# Multiple Solutions for Degenerate Elliptic Systems Near Resonance at Higher Eigenvalues

Authors: Yu-Cheng An; Hong-Min Suo
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study the degenerate semilinear elliptic systems of the form-div(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ω,u|∂Ω=v|∂Ω=0, where Ω⊂RN(N≥2) is an open bounded domain with smooth boundary ∂Ω, the measurable, nonnegative diffusion coefficients h1, h2 are allowed to vanish in Ω (as well as at the boundary ∂Ω) and/or to blow up in Ω¯. Some multiplicity results of solutions are obtained for the degenerate elliptic systems which are near resonance at higher eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory.


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## Body

## 1. Introduction

In this paper, we study a class of degenerate elliptic systems:(1.1)-div⁡(h1(x)∇u)=λ(a(x)u+b(x)v)+Fu(x,u,v),x∈Ω,-div⁡(h2(x)∇v)=λ(d(x)v+b(x)u)+Fv(x,u,v),x∈Ωu∣∂Ω=v∣∂Ω=0,,
where Ω⊂RN(N≥2) is an open bounded domain with smooth boundary ∂Ω, F∈C1(Ω¯×R2,R) satisfies the following sublinear growth condition:
(1.2)lim⁡|s|→∞|∇F(x,s)||s|=0
uniformly in x∈Ω¯, where ∇F=(Fu,Fv) denotes the gradient of F with respect to (u,v)∈R2. The degeneracy of this system is considered in the sense that the measurable, nonnegative diffusion coefficients h1, h2 are allowed to vanish in Ω (as well as at the boundary ∂Ω) and/or to blow up in Ω¯. The consideration of suitable assumptions on the diffusion coefficients will be based on the work [1], where the degenerate scalar equation was studied. We introduce the function space (N)h, which consists of functions h:Ω⊂RN→R, such that h∈L1(Ω), h-1∈L1(Ω), and h-s∈L1(Ω), for some s>N/2.Then for the weight functionsh1, h2 we assume the following hypothesis. (N) there exist functions μ satisfying condition (N)μ, for some sμ, and ν satisfying condition (N)ν, for some sν, such that
(1.3)μ(x)k1≤h1(x)≤k1μ(x),ν(x)k2≤h2(x)≤k2ν(x),
a.e. in Ω, for some constants k1>1 and k2>1.The mathematical modeling of various physical processes, ranging from physics to biology, where spatial heterogeneity plays a primary role, is reduced to nonlinear evolution equations with variable diffusion or dispersion. Also note that problem (1.1) is closely related (see [1]) to the following system:(1.4)-div⁡(h1(x,u,v)∇u)=f(λ,x,u,v,∇u,∇v),x∈Ω,-div⁡(h2(x,u,v)∇v)=g(λ,x,u,v,∇u,∇v),x∈Ω,u∣∂Ω=v∣∂Ω=0.
Problems of such a type have been successfully applied to the heat propagation in heterogeneous materials, to the study of transport of electron temperature in a confined plasma, to the propagation of varying amplitude waves in a nonlinear medium, to the study of electromagnetic phenomena in nonhomogeneous superconductors and the dynamics of Josephson junctions, to electrochemistry, to nuclear reaction kinetics, to image segmentation, to the spread of microorganisms, to the growth and control of brain tumors, and to population dynamics (see [2–4] and the references therein).An example of the physical motivation of the assumptions(N), (N)h may be found in [3]. These assumptions are related to the modeling of reaction diffusion processes in composite materials occupying a bounded domain Ω, which at some point they behave as perfect insulators. When at some point the medium is perfectly insulated, it is natural to assume that h1(x) and/or h2(x) vanish in Ω. For more information we refer the reader to [4] and the references therein.For the perturbed problem, Mawhin and Schmitt [5] first considered the following two point boundary value problem:
(1.5)-u′′-λu=f(x,u)+h(x),u(0)=u(π)=0.
Under the assumption that f is bounded and satisfies a sign condition, if the parameter λ is sufficiently close to λ1 from left, problem (1.5) has at least three solutions, if λ1≤λ<λ2, problem (1.5) has at least one solution, where λ1,λ2 are the first and the second eigenvalues of the corresponding linear problem. Ma et al. [6] considered the boundary value problem Δu+λu+f(x,u)=h(x) defined on a bounded open set Ω⊂RN, no matter whether the boundary conditions are Dirichlet or Neumann condition, as the parameter λ approaches λ1 from left, there exist three solutions. Moreover, existence of three solutions was obtained for the quasilinear problem in bounded domains as the parameter λ approaches λ1 from left. In [7, 8], these results were extended to the perturbed p-Laplacian equation in RN. In [9], Ou and Tang extended above some results to some elliptic systems with the Dirichlet boundary conditions. Especially, de Paiva and Massa in [10] studied the semilinear elliptic boundary value problem in any spatial dimension and by using variational techniques, they showed that a suitable perturbation will turn the almost resonant situation (λ near to λk, i.e., near resonance with a nonprincipal eigenvalue) in a situation where the solutions are at least two. In [11], those results were extended to the cooperative elliptic systems in the bounded domain. Motivated by the idea above, we have the goal in this paper of extending these results in [10, 11] to some degenerate elliptic systems with the Dirichlet boundary conditions.


