Both the parabolic flow and the gradient flow are used

THE GENERALIZED CONLEY INDEX AND MULTIPLE SOLUTIONS OF

SEMILINEAR ELLIPTIC PROBLEMS

E. N. DANCER AND YIHONG DU

Abstract. We establish some framework so that the generalizedConley index can be easily used to study the multiple solution problem of semilinear elliptic boundary value problems. Both the parabolic flow and the gradient flow are used. Some examples are given to compare our approach here with other well-known methods. Our abstract results with parabolic flows may have applications to parabolic problems as well.

1. Introduction

In this paper, we continue our efforts to show how the generalized Conley index developed by Rybakowski can be applied to multiple solution prob- lems ofsemilinear elliptic equations. Our main purpose here is to set up some framework so that the generalized Conley index can be easily used to superlinear problems. To that end, we use both the parabolic flows and the gradient flows ofthe elliptic problems. A simple example with superlinearity is given to illustrate some advantages ofour approach compared with other well-established methods. We also study in some detail a more general ver- sion ofa semilinear elliptic problem with a combined concave and convex nonlinearity, which was studied in [2] and [1] recently. This problem seems to serve as a good example to show when the generalized Conley index ap- proach gives better results and when it does not. Here we also improve some results ofA. Ambrosetti et al [1] and answer affirmatively a question asked therein.

In [26], various applications ofthe generalized Conley index to asymp- totically linear parabolic and elliptic problems can be found. Recently, in [10] and [12], we used the generalized Conley index to study the multiplicity

1991Mathematics Subject Classification. 35J65, 35K20, 47H17, 58E05.

Key words and phrases. Conley index, elliptic equation, critical point theory, fixed point index, superlinear problem.

Both authors are partially supportedby the Australian Research Council.

Received: February 22, 1996.

c

1996 Mancorp Publishing, Inc.

103

and related problems ofsome sublinear and asymptotically linear elliptic equations, where we made use ofthe Morse inequalities ofthe Morse decom- positions together with the order-preserving property ofthe parabolic flow.

The results obtained there seem difficult to get by using the standard topo- logical and variational methods. In this paper, we extend some ofthe ideas in [10] and [12] to superlinear problems. As before, we use Rybakowski’s version ofthe Conley index. We remark that Benci [4] developed a differ- ent version ofthe Conley index and he applied it also to some superlinear problems [4], [5]. He made use ofhis Morse index ofcritical point sets but did not use order structures and the full strength of Morse decompositions.

Our ideas could probably also be carried out by using Benci’s version ofthe Conley index.

We need to establish proper compactness settings in the generalized Con- ley index for superlinear nonlinearities. This is done by making use of the energy functionals, a priori estimates (for the parabolic flow) and P.S. condi- tion (for the gradient flow). We find that while the parabolic flow has wider applications, for example, it can be used to study the existence of connecting orbits ofthe corresponding parabolic equations, the gradient flow seems to provide better compactness result (compare Theorems 2.1 and 3.2). One difficulty in using the gradient flow is that it does not have the strongly order preserving property and the solutions do not improve their regularity as time increases. This is overcome in our applications by the observation that certain entire orbits ofthe gradient flow are compact inC¹(D), and the fact that the flow has a certain invariance property.

To be more precise, we consider the following semilinear elliptic problem:

−∆u=f(u) in D, Bu= 0 on ∂D, (1.1)

where D is a bounded domain in R^N with regular enough boundary ∂D, Bu=u(Dirichlet boundary condition) orBu=∂u/∂n(Neumann boundary condition), where n= n(x) is the unit outward normal to ∂D at x ∈ ∂D, andf :R¹ →R¹ is a locally Lipschitz continuous function. Unless otherwise specified, we suppose that f has a subcritical growth rate, i.e., it satisfies

• (H₁) |f(u)| ≤C(1 +|u|^γ) for all u∈(−∞,∞), whereC, γ are positive constants, and γ <(N+ 2)/(N−2) ifN >2.

In the generalized Conley index approach, as in [26], [10] and [12], the most natural flow to work with is the corresponding parabolic flow of(1.1), that is the local semiflow induced by the following parabolic problem:

u_t−∆u=f(u), t >0, x∈D, Bu= 0, t >0, x∈∂D.

