∞(Ω).

It is, in general, difficult to study the eigenvalues and eigenfunctions of M. Shapiro [12] introduced the concepts of near-related operators and near- eigenvalue ofM.

We first introduce some operators related to this paper.

Definition 2.1. For the quasilinear differential operator M, the two form is

M(u, v) =

N

X

i,j=1

Z

Ωe

h p

1 2

i p

1 2

js

1 2

i (u)s

1 2

j(u)a_ijD_juD_ivi +

Z

Ωe

qs₀(u)a₀uv, u, v ∈H(ee Ω,Γ).

(2.4) Defining

L_xu=−

N

X

i,j=1

D_ih p

1 2

i p

1 2

jb_ijD_jui

+b₀qu, (2.5)

for u∈H_p,q,ρ =H_p,q,ρ(Ω,Γ) (as described in [12]), and Lu=−

N

X

i,j=1

D_ih p

1 2

ip

1 2

jb_ijD_jui

+a₀qu, u∈H(e Ω,e Γ), (2.6) then the bilinear form of L_x is

L_x(u, v) =

N

X

i,j=1

Z

Ω

p

1 2

i p

1 2

jb_ij(x)D_juD_iv+ Z

Ω

b₀uvq, u, v ∈H_p,q,ρ(Ω,Γ), (2.7) and the bilinear form of L is

L(u, v) =

N

X

i,j=1

Z

Ωe

p

1 2

ip

1 2

jb_ij(x)D_juD_iv+ Z

Ωe

b₀uvq, u, v ∈H(ee Ω,Γ). (2.8) We further assume that domain Ω and operator L_x satisfy the so-called V_L(Ω,Γ) conditions [12, 14]:

(V_L-1) There exists a complete orthonormal sequence of functions{ϕ_n}^∞_n=1 inL²_ρ(Ω), such that ϕ_n ∈H_p,q,ρ¹ (Ω,Γ)∩C²(Ω) for all n.

(V_L-2) The uniformly elliptic operatorL_x has a sequence of real eigenvalues {λ_n}^∞_n=1 corresponding to the orthonormal sequence{ϕ_n}^∞_n=1, satisfying

0< λ₁ < λ₂ ≤λ₃ ≤ · · · ≤λ_n → ∞ as n→ ∞, and

Lx(ϕn, v) =λnhϕn, viρ, ∀v ∈H_p,q,ρ¹ (Ω,Γ) andn ≥1.

Alsoϕ₁ >0 in Ω.

Here hu, vi_ρ =R

Ωeuvρ. For the sake of simplicity, in the following, we will denotehu, vi_ρ as hu, vi.

Examples of operators and domains for which theV_L(Ω,Γ) conditions hold can be found in [12](pp. 20-26). The VL(Ω,Γ) conditions play a key role in our study of the resonance problem of singular quasilinear elliptic equations.

Definition 2.2. OperatorMis said to be near-related to operatorL(denoted as M ∼ L for convenience), if, for any v ∈H,e

||u||lim

He→∞

M(u, v)− L(u, v)

||u||_He = 0. (2.9)

Definition 2.3. Assume M ∼ L in H.e λ is called a near-eigenvalue of M if

(1) λ is an eigenvalue of L_x; (2) lim||u||

He→∞M(u,P_λu)−L(u,P_λu)

||u||

He

= 0,

where P_λ is the orthogonal projection from L²_ρ(Ω) onto the eigenspace of L_x

corresponding to the eigenvalue λ.

We now state the main result of this paper:

Theorem 2.4. Let Ω ⊂ R^N(N ≥ 1), T = (−π, π), Ω = Ωe × T, p = (p₁,· · · , p_N), ρ and p_i(i= 1,· · · , N) be positive functions in C⁰(Ω) satisfying (2.1), q ∈ C⁰(Ω) be a nonnegative function satisfying (2.2), and Γ ⊂ ∂Ω be a closed set. Let M and L be given by (1.1) and (2.6) satisfying (S1)-(S2), (A1)-(A3) respectively andL_x satisfies the conditions of V_L(Ω,Γ). If M ∼ L, λj0 is a near-eigenvalue of M of multiplicity j1, (B1) and (f1)-(f2) hold, and G∈(H)e ^∗, then the problem (P) has at least one weak solution; i.e., there exits u^∗ ∈He such that

hD_tu^∗, vi_ρ+M(u^∗, v) =λ_j0hu^∗, vi_ρ+hf(x, t, u^∗)+g(x, t, u^∗), vi_ρ−G(v), ∀v ∈H.e (2.10) Here, we will introduce some lemmas and concepts which will be used later.

