# On Global Attractors for a Class of Reaction-Diffusion Equations on Unbounded Domains with Some Strongly Nonlinear Weighted Term

Authors: Jin Zhang; Chengkui Zhong
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We consider the existence and properties of the global attractor for a class of reaction-diffusion equation∂u/∂t-Δu-u+κ(x)|u|p-2u+f(u)=0,  in  Rn×R+;  u(x,0)=u0(x),  in  Rn. Under some suitable assumptions, we first prove that the problem has a global attractor A in L2(Rn). Then, by using the Z2-index theory, we verify that A is an infinite dimensional set and it contains infinite distinct pairs of equilibrium points.


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## Body

## 1. Introduction

In this paper, we are mainly concerned with the long-time behaviour of solutions for the following reaction-diffusion equation:(1)∂u∂t-Δu-u+κxup-2u+fu=0,inRn×R+,ux,0=u0x,inRn,where p>2 and κ(x)≥α0>0 satisfies(2)∫Rn1κx2/p-2dx<∞(e.g., κ(x)=(1+|x|)r with r>(p-2)n/2). The function f:R→R is continuous and satisfies the following assumptions: (f1)
f(s) is odd; that is, f(-s)=-f(s), for all s∈R;(f2)
there exists a constantα, 0<α<p-1, such that (3)sup⁡lims→∞⁡fssα=0,sup⁡lims→0⁡fss=0;(f3)
there exists a positive constantl such that f′(s)≥-l;(f4)
there exists a positive constantβ such that 0≤βF(s)≤f(s)s, where (4)Fs≔∫0sfsds.As we know, the basic problems to consider the long-time behaviour of solutions for the above equation are to prove the existence of global attractors for the semigroup of solutions and discuss some properties of the global attractors, such as the dimension property and the existence of multiple equilibrium points.Those problems for the equations in bounded domains have been studied extensively by many authors and have been rather well understood; see, for example, [1–7]. However, the solution for the equation is different in unbounded domain. The main difference is the fact that, in contrast to the case of bounded domains, the global attractors for the reaction-diffusion equations in unbounded domains admit finite dimension under some specific assumptions and infinite dimension under general assumptions.For the kind of equation(5)dudt-Δu+fx,u=hin unbounded domains. In pioneering work [8], the authors used weighted spaces instead of the usual spaces to prove the existence of the global attractors; further details can be found in [9–11]. In [12–15], the authors have developed some new ideas and methods to deal with more general cases in unbounded domains, including uniformly local Sobolev spaces, locally compact attractor, and the so-called entropy theory, and have obtained the existence of the locally compact global attractors for the semigroups associated with the equations. Under some structural assumptions on the term f (i.e., f(x,u)u≥0 or f(x,u)u≥α|u|p+ψ(x)), the authors in [16, 17] prove the existence of global attractor for the equation in unbounded domain in usual space L2(Rn).On the other hand, we have noticed that theZ2-index is a powerful method to find multiple critical points of some even functional. The authors in [18] used the Z2-index to obtain the existence of infinite dimensional global attractor for a class of p-Laplacian equation in bounded domain, for which p>2 is necessary. Additional information about other attractor problems can be found in [19–23].Motivated by the above papers, in this paper, we are interested in finding a semigroup associated with a reaction-diffusion equation in unbounded domain, such that the semigroup has a global attractor in the usual space; furthermore the dimension of the global attractor is infinite.The main results of this paper can be stated as follows.Theorem 1.
Assuming thatn≥3, p>2, and κ(x)≥α0>0 satisfies condition (2) and the nonlinear term f satisfies (f1)~(f4), then reaction-diffusion equation (1) has a global attractor A in L2(Rn).Theorem 2.
Assume thatn≥3, p>2, and κ(x)≥α0>0 satisfies condition (2) and the nonlinear term f satisfies (f1)~(f4). Let A be the global attractor of (1). Then, for any m∈N+, there exists a neighborhood O(0) of the origin, such that γ(A∖O(0))≥m, where γ(A∖O(0)) denotes the Z2-index of the set A∖O(0).We recall that, from [24], any compact set E, with fractal dimension dim(E)=n, can be mapped into spaces R2n+1 by a linear odd Hölder continuous one-to-one projector. Thus, we obtained the following corollary.Corollary 3.