## 2. Preliminaries and Main Results

Leth(x) be a nonnegative weight function in Ω which satisfies condition (N)h. We consider the weighted Sobolev space D01,2(Ω,h) to be defined as the closure of C0∞(Ω) with respect to the following norm:
(2.1)‖u‖h=(∫Ωh(x)|∇u|2dx)1/2,
and the following scalar product:
(2.2)〈u,v〉h=∫Ωh(x)∇u∇vdx,
for all u,v∈D01,2(Ω,h). The space D01,2(Ω,h) is a Hilbert space. For a discussion about the space setting we refer to [1] and the references therein. Let
(2.3)2s*=2NsN(s+1)-2s.Lemma 2.1.
Assume thatΩ is a bounded domain in RN and the weight h satisfies (N)h. Then the following embeddings hold:(i)
D01,2(Ω,h)↪L2s*(Ω) continuously,(ii)
D01,2(Ω,h)↪Lr(Ω) compactly for any r∈[1,2s*).
In the sequel one denotes by2μ* and 2ν* the quantities 2sμ* and 2sν*, respectively, where sμ and sν are induced by condition (N), recall that h1, h2 satisfy (N). The assumptions concerning the coefficient functions of systems (1.1) are as follows.
(AD)
The functionsa,d∈C(Ω¯,R) and there exists x0∈Ω, such that a(x0)>0, d(x0)>0.(B)
The functionb∈C(Ω¯,(0,+∞)).The space setting for our problem is the product spaceH=D01,2(Ω,h1)×D01,2(Ω,h2) equipped with the following norm:
(2.4)‖z‖=(‖u‖h12+‖v‖h22)1/2,z=(u,v)∈H,
and the following scalar product:
(2.5)〈z,ϕ〉=〈u,ξ〉h1+〈v,τ〉h2,
for all z=(u,v),ϕ=(ξ,τ). Observe that inequalities (1.3) in condition (N) imply that the functional spaces D01,2(Ω,h1)×D01,2(Ω,h2) and D01,2(Ω,μ)×D01,2(Ω,ν) are equivalent. Especially, by Lemma 2.1 we know that for any 1≤δ≤min{2μ*,2ν*}, the embedding H↪Lδ(Ω)×Lδ(Ω) is continuous and there is a positive constant S=S(δ,N,Ω) such that ∫Ω|z|δdx≤Sδ∥z∥δ for all z∈H. Moreover, if δ<min{2μ*,2ν*}, the embedding above is also compact. Let
(2.6)H(x)=diag⁡(h1(x),h2(x)),A(x)=(a(x)b(x)b(x)d(x)).
Assume that hypothesis (N) is satisfied and the coefficient functions a, d and b satisfy conditions (AD) and (B), respectively.We consider the eigenvalue problem with weightA(x),
(2.7)-div⁡(h1(x)∇u)=λ(a(x)u+b(x)v),x∈Ω,-div⁡(h2(x)∇v)=λ(d(x)v+b(x)u),x∈Ω,u∣∂Ω=v∣∂Ω=0.
A simple calculation shows that λ is an eigenvalue of (2.7) if and only if
(2.8)TAz=λ-1z,
where TA:H→H is the symmetric bounded linear operator defined by
(2.9)〈TAz,ϕ〉=∫Ω(A(x)z,ϕ)dx,∀z,ϕ∈H.Since the coefficient of A are continuous functions and  the embedding H↪L2(Ω)×L2(Ω) is compact, we can check that the operator TA is also compact. Thus, we may invoke the spectral theory for compact operators to conclude that H possesses a Hilbertian basis formed by eigenfunctions of (2.7).Let us denotez=(u,v) and
(2.10)λ1-1=μ1=sup⁡{〈TAz,z〉:‖z‖=1}.
Recalling that A satisfies (AD) and (B), we can use [2, Theorem  1.1] (p=q=2,α=β=0) to conclude that the eigenvalue μ1 is positive, simple, and isolated in the spectrum of TA. Moreover, if we denote by ϕ1 the normalized eigenfunction associated to λ1, we can suppose that the ϕ1 is positive on Ω. By using induction, if we suppose that μ1>μ2≥…≥μk-1 are the k-1 first eigenvalues of TA and {ϕi}i=1k-1 are the associated normalized eigenfunctions, we can define
(2.11)λk-1=μk=sup⁡{〈TAz,z〉≔‖z‖=1,z∈(span⁡{ϕ1,…,ϕk-1})⊥}.
It is proved in [11, Proposition  1.3], that, if μk>0, then it is an eigenvalue of TA with associated normalized eigenfunction ϕk. In view of the condition (AD), we can argue as in the proof of [11, Proposition  1.11(c)], and conclude that μk>0. Thus, we obtain a sequence of eigenvalues for (2.7). 						(2.12)0<λ1<λ2≤⋯≤λk≤⋯
such that λk→∞ as k→∞. We denote by Ek the eigenfunction space corresponding to λk. Moreover, if we set Vk=span{ϕ1,…,ϕk}, we can decompose H=Vk⊕Vk⊥. Moreover, the following inequalities hold:
(2.13)∫Ω(H(x)∇z,∇z)dx≤λk∫Ω(A(x)z,z)dx,∀z∈Vk,(2.14)∫Ω(H(x)∇z,∇z)dx≥λk+1∫Ω(A(x)z,z)dx,∀z∈Vk⊥.Lemma 2.2 (from Lemma  4.6 of [10]).
LetX be a Hilbert space with orthonormal direct sum splitting X=V⊕Z⊕W. Moreover, let dim(V⊕Z)<∞. For ρ>R>0, set
(2.15)A={u∈W:‖u‖≥R}∪{u∈Z⊕W:‖u‖=R},B={u∈V⊕Z:‖u‖=ρ}.
Then A links with B.Lemma 2.3 (from Theorem  8.1 of [13]).
LetH=X1⊕X2 be a Hilbert space where X1 has finite dimension, J∈C1(H,R) satisfying the (P.S.) condition and such that, for given ρ1,ρ2>0,
(2.16)sup⁡z∈ρ1S1J(z)<a=inf⁡z∈ρ2B2J(z)≤b=sup⁡z∈ρ1B1J(z)<inf⁡z∈ρ2S2J(z),
where Bi and Si represent the unit ball and the unit sphere in Xi:i=1,2.
Then there exists a critical pointz0 such that J(z0)∈[a,b].Next, in order to state our main results, we introduce the following assumptions on the nonlinear term:(F1)
lim|s|→∞(∇F(x,s),s)/|s|=+∞ uniformly with respect to x∈Ω.(F2)
lim|s|→∞F(x,s)=+∞ uniformly with respect to x∈Ω.(F3)
lim|s|→∞(∇F(x,s),s)/|s|=-∞ uniformly with respect to x∈Ω.(F4)
lim|s|→∞F(x,s)=-∞ uniformly with respect to x∈Ω.Our main results are given by the following theorems.Theorem 2.4.
Letλk(k≥2) be an eigenvalue of multiplicity m. Suppose that condition (N) and the coefficient functions a, d and b satisfy conditions (AD) and (B), respectively. Assume, in addition, that F satisfies (1.2) and ∇F(x,0)=0 for all x∈Ω, and one of the sets of hypotheses (F1) or (F2). Then there exists δ0>0 such that for λ∈(λk-δ0,λk) problem (1.1) has at least two solutions.Theorem 2.5.
Letλk(k≥2) be an eigenvalue of multiplicity m. Suppose that conditions (N) and the coefficient functions a, d and b satisfy conditions (AD) and (B), respectively. Assume, in addition, that F satisfies (1.2) and ∇F(x,0)=0 for all x∈Ω, and one of the sets of hypotheses (F3) or (F4). Then there exists δ1>0 such that for λ∈(λk,λk+δ1) problem (1.1) has at least two solutions.


## 3. Proof of Theorems

Consider theC1 functional J:H→R, ∀z∈H,
(3.1)J(z)=12∫Ω(H(x)∇z,∇z)dx-λ2∫Ω(A(x)z,z)dx-∫ΩF(x,z)dx.
Since the problems in Theorems 2.4 and 2.5 are not resonant, J satisfies the Palais-Smale condition of compactness (see, e.g., in [14]). In addition, z∈H is a weak solution of problem (1.1) if and only if z is a critical point of J.We set(3.2)V=span⁡{ϕ1,…,ϕk-1},Z=span⁡{ϕk,…,ϕk+m-1}=Ek,W=(V⊕Z)⊥,
and we define
(3.3)BV={z∈V:‖z‖≤1},BVZ={z∈V⊕Z:‖z‖≤1},BZW={z∈Z⊕W:‖z‖≤1},
and SV,SVZ,SZW, respectively, their relative boundaries.Theorems2.4 and 2.5 will be a consequence of the geometry in Propositions 3.1 and 3.2, whose proofs will be postponed to Sections 4 and 5.Proposition 3.1.
Ifλ∈(λk-1,λk) and hypothesis (1.2) is satisfied, then there exist constants Dλ and ρλ such that
(3.4)J(z)≥Dλ,forz∈Z⊕W,(3.5)J(z)<Dλ,forz∈ρλSV.
Moreover, if one of the sets of hypotheses (F1) or (F2) is satisfied, then there exists δ0 such that for λ∈(λk-δ0,λk) there exist DW,Dλ∈R,ρλ>R1>0 such that, in addition to (3.4) and (3.5),
(3.6)J(z)≥DW,forz∈W,(3.7)J(z)<DW,forz∈R1SVZ,(3.8)J(z)<DW,forz∈V,‖z‖≥R1.
(The values with index λ depend on λ, the others may be fixed uniformly.)
Based on this geometry one gives the following proof.Proof of Theorem2.4.
Since the functionalJ satisfies the (P.S.) condition, we can apply the saddle point theorem (see, e.g., in [15]) for two times, let
(3.9)Γk-1={γ∈C0(ρλBV;H)s.t.γ∣ρλSV=id},Γk={γ∈C0(R1BVZ;H)s.t.γ∣R1SVZ=id}.
The first solution, that we denote byzk-1 and may be obtained for any λ∈(λk-1,λk) with just hypothesis (1.2), corresponds to a critical point at the level
(3.10)ck-1=inf⁡γ∈Γk-1sup⁡w∈ρλBVJ(γ(w)),
the criticality of this level is guaranteed by the estimates (3.4) and (3.5), since ρλSV andZ⊕W link, that is, the image of any map in Γk-1 intersects Z⊕W.
The second solution, that we denote byzk, corresponds to a critical point at the critical level
(3.11)ck=inf⁡γ∈Γksup⁡w∈R1BVZJ(γ(w)),
actually, this is a critical level because of the estimates (3.6) and (3.7), since R1SVZ and W link.
To conclude the proof, we need to show that these two solutions are distinct.
We observe first that by estimate (3.6) we have that ck≥DW, then we observe that we may build a map γ0∈Γk-1 in such a way that its image is the union between the annulus {z∈V:∥z∥∈[R1,ρλ]} and the image of a (k-1)-dimensional ball in R1SVZ whose boundary is R1SV. By the estimates (3.7) and (3.8), we deduce that supw∈ρλBVJ(γ0(w))<DW, and as a consequence ck-1<DW, proving that the two solutions are distinct, for being at different critical levels.Proposition 3.2.
Ifλ∈(λk,λk+m) and hypothesis (1.2) is satisfied, then there exist constants Kλ and βλ such that
(3.12)J(z)≥Kλ,forz∈W,(3.13)J(z)<Kλ,forz∈βλSVZ.
Moreover, if one of the sets of hypotheses (F3) or (F4) is satisfied, then there exists δ1 such that for λ∈(λk,λk+δ1) there exist Kλ,KV,E∈R, βλ>R2>0, ξ>0 such that, in addition to (3.12) and (3.13),
(3.14)J(z)<KV,forz∈V,(3.15)J(z)>KV,forz∈R2SZW,(3.16)J(z)>KV,forz∈W,‖z‖≥R2,(3.17)J(z)>E,forz∈R2BZW,(3.18)J(z)<E,forz∈ξSV.
The values with index λ depend on λ, the others may be fixed uniformly.
This geometry, along with Lemma2.2, allows one to give the following.Proof of Theorem2.5.
Since the functionalJ satisfies the (P.S.) condition, we can apply the saddle point theorem and Lemma 2.3.
The first solution that we denote bywk and may be obtained for any λ∈(λk,λk+m) with just hypothesis (1.2) is again obtained through the saddle point theorem and corresponds to a critical point at the critical level
(3.19)dk=inf⁡γ∈Γksup⁡w∈βλBVZJ(γ(w)),
where now
(3.20)Γk={γ∈C0(βλBVZ;H)s.t.γ∣βλSVZ=id},
the criticality is guaranteed by estimates (3.12) and (3.13), since βλSVZ and W link.
The second solution that we denote bywk-1 comes from Lemma 2.3, where we set X1=V and X2=Z⊕W, actually we have the following structure:
(3.21)sup⁡ξSVJ(z)<E=inf⁡R2BZWJ(z)≤sup⁡ξBVJ(z)<KV<inf⁡R2SZWJ(z),
and then we have a critical point wk-1 at the level dk-1≤KV.
Finally, in order to prove that these two solutions are distinct, we need a sharper estimate fordk than that given by (3.13). For this we use Lemma 2.2 to guarantee that for any map γ∈Γk, since βλ>R2, one has that the image of γ either intersects R2SZW or has a point z∈W with ∥z∥≥R2. This implies that supw∈βλBVZJ(γ(w))>KV, by estimates (3.15) and (3.16), and then dk>KV proving that the two solutions are distinct, for being at different critical levels.


