# Existence and Multiplicity of Nontrivial Solutions for a Class of Fourth-Order Elliptic Equations

Authors: Chun Li; Zeng-Qi Ou; Chun-Lei Tang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Using the Fountain theorem and a version of the Local Linking theorem, we obtain some existence and multiplicity results for a class of fourth-order elliptic equations.


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## Body

## 1. Introduction and Main Results

Consider the fourth-order Navier boundary value problem(1)Δ2u+cΔu+a(x)u=f(x,u),inΩ,u=Δu=0,on∂Ω,
where Ω⊂ℝN(N>4) is a bounded smooth domain, a∈L∞(Ω),c∈ℝ, and f∈C(Ω×ℝ,ℝ). Δ is the Laplace operator and Δ2 is the biharmonic operator.Let0<λ1<λ2≤λ3≤⋯≤λk≤⋯ be the eigenvalues of -Δ in H01(Ω). The eigenvalue problem
(2)Δ2u+cΔu=μu,inΩ,u=Δu=0,on∂Ω,
has infinitely many eigenvalues μi=λi(λi-c),i=1,2,….We will always assumec<λ1. Let E be the Hilbert space H2(Ω)⋂‍H01(Ω).  E is equipped with the inner product
(3)〈u,v〉=∫Ω‍(ΔuΔv-c∇u∇v)dx
and the norm
(4)∥u∥=(∫Ω(|Δu|2-c|∇u|2)dx)1/2.A weak solution of problem (1) is any u∈E such that
(5)∫Ω(ΔuΔv-c∇u·∇v+a(x)uv)dx-∫Ωf(x,u)vdx=0
for any v∈E.Letφ:E→ℝ be the functional defined by
(6)φ(u)=12∫Ω‍(|Δu|2-c|∇u|2+a(x)u2)dx-∫ΩF(x,u)dx,
where F(x,u)=∫0u‍f(x,s)ds. And, one has
(7)〈φ′(u),v〉=∫Ω(ΔuΔv-c∇u·∇v+a(x)uv)dx-∫Ωf(x,u)vdx
for any u,v∈E, so that a critical point of the functional φ in E corresponds to a weak solution of problem (1).In recent years, fourth-order problems have been studied by many authors. In [1], Lazer and McKenna have pointed out that problem (1) furnishes a model to study travelling waves in suspension bridges if f(x,u)=b((u+1)+-1), where u+=max{u,0} and b∈ℝ. Since then, more general nonlinear fourth-order elliptic boundary value problems have been studied.In [2, 3], Micheletti and Pistoia proved that problem
(8)Δ2u+cΔu=f(x,u),inΩ,u=Δu=0,onΩ,
admits two or three solutions by variational method. In [4], Zhang obtained the existence of weak solutions for problem (8) when f(x,u) is sublinear at ∞. In [5], Zhang and Li showed that problem (8) has at least two nontrivial solutions by means of Morse theory and local linking. When f(x,u) is asymptotically linear at infinity, the existence of three nontrivial solutions has been obtained in [6] by using Morse theory. In [7], by using the mountain pass theorem, An and Liu gave the existence result for nontrivial solutions for a class of asymptotically linear fourth-order elliptic equations. In [8], Zhou and Wu got the existence of four sign-changing solutions or infinitely many sign-changing solutions for (8) by using the sign-changing critical point theorems. In [9], Yang and Zhang showed new results on invariant sets of the gradient flows of the corresponding variational functionals and proved the existence of positive, negative, and sign-changing solutions for some fourth-order semilinear elliptic boundary value problems. In [10], by using the variational method, Liu and Huang obtained an existence result of sign-changing solutions as well as positive and negative solutions for a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity.In this paper, we will study the existence of nontrivial solutions of problem (1). Our main results are the following theorems.Theorem 1.
Assume thatF is even in u and the following conditions hold:  (F1)
(9)F(x,u)u2⟶+∞as|u|⟶∞uniformly inx∈Ω;(F2)
there exist two constants2<θ<2N/(N-4)=2** and k1>0 such that
(10)|f(x,u)|≤k1(1+|u|θ-1)uniformly for allx∈Ω;(F3)
there exists a constantγ>N(θ-2)/4 such that
(11)liminf|u|→∞f(x,u)u-2F(x,u)|u|γ>0,uniformly for allx∈Ω.Then problem (1) has infinitely many nontrivial solutions.Theorem 2.