Under the assumptions of Theorem2, the fractal dimension of the global attractor A is infinite.Theorem 4.
Assume thatn≥3, p>2, and κ(x)≥α0>0 satisfies condition (2) and the nonlinear term f satisfies (f1)~(f4). Let A be the global attractor of (1). Then, A contains infinite distinct pairs of equilibrium points.The proofs of the above theorems are, respectively, given in Sections3 and 4. Some preliminaries and associate lemmas can be found in Section 2.


## 2. Some Preliminaries

Initially, backgrounds about global attractors andZ2-index theory are reviewed. Proofs are then given for the lemmas and the existence of solution for (1).In this paper, we define the following space:(6)Hκ1Rn≜D1,2Rn∩LκpRn=u∣up,κ+uD1,2<+∞,with the corresponding norm (7)uHκ1=up,κ+uD1,2,where (8)uD1,2=∫Rn∇u2dx1/2,up,κ=∫Rnκxupdx1/p.LetV be a Banach space, and define Σ={A⊂V∣A closed, A=-A} as the class of closed symmetric subsets of V. Based on this, the formal definition of Z2-index can be given.Definition 5 (see [25]).
LetA∈Σ, A≠∅. The Z2-index or Krasnoselskii genus γ(A) of A is defined by (9)γA=inf⁡m:∃h∈C0A,Rm∖0,h-u=-hu,∞,ifm:∃h∈C0A,Rm∖0,h-u=-hu=∅.In particular, if 0∈A, γ(A)=∞, then define γ(∅)=0.The properties ofZ2-index γ(A) are provided in the following lemma.Lemma 6 (see [25]).
Leth∈C0(V,V) be an odd map and A,A1,A2∈Σ. Then the Z2-index γ(A) on V satisfies the following properties: (A1)
γA≥0, γA=0⇔A=∅;(A2)
A1⊂A2⇒γ(A1)≤γ(A2);(A3)
γA1∪A2≤γA1+γA2;(A4)
γ(A)≤γ(h(A)¯),  for all  A∈Σ, and h:V→V is odd and continuous;(A5)
IfA∈Σ is compact and 0∉A, then γ(A)<∞ and there exists a neighborhood N of A in V such that N¯∈Σ and γ(A)=γ(N¯);(A6)
For any bounded symmetric neighborhoodΩ of the origin in Rn there holds γ(∂Ω)=n.Applying the index theory to an even functionalE on some Banach space V, we can obtain a sequence of minimax values. Moreover, if E satisfies the (P.S.) condition, the sequence of minimax values must be the critical values of the functional E.Definition 7 (see [25]).
LetV be a Banach space, E∈C1(V,R), and d∈R. The functional E is said to satisfy the (P.S.) condition if any sequence {un}⊂V such that (10)Eunis  bounded  andE′un⟶0has a convergent subsequence.Lemma 8 (see [26]).
SupposeV is a Banach space and suppose E∈C1(V,R), E(u)=E(-u) for all u. E satisfies the following conditions: (B1)
there exists a subspaceVr⊂V with dimVr=r and ρ>0, such that (11)supu∈Vr∩Sρ⁡Eu<E0,whereSρ={u∈V∣u=ρ};(B2)
there exists a closed subspaceW⊂V with codimW=s, such that (12)infu∈W⁡Eu>-∞;(B3)
E(u) satisfies the (P.S.) condition.Then if r>s, the functional E possesses at least r-s pairs of critical points.Following the proof in [8], we will prove the existence of a unique weak solution of (1) for any initial data u0∈L2(Rn).Firstly, we consider the problem in the bounded domain. We denoteΩR={x∈Rn:|x|<R} and the function ΨR∈C∞(Rn) with 0≤ΨR(x)≤1, |ΨR′(x)|≤2, satisfying(13)ΨRx=1,x≤R-1,0,x≥R.It is well known that (see, e.g., [5, 6])(14)∂u∂t-Δu-u+κxup-2u+fu=0inΩR×R+,u0=u0,RinΩR,ut∂ΩR=0t∈0,∞,has a unique solution uR, where u0,R=ΨRu0. And for every T>0 it satisfies (15)uR∈C0,T,L2ΩR∩L20,T,H01ΩR∩Lp0,T,LκpΩR.The following lemmas give some estimates for solutionuR of the bounded problem (14).Lemma 9.