If (A1)-(A3) and the conditions ofV_L(Ω,Γ) hold, we have

{ϕe^c_jk}^∞,∞_j=1,k=0∪ {ϕe^s_jk}^∞,∞_j=1,k=1 is a CONS for Le²_ρ, (2.11) where

ϕe^c_jk(x, t) =

( _ϕj(x)

√

2π, k = 0, j = 1,2,· · · ,

ϕj(x) cos(kt)√

π , k, j= 1,2,· · · , (2.12) and

ϕe^s_jk(x, t) = ϕ_j(x) sin(kt)

√π , k, j = 1,2,· · ·. (2.13) Obviously, bothϕe^c_jk and ϕe^s_jk are in H(e Ω,e Γ).

Lemma 2.5. If{ϕe^c_jk}^∞,∞_j=1,k=0∪ {ϕe^s_jk}^∞,∞_j=1,k=1 is aCONS for L²_ρ(eΩ)defined by (2.11), setting

τ_n(v) =

n

X

j=1

bv^c(j,0)ϕe^c_j0 +

n

X

j=1 n

X

k=1

vb^c(j, k)ϕe^c_jk+bv^s(j, k)ϕe^s_jk

, (2.14) we have

n→∞lim ||τ_n(v)−v||_He = 0, for all v ∈H.e (2.15)

Lemma 2.6. (i) If v ∈H,e then L₁(v, v) +||D_tv||²_ρ=

∞

X

j=1

|bv^c(j,0)|²(λ_j + 1)

+

∞

X

j=1

∞

X

k=1

|bv^c(j, k)|²+|bv^s(j, k)|²

λj + 1 +k² .

(2.16)

(ii) If v ∈ L²_ρ(eΩ) and L₁(v, v) +||D_tv||²_ρ < ∞, then v ∈ H. Heree L₁(v, v) = L(v, v)+< v, v > .

Lemma 2.7. Let Ω, ρ, p, q,e and L be as in the hypothesis of Theorem 2.1 and assume that(Ω,Γ)is aV_L(Ω,Γ).ThenHe is compactly imbedded inL²_ρ(eΩ).

The proofs of Lemmas 2.1-2.3 can be found in [12]. We define the set S_n=

(

v ∈He :v =

n

X

j=1

η^c_j0ϕe^c_j0+

n

X

j=1 n

X

k=1

η_jk^c ϕe^c_jk +η_jk^s ϕe^s_jk, η^c_jk, η^s_jk ∈R )

. (2.17) Remark 2.8. (1) If un ∈ Sn, then M(un, Dtun) = 0; (2) hDt(αϕe^c_jk + βϕe^s_jk), αϕe^c_jk+βϕe^s_jki= 0, j, k ≥1, α, β ∈R.

3 Proof of Theorem 2.1

The proof of Theorem 2.1 can be divided into three steps. The first step is to construct a set of approximate solutions {u_n} of (2.10) in H, wheree u_n ∈ S_n and Sn is defined as in (2.17). Then we show in the second step that {un} is bounded in H. Finally, we showe {u_n} converges to a weak solution u^∗ ∈He of (2.10).

Lemma 3.1. Assume that all the conditions in the hypothesis of Theorem 2.1 hold. Let S_n be the subspace of He defined by (2.17). Taking n₀ = j₀+j₁ andγ₀ = ¹₂(λ_j0+j1−λ_j0), then forn ≥n₀, there is a function u_n∈S_n with the property that

hD_tu_n, vi+M(u_n, v) =(λ_j0 +γ₀n⁻¹)hu_n, vi+hb(x, t, u_n)(u_n)⁻, vi

+(1−n⁻¹)hf(x, t, un), vi −G(v), ∀v ∈Sn. (3.1) Proof. Let{ψ_i}²ⁿ_i=1²⁺ⁿbe an enumeration of{ϕe^c_jk}^n,n_j=1,k=0∪ {ϕe^s_jk}^n,n_j=1,k=1,and set n^∗ = (j₀+j₁−1)(2n+ 1). (3.2) So{ψ_i}ⁿ_i=1^∗ is an enumeration of {ϕe^c_jk}^j_j=1,k=0^0+j1−1,n∪ {ϕe^s_jk}^j_j=1,k=1^0+j1−1,n, wheren ≥n₀.