(1.2)

To be more precise, we choose p > N, θ ∈ (1/2 +N/(2p),1) and let X = L^p(D). By well known results (see, e.g., [20]), (1.2) defines a local semiflow π(t) : X^θ → X^θ by π(t)u(0,·) = u(t,·), where X^θ is the fractional power space induced by (1.2) and u(t, x) is a solution of(1.2). We will callπ the parabolic flow for (1.1) and denote π(t, u(0,·)) = π(t)u(0,·) = u(t,·). By our choice of p and θ, X^θ is continuously imbedded into C¹(D) (see [20]).

Clearly, solutions of(1.1) are equilibria ofπ and vice versa.

Ifthe nonlinearity f is superlinear, one difficulty in using the generalized Conley index theory with the above parabolic flow is that solutions of(1.2) may blow up in finite time. To overcome this, or more precisely, to meet the more demanding compactness requirements (i.e., the π-admissibility con- dition) in the generalized Conley index theory, we make use ofits energy functional J :W_B^1,2(D)→R,

J(u) =

D[|∇u|²/2−F(u)]dx,

whereF(u) =₀^uf(s)ds, and a priori estimates for parabolic equations. Here and throughout this paper, we understand thatW_B^1,2=W₀^1,2 ifBu=u(i.e., the Dirichlet case), and W_B^1,2 = W^1,2 if Bu = ∂u/∂n (i.e., the Neumann case). We find that the energy level sets are very convenient to work with for this purpose. We use ideas of F. Rothe [25] on a priori estimates for parabolic systems, and energy estimates for parabolic equations as in [7]

and [19], and show in particular that, if f satisfies (H₁), and

• (H2) For some M > 0 and q > 1, 0 < (1 +q)F(u) ≤ uf(u) for all

|u| ≥M,

then the set ofpoints between any two energy levels (i.e., the set{u∈X^θ : a ≤ J(u) ≤ b}) is strongly π-admissible provided that a further condition γ <1+(4q+8)/(3N) is met . This compactness result enables us to use the generalized Conley index forπ for superlinear problems. Note that condition (H2) implies thatF(u)≥C1(|u|^q+1−1) for someC1 >0 and allu. Therefore we necessarily have q≤γ iff satisfies both (H₁) and (H₂). It then follows from γ <1 + (4q+ 8)/(3N) thatγ <(3N + 8)/(3N −4) ifN >1.

It is well known that conditions (H₁) and (H₂) guarantee that the energy functional J satisfies the P.S. condition on W_B^1,2, i.e., {u_n} has a conver- gent subsequence in W_B^1,2 whenever {J(u_n)} is bounded and J(u_n) → 0.

Therefore, standard variational methods are applicable to (1.1) under these conditions only, but our generalized Conley index setting is not. We suspect that our extra restriction γ <1 + (4q+ 8)/(3N) is not necessary.

Another natural flow associated to (1.1) is a slight variant ofthe negative gradient flow ofJ, that is, the global flowη :R×W_B^1,2(D)→W_B^1,2(D) given by η(t, u) =η(t), where η(t) is the unique solution ofthe problem

η_t=−σ(η)J(η), η(0) =u, (1.3)

where σ(u) = 1/max{1,J(u)}. Since critical points of J are solutions of (1.1), it is easy to see that equilibria of η are solutions of(1.1), and vice versa.

Here we do not have blow-ups, but we still need η-admissibility. We find that η possesses this compactness property under (H₁) and (H₂) only. In fact, we will show that, under condition (H₁), J satisfies the P.S. condition ifand only if{u ∈ W_B^1,2(D) : a ≤ J(u) ≤ b} is strongly η-admissible for any finite numbersa < b. Hence, in almost all the cases where the standard variational method can be used to (1.1), the generalized Conley index theory

with the gradient flow works. One disadvantage ofηcompared withπis that it does not have the order preserving property that the parabolic flow enjoys.

Moreover, unlike the parabolic flow, solutions alongη do not improve their regularity when time goes from zero to positive. These poses some difficulties in applications. We overcome these by using an invariance property ofηused by Hofer [21], and an observation that any entire orbit lying between two energy levels is precompact inC¹(D). This makes it possible to use upper and lower solution arguments in our generalized Conley index setting. These properties ofη come from the fact thatJ(u) =u−AuwhereAis compact, order preserving, and improves regularity.