Assume thatF satisfies (F1) and  (F4)
there exist three positive constantsL, m1, and m2 such that
(j1)
f(x,u)u-2F(x,u)≥m1|u|2, if |u|≥L;(j2)
|f(x,u)|σ/|u|σ≤m2(f(x,u)u-2F(x,u)), if |u|≥L, where σ>(N-2)/2.If F is even in u, problem (1) has infinitely many nontrivial solutions.Remark 3.
For Schrödinger equation, the corresponding condition (F4) is due to Ding and Luan [11]. The condition (F4) is weaker than the usual Ambrosetti-Rabinowitz-type condition (see [11, 12]).Theorem 4.
Assume thatF satisfies (F1), (F2), (F3), and (F5)
(12)F(x,u)u2⟶0asu⟶0uniformly inx∈Ω.If 0 is an eigenvalue of Δ2+cΔ+a (with Navier boundary condition), assume also the condition that  (F6)
there existsδ>0 such that
(i)
F(x,u)≥0,  for all |u|≤δ,x∈Ω; or(ii)
F(x,u)≤0,  for all |u|≤δ,x∈Ω. Then problem (1) has at least one nontrivial solution.Theorem 5.
Suppose thatF satisfies (F1), (F4), and (F5). If 0 is an eigenvalue of Δ2+cΔ+a (with Navier boundary condition), assuming also (F6), then problem (1) has at least one nontrivial solution.Here, we have(13)φ(u)=12∫Ω‍(|Δu|2-c|∇u|2+a(x)u2)dx-∫ΩF(x,u)dx=12(∥u+∥2-∥u-∥2)-∫ΩF(x,u)dx,
where u-∈E-,u+∈E+, and E-(E+) is the space spanned by the eigenvectors corresponding to negative (positive) eigenvalues of Δ2+cΔ+a. It is easy to know that φ∈C1(E,ℝ) under the conditions of our theorems.It is well known thatE is continuously embedded in Lλ(Ω) for every λ∈[1,2N/(N-4)]. If 1≤λ<2N/(N-4), the embedding is compact. It follows from (F2), (F3), and (F4) that
(14)θ<2NN-4,θN-4γN-4<2NN-4,2σσ-1<2NN-4.
Hence, there is a positive constant K such that
(15)∥u∥Lλ≤K∥u∥,∀u∈E
for λ=1,2,θ,2σ/(σ-1), 2N/(N-4)=2**, where ∥·∥Lλ denotes the norm of Lλ(Ω).


## 2. Proof of Main Results

In this paper, we will use the Fountain Theorem of Bartsch ([13, Theorem 2.5], [14, Theorem 3.6]) to prove our Theorems 1 and 2. And, we will prove Theorems 4 and 5 by using a version of Local Linking theorem [12, Theorem 2.2] which extends theorems given by Li and Willem [15], Li and Szulkin [16].In [13, 14], Bartsch established the Fountain Theorem under the (PS)c condition. Since the Deformation Theorem is still valid under the Cerami condition, the Fountain Theorem is true under the Cerami condition. So, we have the following Fountain Theorem.LetX be a reflexive and separable Banach space. It is well known that there exist {vn}n∈ℕ⊂X,{ψn}n∈ℕ⊂X* such that(1)
〈ψn,vm〉=δn,m where δn,m=1 for n=m and δn,m=0 for n≠m.(2)
span¯{vnn∈ℕ}=X, span¯ω*{ψn∣n∈ℕ}=X*.LetXj=ℝvj; then X=⨁j≥1Xj¯. We define
(16)Yk=⨁kj=1Xj,Zk=⨁∞j=kXj¯.Theorem A (Fountain theorem).
Assume thatφ∈C1(X,ℝ) satisfies the Cerami condition (C), φ(-u)=φ(u). If for almost every k∈ℕ, there exist ρk>rk>0 such that  (A1)
(17)ak:=maxu∈Yk,∥u∥=ρkφ(u)≤0,(A2)
(18)bk:=infu∈Zk,∥u∥=rkφ(u)⟶∞,k⟶∞,thenφ has an unbounded sequence of critical values.For the reader's convenience, we state the following Local Linking theorem [12, Theorem 2.2]. Let X be a real Banach space with X=X1⊕X2 and X0j⊂X1j⊂X2j⊂⋯⊂Xj such that Xj=⋃n∈ℕ‍Xnj¯,j=1,2. For every multi-index α=(α1,α2)∈ℕ2, let Xα=Xα11⊕Xα22. We define that α≤β⇔α1≤β1,α2≤β2. A sequence {αn}⊂ℕ2 is admissible if for every α∈ℕ2 there is m∈ℕ such that n≥m⇒αn≥α. We say that φ∈C1(X,ℝ) satisfies the (C*) condition if every sequence {uαn} such that {αn} is admissible and satisfies
(19)uαn∈Xαn,supnφ(uαn)<∞,(1+∥uαn∥)φαn′(uαn)⟶0
contains a subsequence which converges to a critical point of φ, where φα=φ|Xα.Theorem B (see [12, Theorem 2.2]).