With this enumeration defined, for α= (α₁,· · · , α_2n²_+n), we set u=

2n²+n

X

i=1

α_iψ_i, eu=

2n²+n

X

i=1

δ_iα_iψ_i, (3.3) whereδ_i =−1, if 1≤i≤n^∗; δ_i = 1, if n^∗+ 1≤i≤2n²+n, and define

F_i(α) =hD_tu, δ_iψ_ii+M(u, δ_iψ_i)−(λ_j0 +γ₀n⁻¹)hu, δ_iψ_ii

−hb(x, t, u)u⁻, δ_iψ_ii −(1−n⁻¹)hf(x, t, u), δ_iψ_ii+G(δ_iψ_i). (3.4) It is clear from orthogonality that hD_tu,uie = 0. From (3.3) and (3.4) we get

2n²+n

P

i=1

F_i(α)α_i =M(u,u)e −(λ_j0 +γ₀)hu,eui

−hb(x, t, u)u⁻,eui −(1−n⁻¹)hf(x, t, u)−γ₀u,eui+G(eu).

(3.5) Then

2n²+n

X

i=1

F_i(α)α_i =I(α) +II(α), (3.6) where

I(α) =L(u,eu)−(λ_j0 +γ₀)hu,ui − hb(x, t, u)ue ⁻,eui

−(1−n⁻¹)hf(x, t, u)−γ₀u,eui+G(eu), II(α) =M(u,eu)− L(u,eu).

Consider I(α) in (3.6) first. Note that γ₀ = ¹₂(λ_j0+j1 −λ_j0) and δ_j(λ_j − λ_j0 −γ₀)≥γ₀(j = 1,2,· · · , n), then

L(u,eu)−(λ_j0 +γ₀)hu,uie > γ₀|α|². (3.7) By condition (B1), we have

|hb(x, t, u)u⁻,ui˜ ρ| ≤ Z

Ω∩{|u|≤γe ₁}

|u|²|˜u|ρ+δγ1

Z

Ω∩{|u|>γe ₁}

|u||˜u|ρ (|u|+ 1−γ₁)^m

≤c|α|.

(3.8) From (f1), H¨older inequality and Minkowski inequality, we have

|hf(x, t, u)−γ0u, ui| ≤˜ γ0|α|²+||f0||ρ|α|. (3.9) Note thatG∈(H)e ^∗. It follows from Lemma 2.3 that, for each givenn≥j₀+j₁,

|G(˜u)| ≤c|α|. (3.10)

Thus, it follows from (3.7)-(3.10) that I(α)> 1

nγ₀|α|² −c|α|. (3.11)

ByM ∼ L and ||u||²_ρ=||eu||²_ρ=|α|², we have lim

|α|→∞

II(α)

|α|² = lim

|α|→∞

M(u,u)˜ − L(u,u)˜

|α|² = 0. (3.12)

Thus it follows from (3.6), (3.11) and (3.12) that, for any given n ≥ j₀+j₁, there exists A₀ > 0 such that Pn

i=1F_i(α)α_i > 0 for |α| ≥ A₀. Under the assumptions of Theorem 2.1, it is straightforward to verify thatFi :Rⁿ→Ris continuous for 1≤i≤ n. By applying the Brouwer’s fixed-point theorem [5], there existsα^∗ = (α^∗₁, α^∗₂,· · · , α^∗_n)∈Rⁿsuch thatF_i(α^∗) = 0 for 1 ≤i≤n. Let u^∗_n=Pn

i=1α^∗_iϕi ∈Sn. It follows from (3.4) that u^∗_n is a solution of (3.1).