As a simple example to show that the generalized Conley index approach may give better results than the other well established methods, we apply our general results on the parabolic and gradient flows to a special case of (1.1), i.e., the following Neumann problem:

−∆u=f(u), _∂n^∂u|_∂D= 0, (1.4)

wheref is C¹ and satisfies (H1), (H2) and

• (H3) There exist real numbers α andβ withα < β such that

f(α) =f(β) = 0, f(α)>0, f(β)>0,f(x)<0 f or x < α,f(x)>0 f or x > β.

Recall that condition (H₂) implies that f(u) is superlinear near infinity. It follows from condition (H3) that u = α and u = β are the only constant solutions of(1.4) outside the interval (α, β). We are interested in establishing a relationship between the position ofthe point (f(α), f(β)) in R² and the number ofnonconstant solutions of(1.4) which are outside (α, β). Here we say u = u(x) is outside (α, β) iffor some x₀ ∈ D, u(x₀) ∈ (α, β), that is, in the sense that (α, β) is regarded as an order interval in a proper function space. Note that it follows from the continuity of f that (1.4) has at least one constant solution in (α, β). Moreover, since we have no restriction on f for u ∈(α, β) except requiring it to be C¹, by varying f in (α, β) one can produce many constant and nonconstant solutions of(1.4) which lie inside this interval. The point ofour method is that it does not depend on howf varies in (α, β).

Let 0 ≤ λ₁ < λ₂ ≤ λ₃ ≤ ... ≤ λ_k ≤ λ_k+1 ≤ ... be all the eigenvalues of −∆u = λu, Bu|_∂D = 0, counting multiplicity. Note that {λ_k} with Dirichlet boundary conditions is different from that with Neumann boundary conditions. In concrete problems, we always understand that{λ_k} refers to the one with the boundary condition ofthe problem under consideration.

Define the sets G₁, G₂ and G₃ inR² as follows:

G₁ = [(λ₁, λ₂)×(λ₁, λ₂)]∪[(λ₁, λ₂)×(λ₂, λ₃)]∪[(λ₂, λ₃)×(λ₁, λ₂)].

G2 = [(λ1, λ2)×(λ3,∞)]∪[(λ3,∞)×(λ1, λ2)].

G₃ = [(λ₂, λ₃)×(λ₂, λ₃)]∪[(λ₃,∞)×(λ₃,∞)]∪ {(λ₂, λ₃)×[(λ₃,∞)\ {λ_k: k >3}]}

∪{[(λ₃,∞)\ {λ_k :k >3}]×(λ₂, λ₃)}.

Note that the union ofthe closure ofG1, G2 and G3 is the entire first quadrant in R² (where the Neumann boundary condition is used).

We have the following result.

Proposition 1.1. Suppose that f satisfies (H1)−(H3). Then (1.4) has at least inonconstant solutions outside(α, β)if (f(α), f(β))∈G_i, i= 1,2,3.

This result is proved by using the generalized Conley index theory for the gradient flow of(1.4). It seems difficult to obtain by using the standard topological and variational methods. We will also study a slightly more general version ofa problem studied by [1] and [2] recently, i.e., the following Dirichlet problem:

−∆u=λ|u|^r−1u+g(u), u|_∂D= 0, (1.5)

where λ >0 and r∈(0,1) are constants, g:R¹ →R¹ isC¹ and

• (H₄) g(u)u≥0, ∀u; lim_u→0g(u)/u= 0.

We will show that ifg is asymptotically linear, then our generalized Conley index approach as developed in [12] gives better results than other well- known methods; however, ifgis superlinear, the degree method used in [11]

seems to give the best result.

We prove in particular the following.