## 4. Proof of Estimates

In this section we will prove all the estimates in Propositions3.1 and 3.2.From (1.2) and the continuity of the potential F, for any ε>0, there exists a positive constant Mε=M(ε) such that
(4.1)|∇F(x,s)|≤ε|s|+Mε,
for all (x,s)∈Ω¯×R2. By (4.1), Hölder’s inequality, we have
(4.2)|∫ΩF(x,z)dx|≤∫Ω|∫01(∇F(x,tz),z)dt|dx≤∫Ω∫01|∇F(x,tz)||z|dtdx≤∫Ω(ε|z|2+Mε|z|)dx≤ε‖z‖L22+Mε|Ω|1/2‖z‖L2≤εS2‖z‖2+MεS|Ω|1/2‖z‖,
where S is the best embedding constant.


### 4.1. Estimates of the Saddle Geometry

Lemma 4.1.
Under hypothesis (1.2), one gets the following:(i)
forλ∈(λk-1,λk), there exists Dλ satisfying (3.4) and DW∈R satisfying (3.6);(ii)
forλ∈(λk,λk+m):
(a)
there existsKλ∈R satisfying (3.12),(b)
for a givenR2>0, there exists E∈R satisfying (3.17).Proof.
Letz∈W: using estimates (4.2) and (2.14) we get
(4.3)J(z)≥(λk+m-λ2λk+m-εS2)‖z‖2-Mε|Ω|1/2S‖z‖.
Forλ∈(λk-1,λk), letting ε<(λk+m-λk)/2S2λk+m<(λk+m-λ)/2S2λk+m, it follows that J is bounded below in W, that is, there exists a DW as in (3.6).
Forλ∈(λk,λk+m), then the same estimate holds but the constant cannot be made independent of λ, giving (3.12).
In the same way, letz∈Z⊕W and set δ=λk-λ>0, we get
(4.4)J(z)≥(λk-λ2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖≥(δ2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖.
Letting ε<δ/2S2λk, it follows that J is bounded below in Z⊕W, that is, there exists a Dλ such that for all z∈Z⊕W we have (3.4), where again the constant Dλ depends on δ, that is, on λ.
Finally, (4.4) with λ∈(λk,λk+m) implies
(4.5)J(z)≥(λk-λk+m2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖,
then, no matter the value of λ, J is bounded from below in any bounded subset of Z⊕W, giving (3.17) for a suitable value of E.Lemma 4.2.
Under hypothesis (1.2), one gets the following:(i)
forλ∈(λk-1,λk), given the constant Dλ∈R, there exists ρλ>0 satisfying (3.5);(ii)
forλ∈(λk,λk+m):
(a)
there existsKV∈R satisfying (3.14),(b)
for a givenKλ∈R, there exists βλ>0 satisfying (3.13),(c)
for a givenE∈R, there exists ξ>0 satisfying (3.18).
Moreover, given the valuesR1, R2, one may always choose ρλ>R1, βλ>R2 as claimed in Propositions 3.1 and 3.2.Proof.
Letz∈V, by estimates (4.2) and (2.13) we get
(4.6)J(z)≤(λk-1-λ2λk-1+εS2)‖z‖2+Mε|Ω|1/2S‖z‖.
Forλ∈(λk-1,λk), letting ε<(λ-λk-1)/2S2λk-1, then one obtains (3.5) for suitably large ρλ>R1.
Forλ∈(λk,λk+m), letting ε<(λk-λk-1)/2S2λk-1, one obtains, for suitable KV and ξ>0, (3.14) and (3.18).
Finally, letz∈V⊕Z and set δ=λ-λk>0, we get
(4.7)J(z)≤(λk-λ2λk+εS2)‖z‖2+Mε|Ω|1/2S‖z‖≤(-δ2λk+εS2)‖z‖2+Mε|Ω|1/2S‖z‖.
Letting ε<δ/2S2λk, it is clear that (once that δ is fixed) this goes to -∞ and then we may find the claimed βλ>R2 such that (3.13) holds.
Observe thatKV and E can be chosen uniformly for λ∈(λk,λk+m), while ρλ, βλ in fact depend on λ.


### 4.2. Estimating the Effect of the Nontrivial Perturbation

In this section we will prove the remaining inequalities in Propositions3.1 and 3.2, which rely on the hypotheses (F1), or (F2), or (F3), or (F4), which, roughly speaking, say that the perturbation F is nontrivial in such a way that a new solution arises when it is sufficiently near to the eigenvalue λk. The proof is simpler for Theorem 2.4, since we need to estimate the functional in the compact set SVZ, while for Theorem 2.5 the same kind of estimate is required in the noncompact set SZW.