Suppose thatφ∈C1(X,ℝ) satisfies the following assumptions:  (i1)
X≠{0} and φ has a local linking at 0; that is, for some r>0,
(20)φ(u)≥0,∀u∈X1with∥u∥≤r,φ(u)≤0,∀u∈X2with∥u∥≤r;(i2)
φ satisfies (C*) condition;(i3)
φ maps bounded sets into bounded sets;(i4)
for everym∈ℕ,φ(u)→-∞ as |u|→∞, on u∈Xm1⊕X2. Then φ has at least one nonzero critical point.Now, we can give the proof of our theorems.Proof of Theorem1.
At first, we claim thatφ satisfies the Cerami condition (C). Consider a sequence {un} such that φ(un) is bounded and ∥φ′(un)∥(1+∥un∥)→0 as n→∞. Then there exists a constant M1>0 such that
(21)|φ(un)|≤M1,(1+∥un∥)∥φn′(un)∥≤M1.
By a standard argument, we only need to prove that {un} is a bounded sequence in E. Otherwise, going if necessary to a subsequence, we can assume that ∥un∥→∞ as n→∞. From (F3), there exist two constants k2,k3>0 such that
(22)f(x,u)u-2F(x,u)≥k2|u|γ-k3,∀(x,u)∈Ω×ℝ.
So, by (21) and (22), we have
(23)3M1≥2φ(un)-〈φ′(un),un〉=∫Ω‍(f(x,un)un-2F(x,un))dx≥k2∫Ω‍|un|γdx-k3|Ω|
which implies that
(24)∫Ω‍|un|γdx<k4
for all n∈ℕ and some positive constant k4.
Since(25)γ>N4(θ-2),N4(θ-2)<2NN+4(θ-1)<2NN-4.
On the one hand, we consider the case
(26)N4(θ-2)<γ<2NN+4(θ-1).
Putting(27)α=2(θ-1)N-(N+4)γ2N-(N-4)γ,
one has 0<α<1. Let
(28)p=γθ-1-α>1,un=un++un-+un0∈E+⊕E-⊕E0,
where E0=ker(Δ2+cΔ+a). We can obtain from Hölder's inequality, (15), and (24) that
(29)∫Ω‍|un|θ-1|un+|dx=∫Ω‍|un|θ-1-α|un|α|un+|dx=∫Ω‍|un|γ/p|un|α|un+|dx≤(∫Ω‍(|un|γ/p)pdx)1/p(∫Ω‍(|un|α|un+|)qdx)1/q≤(∫Ω‍|un|γdx)1/p(∫Ω‍(|un|qα)2**/(qα)dx)α/2**×(∫Ω‍(|un+|q)2**/qdx)1/2**≤k41/p∥un∥L2**α∥un+∥L2**≤k41/pKα+1∥un∥α∥un+∥
for all n, where q=p/(p-1)=2**/(α+1).
By (F2) and (29), one has
(30)〈φ′(un),un+〉=∥un+∥2-∫Ω‍f(x,un)un+dx≥∥un+∥2-∫Ω‍|f(x,un)||un+|dx≥∥un+∥2-k1∫Ω‍(|un|θ-1|un+|+|un+|)dx=∥un+∥2-k1∫Ω‍|un|θ-1|un+|dx-k1∫Ω‍|un+|dx≥∥un+∥2-k1k41/pKα+1∥un∥α∥un+∥-k1K∥un+∥
for all n. Since α<1, we have
(31)∥un+∥∥un∥⟶0,asn⟶∞.
On the other hand, ifγ satisfies,
(32)2NN+4(θ-1)≤γ<2NN-4.
then one sees 1≤γ/(γ-θ+1)≤2N/(N-4). So, we get
(33)∥u∥Lγ/(γ-θ+1)≤K∥u∥,∀u∈E.
It follows from (24) that
(34)∫Ω‍|un|γdx∥un∥⟶0,asn⟶∞.