In next step, we will prove that {u^∗_n}^∞_n=j0+j1 is bounded in H.e

Lemma 3.2. Assume the conditions in Lemma 3.1 hold, and {u^∗_n}^∞_n=j0+j1 ⊂ He is the sequence of solutions obtained in Lemma 3.1. Assume further G ∈ (H)e ^∗ satisfies the following generalized Landesman-Lazer condition:

G(w)<

Z

Ωe1

f⁺(x, t)w(x)ρ+ Z

Ωe2

f⁻(x, t)w(x)ρ(x), (3.13) for every nontrivial λ_j0-eigenfunction w of L_x, where Ωe_i = Ω_i ×(−π, π)(i = 1,2), Ω₁ = {x ∈ Ω;w(x) > 0} and Ω₂ = {x ∈ Ω;w(x) < 0}. Then {u^∗_n} is bounded in H.e

Proof. For simplicity of notation, we denote {u^∗_n}^∞_n=j0+j1 by {u_n}^∞_n=j0+j1. It follows from Lemma 3.1 that u_n∈S_n and u_n satisfies

hDtun, vi+M(un, v) = (λj0 +γ0n⁻¹)hun, vi+hb(x, t, un)(un)⁻, vi

+(1−n⁻¹)hf(x, t, u_n), vi −G(v), ∀v ∈S_n, (3.14) whereγ₀ = (λ_j0+j1−λ_j0)/2, and n≥n₀ =j₀+j₁.

In order to prove Lemma 3.2, we only need to prove that there exists a constant such that{u_n} obtained by Lemma 3.1 satisfies

ku_nk

He ≤K. (3.15)

Assume that (3.15) dose not hold. Then there exists a subsequence of{u_n}, denoted again by{u_n}, such that

n→∞lim ku_nk_He =∞. (3.16)

Lettingv =D_tu_n in (3.14), by (f2), hD_tu_n, u_ni= 0 and M(D_tu_n, u_n) = 0, we have

|hb(x, t, u_n)u⁻_n, D_tu_ni| ≤ Z

Ω∩{|ue n|≤γ1}

|u_n|²|D_tu_n|ρ +δγ₁

Z

Ω∩{|ue n|>γ1}

|u_n| · |D_tu_n|ρ (|u_n|+ 1−γ₁)^m

≤c(δ, γ1,|eΩ|)kDtunkρ , and we can conclude that there existsK >0 such that

kD_tu_nkρ≤K. (3.17)

Under conditions (B1) and (S2), it follows from (1.1) that M(u_n, u_n)≥c₀(

N

X

i=1

kD_iu_nk²_pi +ku_nk²_q), wherec0 is a positive constant. Then we have

c₁ku_nk²

He ≤ M(u_n, u_n) +c₂(ku_nk²_ρ+kD_tu_nk²_ρ). (3.18) Now by lettingv =u_n in (3.14), and the proof of (3.9), we have

|hf(x, t, u_n)−γ₀u_n, u_ni| ≤γ₀ku_nk²_ρ+Kku_nk_ρ. (3.19) From (B1) and H¨older inequality, we have

|hb(x, t, u_n)u⁻_n, u_ni| ≤ Z

Ω∩{|ue n|≤γ1}

δ|u_n|³ρ+δγ₁ Z

Ω∩{|ue n|>γ1}

|u_n|²ρ (|u_n|+ 1−γ₁)^m

≤c^∗₂(δ, γ₁,|Ω|)kue _nk^2−m_ρ +c^∗₃(δ, γ₁,|eΩ|).

(3.20) Then by (3.19), (3.20) andhD_tu_n, u_ni= 0, we have

c1kunk²

He ≤(λj0 +γ0)hun, uni+hb(x, t, un)u⁻_n, uni

+ (1−n⁻¹)hf(x, t, u_n)−γ₀u_n, u_ni −G(u_n) +c₁(ku_nk²_ρ+kD_tu_nk²_ρ)

≤K₄ku_nk²_ρ+Kku_nk_He +c^∗₂(δ, γ₁,|Ω|)kue _nk^2−m_ρ +c^∗₃(δ, γ₀,|eΩ|),

where K₄ =λ_j0 + 2γ₀ +c₁, and m > 1. Dividing both sides of the above in- equalities byku_nk²

He and then by (3.16), we know that there existsn₁(n₁ ≥n₀) such that

0< c₁

K₄ ≤ ku_nk²_ρ ku_nk²

He

≤1, ∀n≥n₁.