Letn≥3, p>2, κ(x)≥α0>0 satisfy condition (2), and f satisfies (f1)~(f4); let uR be a solution of problem (14). Then, for any T>0, the following estimates hold:(16)∫ΩRuRt2dx≤C1,t∈0,T,(17)∫0TuRH01ΩR2dt+∫0TuRLκpΩRpdt≤C2,where the constants C1, C2 depend on data T, u0,R, and κ but are independent of R.Proof.
Firstly, for anyu∈Lκp(ΩR), utilizing Hölder inequality, we have(18)∫ΩRu2dx=∫ΩR1κ2/p·κ2/pu2dx≤∫ΩR1κ2/p·p/p-2dxp-2/p·∫ΩRκupdx2/p≤∫Rn1κ2/p-2dxp-2/p·∫ΩRκupdx2/p=M1κuLκpΩR2.Then utilizing Young’s inequality, we have(19)∫ΩRu2dx≤M2κ+1puLκpΩRp,where (20)M1κ=∫Rn1κ2/p-2dxp-2/p,M2κ=22/p·p-2p∫Rn1κ2/p-2dx.Multiplying (14) by uR and integrating over ΩR, we have(21)12ddt∫ΩRuR2dx+∫ΩR∇uR2dx-∫ΩRuR2dx+∫ΩRκuRpdx+∫ΩRfuRuRdx≤0,and it follows from (19) and f(uR)uR≥0 that (22)12ddt∫ΩRuR2dx+∫ΩR∇uR2dx+p-1p∫ΩRκuRpdx≤M2κ.Integrating t between 0 and T yields (23)12uRTL2ΩR2-12u0,RL2ΩR2+∫0TuRH01ΩR2dt+p-1p∫0TuRLκpΩRpdt≤TM2κ.It follows that (24)∫0TuRH01ΩR2dt+p-1p∫0TuRLκpΩRpdt≤TM2κ+12u0,RL2ΩR2,which implies second estimate (17).
On the other hand, it follows from (21) and f(uR)uR≥0 that (25)12ddt∫ΩRuR2dx≤∫ΩRuR2dx.Referring to Gronwall’s inequality, first estimate (16) can be easily obtained.Lemma 10.
Letn≥3, p>2, and κ(x)≥α0>0 satisfy condition (2), and f satisfies (f1)~(f4); let uR be a solution of problem (14). Then, for any T>0, the following estimate holds:(26)∂uR∂tLp/p-10,T,Hκ1ΩR∗≤C3,where (Hκ1(ΩR))∗≜H01(ΩR)∩Lκp(ΩR)∗ and the constant C3 depends on data T, u0,R, and κ but independent of R.Proof.
For anyv∈H01(ΩR)∩Lκp(ΩR), we have(27)∂uR∂t,v≤∫ΩR∇uR·∇vdx+∫ΩRuRvdx+∫ΩRκuRp-1vdx+∫ΩRfuRvdx.By (f2) we find that (28)fu≤Cu+up-1,so we get (29)∫ΩRfuRvdx≤C∫ΩRuRvdx+C∫ΩRuRp-1vdx≤C∫ΩRuRvdx+Cα0∫ΩRκuRp-1vdx.Applying the Hölder inequality to each term, it follows that (30)∫ΩR∇uR·∇vdx≤uRH01ΩRvH01ΩR,∫ΩRuRvdx≤uRL2ΩRvL2ΩR≤M12κuRLκpΩRvLκpΩR,∫ΩRκuRp-1vdx≤uRLκpΩRp-1vLκpΩR.Substituting into inequality (27), there exists a constant C~3>0, such that (31)∂uR∂tHκ1ΩR∗≤C~3uRH01ΩR+uRLκpΩR+uRLκpΩRp-1.Then, referring to Lemma 9, the estimate (32)∫0T∂uR∂tHκ1ΩR∗p/p-1dtp-1/p≤C3yields the conclusion.It is worth noting that both estimates in Lemmas9 and 10 are independent of R, so we let R→+∞, providing the existence and uniqueness of the solution of problem (1). Before giving the proof of the existence theorem, we first state the following two lemmas.Lemma 11 (see [5]).