#### 4.2.1. EstimatingJ in SVZ

For the next estimates, we will need the following lemma.Lemma 4.3.
Hypotheses(F2) implies that there exists a nondecreasing function D:(0,+∞)→R such that
(4.8)lim⁡R→+∞D(R)=+∞,inf⁡z∈RSVZ∫ΩF(x,z)dx>D(R).Proof.
First we claim that there exists a constantη>0 such that the sets Ωz={x∈Ω:|z(x)|>η} have measure |Ωz|>η, for all z∈SVZ.
Actually,V⊕Z is a finite-dimensional subspace and the functions z∈SVZ are smooth, they are uniformly bounded, that is, there exists M>0 such that |z(x)|≤M for all x∈Ω. Suppose that for ηn→0(ηn<1) there exists {zn}⊂SVZ such that |Ωzn|≤ηn.
On one hand, by (2.13), one obtains
(4.9)1λk≤∫Ω(A(x)zn,zn)dx.
On the other hand,(4.10)∫Ω(A(x)zn,zn)dx=∫Ω(a(x)un2+2b(x)unvn+d(x)vn2)dx≤∫Ω(a(x)+b(x))un2+∫Ω(b(x)+d(x))vn2dx≤M¯∫Ω|zn|2dx=M¯(∫Ωzn|zn|2dx+∫Ω-Ωzn|zn|2dx)≤M¯(M2|Ωzn|+ηn2|Ω-Ωzn|)≤ηnC⟶0,
where M¯=maxx∈Ω{|a(x)+b(x)|,|b(x)+d(x)|}, zn=(un,vn), C=M¯(M2+|Ω|). That is a contradiction.
Now for anyH>0, we will show that we can find an R̃ large enough so that ∫ΩF(x,Rz)dx≥H for any z∈SVZ and R≥R̃, which means that
(4.11)lim⁡R→∞inf⁡z∈RSVZ∫ΩF(x,z)dx=+∞.
Actually, lettingM=(H+|Ω|CF)η-1, by (F2) we have that there exists s0 such that F(x,s)>M for |s|>s0.
ForR>s0/η, one has Ωz⊆{x∈Ω:|Rz(x)|>s0}, and then one gets
(4.12)∫|Rz|≥s0F(x,Rz)dx≥Mη.
ForR≤s0/η, by (F2) and F∈C1(Ω¯×R2,R), there exists CF>0 such that F(x,s)≥-CF, for all (x,s)∈(Ω,R2).
One finally obtains(4.13)∫ΩF(x,Rz)dx=∫|Rz|≥s0F(x,Rz)dx+∫|Rz|≤s0F(x,Rz)dx≥Mη-|Ω|CF=H,
it is elementary that
(4.14)D(R)=inf⁡ρ≥Rinf⁡z∈RSVZ∫ΩF(x,z)dx
is well defined and satisfies the claim.Now we may prove the following.Lemma 4.4.
Consider Theorem2.4 with one of the sets of hypotheses (F1) or (F2). Given the constant DW∈R, there exist R1, δ0>0 such that, for any λ∈(λk-δ0,λk), (3.7) and (3.8) hold.Proof.
We consider the two sets of hypotheses separately.
(i) In case(F1), assuming (1.2) and (F1) hold, we claim that there exists CM such that
(4.15)F(x,s)≥M|s|-CM,∀M∈R,
uniformly in x∈Ω, in particular we set M=1.
In fact, by(F1), there exits R0>0 such that for |s|≥R0, (∇F(x,s),s)≥|s| uniformly in x∈Ω. For any s∈R2(s≠0), from (4.1) (letting ε=2) we have
(4.16)F(x,s)=∫01(∇F(x,ts),s)dt=∫R0/|s|1(∇F(x,ts),s)dt+∫0R0/|s|(∇F(x,ts),s)dt≥∫R0/|s|1|s|dt+∫0R0/|s|(∇F(x,ts),s)dt≥|s|-R0-∫0R0/|s|(2t|s|2+|M2||s|)dt=|s|-R0-R02-M2R0=|s|-C1,
where C1=R0+R02+M2R0.
Letz∈RSVZ, for being in a finite-dimensional subspace, all the norms are equivalent, so that (set δ=λk-λ>0 and uses estimates (4.15) and (2.13))
(4.17)J(z)≤λk-λ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λkR2-R+C1|Ω|.
(ii) In case (F2), let D(R) be as in Lemma 4.3, for ∥z∥=R, let z=w+ϕ with w∈V and ϕ∈Z=Ek,
(4.18)J(z)=12∫Ω(H(x)∇z,∇z)dx-λ2∫Ω(A(x)z,z)-∫ΩF(x,z)dx≤λk-1-λ2λk-1‖w‖2+λk-λ2λk‖ϕ‖2-∫ΩF(x,z)dx≤λk-1-λk+δ2λk-1‖w‖2+δ2λk‖ϕ‖2-∫ΩF(x,z)dx≤δ2λk‖ϕ‖2-λk-λk-14λk-1‖w‖2-∫ΩF(x,z)dx.
Assume that δ≤(λk-λk-1)/2, it is easy to see that(-(λk-λk-1)/4λk-1)∥w∥2≤C for some constant C, we estimate
(4.19)J(z)≤δ2λk‖ϕ‖2+C-D(R)≤δ2λkR2-D(R)+C.
Considering (4.17) and (4.19), we see that since limR→∞D(R)=+∞ by Lemma 4.3, we may fix R1 so that C-D(R1)<DW-1 (or DM-R1<DW-1 for the case (F1)) and then for 0<δ<min{2λk/R12,(λk-λk-1)/2} one gets (3.7).
To obtain (3.8), we observe that (since λ>λk-1) if ϕ=0, that is, if z∈V, then in estimates (4.17) and (4.19) we may avoid the term (δ/2λk)R2 so that (remember that D(R) is nondecreasing) J(z)<DW-1 for ∥z∥>R1.


#### 4.2.2. EstimatingJ in SZW

We consider the corresponding of the previous lemma, for Theorem2.5.Lemma 4.5.
Considering Theorem2.5 with one of the sets of hypotheses (F3) or (F4). Given the constant KV∈R, there exists R2, δ1>0 such that, for any λ∈(λk,λk+δ1), (3.15) and (3.16) hold.Proof.
Lettingλ=λk+δ, we see from (4.3), that property (3.16) will be satisfied provided that R2 is large enough (say R2>R̃) and observing that this value can be made independent from λ once that δ is small enough.
Now we consider the two sets of hypotheses separately.
(i) In case(F3), suppose z∈Ek⊕W, we can assume that z=w+ϕ, with w∈W and ϕ∈Ek. Since Ek is a finite dimension subspace, all the norms are equivalent, so that there exists C>0 such that for all ϕ∈Ek we have ∥ϕ∥≤C∥ϕ∥L1. By (F3), from the proof of (4.15), we also have the similar inequality: there exists C2 such that
(4.20)-F(x,s)≥C|s|-C2,
uniformly in x∈Ω. So by (2.14) and (4.20),
(4.21)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖w+ϕ‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖ϕ‖L1-C‖-w‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+‖ϕ‖-C3‖w‖-C4,
where C3=|Ω|1/2SC, C4=C2|Ω|. Since
(4.22)(1-δ2λk‖z‖)‖z‖≤‖w‖+‖ϕ‖-δ2λk(‖ϕ‖2+‖w‖2)≤(1-δ2λk‖ϕ‖)‖ϕ‖+‖w‖,
suppose δ≤(λk+m-λk)/2, (4.21) becomes
(4.23)J(w+ϕ)≥λk+m-(λk+δ)2λk+m‖w‖2-C3‖w‖-C4-‖w‖+(1-δ2λk‖z‖)‖z‖≥λk+m-λk4λk+m‖w‖2-(C3+1)‖w‖-C4+(1-δ2λk‖z‖)‖z‖,
since (λk+1-λk)/4λk+1>0, so ((λk+1-λk)/4λk+1)∥w∥2-(C3+1)∥w∥-C4 is bounded below for all w∈W, that is, there exists C5∈R such that
(4.24)λk+m-λk4λk+m‖w‖2-C3‖w‖-C4≥C5,
by (4.23) one gets
(4.25)J(z)≥(1-δ2λk‖z‖)‖z‖+C5=-δ2λk‖z‖2+‖z‖+C5.
(ii) In case (F4), first we give some conclusions which are similar to Lemma  3 of [16]. Under the property of F, there exists a constant C, and G∈C(R2,R) which is subadditive, that is,
(4.26)G(s+t)≤G(s)+G(t),
for all s,t∈R2, and coercive, that is,
(4.27)G(s)⟶+∞,
as |s|→∞, and satisfies that
(4.28)G(s)≤|s|+4,
for all s∈R2, such that
(4.29)-F(x,s)≥G(s)-C,
for all s∈R2 and x∈Ω¯.
In fact, since-F(x,s)→+∞ as |s|→∞ uniformly for all x∈Ω¯, there exists a sequence of positive integers nk with nk+1>2nk for all positive integers k such that
(4.30)-F(x,s)≥k,
for all |s|≥nk and all x∈Ω¯. Let n0=0 and define
(4.31)G(s)=k+2+|s|-nk-1nk-nk-1,
for nk-1≤|s|<nk, where k∈N.
By the definition ofG we have
(4.32)k+2≤G(s)≤k+3,
for all nk-1≤|s|<nk. By (F4) and F∈C1(Ω¯×R2,R), there exists CF>0 such that
(4.33)-F(x,s)≥-CF,∀(x,s)∈(Ω,R2).
It follows that
(4.34)-F(x,s)≥G(s)-C,
where C=CF+4. In fact, when nk-1≤|s|<nk for some k≥2, one has, by (4.30) and (4.32),
(4.35)-F(x,s)≥k-1≥G(s)-4≥G(s)-C,
for all x∈Ω¯. When |s|<n1, we have, by (4.32) and (4.33),
(4.36)-F(x,s)≥-CF=4-C≥G(s)-C,
for all x∈Ω¯.
It is obvious thatG is continuous and coercive. Moreover one has
(4.37)G(s)≤|s|+4,
for all s∈R2. In fact, for every s∈R2 there exists k∈N such that
(4.38)nk-1≤|s|<nk,
which implies that
(4.39)G(s)≤(k-1)+4≤nk-1+4≤|s|+4,
for all s∈R2 by (4.32) and the fact that nk≥k for all integers k≥0.
Now we only need to prove the subadditivity ofG. Let
(4.40)nk-1≤|s|<nk,nj-1≤|t|<nj,
and m=max{k,j}. Then we have
(4.41)|s+t|≤|s|+|t|<nk+nj≤2nm<nm+1.
Hence we obtain, by (4.32),
(4.42)G(s+t)≤m+4≤k+2+j+2≤G(s)+G(t),
which shows that G is subadditive.
Forz∈=Ek⊕W, assuming that z=w+ϕ, with w∈W and ϕ∈Ek, and letting 0<δ<(λk+m-λk)/2, by (2.13), (4.29), (4.26), and (4.28), one gets
(4.43)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ+w)dx-C|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫ΩG(-w)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫Ω(|w|+4)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖z‖2+∫ΩG(ϕ)dx-S|Ω|‖w‖-C1=g(w)+∫ΩG(ϕ)dx-δ2λk‖z‖2,
where g(w)=((λk+m-λk)/4λk+m)∥w∥2-S|Ω|∥w∥-C1, C1=(4+C)|Ω|. Since ϕ∈Ek, Ek is a finite-dimensional subspace, and G is coercive, from the proof of (4.8), one can get
(4.44)lim⁡∥ϕ∥→∞∫ΩG(ϕ)dx=+∞,
that is, ∫ΩG(ϕ)dx is coercive on Ek. Since (λk+m-λk)/4λk+m>0, so g(w) is coercive on W, and ∫ΩG(ϕ)dx and g(w) is bounded below, it is obvious that
(4.45)lim⁡∥z∥→∞(g(w)+∫ΩG(ϕ)dx)=+∞,
for all z∈Z⊕W.
Considering (4.25), (4.43), and (4.45), we can choose R2 large enough such that for all ∥z∥≥R2 one gets
(4.46)g(w)+∫ΩG(ϕ)dx>KV+1,
(or R2+C5>KV+1 for the case (F3)) and property (3.16) holds, then for 0<δ<min{2λk/R22,(λk+m-λk)/2}=δ1 and z∈R2SZW one gets J(z)>KV, that is, the property (3.15) holds.