Proposition 1.2. Suppose thatg satisfies(H4) and either

• (H₅) lim_|u|→∞g(u)/u=a∈(λ_k,+∞)\ {λ_i}, k≥2, or

• (H6) lim_|u|→∞g(u)/|u|^γ−1u=b >0,

where 1 < γ <(N + 2)/(N −2) if N >2, 1 < γ < ∞ if N = 1,2. Then there existsΛ⁺,Λ⁻>0 such that

(i) for λ >Λ⁺(resp. Λ⁻),(1.5) has no positive (resp. negative) solution;

(ii) for 0< λ <Λ⁺ (resp. Λ⁻),(1.5)has at least two positive (resp. nega- tive) solutions;

(iii) for 0 < λ < max{Λ⁺,Λ⁻}, (1.5) has at least two sign-changing solu- tions.

Remarks. 1. The above result (essentially part (iii)) improves a result in [1] where they proved that under conditions similar to but slightly more restrictive than (H₄) and (H₆) f or g, for all small positive λ , (1.5) has at least 6 nontrivial solutions. Parts (i) and (ii) above follow essentially from [2].2. In fact we will prove more than Proposition 1.2 in section 4 later. In particular, we will answer affirmatively a question in [1] (see Remark 4.1 for details).

3. If g(u) =|u|^p−1u, wherep >1 andp <(N+ 2)/(N −2) when N >2, then the above result follows directly from [2]. In fact, in this case, using the oddness ofthe nonlinearity, Λ⁺= Λ⁻and (iv) can be improved significantly, i.e., there are infinitely many nontrivial solutions for any real λ, see [2] and the note added in proofat the end of[2].

4. Conditions (H5) and (H6) can be relaxed considerably. For example, (H₅) can be replaced by

u→±∞lim g(u)/u=a^±∈(λk,+∞)\ {λi}, k≥2,

and (H6) can be replaced by

u→±∞lim g(u)/|u|^γ±−1u=b^±>0,

where γ^± satisfies the same condition as γ. See also Remark 4.4 at the end ofsection 4.

Our results on the parabolic and gradient flows, especially that on the par- abolic flow, may have many other applications. For example, many equation systems have certain single equations as limiting problems, and solutions of the single equation problems often induce solutions to the original systems (some such examples can be found in [13], [15] and [16]). Since the gen- eralized Conley index possesses continuation stability, it may be useful to consider the flow π for a single limiting equation as a limit of the flow π generated by the original system and use results on π to study π. In this case, the flowη is difficult to use as there is in general no gradient flow for systems due to the fact that most natural reaction-diffusion systems lack variational structures. Another use ofthe flowπ is to study the existence of connecting orbits (see Remark 2.3). We should point out that our results on the gradient flow are sufficient for the applications in this paper. The main point to include the results on the parabolic flow here is for comparison with the gradient flow (which justifies our use ofthe less natural gradient flow) and for possible future applications.

Most ofour results on the flows π andη hold true for much more general flows, where they are induced by more general uniformly elliptic operators, and the nonlinearityf can also bex dependent.

The rest ofthis paper is organized as follows. In section 2, we show how a framework for π can be established so that the generalized Conley index works. A weaker version ofProposition 1.1 is proved using the flow π. In section 3, we study the gradient flowη and prove Proposition 1.1. Section 4 is devoted to the study of(1.5). Section 5 is an appendix, where we give proofs for the a priori estimates used in section 2.

2. The Parabolic Flow

In this section, we study (1.1) by making use ofits parabolic flow and the generalized Conley index. We consider only the superlinear case since the sublinear and asymptotically linear cases are relatively well understood (see, e.g., [26], [10] and [12]).

Let J1=J|_X^θ and for any interval Λ in R, define

J₁⁻¹Λ ={u∈X^θ :J1(u)∈Λ}, J⁻¹Λ ={u∈W_B^1,2 :J(u)∈Λ}.

The following result plays an important role in this section.

Theorem 2.1. Let a < b be real numbers, and (H₁),(H₂) be satisfied. If further γ <1 + (4q+ 8)/(3N), then J₁⁻¹[a, b]is strongly π-admissible.

Recall that a set N ∈X^θ is called stronglyπ-admissible ([26]) if(i) N is closed; (ii) π does not explode in N, i.e., u ∈X^θ, {t_n} ⊂R⁺ bounded and {π(t, u) :t∈[0, t_n]} ⊂N imply {π(t_n, u)}is bounded; (iii) for anyu_n∈X^θ,

{tn} ⊂ R⁺ with tn → ∞ and {π(t, un) : t∈[0, tn]} ⊂ N,{π(tn, un)} has a convergent subsequence.