LetV⊂⊂H⊂Y be Banach spaces, with V reflexive. Suppose that {un} is a sequence uniformly bounded in L2(0,T,V) and {dun/dt} is uniformly bounded in Lp(0,T,Y), for p>1. Then there is a subsequence that converges strongly in L2(0,T,H).Lemma 12 (see [27]).
Letx,y∈RN, p≥2, and 〈·,·〉 be the standard scalar product in RN. Then, there exists a constant α>0 such that (33)xp-2x-yp-2y,x-y≥αx-yp.Theorem 13.
Letn≥3, p>2, κ(x)≥α0>0 satisfy condition (2), and f satisfies (f1)~(f4), then for any u0∈L2(Rn) and T>0, there exists a unique weak solution u(x,t) of (1) which satisfies (34)u∈C0,T,L2Rn∩L20,T,D1,2Rn∩Lp0,T,LκpRn.Furthermore, u0↦u(t) is continuous on L2(Rn).Proof.
We chooseRk such that Rk→+∞ as k→+∞. Denote that uRkk=1,2,… are the solutions of boundary problem (14) in the domain ΩRk.
Now, we extend the functionsuRk(k=1,2,…) from L2(ΩR) into L2(Rn). For each k∈N, define the function uRk as zero for |x|≥Rk and multiply by ΨRk(x), where ΨRk(x) is defined by (13). For simplicity, we denote uk=ΨRkuRk by the extended functions and Ωk≜ΩRk. Since (35)∫Rnuk2dx=∫RnΨRkuRk2dx≤∫ΩkuRk2dx,∫Rn∇uk2dx≤2∫Rn∇ΨRk2uRk2dx+2∫RnΨRk2∇uRk2dx≤8∫ΩkuRk2dx+2∫Ωk∇uRk2dx,∫Rnκukpdx≤∫ΩkκuRkpdx,we obtain that Lemmas 9 and 10 are still valid. It follows that (36)ukis  uniformly  bounded  inL∞0,T,L2Rn,ukis  uniformly  bounded  inL20,T,D1,2Rn,ukis  uniformly  bounded  inLp0,T,LκpRn,∂uk∂tis  uniformly  bounded  inLp/p-10,T,Hκ1Rn∗.Hence, taking a subsequence of {uk} if necessary there exists u∈L∞[0,T],L2(Rn)∩L20,T,D1,2(Rn)∩Lp0,T,Lκp(Rn) such that (37)uk⇀u,inL20,T,D1,2Rn,uk⇀u,inLp0,T,LκpRn,∂uk∂t⇀∂u∂tinLp/p-10,T,Hκ1Rn∗.Similarly to the proof in [5, 6], we can obtain (38)fuk⇀fuinLp/p-10,T,LκpRn∗.In addition, referring to Lemma 11, we have (39)uk⟶uinL20,T,L2Rn.Therefore, for any v∈C0∞[0,T],Hκ1(Rn), (40)0=∫0T∫Rn∂uk∂t-Δuk-uk+κukp-2uk+fuk·vdxdt⟶∫0T∫Rn∂u∂t-Δu-u+κup-2u+fuvdxdt.Thus, u is the weak solution of (1).