## 4.1. Estimates of the Saddle Geometry

Lemma 4.1.
Under hypothesis (1.2), one gets the following:(i)
forλ∈(λk-1,λk), there exists Dλ satisfying (3.4) and DW∈R satisfying (3.6);(ii)
forλ∈(λk,λk+m):
(a)
there existsKλ∈R satisfying (3.12),(b)
for a givenR2>0, there exists E∈R satisfying (3.17).Proof.
Letz∈W: using estimates (4.2) and (2.14) we get
(4.3)J(z)≥(λk+m-λ2λk+m-εS2)‖z‖2-Mε|Ω|1/2S‖z‖.
Forλ∈(λk-1,λk), letting ε<(λk+m-λk)/2S2λk+m<(λk+m-λ)/2S2λk+m, it follows that J is bounded below in W, that is, there exists a DW as in (3.6).
Forλ∈(λk,λk+m), then the same estimate holds but the constant cannot be made independent of λ, giving (3.12).
In the same way, letz∈Z⊕W and set δ=λk-λ>0, we get
(4.4)J(z)≥(λk-λ2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖≥(δ2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖.
Letting ε<δ/2S2λk, it follows that J is bounded below in Z⊕W, that is, there exists a Dλ such that for all z∈Z⊕W we have (3.4), where again the constant Dλ depends on δ, that is, on λ.
Finally, (4.4) with λ∈(λk,λk+m) implies
(4.5)J(z)≥(λk-λk+m2λk-εS2)‖z‖2-Mε|Ω|1/2S‖z‖,
then, no matter the value of λ, J is bounded from below in any bounded subset of Z⊕W, giving (3.17) for a suitable value of E.Lemma 4.2.
Under hypothesis (1.2), one gets the following:(i)
forλ∈(λk-1,λk), given the constant Dλ∈R, there exists ρλ>0 satisfying (3.5);(ii)
forλ∈(λk,λk+m):
(a)
there existsKV∈R satisfying (3.14),(b)
for a givenKλ∈R, there exists βλ>0 satisfying (3.13),(c)
for a givenE∈R, there exists ξ>0 satisfying (3.18).
Moreover, given the valuesR1, R2, one may always choose ρλ>R1, βλ>R2 as claimed in Propositions 3.1 and 3.2.Proof.
Letz∈V, by estimates (4.2) and (2.13) we get
(4.6)J(z)≤(λk-1-λ2λk-1+εS2)‖z‖2+Mε|Ω|1/2S‖z‖.
Forλ∈(λk-1,λk), letting ε<(λ-λk-1)/2S2λk-1, then one obtains (3.5) for suitably large ρλ>R1.
Forλ∈(λk,λk+m), letting ε<(λk-λk-1)/2S2λk-1, one obtains, for suitable KV and ξ>0, (3.14) and (3.18).
Finally, letz∈V⊕Z and set δ=λ-λk>0, we get
(4.7)J(z)≤(λk-λ2λk+εS2)‖z‖2+Mε|Ω|1/2S‖z‖≤(-δ2λk+εS2)‖z‖2+Mε|Ω|1/2S‖z‖.
Letting ε<δ/2S2λk, it is clear that (once that δ is fixed) this goes to -∞ and then we may find the claimed βλ>R2 such that (3.13) holds.
Observe thatKV and E can be chosen uniformly for λ∈(λk,λk+m), while ρλ, βλ in fact depend on λ.


## 4.2. Estimating the Effect of the Nontrivial Perturbation

In this section we will prove the remaining inequalities in Propositions3.1 and 3.2, which rely on the hypotheses (F1), or (F2), or (F3), or (F4), which, roughly speaking, say that the perturbation F is nontrivial in such a way that a new solution arises when it is sufficiently near to the eigenvalue λk. The proof is simpler for Theorem 2.4, since we need to estimate the functional in the compact set SVZ, while for Theorem 2.5 the same kind of estimate is required in the noncompact set SZW.