Noticing (3.16), the above inequalities establish if and only if

n→∞lim ku_nk_ρ=∞, (3.21)

that is, there existsK >0 such that

ku_nk_He ≤Kku_nk_ρ, ∀n≥n₁. (3.22) Rewriteu_n as u_n=u_n1+u_n2+u_n3, and let ˜u_n=−u_n1−u_n2+u_n3, where

















 un1 =

j0−1

P

j=1 ub^c_n(j,0)ϕe^c_j0+

j0−1

P

j=1 n

P

k=1

(bu^c_n(j, k)ϕe^c_jk +bu^s_n(j, k)ϕe^s_jk), u_n2 =

j0+j1−1

P

j=j0

ub^c_n(j,0)ϕe^c_j0+

j0+j1−1

P

j=j0

n

P

k=1

(ub^c_n(j, k)ϕe^c_jk+ub^s_n(j, k)ϕe^s_jk), u_n3 =

n

P

j=j0+j1

bu^c_n(j,0)ϕe^c_j0+

n

P

j=j0+j1

n

P

k=1

(ub^c_n(j, k)ϕe^c_jk+ub^s_n(j, k)ϕe^s_jk).

(3.23)

First, for given anyn≥n₁, we can prove the following conclusion

n→∞lim ku_n1k

He +ku_n3k

He

ku_nk_ρ = 0. (3.24)

As a result, from (3.14) withv =eu_n, we have

hb(x, t, u_n)(u_n)⁻,u˜_ni+ (1−n⁻¹)hf(x, t, u_n)−γ₀u_n,u˜_ni

−G(˜un) +L(un,u˜n)− M(un,u˜n)

=

n

X

j=1

δ_j(λ_j−λ_j0 −γ₀)|ˆu^c_n(j,0)|²

+

n

X

j,k=1

δj(λj−λj0)[|ˆu^c_n(j, k)|²+|ˆu^s_n(j, k)|²].

(3.25)

Since

(3.25)_R =γ₀ku_nk²_ρ+

j0+j1−1

X

j=1

(λ_j0 −λ_j)|uˆ^c_n(j,0)|² +

n

X

j=j0+j1

(λ_j−λ_j0 −2γ₀)|ˆu^c_n(j,0)|²

+

j0+j1−1

X

j=1 n

X

k=1

(λ_j0 −λ_j)[|ˆu^c_n(j, k)|²+|ˆu^s_n(j, k)|²] +

n

X

j=j0+j1

n

X

k=1

(λ_j −λ_j0 −2γ₀)[|ˆu^c_n(j, k)|²+|uˆ^s_n(j, k)|²], by (3.8) and the proof of (3.9), we get

(3.25)_L≤γ₀ku_nk²_ρ+c^∗(δ, γ₁,|Ω|, K)kue _nk_ρ+L(u_n,u˜_n)− M(u_n,u˜_n).

In this way, it follows from (3.25) that

(3.25)R≤γ0kunk²_ρ+c^∗(δ, γ1,|Ω|, K)kue nkρ+L(un,u˜n)− M(un,u˜n). (3.26) For fixedn, there exists a constantγ⁰ >0 such that

γ⁰(1 +λ_k)≤λ_j0 −λ_k, k= 1,2,· · ·j₀−1, γ⁰(1 +λ_k)≤λ_k−λ_j0 −2γ₀, k ≥j₀+j₁. Since

L₁(u_n, u_n) =

n

X

j=1

(1 +λ_j)ˆu^c_n(j,0)ϕe^c_j0+

n

X

j=1 n

X

k=1

(1 +λ_j)[ˆu^c_n(j, k)ϕe^c_jk+ ˆu^s_n(j, k)ϕe^s_jk], by (3.26) and the above inequalities, there existsγ^∗ >0 such that

γ^∗(kun1k_He2 +kun3k_He2)≤c^∗kunkρ+L(un,u˜n)− M(un,u˜n) +K.

Dividing both sides of the above inequality by ku_nk²_ρ and taking the limit as n→ ∞, it follows from (3.21) and M∼L that (3.23) establishes.