In the following, we will prove uniqueness of solution and the continuous dependence. Letu, v be any two solutions of (1) with initial data u0, v0; setting w=u-v, we have (41)∂w∂t-Δw-w+κxup-2u-κxvp-2v+fu-fv=0,with initial data w(0)=u0-v0. Multiplying by w and integrating on Rn, we obtain (42)12ddt∫Rnw2dx+∫Rn∇w2dx-∫Rnw2dx+∫Rnfu-fvu-vdx+∫Rnκup-2u-κvp-2vu-vdx=0,and it follows from Lemma 12 that (43)∫Rnκup-2u-κvp-2vu-vdx≥α∫Rnκu-vpdx≥0,and by condition (f3) we find that (44)∫Rnfu-fvu-vdx=∫Rn∫vxuxf′sdsux-vxdx≥-l∫Rnu-v2dx=-l∫Rnw2dx.Then note that ∫Rn‍|∇w|2dx≥0; we have (45)12ddt∫Rnw2dx≤l+1∫Rnw2dx,and integrating this gives(46)ut-vtL2≤e2l+1tu0-v0L2,which implies uniqueness if u0=v0 and the continuous dependence on initial data.The following theorem shows the existence of global attractors when an absorbing set exists.Lemma 14 (see [5, 6]).
If a continuous semigroupS(t) has a compact absorbing set B, then there exists a global attractor A=ω(B), where ω(B) is the ω-limit set of the set B.


## 3. The Existence of a Global Attractor

In this section, we will prove Theorem1. Before the proof, we first give the following lemma.Lemma 15.
Assuming thatn≥3, p>2, and κ(x)≥α0>0 satisfies assumption (2), then Hκ1(Rn) is compactly embedded in L2(Rn).Proof.
Assume that{un} is a bounded sequence in Hκ1(Rn). Then there exists a constant C>0, such that (47)unD1,2Rn≤C,unLκpRn≤C,so it has a subsequence {unk} satisfying (48)unk⇀u0inD1,2Rn,unk⇀u0inLκpRn.For arbitrary ɛ>0, choose the constant R sufficiently large, such that(49)∫Rn∖ΩR1κ2/p-2dxp-2/p<ɛC2.
Note thatΨR+1unk⇀ΨR+1u0 in H01(ΩR+1) and due to the boundedness of the domain ΩR+1, the Sobolev embedding theorem can be used, yielding (50)ΨR+1unk⟶ΨR+1u0inL2ΩR+1.Then there exists K>0 sufficiently large such that, for all nk≥K, we have(51)∫ΩR+1ΨR+1unk-ΨR+1u02dx<ɛ,and it follows that (52)unk-u0L2Rn2=∫ΩRunk-u02dx+∫Rn∖ΩRunk-u02dx≤∫ΩR+1ΨR+1unk-ΨR+1u02dx+∫Rn∖ΩRunk-u02dx≤ɛ+∫Rn∖ΩRunk-u02dx.Utilizing the Hölder inequality and inequality (49), we have (53)∫Rn∖ΩRunk-u02dx=∫Rn∖ΩR1κ2/p·κ2/punk-u02dx≤∫Rn∖ΩR1κ2/p·p/p-2dxp-2/p·∫Rn∖ΩRκ2/punk-u02p/2dx2/p=∫Rn∖ΩR1κ2/p-2dxp-2/p·∫Rn∖ΩRκunk-u0pdx2/p<ɛC2·unk-u0LκpRn2<ɛC2·4C2=4ɛ,which implies (54)unk-u0L2Rn2<5ɛ,fornk≥K.This completes the proof of Lemma 15.Proof of Theorem1.
In order to prove that (1) has a global attractor, referring to Lemma 14, it is sufficient to show the existence of a compact absorbing set in L2(Rn).