### 4.2.1. EstimatingJ in SVZ

For the next estimates, we will need the following lemma.Lemma 4.3.
Hypotheses(F2) implies that there exists a nondecreasing function D:(0,+∞)→R such that
(4.8)lim⁡R→+∞D(R)=+∞,inf⁡z∈RSVZ∫ΩF(x,z)dx>D(R).Proof.
First we claim that there exists a constantη>0 such that the sets Ωz={x∈Ω:|z(x)|>η} have measure |Ωz|>η, for all z∈SVZ.
Actually,V⊕Z is a finite-dimensional subspace and the functions z∈SVZ are smooth, they are uniformly bounded, that is, there exists M>0 such that |z(x)|≤M for all x∈Ω. Suppose that for ηn→0(ηn<1) there exists {zn}⊂SVZ such that |Ωzn|≤ηn.
On one hand, by (2.13), one obtains
(4.9)1λk≤∫Ω(A(x)zn,zn)dx.
On the other hand,(4.10)∫Ω(A(x)zn,zn)dx=∫Ω(a(x)un2+2b(x)unvn+d(x)vn2)dx≤∫Ω(a(x)+b(x))un2+∫Ω(b(x)+d(x))vn2dx≤M¯∫Ω|zn|2dx=M¯(∫Ωzn|zn|2dx+∫Ω-Ωzn|zn|2dx)≤M¯(M2|Ωzn|+ηn2|Ω-Ωzn|)≤ηnC⟶0,
where M¯=maxx∈Ω{|a(x)+b(x)|,|b(x)+d(x)|}, zn=(un,vn), C=M¯(M2+|Ω|). That is a contradiction.
Now for anyH>0, we will show that we can find an R̃ large enough so that ∫ΩF(x,Rz)dx≥H for any z∈SVZ and R≥R̃, which means that
(4.11)lim⁡R→∞inf⁡z∈RSVZ∫ΩF(x,z)dx=+∞.
Actually, lettingM=(H+|Ω|CF)η-1, by (F2) we have that there exists s0 such that F(x,s)>M for |s|>s0.
ForR>s0/η, one has Ωz⊆{x∈Ω:|Rz(x)|>s0}, and then one gets
(4.12)∫|Rz|≥s0F(x,Rz)dx≥Mη.
ForR≤s0/η, by (F2) and F∈C1(Ω¯×R2,R), there exists CF>0 such that F(x,s)≥-CF, for all (x,s)∈(Ω,R2).
One finally obtains(4.13)∫ΩF(x,Rz)dx=∫|Rz|≥s0F(x,Rz)dx+∫|Rz|≤s0F(x,Rz)dx≥Mη-|Ω|CF=H,
it is elementary that
(4.14)D(R)=inf⁡ρ≥Rinf⁡z∈RSVZ∫ΩF(x,z)dx
is well defined and satisfies the claim.Now we may prove the following.Lemma 4.4.
Consider Theorem2.4 with one of the sets of hypotheses (F1) or (F2). Given the constant DW∈R, there exist R1, δ0>0 such that, for any λ∈(λk-δ0,λk), (3.7) and (3.8) hold.Proof.
We consider the two sets of hypotheses separately.
(i) In case(F1), assuming (1.2) and (F1) hold, we claim that there exists CM such that
(4.15)F(x,s)≥M|s|-CM,∀M∈R,
uniformly in x∈Ω, in particular we set M=1.
In fact, by(F1), there exits R0>0 such that for |s|≥R0, (∇F(x,s),s)≥|s| uniformly in x∈Ω. For any s∈R2(s≠0), from (4.1) (letting ε=2) we have
(4.16)F(x,s)=∫01(∇F(x,ts),s)dt=∫R0/|s|1(∇F(x,ts),s)dt+∫0R0/|s|(∇F(x,ts),s)dt≥∫R0/|s|1|s|dt+∫0R0/|s|(∇F(x,ts),s)dt≥|s|-R0-∫0R0/|s|(2t|s|2+|M2||s|)dt=|s|-R0-R02-M2R0=|s|-C1,
where C1=R0+R02+M2R0.
Letz∈RSVZ, for being in a finite-dimensional subspace, all the norms are equivalent, so that (set δ=λk-λ>0 and uses estimates (4.15) and (2.13))
(4.17)J(z)≤λk-λ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λkR2-R+C1|Ω|.
(ii) In case (F2), let D(R) be as in Lemma 4.3, for ∥z∥=R, let z=w+ϕ with w∈V and ϕ∈Z=Ek,
(4.18)J(z)=12∫Ω(H(x)∇z,∇z)dx-λ2∫Ω(A(x)z,z)-∫ΩF(x,z)dx≤λk-1-λ2λk-1‖w‖2+λk-λ2λk‖ϕ‖2-∫ΩF(x,z)dx≤λk-1-λk+δ2λk-1‖w‖2+δ2λk‖ϕ‖2-∫ΩF(x,z)dx≤δ2λk‖ϕ‖2-λk-λk-14λk-1‖w‖2-∫ΩF(x,z)dx.
Assume that δ≤(λk-λk-1)/2, it is easy to see that(-(λk-λk-1)/4λk-1)∥w∥2≤C for some constant C, we estimate
(4.19)J(z)≤δ2λk‖ϕ‖2+C-D(R)≤δ2λkR2-D(R)+C.
Considering (4.17) and (4.19), we see that since limR→∞D(R)=+∞ by Lemma 4.3, we may fix R1 so that C-D(R1)<DW-1 (or DM-R1<DW-1 for the case (F1)) and then for 0<δ<min{2λk/R12,(λk-λk-1)/2} one gets (3.7).
To obtain (3.8), we observe that (since λ>λk-1) if ϕ=0, that is, if z∈V, then in estimates (4.17) and (4.19) we may avoid the term (δ/2λk)R2 so that (remember that D(R) is nondecreasing) J(z)<DW-1 for ∥z∥>R1.


### 4.2.2. EstimatingJ in SZW

We consider the corresponding of the previous lemma, for Theorem2.5.Lemma 4.5.
Considering Theorem2.5 with one of the sets of hypotheses (F3) or (F4). Given the constant KV∈R, there exists R2, δ1>0 such that, for any λ∈(λk,λk+δ1), (3.15) and (3.16) hold.Proof.
Lettingλ=λk+δ, we see from (4.3), that property (3.16) will be satisfied provided that R2 is large enough (say R2>R̃) and observing that this value can be made independent from λ once that δ is small enough.
Now we consider the two sets of hypotheses separately.
(i) In case(F3), suppose z∈Ek⊕W, we can assume that z=w+ϕ, with w∈W and ϕ∈Ek. Since Ek is a finite dimension subspace, all the norms are equivalent, so that there exists C>0 such that for all ϕ∈Ek we have ∥ϕ∥≤C∥ϕ∥L1. By (F3), from the proof of (4.15), we also have the similar inequality: there exists C2 such that
(4.20)-F(x,s)≥C|s|-C2,
uniformly in x∈Ω. So by (2.14) and (4.20),
(4.21)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖w+ϕ‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖ϕ‖L1-C‖-w‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+‖ϕ‖-C3‖w‖-C4,
where C3=|Ω|1/2SC, C4=C2|Ω|. Since
(4.22)(1-δ2λk‖z‖)‖z‖≤‖w‖+‖ϕ‖-δ2λk(‖ϕ‖2+‖w‖2)≤(1-δ2λk‖ϕ‖)‖ϕ‖+‖w‖,
suppose δ≤(λk+m-λk)/2, (4.21) becomes
(4.23)J(w+ϕ)≥λk+m-(λk+δ)2λk+m‖w‖2-C3‖w‖-C4-‖w‖+(1-δ2λk‖z‖)‖z‖≥λk+m-λk4λk+m‖w‖2-(C3+1)‖w‖-C4+(1-δ2λk‖z‖)‖z‖,
since (λk+1-λk)/4λk+1>0, so ((λk+1-λk)/4λk+1)∥w∥2-(C3+1)∥w∥-C4 is bounded below for all w∈W, that is, there exists C5∈R such that
(4.24)λk+m-λk4λk+m‖w‖2-C3‖w‖-C4≥C5,
by (4.23) one gets
(4.25)J(z)≥(1-δ2λk‖z‖)‖z‖+C5=-δ2λk‖z‖2+‖z‖+C5.
(ii) In case (F4), first we give some conclusions which are similar to Lemma  3 of [16]. Under the property of F, there exists a constant C, and G∈C(R2,R) which is subadditive, that is,
(4.26)G(s+t)≤G(s)+G(t),
for all s,t∈R2, and coercive, that is,
(4.27)G(s)⟶+∞,
as |s|→∞, and satisfies that
(4.28)G(s)≤|s|+4,
for all s∈R2, such that
(4.29)-F(x,s)≥G(s)-C,
for all s∈R2 and x∈Ω¯.
In fact, since-F(x,s)→+∞ as |s|→∞ uniformly for all x∈Ω¯, there exists a sequence of positive integers nk with nk+1>2nk for all positive integers k such that
(4.30)-F(x,s)≥k,
for all |s|≥nk and all x∈Ω¯. Let n0=0 and define
(4.31)G(s)=k+2+|s|-nk-1nk-nk-1,
for nk-1≤|s|<nk, where k∈N.
By the definition ofG we have
(4.32)k+2≤G(s)≤k+3,
for all nk-1≤|s|<nk. By (F4) and F∈C1(Ω¯×R2,R), there exists CF>0 such that
(4.33)-F(x,s)≥-CF,∀(x,s)∈(Ω,R2).
It follows that
(4.34)-F(x,s)≥G(s)-C,
where C=CF+4. In fact, when nk-1≤|s|<nk for some k≥2, one has, by (4.30) and (4.32),
(4.35)-F(x,s)≥k-1≥G(s)-4≥G(s)-C,
for all x∈Ω¯. When |s|<n1, we have, by (4.32) and (4.33),
(4.36)-F(x,s)≥-CF=4-C≥G(s)-C,
for all x∈Ω¯.
It is obvious thatG is continuous and coercive. Moreover one has
(4.37)G(s)≤|s|+4,
for all s∈R2. In fact, for every s∈R2 there exists k∈N such that
(4.38)nk-1≤|s|<nk,
which implies that
(4.39)G(s)≤(k-1)+4≤nk-1+4≤|s|+4,
for all s∈R2 by (4.32) and the fact that nk≥k for all integers k≥0.
Now we only need to prove the subadditivity ofG. Let
(4.40)nk-1≤|s|<nk,nj-1≤|t|<nj,
and m=max{k,j}. Then we have
(4.41)|s+t|≤|s|+|t|<nk+nj≤2nm<nm+1.
Hence we obtain, by (4.32),
(4.42)G(s+t)≤m+4≤k+2+j+2≤G(s)+G(t),
which shows that G is subadditive.
Forz∈=Ek⊕W, assuming that z=w+ϕ, with w∈W and ϕ∈Ek, and letting 0<δ<(λk+m-λk)/2, by (2.13), (4.29), (4.26), and (4.28), one gets
(4.43)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ+w)dx-C|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫ΩG(-w)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫Ω(|w|+4)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖z‖2+∫ΩG(ϕ)dx-S|Ω|‖w‖-C1=g(w)+∫ΩG(ϕ)dx-δ2λk‖z‖2,
where g(w)=((λk+m-λk)/4λk+m)∥w∥2-S|Ω|∥w∥-C1, C1=(4+C)|Ω|. Since ϕ∈Ek, Ek is a finite-dimensional subspace, and G is coercive, from the proof of (4.8), one can get
(4.44)lim⁡∥ϕ∥→∞∫ΩG(ϕ)dx=+∞,
that is, ∫ΩG(ϕ)dx is coercive on Ek. Since (λk+m-λk)/4λk+m>0, so g(w) is coercive on W, and ∫ΩG(ϕ)dx and g(w) is bounded below, it is obvious that
(4.45)lim⁡∥z∥→∞(g(w)+∫ΩG(ϕ)dx)=+∞,
for all z∈Z⊕W.
Considering (4.25), (4.43), and (4.45), we can choose R2 large enough such that for all ∥z∥≥R2 one gets
(4.46)g(w)+∫ΩG(ϕ)dx>KV+1,
(or R2+C5>KV+1 for the case (F3)) and property (3.16) holds, then for 0<δ<min{2λk/R22,(λk+m-λk)/2}=δ1 and z∈R2SZW one gets J(z)>KV, that is, the property (3.15) holds.