Next, taking use of the notation of (3.23) and letting w_n= u_n

ku_nk_ρ, w_ni = u_ni

ku_nk_ρ, i= 1,2,3, (3.27) thus by (3.22), there existsK >0 such that

kw_nk

He ≤K and kw_nik

He ≤K, i= 1,2,3, ∀n ≥n₁, (3.28) that is,kw_nk_He is a bounded sequence inH. Ase He is a separable Hilbert space, by Lemma 2.3 and (3.28), there exists a subsequence ofwn( denoted again by w_n) andw∈He such that











(1) lim

n→∞||w_n−w||

He = 0;

(2) ∃w^∗ ∈Le²_ρ, s.t.|w_n(x, t)| ≤w^∗(x, t), a.e. (x, t)∈Ω;e (3) lim

n→∞w_n(x, t) =w(x, t), a.e. (x, t)∈Ω.e

(3.29)

SinceM∼L, we get from (3.28) that

n→∞lim

M(u_n, w_ni)− L(u_n, w_ni)

ku_nk_ρ = 0, i= 1,2,3.

We observe from (3.24) that lim

n→∞kw_n3k_ρ= 0. Hence, if n → ∞, then hw_n,ϕe^c_jki=hw_n3,ϕe^c_jki →0, j ≥j₀+j₁.

Now by (3.29), we get ˆw^c(j, k) = 0, forj ≥j₀+j₁ and allk. Similarly, we have ˆ

w^s(j, k) = 0, for j ≥ j₀ +j₁ and all k. By (3.24), we gain lim

n→∞kw_n1k_ρ = 0, similarly, we can obtain ˆw^c(j, k) = 0 and ˆw^s(j, k) = 0, for 1≤j ≤j₀ −1 and allk. Thus, we get

(wˆ^c(j, k) = 0 and ˆw^s(j, k) = 0, forj ≥j₀+j₁ and allk;

ˆ

w^c(j, k) = 0 and ˆw^s(j, k) = 0, for 1≤j ≤j₀−1 and all k. (3.30) Hence, lettingv =D_tu_nin (3.14), and byM(u_n, D_tu_n) = 0, Schwarz inequality and G∈(H)e ^∗, we get

kD_tu_nk_ρ≤ kf(x, t, u_n)k_ρ+c(δ, γ₁,|Ω|).e Therefore, we have

n→∞lim

kD_tu_nk²_ρ ku_nk²_ρ = 0, that is,

n→∞lim kD_tw_nk²_ρ = 0. (3.31) On the other hand, for k ≥1 and j₀ ≤j ≤j₀+j₁−1, from (2.12), (2.13) and (3.31), we know

kwˆ^c(j, k) = − lim

n→∞

Z

Ωe

D_tw_n(x, t)ϕ^s_jk(x, t)ρ(x)dxdt= 0.

A similar situaion prevails forkwˆ^s(j, k) = 0. So we have ˆ

w^c(j, k) = 0 and ˆw^s(j, k) = 0,

for k ≥1 and ₀ ≤j ≤j₀+j₁ −1. Hence, we know thatw(x, t) is a function unrelated tot; i.e.,

w(x, t)≡w(x) =

j0+j1−1

X

j=j0

wb^c(j,0)ϕe^c_j0(x). (3.32) Replacing v byu_n2 in (3.14), and by (V_L−2), for ∀n≥n₁, we have

(1−n⁻¹)hf(x, t, u_n), u_n2i −G(u_n2) +L(u_n, u_n2)− M(u_n, u_n2)

≤ −γ₀n⁻¹ku_n2k²_ρ+|hb(x, t, u_n)(u_n)⁻, u_n2i| ≤ h|b(x, t, u_n)(u_n)⁻, u_n2i|. (3.33) On the other hand, we have

|hb(x, t, un)(un)⁻, un2i| ≤ int_Ωe|b(x, t, un)| · |un|²ρ +

Z

Ωe

|b(x, t, u_n)u_n(u_n1+u_n3)|ρ. (3.34)

By (B1) and the computing method of (3.20), we can get Z

Ωe

|b(x, t, u_n)| · |u_n|²ρ≤δγ₁⁴|eΩ|+ (δγ₁)² Z

Ω∩{|ue _n|>γ₁}

ρ=c₄(δ, γ₁,|eΩ|). (3.35) So, by (3.35), we can obtain

Z

Ωe

|b(x, t, u_n)u_n(u_n1+u_n3)|ρ≤c^∗₄(δ, γ₁,Ω)kue _n1+u_n3k_ρ. (3.36) By using of (3.34)-(3.36), then it follows from (3.33) that

(1−n⁻¹)hf(x, t, u_n), un2i −G(un2) +L(u_n, un2)− M(u_n, un2)