Letu be the solution of (1); multiplying the first equation of (1) by u and integrating on Rn, it follows that(55)12ddt∫Rnu2dx+∫Rn∇u2dx-∫Rnu2dx+∫Rnκxupdx+∫Rnfuudx=0.Since ∫Rn‍|∇u|2dx≥0 and f(u)u≥0, we have (56)12ddt∫Rnu2dx+∫Rnu2dx≤2∫Rnu2dx-∫Rnκxupdx.Similar to estimate (19), we have(57)∫Rnu2dx≤Cκ+1p∫Rnκxupdx,and thus (58)12ddt∫Rnu2dx+∫Rnu2dx≤2Cκ-1-2p∫Rnκxupdx.Note that p>2 yielded (59)ddt∫Rnu2dx+2∫Rnu2dx≤4C,and then Gronwall’s inequality can be applied, yielding(60)utL2Rn2≤4C,whent≥t0u0L2,where t0=(1/2)ln⁡(u0L22/2C). Now, combining estimates (55) and (57), we have(61)12ddt∫Rnu2dx+12∫Rn∇u2dx+1p∫Rnκxupdx+∫Rnfuudx≤C.Integrating between t and t+1, it follows that (62)∫tt+112∫Rn∇us2dx+1p∫Rnκxuspdx+∫Rnfususdxds≤C+12utL22.When t>t0, it follows from (60) that(63)∫tt+112∫Rn∇us2dx+1p∫Rnκxuspdx+∫Rnfususdxds≤3C.Now, multiplying the first equation of (1) by ut and integrating on Rn, we obtain(64)ddt12∫Rn∇u2dx-12∫Rnu2dx+1p∫Rnκxupdx+∫RnFudx=-∫Rnut2dx≤0,so it follows from (61) and (64) that (65)ddt12∫Rn∇u2dx+1p∫Rnκxupdx+∫RnFudx≤C.Integrating between s and t (t-1≤s<t), it holds that(66)12∫Rn∇ut2dx+1p∫Rnκutpdx+∫RnFutdx≤C+12∫Rn∇us2dx+1p∫Rnκuspdx+∫RnFusdx≤C+12∫Rn∇us2dx+1p∫Rnκuspdx+1β∫Rnfususdx.Then integrating the equation with respect to s between t-1 and t again, we have (67)12∫Rn∇ut2dx+1p∫Rnκutpdx+∫RnFutdx≤C+∫tt+112∫Rn∇us2dx+1p∫Rnκuspdx+1β∫Rnfususdxds,when t>t0, so it follows from (63) and ∫Rn‍F(u(t))dx≥0 that (68)12∫Rn∇ut2dx+1p∫Rnκutpdx≤C+max⁡3,3βC,which implies that there exists a constant C~, such that (69)utHκ1≤C~,∀t>t0.
Finally, referring to Lemma15, Hκ1(Rn) is compactly embedded in L2(Rn), and we obtain a compact absorbing set in L2(Rn) which concludes the proof of Theorem 1.


## 4. The Dimension of the Global Attractor and the Equilibrium Points

Next we will estimate theZ2-index of the global attractor obtained in Theorem 1. Before the formal proof of Theorem 2, we first consider the energy function(70)Eu=12∫Rn∇u2-u2dx+1p∫Rnκxupdx+∫RnFudx.It is well known that functional (70) has an infinite dimensional negative subspace H- of Hκ1(Rn); that is, there exists linearly independent nonzero functions u1,u2,…,um,…∈H-, satisfying(71)∫Rn∇um2dx-∫Rnum2dx<0,m=1,2,….LetHm≜span{u1,u2,…,um} be a subspace of H- with dim(Hm)=m, where u1,u2,…,um are orthogonal in both D1,2(Rn) and L2(Rn).Now, we give the proof of Theorem2.Proof of Theorem2.
For arbitrarym∈N+, we first prove that there exists a set Bm∈Hκ1(Rn) with γ(Bm)≥m and a neighborhood O of the origin, such that (72)ωBm⊂A∖O.It follows from (64) that (73)ddtEu=-∫Rnut2dx≤0;that is, for any u0∈Hκ1(Rn), the function t→E(u(t)) is nonincreasing. For arbitrary m>0 and u∈Hm∖{0}, we have (74)∫Rn∇u2dx-∫Rnu2dx<0.Denoting Am≜Hm∩S1={u∈Hm:uHκ1=1}, then Am is compact in Hκ1(Rn); thus, there exists δ>0, such that for all u∈Am(75)∫Rn∇u2dx-∫Rnu2dx<-δ<0.Referring to Lemma 6  (A6), for every constant ɛ>0, we have(76)γɛAm=γAm=m,where ɛAm={ɛu:u∈Am}. Thus, for v=ɛu∈ɛAm, it follows that (77)Ev=ε22∫Rn∇u2dx-ε22∫Rnu2dx+εpp∫Rnκxupdx+∫RnFɛudx.Recalling p>2 and condition (f2), when ɛ is sufficiently small, we have (78)Ev≤-ε2δ2+εpp+oε2≤-δ1<0,∀v∈ɛAm.In addition, since E(0)=0 and t→E(u(t)) is nonincreasing, then ω(ɛAm)⊂A∖{0}. Since ω(ɛAm) is closed and compact, there exists open neighborhood O of 0, such that (79)ωɛAm⊂A∖O.Let Bm=ɛAm; we have completed the proof of the first step.