By (F2) and (33), we obtain
(35)〈φ′(un),un+〉=∥un+∥2-∫Ω‍f(x,un)un+dx≥∥un+∥2-∫Ω‍|f(x,un)||un+|dx≥∥un+∥2-k1∫Ω‍(|un|θ-1|un+|+|un+|)dx=∥un+∥2-k1∫Ω‍|un|θ-1|un+|dx-k1∫Ω‍|un+|dx≥∥un+∥2-k1(∫Ω‍(|un|θ-1)γ/θ-1dx)(θ-1)/γ×(∫Ω‍|un+|γ/(γ-θ+1)dx)(γ-θ+1)/γ-k1K∥un+∥=∥un+∥2-k1∥un∥Lγθ-1·∥un+∥Lγ/(γ-θ+1)-k1K∥un+∥≥∥un+∥2-k1K∥un∥Lγθ-1·∥un+∥-k1K∥un+∥
for all n. Note that γ≥2N(θ-1)/(N+4) and N>4 imply that γ>θ-1. So, it follows from (34) and the above expression that
(36)∥un+∥∥un∥⟶0,asn⟶∞.
Hence, we conclude from (31) and (36) that
(37)∥un+∥∥un∥⟶0,asn⟶∞.
Similarly for un-, we get
(38)∥un-∥∥un∥⟶0,asn⟶∞.
It follows from the equivalence of the norms on the finite dimensional spaceE0 that there exists K1>0 such that
(39)∥u∥∞≤K1∥u∥,∥u∥≤K1∥u∥L1,∥u∥≤K1∥u∥L2,∀u∈E0.
Putting τ=2N+(5-N)γ+1, one has
(40)1<2τ-γτ-1<2NN-4.
It follows from (24), (39), and Hölder's inequality that
(41)1K12∥un0∥2≤∫Ω‍|un0|2dx≤∫Ω‍|un|2dx=∫Ω‍|un|γ/τ|un|2-γ/τdx≤(∫Ω‍|un|γdx)1/τ(∫Ω‍|un|(2τ-γ)/(τ-1)dx)(τ-1)/τ≤k41/τK(2τ-γ)/τ∥un∥(2τ-γ)/τ
and consequently
(42)∥un0∥∥un∥⟶0,asn⟶∞.
Hence, by (37), (38), and (42), one sees
(43)1=∥un∥∥un∥≤∥un+∥+∥un0∥+∥un-∥∥un∥⟶0
as n→∞, which is a contradiction. So, we obtain that {un} is bounded in E. By a standard argument, we get that φ satisfies the condition (C).
LetE=⊕j≥1Xj¯ with dimXj<∞ for any j≥1. Set
(44)Yk=⨁kj=1Xj,Zk=⨁∞j=kXj¯.
Since dim(Yk)<∞, all the norms are equivalent. For u∈Yk, there exists a constant K2>0 such that
(45)∥u∥≤K2∥u∥L2.
From condition (F1), there exists k5>0 such that
(46)F(x,u)≥K22|u|2-k5,∀(x,u)∈Ω×ℝ.
For u∈Yk, it follows from (45) and (46) that
(47)φ(u)=12(∥u+∥2-∥u-∥2)-∫Ω‍F(x,u)dx≤12(∥u+∥2-∥u-∥2)-K22∥u∥L22+k5|Ω|≤12(∥u+∥2-∥u-∥2)-(∥u+∥2+∥u0∥2)+k5|Ω|=-12∥u+∥2-12∥u-∥2-∥u0∥2+k5|Ω|≤-12∥u∥2+k5|Ω|
which implies that
(48)φ(u)⟶-∞as∥u∥⟶∞,inYk.
So, (A1) of Theorem A is satisfied for every ρk>0 large enough.
Here, we obtain from (F2) that there exists a positive constant k6 such that
(49)|F(x,u)|≤k6(1+|u|θ)
uniformly for all x∈Ω. Let us define
(50)βk=supu∈Zk,∥u∥=1∥u∥Lθ.
For k large enough, one has Zk⊂E+. By (49), on Zk, we have
(51)φ(u)=∥u∥22-∫Ω‍F(x,u)dx≥∥u∥22-k6∥u∥Lθθ-k6|Ω|≥∥u∥22-k6βkθ∥u∥θ-k6|Ω|.
Choosing rk=(k6θβkθ)1/(2-θ), we obtain, if u∈Zk and ∥u∥=rk,
(52)φ(u)≥(12-1θ)(k6θβkθ)2/(2-θ)-k6|Ω|.