≤c^∗₂(δ, γ₁,|eΩ|)ku_nk^2−m_ρ +c^∗₃(δ, γ₁,|eΩ|) +c^∗₄(δ, γ₁,|eΩ|)ku_n1+u_n3k_ρ. (3.37) Dividing byku_nk_ρ on both sides of (3.37), we get

(1−n⁻¹)hf(x, t, u_n), w_ni −G(w_n) + (L(u_n, u_n2)− M(u_n, u_n2))/ku_nk_ρ

≤c^∗₂(δ, γ₁,|eΩ|)ku_nk^1−m_ρ +c^∗₃(δ, γ₁,|eΩ|)/ku_nk_ρ +c^∗₄(δ, γ₁,|eΩ|)ku_n1+u_n3k_ρ/ku_nk_ρ.

(3.38) From (f2) and (3.29)(2), there exists K such that

Z

Ωe

f(x, t, u_n)w_nρ≤ kh(x, t)k_ρkw^∗(x, t)k_ρ≤K. (3.39) Because of M ∼ L, by (3.21) and (3.22), we have

lim

kunkρ→∞

|L(u_n, u_n2)− M(u_n, u_n2)|

kunkρ

= 0. (3.40)

Taking the limit in (3.38) asn → ∞, and by (3.21), (3.24), (3.39), (3.40) and (3.29)(3), we get

lim sup

n→∞

Z

Ωe

f(x, t, u_n)w_nρ≤G(w). (3.41) Setting

Ωe₁ ={(x, t)∈Ω :e w(x)>0}, Ωe₂ ={(x, t)∈Ω :e w(x)<0}, it follows from (3.39) and (3.41) that

lim inf

n→∞

Z

Ωe1

f(x, t, u_n)w_nρ+ lim inf

n→∞

Z

Ωe2

f(x, t, u_n)w_nρ≤G(w). (3.42) By (3.21) and (3.29)(1)(3), we have

n→∞lim u_n(x, t) = +∞, a.e. (x, t)∈Ωe₁;

n→∞lim u_n(x, t) = −∞, a.e. (x, t)∈Ωe₂. Next, it follows from (f2) and (3.29)(3) that







f⁺wρ= lim inf

n→∞ f(x, t, u_n)w_nρ, a.e. (x, t)∈Ωe₁; f⁻wρ= lim inf

n→∞ f(x, t, u_n)w_nρ, a.e. (x, t)∈Ωe₂. (3.43) And by (3.42), (3.43) and Fatou Lemma, we obtain

Z

Ωe1

f⁺(x, t)w(x)ρ+ Z

Ωe2

f⁻(x, t)w(x)ρ≤G(w).

By (3.24) and (3.27), we knowkwk_ρ= 1, thus, wis a nontrivial eigenfunction and satisfies (3.13). But it forms a contradiction between (3.13) and the above inequalities. Therefore, (3.15) is established and we complete the proof of Lemma 3.2.

We are now ready to prove Theorem 2.1.

Proof of Theorem 2.1. Since H(ee Ω,Γ) is a separable Hilbert space, we see from (3.15) and Lemma 2.3 that there exists a subsequence (For the sake of simplicity, we take to be a full sequence{un}) and a function u^∗ ∈ H(ee Ω,Γ) with the following properties:























(1) lim

n→∞||un−u^∗||ρ= 0;

(2) ∃k(x, t)∈Le²_ρ, s.t. |u_n(x, t)| ≤k(x, t), a.e. (x, t)∈Ω,e ∀n;

(3) lim

n→∞u_n(x, t) =u^∗(x, t), a.e. (x, t)∈Ω;e (4) lim

n→∞hD_iu_n, vi_pi =hD_iu^∗, vi_pi, for all v ∈Le²_p

i, i= 1,· · · , N; (5) lim

n→∞ha₀(x)u_n, vi_q =ha₀(x)u^∗, vi_q, for all v ∈Le²_q.

(3.44) Sinces_i(u) satisfies (S1), we have

n→∞lim s_i(u_n) = s_i(u^∗), i= 0,1,2,· · · , N.