The proofofTheorem 2.1 relies on some a priori estimates for solutions of(1.2).

Lemma 2.1. Let the conditions of Theorem 2.1 be satisfied and u=u(t, x) be a solution of (1.2) satisfying u(t,·) ∈J₁⁻¹[a, b] for t∈[T₁, T₂]. Then for any r ∈ (1,(2q + 4)/3), there exists M = Mr independent of u and T1, T2

such that

t∈[Tsup1,T2]u(t,·)L^r ≤M.

Lemma 2.1 follows from some well known energy estimates and interpo- lation inequalities, which were used, for example, in [7] and [19]. For the convenience ofthe readers, we will give the proofofLemma 2.1 in the Ap- pendix.

Lemma 2.2. Under the conditions of Lemma 2.1, there exist positive con- stants M andδ independent of u and T₁, T₂ such that

u(t,·)_L^∞ ≤[m(t−T₁)]^−δM,∀t∈(T₁, T₂].

Here m(s) = min{s,1}.

The proofofLemma 2.2 is based on some well known methods ofF. Rothe [25] and is rather technical. Since the ideas needed in proving Lemma 2.2 are scattered in [25], we will give a complete proofofLemma 2.2 in the Appendix.

Proof of Theorem 2.1. For convenience ofnotations, we will denote N =J₁⁻¹[a, b].

(i) It follows from the continuity ofJ₁ thatN is closed.

(ii) Suppose thatπ explodes inN. We are going to derive a contradiction.

Let u₀ ∈ X^θ be such that π(t, u₀) ∈N for all t ∈[0, T) with 0 < T < ∞, and for some tn ∈ (0, T), tn → T, π(tn, u0)_X^θ → ∞. By Lemma 2.2, π(t, u₀)_L^∞ ≤ M for all t ∈ [T/2, t_n], n ≥ 1. Hence, by Theorem 4.3 of [26],π(t_n, u₀)_X^σ ≤M₁ for all t∈[2T/3, t_n], n≥1 with someM₁ >0 and σ ∈(θ,1). Since X^σ imbeds continuously (in fact, compactly) intoX^θ, we arrive at a contradiction. Thus,π does not explode inN.

(iii) Let u_n ∈X^θ, t_n ∈ R⁺ satisfy t_n → ∞ and {π(t, u_n) : t∈ [0, t_n]} ⊂ N. We may assume that tn > 1 for all n. Then using Lemma 2.2 and Theorem 4.3 of[26] as in (ii), we deduce thatπ(t, u_n)_X^σ ≤M₁ for all n and t ∈ [1, t_n], where M₁ is a positive number and σ ∈ (θ,1). Since X^σ imbeds compactly intoX^θ, {π(t_n, u_n)} is precompact in X^θ and hence has a convergent subsequence.

The proofofTheorem 2.1 is complete.

Theorem 2.2. Suppose that (H₁), (H₂) are satisfied, and γ < 1 + (4q + 8)/(3N). Then there exists a0 < 0 such that J has no critical point in J⁻¹(−∞, a₀]. If further J has no critical point in J⁻¹[b₀,∞) for someb₀ >

a0, γ < 1 + (4q + 8)/(3N) and K is the maximal invariant set of π in J₁⁻¹[a₀, b₀], then the generalized Morse index h(π, K) is well-defined and

H_q(h(π, K)) = 0, q= 0,1,2, ....

Proof. Let S^∞ ={u ∈W_B^1,2(D) : u_W^1,2

B = 1}. Then it follows from (H₂) that for any u ∈ S^∞, J(tu) → −∞ as t → ∞. Moreover, there exists a0 < 0 such that t > 0, u ∈ S^∞ and J(tu) ≤ a0 imply (d/dt)J(tu) < 0.

A detailed proofofthese facts can be found in [27] where the Dirichlet boundary condition is considered, but the prooffor the Neumann problem is the same. As in [27], one finds easily from these two facts that, fora≤a0, (2.1) H_q(W_B^1,2, J⁻¹(−∞, a]) =H_q(W_B^1,2, S^∞) = 0, q= 0,1,2....