Since, by Lemma 3.8 of [14], βk→0 as k→∞, (A2) is proved. Hence, the proof is completed by using Fountain theorem.Proof of Theorem2.
Firstly, we claim thatφ satisfies the Cerami condition (C). Consider a sequence {un} such that φ(un) is bounded from above and ∥φ′(un)∥(1+∥un∥)→0 as n→∞. By a standard argument, we only need to prove that {un} is a bounded sequence in E. For otherwise, we can assume that ∥un∥→∞ as n→∞.
From assumption (F4), there exist two positive constants m3 and m4, such that
(53)m3≥2φ(un)-〈φ′(un),un〉=∫Ω‍(f(x,un)un-2F(x,un))dx≥∫Ω‍m1|un|2dx-m4|Ω|.
So, one has
(54)∫Ω‍|un|2dx≤m5
for all n and some positive constant m5.
Letvn=un/∥un∥; then ∥vn∥=1 and ∥vn∥Lr≤Cr for all r∈[1,2N/(N-4)). By (54), we have
(55)∫Ω‍vn2dx=1∥un∥2∫Ω‍|un|2dx≤m5∥un∥2⟶0
as n→∞. So, for r∈(2,(2N-4)/(N-4)), it follows from Hölder's inequality and the above expression that
(56)∫Ω‍|vn|rdx≤(∫Ω‍|vn|2(r-1)dx)1/2(∫Ω‍|vn|2dx)1/2⟶0
as n→∞. It follows from (39) that
(57)〈φ′(un),un+-un-〉=∥un+-un-∥2-∫Ω‍f(x,un)(un+-un-)dx=∥un∥2-∥un0∥2-∫Ω‍f(x,un)(un+-un-)dx=∥un∥2(1-∫Ω‍f(x,un)(un+-un-)∥un∥2dx)-∥un0∥2≥∥un∥2(1-∫Ω‍f(x,un)(un+-un-)∥un∥2dx)-K12∥un0∥L22.
Hence, we get
(58)1-∫Ω‍f(x,un)(un+-un-)∥un∥2dx=o(1).
From (F4), (53), and (56), there exists a positive constant m6 such that
(59)|∫Ω‍f(x,un)(un+-un-)∥un∥2dx|≤2∫Ω‍|f(x,un)||un||vn|2dx≤2(∫Ω‍(|f(x,un)||un|)σdx)1/σ×(∫Ω‍|vn|2σ′dx)1/σ′≤m6(∫Ω‍|vn|2σ′dx)1/σ′⟶0
as n→∞; therefore, 1=o(1), which is a contradiction. Hence, {un} is bounded.
In a way similar to the proof of Theorem1, we can obtain that φ satisfies (A1) of Theorem A.
It follows from (F1) that there is L1>0 such that
(60)F(x,u)≥|u|2,∀|u|≥L1.
Then, by (F4) and (60), for |u|≥L2=max{L,L1},x∈Ω, one has
(61)|f(x,u)|σ≤m2(f(x,u)u-2F(x,u))|u|σ≤m2|f(x,u)||u|σ+1
which implies that
(62)|f(x,u)|≤m21/(σ-1)|u|(σ+1)/(σ-1)
for |u|≥L2.
Therefore, there exist two positive constantsm7 and m8 such that
(63)|F(x,u)|≤m7+m8|u|2σ/(σ-1),∀(x,u)∈Ω×ℝ,
where 2σ/(σ-1)<2N/(N-4). As the proof of Theorem 1, we can get (A2). Therefore, Theorem 2 holds.Proof of Theorem4.
The proof of this theorem is divided in several steps.
Step  1. We claim that φ has a local linking at zero with respect to (X1,X2).
By (F5), for any ε>0, there exists δ1>0 such that
(64)|F(x,u)|≤εu2,∀|u|≤δ1.
We obtain from the above expression and (49) that
(65)|F(x,u)|≤εu2+M2|u|θ,∀(x,u)∈Ω×ℝ,
where M2=k6+k6δ1-θ. Hence, we can get from (15) and (65) that
(66)|∫Ω‍F(x,u)dx|≤∫Ω‍εu2dx+M2∫Ω‍|u|θdx=ε∥u∥L22+M2∥u∥Lθθ≤K2ε∥u∥2+KθM2∥u∥θ
for all u∈E.
Here, we consider only the case where0 is an eigenvalue of Δ2+cΔ+a and case (ii) of (F6) holds. The case (i) is similar.