Letv ∈He and τ_J(v) be defined by (2.14). Then τ_J(v)∈S_J(J ≥n₀) and from (3.44)(1)(4)(5) we have that

n→∞lim M(u_n, τJ(v)) + lim

n→∞hD_tun, τJ(v)i=M(u^∗, τJ(v)) +hD_tu^∗, τJ(v)i. (3.45) Next from (f1)-(f2), (3.44)(2)(3) and the Lebesgue dominated convergence theorem, we obtain

n→∞limhf(x, t, u_n), τ_J(v)i=hf(x, t, u^∗), τ_J(v)i, a.e. (x, t)∈Ω.e (3.46)

And from (B1), (3.44)(2)(3) and the Lebesgue dominated convergence theo- rem, we get

n→∞limhb(x, t, u_n)(u_n)⁻, τ_J(v)i=hb(x, t, u^∗)(u^∗)⁻, τ_J(v)i, a.e. (x, t)∈Ω.e (3.47) It follows from (3.44)-(3.47) that

hD_tu^∗, τ_J(v)i+M(u^∗, τ_J(v)) = λ_j0hu^∗, τ_J(v)i+hb(x, t, u^∗)(u^∗)⁻, τ_J(v)i +hf(x, t, u^∗), τ_J(v)i −G(τ_J(v)).

(3.48) Passing to the limit asJ → ∞ on both sides of (3.48), we have

hD_tu^∗, vi+M(u^∗, v) =λ_j0hu^∗, vi+hb(x, t, u^∗)(u^∗)⁻, vi+hf(x, t, u^∗), vi −G(v).

Thus we complete the proof of Theorem 2.1. .

Competing Interests: The authors declare that they have no competing interests.

Authors Contributions: We declare that all authors collaborated and dedicated the same amount of time in order to perform this article.

Acknowledgements: This work has been supported by the Natural Sci- ence Foundation of China (11171220) and Shanghai Leading Academic Disci- pline Project (XTKX2012) and Hujiang Foundation of China (B14005).

References

[1] H. Brezis and L. Nirenberg, Characterization of ranges of some nonlinear operators and applications of boundary value problems,Ann. Scuo. Norm.

Sup. Pisa, 5(1978), 225-326.

[2] G. Jia, M.L. Zhao and F.L. Li, Eigenvalue problems for a class of singular quasilinear elliptic equations in weighted spaces,Elec. J. Qual. Theo. Diff.

Equa., 71(2012), 1-10.

[3] G. Jia and D. Sun, Existence of solutions for a class of singular quasilinear elliptic resonance problems,Nonlinear Analysis, 74(2011), 3055-3064.

[4] G. Jia, X.J. Zhang and L.N. Huang, Existence of solutions for quasilinear parabolic equations at resonance,Elec. J. Diff. Equa., 13(2013), 1-16.

[5] S. Kesavan, Topics in Functional Analysis and Applications, John Wiley and Sons, New York, (1989).

[6] C.C. Kuo, Solvability of a quasilinear elliptic resonance equation on the N-torus,Appl. Anal., 80(2001), 205-215.

[7] C.C. Kuo, On the solvability of a quasilinear parabolic partial differential equation at resonance, J. Math. Anal. Appl., 275(2002), 13-937.

[8] E.M. Landesman and A.C. Lazer, Nonlinear perturbations of a linear elliptic boundary value problem at resonance,J. Math. Mech., 19(1970), 609-623.

[9] L. Lefton and V.L. Shapiro, Resonance and quasilinear parabolic differ- ential equations,J. Diff. Equa., 101(1993), 148-177.

[10] M. Legner and V.L. Shapiro, Time-periodic quasilinear reaction-diffusion equations,SIAM J. Math. Anal., 26(1996), 135-169.

[11] A. Rumbos and V.L. Shapiro, Jumping nonlinearities and weighted Sobolev spaces, J. Diff. Equ., 214(2005), 326-357.

[12] V.L. Shapiro, Singular Quasilinearity and Higher Eigenvalues, Memoirs of the AMS, Providence, Rhode Island, (2001).

[13] V.L. Shapiro, Resonance, distributions and semilinear elliptic partial dif- ferential equations,Nonlinear Analysis, TMA, 8(1984), 857-871.

[14] V.L. Shapiro, Special functions and singular quasilinear partial differential equations,SIAM J. Math. Anal., 22(1991), 1411-1429.

[15] E. da Silva, Quasilinear elliptic problems under strong resonance condi- tions, Nonlinear Anal., 73(2010), 2451-2462.