It is well known that under (H1) and (H2), J satisfies the P.S. condition.

Hence ifJhas no critical point inJ⁻¹[b₀,∞), then for anyb≥b₀,J⁻¹(−∞, b]

is a strong deformation retractor of W_B^1,2. Thus, by (2.1), fora≤a₀, b≥b₀, Hq(J⁻¹(−∞, b], J⁻¹(−∞, a]) =Hq(W_B^1,2, J⁻¹(−∞, a])

= 0, q= 0,1,2. . . . (2.2)

Now let a < a1 < a0, b1 > b > b0. Then J⁻¹(−∞, a] and J⁻¹(−∞, b] are strong deformation retractors ofJ⁻¹(−∞, a₁) andJ⁻¹(−∞, b₁) respectively.

Therefore,

H_q(J⁻¹(−∞, b], J⁻¹(−∞, a]) =H_q(J⁻¹(−∞, b₁), J⁻¹(−∞, a₁)).

By Palais’ theorem (see, Theorem 3.2 in [8]),

Hq(J₁⁻¹(−∞, b1), J₁⁻¹(−∞, a1)) =Hq(J⁻¹(−∞, b1), J⁻¹(−∞, a1)).

Thus, using (2.2), we have

Hq(J₁⁻¹(−∞, b1), J₁⁻¹(−∞, a1)) = 0, q= 0,1,2, ....

(2.3)

By Theorem 2.1, for any a < a₀ and b > b₀,N ≡J₁⁻¹[a, b] is strongly π- admissible. We show thatN is an isolating neighborhood ofK. It suffices to show thatKis the maximal invariant set ofπinNand thatK⊂J₁⁻¹(a₀, b₀).

Let K1 be the maximal invariant set of π inN and u∈K1 (If K1 is empty, then the conclusion ofthe theorem holds trivially). Thenu=u(t, x) is a f ull solution of(1.2). Since N is strongly π-admissible, it is easy to see that the omega limit set ω(u) and the alpha limit setα(u) are both nonempty. Since J is a Lyapunov functional for (1.2), ω(u) and α(u) consist ofequilibria of(1.2), i.e., solutions of(1.1). But solutions of(1.1) are critical points of J. Therefore, by the assumptions, for any u₁ ∈ α(u) and u₂ ∈ ω(u), J(u1), J(u2) ∈ (a0, b0). It follows that J(u(t,·)) ∈ [J(u1), J(u2)] ∈ (a0, b0) for any t ∈(−∞,∞). This shows that u ∈J₁⁻¹(a0, b0). Thus K =K1 and N is an isolating neighborhood ofK. Thereforeh(π, K) is well-defined.

Since J1 is a quasi-potential ofπ, by Corollary 4.7 of[26],

(2.4) H_q(h(π, K)) =H_q(J₁⁻¹(−∞, a], J₁⁻¹(−∞, b]), q= 0,1,2, ....

A simple variant ofthe argument on page 59 of[8] shows that, ifa < a1≤a0

and b₁ > b ≥b₀, thenJ₁⁻¹(−∞, a] and J₁⁻¹(−∞, b] are strong deformation retractors of J₁⁻¹(−∞, a₁) and J₁⁻¹(−∞, b₁) respectively. Thus

H_q(J₁⁻¹(−∞, a], J₁⁻¹(−∞, b]) =H_q(J₁⁻¹(−∞, a₁), J₁⁻¹(−∞, b₁)).

This and (2.3), (2.4) imply

H_q(h(π, K)) = 0, q = 0,1,2, ....

The proofis complete.

Remark 2.1. All our above results in this section are true for much more general problems than (1.2). We could replace ∆ by a general second order self-adjoint uniformly elliptic operator. f can also be dependent on x. Of course, the functional J should be adjusted accordingly.

Now we are able to prove a weaker version ofProposition 1.1. We remark that though we give a unified generalized Conley index approach, a number ofsubcases in Proposition 2.1 can be proved by the standard topological and variational methods.

Proposition 2.1. Suppose that (H₁)−(H₃) are satisfied andγ <1 + (4q+ 8)/(3N). Then (1.4) has at least i nonconstant solutions outside (α, β) if (f(α), f(β))∈G_i, i= 1,2,3.