LetX=E,X1=E+⊕E0, and X2=E-, where E0=ker(Δ2+cΔ+a). Choose a Hilbertian basis {en}n≥0 for X1 and define
(67)Xn1=span{e0,…,en},n∈ℕ,Xn2=X2,n∈ℕ,X1=⋃n‍Xn1¯.
Now, by (66), for each u∈X2=E-, one has
(68)φ(u)=-12∥u∥2-∫Ω‍F(x,u)dx≤-12∥u∥2+K2ε∥u∥2+KθM2∥u∥θ.
Letting ε=1/(8K2) and by θ>2, we have
(69)φ(u)≤0,∀u∈X2with∥u∥≤δ2
for δ2>0 small enough.
Letu=u0+u+∈E0⊕E+=X1 be such that ∥u∥≤δ3:=δ/(2K1). Put
(70)Ω1={x∈Ω∣|u+(x)|≤δ2},Ω2=Ω∖Ω1.
Then, for all ∥u∥≤δ3 and x∈Ω, by (39), one sees
(71)|u0(x)|≤∥u0∥∞≤K1∥u0∥≤K1∥u∥≤δ2.
On one hand, from above expression, for any x∈Ω1, we have
(72)|u(x)|≤|u0(x)|+|u+(x)|≤∥u0∥∞+δ2≤δ.
Hence, by condition (ii) of (F6), we get
(73)∫Ω1‍F(x,u)dx≤0.
On the other hand, for any x∈Ω2, one has
(74)|u(x)|≤|u0(x)|+|u+(x)|≤δ2+|u+(x)|≤2|u+(x)|.
Hence, for all x∈Ω2 and u∈X1 with ∥u∥≤δ3, we can obtain from (65) that
(75)F(x,u)≤εu2+M2|u|θ≤4ε|u+(x)|2+2θM2|u+(x)|θ,
which implies that
(76)∫Ω2‍F(x,u)dx≤4ε∫Ω2‍|u+(x)|2dx+2θM2∫Ω2‍|u+(x)|θdx=4ε∥u+∥L22+2θM2∥u+∥Lθθ≤4K2ε∥u+∥2+(2K)θM2∥u+∥θ.
Letting ε=1/(16K2) in above expression, then for all x∈Ω2 and u∈X1 with ∥u∥≤δ3, we have
(77)φ(u)=12∥u+∥2-∫Ω2‍F(x,u)dx-∫Ω1‍F(x,u)dx≥12∥u+∥2-4K2ε∥u+∥2-(2K)θM2∥u+∥θ≥14∥u+∥2-(2K)θM2∥u+∥θ,
which implies that
(78)φ(u)≥0,∀u∈X1with∥u∥≤δ4
for δ4>0 small enough. Hence, φ has a local linking at zero with respect to (X1,X2) for δ5=min{δ2,δ4} small enough.
Step  2. In a way similar to the proof of Theorem 1, we can get that φ satisfies the (C*) condition.
Step  3. Now, we claim that for each m∈ℕ, one has
(79)φ(u)⟶-∞,as∥u∥⟶∞,inXm1⊕X2.
Since dim(E0)<∞ and dim(Xm1)<∞, all the norms are equivalent. For u∈Xm1⊕X2, there exists a constant k7>0 such that
(80)∥u∥≤k7∥u∥L2.
From condition (F1), there exists a constant k8>0 such that
(81)F(x,u)≥k72|u|2-k8,∀(x,u)∈Ω×ℝ.
For u∈Xm1⊕X2, it follows from (80) and (81) that
(82)φ(u)=12(∥u+∥2-∥u-∥2)-∫Ω‍F(x,u)dx≤12(∥u+∥2-∥u-∥2)-k72∥u∥L22+k8|Ω|≤12(∥u+∥2-∥u-∥2)-k72(∥u+∥L22+∥u0∥L22)+k8|Ω|≤12(∥u+∥2-∥u-∥2)-(∥u+∥2+∥u0∥2)+k8|Ω|=-12∥u+∥2-12∥u-∥2-∥u0∥2+k8|Ω|≤-12∥u∥2+k8|Ω|
which implies that
(83)φ(u)⟶-∞as∥u∥⟶∞,inXm1⊕X2.
Hence, all the assumptions of Theorem B are verified. Then, the proof of Theorem 4 is completed.Proof of Theorem5.
In a way similar to the proof of Theorems2 and 4, we can obtain that φ satisfies (i1), (i2), (i3), and (i4) of Theorem B. Therefore, Theorem 5 holds.



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*Source: 597193-2013-12-05.xml*