## 4.2.1. EstimatingJ in SVZ

For the next estimates, we will need the following lemma.Lemma 4.3.
Hypotheses(F2) implies that there exists a nondecreasing function D:(0,+∞)→R such that
(4.8)lim⁡R→+∞D(R)=+∞,inf⁡z∈RSVZ∫ΩF(x,z)dx>D(R).Proof.
First we claim that there exists a constantη>0 such that the sets Ωz={x∈Ω:|z(x)|>η} have measure |Ωz|>η, for all z∈SVZ.
Actually,V⊕Z is a finite-dimensional subspace and the functions z∈SVZ are smooth, they are uniformly bounded, that is, there exists M>0 such that |z(x)|≤M for all x∈Ω. Suppose that for ηn→0(ηn<1) there exists {zn}⊂SVZ such that |Ωzn|≤ηn.
On one hand, by (2.13), one obtains
(4.9)1λk≤∫Ω(A(x)zn,zn)dx.
On the other hand,(4.10)∫Ω(A(x)zn,zn)dx=∫Ω(a(x)un2+2b(x)unvn+d(x)vn2)dx≤∫Ω(a(x)+b(x))un2+∫Ω(b(x)+d(x))vn2dx≤M¯∫Ω|zn|2dx=M¯(∫Ωzn|zn|2dx+∫Ω-Ωzn|zn|2dx)≤M¯(M2|Ωzn|+ηn2|Ω-Ωzn|)≤ηnC⟶0,
where M¯=maxx∈Ω{|a(x)+b(x)|,|b(x)+d(x)|}, zn=(un,vn), C=M¯(M2+|Ω|). That is a contradiction.
Now for anyH>0, we will show that we can find an R̃ large enough so that ∫ΩF(x,Rz)dx≥H for any z∈SVZ and R≥R̃, which means that
(4.11)lim⁡R→∞inf⁡z∈RSVZ∫ΩF(x,z)dx=+∞.
Actually, lettingM=(H+|Ω|CF)η-1, by (F2) we have that there exists s0 such that F(x,s)>M for |s|>s0.
ForR>s0/η, one has Ωz⊆{x∈Ω:|Rz(x)|>s0}, and then one gets
(4.12)∫|Rz|≥s0F(x,Rz)dx≥Mη.
ForR≤s0/η, by (F2) and F∈C1(Ω¯×R2,R), there exists CF>0 such that F(x,s)≥-CF, for all (x,s)∈(Ω,R2).
One finally obtains(4.13)∫ΩF(x,Rz)dx=∫|Rz|≥s0F(x,Rz)dx+∫|Rz|≤s0F(x,Rz)dx≥Mη-|Ω|CF=H,
it is elementary that
(4.14)D(R)=inf⁡ρ≥Rinf⁡z∈RSVZ∫ΩF(x,z)dx
is well defined and satisfies the claim.Now we may prove the following.Lemma 4.4.
Consider Theorem2.4 with one of the sets of hypotheses (F1) or (F2). Given the constant DW∈R, there exist R1, δ0>0 such that, for any λ∈(λk-δ0,λk), (3.7) and (3.8) hold.Proof.
We consider the two sets of hypotheses separately.
(i) In case(F1), assuming (1.2) and (F1) hold, we claim that there exists CM such that
(4.15)F(x,s)≥M|s|-CM,∀M∈R,
uniformly in x∈Ω, in particular we set M=1.
In fact, by(F1), there exits R0>0 such that for |s|≥R0, (∇F(x,s),s)≥|s| uniformly in x∈Ω. For any s∈R2(s≠0), from (4.1) (letting ε=2) we have
(4.16)F(x,s)=∫01(∇F(x,ts),s)dt=∫R0/|s|1(∇F(x,ts),s)dt+∫0R0/|s|(∇F(x,ts),s)dt≥∫R0/|s|1|s|dt+∫0R0/|s|(∇F(x,ts),s)dt≥|s|-R0-∫0R0/|s|(2t|s|2+|M2||s|)dt=|s|-R0-R02-M2R0=|s|-C1,
where C1=R0+R02+M2R0.
Letz∈RSVZ, for being in a finite-dimensional subspace, all the norms are equivalent, so that (set δ=λk-λ>0 and uses estimates (4.15) and (2.13))
(4.17)J(z)≤λk-λ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λk‖z‖2-‖z‖+C1|Ω|≤δ2λkR2-R+C1|Ω|.
(ii) In case (F2), let D(R) be as in Lemma 4.3, for ∥z∥=R, let z=w+ϕ with w∈V and ϕ∈Z=Ek,
(4.18)J(z)=12∫Ω(H(x)∇z,∇z)dx-λ2∫Ω(A(x)z,z)-∫ΩF(x,z)dx≤λk-1-λ2λk-1‖w‖2+λk-λ2λk‖ϕ‖2-∫ΩF(x,z)dx≤λk-1-λk+δ2λk-1‖w‖2+δ2λk‖ϕ‖2-∫ΩF(x,z)dx≤δ2λk‖ϕ‖2-λk-λk-14λk-1‖w‖2-∫ΩF(x,z)dx.
Assume that δ≤(λk-λk-1)/2, it is easy to see that(-(λk-λk-1)/4λk-1)∥w∥2≤C for some constant C, we estimate
(4.19)J(z)≤δ2λk‖ϕ‖2+C-D(R)≤δ2λkR2-D(R)+C.
Considering (4.17) and (4.19), we see that since limR→∞D(R)=+∞ by Lemma 4.3, we may fix R1 so that C-D(R1)<DW-1 (or DM-R1<DW-1 for the case (F1)) and then for 0<δ<min{2λk/R12,(λk-λk-1)/2} one gets (3.7).
To obtain (3.8), we observe that (since λ>λk-1) if ϕ=0, that is, if z∈V, then in estimates (4.17) and (4.19) we may avoid the term (δ/2λk)R2 so that (remember that D(R) is nondecreasing) J(z)<DW-1 for ∥z∥>R1.