Next, we only need to proveγ(A∖O)≥γ(Bm). Referring to Lemma 6  (A5), there exists N(A∖O), which is a neighborhood of A∖O, satisfying (80)γNA∖O=γA∖O.In addition, referring to the definition of ω-limit set ω(Bm), there exists t0>0 such that (81)StBm¯⊂NωBm⊂NA∖O.Thus, (82)γA∖O=γNA∖O≥γStBm¯.
It is obvious thatS(t) is odd since S(t)(-u0)=-u(t)=-S(t)u0. Then, referring to Lemma 6  (A4), we have γ(S(t)Bm¯)≥γ(Bm), and then γ(A∖O(0))≥γ(Bm). The proof is complete.At last, we want to investigate existence of the multiple equilibrium points of the equation, that is, solutions of the following elliptic equation:(83)-Δu-u+κxup-2u+fu=0inRn.We consider the critical values of the energy functionalE(u) defined by (70). In order to obtain infinite critical values by Lemma 8, we verify that the functional E(u) is bounded from below and satisfies the (P.S.) condition.Lemma 16.
The functionalE(u) defined by (70) is bounded from below.Proof.
The functional is(84)Eu=12∫Rn∇u2dx-12∫Rnu2dx+1p∫Rnκxupdx+∫RnFudx.By estimate (57), it is easy to verify that the functional E(u) is bounded from below.Lemma 17.
LetE(u) be a functional defined by (70) and d∈R be a constant, then any sequence {un}⊂Hκ1(Rn) such that (85)Eun≤d∀n∈N,E′un⟶0contains a convergent subsequence.Proof.
SinceE(un)≤d, we obtain (86)12∫Rn∇un2dx+1p∫Rnκxunpdx+∫RnFudx≤d+12∫Rnun2dx.Similarly to estimate (19), we have (87)∫Rnun2dx≤Cκ+1p∫Rnκxunpdx,and combining the above estimates and F(u)≥0, we have (88)12∫Rn∇un2dx+12p∫Rnκxunpdx≤d+Cκ2.It follows that unHκ1 is bounded.
Going if necessary to a subsequence, we assume that(89)un⇀uinHκ1Rn.By Lemma 15, we have un→u in L2(Rn). Observe that (90)E′un-E′u,un-u-∫Rnfun-fuun-udx=∫Rn∇un-u2dx-∫Rnun-u2dx+∫Rnκunp-2un-κup-2uun-udx≥∫Rn∇un-u2dx-∫Rnun-u2dx+α∫Rnκun-updx.Since E′(un)→0 and un→u in L2(Rn), we can obtain that (91)E′un-E′u,un-u⟶0,n⟶∞,∫Rnun-u2dx⟶0,n⟶∞,-∫Rnfun-fuun-udx=-∫Rn∫uunf′sdsun-udx≤l∫Rnun-u2dx⟶0,n⟶∞.Thus we have proved that un-uHκ1→0, n→∞.Now, we prove that the global attractor contains infinite distinct pairs of equilibrium points.Proof of Theorem4.
By the proof of Theorem2, we obtain that, for arbitrary r>0, there exists a subspace Vr⊂V with dimVr=r and ρ>0, such that (92)supu∈Vr∩Sρ⁡Eu<E0=0.The above two lemmas show that E also satisfies conditions (B2) and (B3) of Lemma 8; thus we obtain infinite pairs of critical points, which implies the conclusion.Remark 18.
In this paper, we suppose that the nonlinear termf is continuous. If f(u) is a weak continuous function in space Hκ1(Rn) or f(x,u):Rn×R→R is a Carathéodory mapping of C1 in u, all conclusions in this paper are still valid.



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*Source: 251614-2015-06-14.xml*