Proof. It follows from (H3) that (1.4) has no constant solution outside [α, β].

Moreover, there exist uniqueα₁ > αand β₁ < β such thatα₁≤β₁ and f(α1) =f(β1) = 0, f(u)>0 f or x∈(α, α1), f(u)<0 f or u∈(β1, β).

(2.5)

We observe two facts by using (2.5). First, it follows from (2.5) and the elliptic maximum principle that any nonconstant solution of(1.4) which is outside [α₁, β₁] must be outside [α, β].

Second, for any α ∈(α, α₁) andβ ∈(β₁, β),u≡α andv≡β are strict lower and upper solutions of(1.4) respectively. Fix such a pair and define

N0 = [α, β]_X^θ ≡ {u∈X^θ:α ≤u≤β}.

As in [10], one can easily show that N₀ is a strongly π-admissible isolating neighborhood of K0, where K0 is the maximal invariant set of π in N0. Moreover

H_q(h(π, K₀)) =δ_0,1G.

(2.6)

(2.6) can be proved by either a simple homotopy argument (see Remark 3.2 later) or by calculating h(π, K0) directly (observing that N0 is in fact an isolating block ofK₀).

We have three cases to consider:

(i) (f(α), f(β))∈G₁, (ii) (f(α), f(β))∈G₂, (iii) (f(α), f(β))∈G₃. We give the detailed proof for case (ii) only. The proofs for the other cases are similar or simpler, and therefore are left to the interested readers.

Now suppose that (f(α), f(β)) ∈ G2. We may assume that f(α) ∈ (λ₁, λ₂) andf(β)∈(λ₃,∞) as the remaining case can be proved analogously.

A simple variant ofthe proofofProposition 2 in [12] (where the Dirichlet boundary condition is considered) shows thatu =β₁ is a strict local mini- mum of J restricted to the setS={u∈W^1,2(D) :u≥β₁}, and (1.4) has a solution u1 > β1 which is ofmountain pass type. We may assume that (1.4) has only finitely many solutions inS. Then it follows as in [12] that

C_q(J, u₁) =δ_1,qG.

Here, and in what follows,C_q(J, u) denotes the critical groups ofthe critical point u ofJ. By Proposition 2.1 in [10] (applied to the Neumann boundary condition case),

Hq(h(π,{u1})) =Cq(J, u1).

Thus we have

Hq(h(π,{u1})) =δ1,qG.

(2.7)

Since f(β)∈(λ₃,∞), we can use the shifting theorem (see, e.g., [22]) to calculate the critical groups ofthe critical pointu₂=β, and obtain

C_q(J, u₂) = 0, q= 0,1,2.

Now use this and Proposition 2.1 in [10] as above, we obtain Hq(h(π,{u2})) = 0, q = 0,1,2.

(2.8)

This implies in particular that u2 = u1. As u3 = α ∈ S, we also have u₃ =u₁. Moreover, it follows fromf(α)∈(λ₁, λ₂) that

H_q(h(π,{u₃})) =δ_1,qG.

(2.9)

Now clearly u₁ is a nonconstant solution of(1.4) outside (α, β). We need to show that there is at least one more such solution. Suppose that this is not the case. Then one easily sees that the solution set of(1.4) is compact in X^θ. Hence we can find a < b such that a < J(u) < b for any solution u of (1.4). Moreover, ifK is the maximal invariant set ofπ inJ₁⁻¹[a, b], then by Theorem 2.2, h(π, K) is well-defined and

Hq(h(π, K)) = 0, q = 0,1,2, ....