## 4.2.2. EstimatingJ in SZW

We consider the corresponding of the previous lemma, for Theorem2.5.Lemma 4.5.
Considering Theorem2.5 with one of the sets of hypotheses (F3) or (F4). Given the constant KV∈R, there exists R2, δ1>0 such that, for any λ∈(λk,λk+δ1), (3.15) and (3.16) hold.Proof.
Lettingλ=λk+δ, we see from (4.3), that property (3.16) will be satisfied provided that R2 is large enough (say R2>R̃) and observing that this value can be made independent from λ once that δ is small enough.
Now we consider the two sets of hypotheses separately.
(i) In case(F3), suppose z∈Ek⊕W, we can assume that z=w+ϕ, with w∈W and ϕ∈Ek. Since Ek is a finite dimension subspace, all the norms are equivalent, so that there exists C>0 such that for all ϕ∈Ek we have ∥ϕ∥≤C∥ϕ∥L1. By (F3), from the proof of (4.15), we also have the similar inequality: there exists C2 such that
(4.20)-F(x,s)≥C|s|-C2,
uniformly in x∈Ω. So by (2.14) and (4.20),
(4.21)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖w+ϕ‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+C‖ϕ‖L1-C‖-w‖L1-C2|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+‖ϕ‖-C3‖w‖-C4,
where C3=|Ω|1/2SC, C4=C2|Ω|. Since
(4.22)(1-δ2λk‖z‖)‖z‖≤‖w‖+‖ϕ‖-δ2λk(‖ϕ‖2+‖w‖2)≤(1-δ2λk‖ϕ‖)‖ϕ‖+‖w‖,
suppose δ≤(λk+m-λk)/2, (4.21) becomes
(4.23)J(w+ϕ)≥λk+m-(λk+δ)2λk+m‖w‖2-C3‖w‖-C4-‖w‖+(1-δ2λk‖z‖)‖z‖≥λk+m-λk4λk+m‖w‖2-(C3+1)‖w‖-C4+(1-δ2λk‖z‖)‖z‖,
since (λk+1-λk)/4λk+1>0, so ((λk+1-λk)/4λk+1)∥w∥2-(C3+1)∥w∥-C4 is bounded below for all w∈W, that is, there exists C5∈R such that
(4.24)λk+m-λk4λk+m‖w‖2-C3‖w‖-C4≥C5,
by (4.23) one gets
(4.25)J(z)≥(1-δ2λk‖z‖)‖z‖+C5=-δ2λk‖z‖2+‖z‖+C5.
(ii) In case (F4), first we give some conclusions which are similar to Lemma  3 of [16]. Under the property of F, there exists a constant C, and G∈C(R2,R) which is subadditive, that is,
(4.26)G(s+t)≤G(s)+G(t),
for all s,t∈R2, and coercive, that is,
(4.27)G(s)⟶+∞,
as |s|→∞, and satisfies that
(4.28)G(s)≤|s|+4,
for all s∈R2, such that
(4.29)-F(x,s)≥G(s)-C,
for all s∈R2 and x∈Ω¯.
In fact, since-F(x,s)→+∞ as |s|→∞ uniformly for all x∈Ω¯, there exists a sequence of positive integers nk with nk+1>2nk for all positive integers k such that
(4.30)-F(x,s)≥k,
for all |s|≥nk and all x∈Ω¯. Let n0=0 and define
(4.31)G(s)=k+2+|s|-nk-1nk-nk-1,
for nk-1≤|s|<nk, where k∈N.
By the definition ofG we have
(4.32)k+2≤G(s)≤k+3,
for all nk-1≤|s|<nk. By (F4) and F∈C1(Ω¯×R2,R), there exists CF>0 such that
(4.33)-F(x,s)≥-CF,∀(x,s)∈(Ω,R2).
It follows that
(4.34)-F(x,s)≥G(s)-C,
where C=CF+4. In fact, when nk-1≤|s|<nk for some k≥2, one has, by (4.30) and (4.32),
(4.35)-F(x,s)≥k-1≥G(s)-4≥G(s)-C,
for all x∈Ω¯. When |s|<n1, we have, by (4.32) and (4.33),
(4.36)-F(x,s)≥-CF=4-C≥G(s)-C,
for all x∈Ω¯.
It is obvious thatG is continuous and coercive. Moreover one has
(4.37)G(s)≤|s|+4,
for all s∈R2. In fact, for every s∈R2 there exists k∈N such that
(4.38)nk-1≤|s|<nk,
which implies that
(4.39)G(s)≤(k-1)+4≤nk-1+4≤|s|+4,
for all s∈R2 by (4.32) and the fact that nk≥k for all integers k≥0.
Now we only need to prove the subadditivity ofG. Let
(4.40)nk-1≤|s|<nk,nj-1≤|t|<nj,
and m=max{k,j}. Then we have
(4.41)|s+t|≤|s|+|t|<nk+nj≤2nm<nm+1.
Hence we obtain, by (4.32),
(4.42)G(s+t)≤m+4≤k+2+j+2≤G(s)+G(t),
which shows that G is subadditive.
Forz∈=Ek⊕W, assuming that z=w+ϕ, with w∈W and ϕ∈Ek, and letting 0<δ<(λk+m-λk)/2, by (2.13), (4.29), (4.26), and (4.28), one gets
(4.43)J(w+ϕ)=12∫Ω(H(x)∇(w+ϕ),∇(w+ϕ))dx-λ2∫Ω(A(x)(w+ϕ),w+ϕ)dx-∫ΩF(x,w+ϕ)dx≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ+w)dx-C|Ω|≥λk+m-(λk+δ)2λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫ΩG(-w)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖ϕ‖2+∫ΩG(ϕ)dx-∫Ω(|w|+4)dx-C|Ω|≥λk+m-λk4λk+m‖w‖2-δ2λk‖z‖2+∫ΩG(ϕ)dx-S|Ω|‖w‖-C1=g(w)+∫ΩG(ϕ)dx-δ2λk‖z‖2,
where g(w)=((λk+m-λk)/4λk+m)∥w∥2-S|Ω|∥w∥-C1, C1=(4+C)|Ω|. Since ϕ∈Ek, Ek is a finite-dimensional subspace, and G is coercive, from the proof of (4.8), one can get
(4.44)lim⁡∥ϕ∥→∞∫ΩG(ϕ)dx=+∞,
that is, ∫ΩG(ϕ)dx is coercive on Ek. Since (λk+m-λk)/4λk+m>0, so g(w) is coercive on W, and ∫ΩG(ϕ)dx and g(w) is bounded below, it is obvious that
(4.45)lim⁡∥z∥→∞(g(w)+∫ΩG(ϕ)dx)=+∞,
for all z∈Z⊕W.
Considering (4.25), (4.43), and (4.45), we can choose R2 large enough such that for all ∥z∥≥R2 one gets
(4.46)g(w)+∫ΩG(ϕ)dx>KV+1,
(or R2+C5>KV+1 for the case (F3)) and property (3.16) holds, then for 0<δ<min{2λk/R22,(λk+m-λk)/2}=δ1 and z∈R2SZW one gets J(z)>KV, that is, the property (3.15) holds.


## 5. Proof of the Geometry in Propositions3.1 and 3.2

We finally give the proof of Propositions3.1 and 3.2, which is nothing but a resume of the lemmata above, verifying that all the constants can be chosen sequentially without contradictions.Proof of Proposition3.1.
Under hypothesis (1.2), if we fix a value λ, then we obtain the constant Dλ from Lemma 4.1 and with this we get ρλ from Lemma 4.2. If we also consider one of the two sets of hypotheses (F1) or (F2), then we proceed as follows: first of all, we determine (once for ever) the constant DW from Lemma 4.1, with this we obtain from Lemma 4.4 the values R1 and δ0. Then, for any (now fixed) λ∈(λk-δ0,λk), we obtain from Lemma 4.1 the value Dλ. Finally, we can get from Lemma 4.2 the corresponding value of ρλ>R1.Proof of Proposition3.2.
Under hypothesis (1.2), if we fix a value λ∈(λk,λk+m), then we obtain the constant Kλ from Lemma 4.1 and with this we get βλ from Lemma 4.2. If we also consider one of the two sets of hypotheses (F3) or (F4), then we proceed as follows: first of all, we determine (once for ever) the constant KV from Lemma 4.2, with this we obtain from Lemma 4.5 the values R2 and δ1. Since we have R2, we can get from Lemma 4.1 the constant E and with this obtain ξ from Lemma 4.2.
Finally, for any (now fixed)λ∈(λk,λk+δ1), we obtain from Lemma 4.1 the constant Kλ and with this we get from Lemma 4.2 the corresponding value of βλ>R2.



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*Source: 532430-2012-06-25.xml*