(2.10)

By changing the subscripts in u1, u2 and u3 we can suppose that J(u1) ≤ J(u₂) ≤ J(u₃). We show next that {K₀,{u₁},{u₂},{u₃}} is a Morse de- composition of K. For convenience, we denote Ki = {ui}, i = 1,2,3. It suffices to show that, for any u ∈ K, either u ∈ K_i for some 0 ≤ i ≤ 3 or α(u) ∈ Kj and ω(u) ∈ Ki for some 0 ≤ i < j ≤ 3. Now let u ∈ K and u ∈ ∪³_i=0K_i. Then u cannot be an equilibrium of π. Moreover, since t → J(u(t,·)) is strictly decreasing, α(u) and ω(u) are different sets and consist ofsolutions of(1.4). Since both α(u) andω(u) are connected, each ofthem is contained in one and only oneKi. Choosew∈α(u). If w∈Kj, j > 0, then we necessarily haveα(u) =K_j, and J(ω(u))< J(α(u)). Hence we must have ω(u) ⊂Ki,i < j. It remains to check the case that w∈K0. In this case, w ∈[α₁, β₁]. It follows from the parabolic maximum principle

and the fact thatu=α, u=β are lower and upper solutions of(1.4) that α < u(t, x)< β for all t∈(−∞,∞) and x∈D, i.e., u∈K₀. But this con- tradicts our initial assumption. Therefore the last case never occurs. This proves that {K₀, K₁, K₂, K₃}is a Morse decomposition ofK.

Next we use the Morse equation for this Morse decomposition to deduce a contradiction and thereby complete the proof. We substitute (2.6)-(2.10) into the following Morse equation

Σ^∞_q=0rankHq(h(π, K))t^q

= Σ³_i=0Σ^∞_q=0 rankH_q(h(π, K_i))t^q−(1 +t)Q(t),

where Q(t) = Σ^∞_q=0d_qt^q, d_q≥0, and obtain by comparing the coefficients:

0 = 1−d0,0 = 1 + 1−d1−d0,0 =−d2−d1.

This gives d₀ = d₁ = 1, d₂ = −1, contradicting d₂ ≥ 0. The proofis complete.

Remark 2.2. As in [26], one can use irreducibility properties ofthe gen- eralized Conley index to discuss the existence ofconnecting orbits of(1.2).

For example, under the conditions ofProposition 2.1, corresponding to each isolated solution of(1.4) with non-trivial Conley index, there is a bounded non-equilibrium entire solution of(1.2) (with Neumann boundary condi- tions) having this solution of(1.4) as its alpha or omega limit set.

3. The Gradient Flow

In this section, we set up the framework for the gradient flow η so that the generalized Conley index can be used.

We note first thatσ(η)J(η) is locally Lipschitz inηandσ(η)J(η)_W^1,2

B ≤

1. Therefore η is defined on the wholeR×W_B^1,2. Fix any k >0 and defineA:W_B^1,2→W_B^1,2 by

A(u) = (−∆ +k)⁻¹(f(u) +ku),

where (−∆ +k)⁻¹ denotes the inverse of −∆ +k under the homogeneous boundary conditions Bu= 0. It is well known that, under condition (H1), A is compact, and ifwe use the equivalent norm

u=

D[|∇u|²/2 +ku²]dx in W_B^1,2(D), then

J(u) =u−A(u).

For any u₀ ∈W_B^1,2, let

w(t, u₀) = ^t

0 σ(η(ξ, u₀))dξ.

Then

(3.1) η(t, u0) =e^−w(t,u0)u0+e^−w(t,u0) _t

0 e^w(s,u0)A(η(s, u0))σ(η(s, u0))ds.

Let X₀ = W_B^2,p, p ≥ 2 is such that W_B^2,p imbeds compactly into C¹(D).

Then it is well known (see, e.g., [8] or [21]) that there exist a finite sequence ofBanach spaces X₁, ..., X_k such that

W_B^2,p=X₀9→X₁9→... 9→X_k=W_B^1,2,

the imbedding of Xi−1 intoXi is compact, andA:Xi→Xi−1 is continuous and maps bounded sets into bounded sets, i= 1, ..., k.

Using this and (3.1), one sees that if η(t, u0) is a solution of(1.3), then η(t, u₀) ∈ X_i for all t if u₀ ∈X_i, t→ η(t, u₀) is continuously differentiable inX_i and (t, u₀)→η(t, u₀) is continuous inR×X_i.

Lemma 3.1. Let (H1)be satisfied, {un} ⊂Xi for some fixed 1≤i≤k and 0< T ≤ ∞. Suppose that

M₁ = sup{η(t, u_n)_Xk : 0≤t < T, n≥1}<∞ and M₂ = sup{u_nXi :n≥1}<∞.

Then there exists C depending only on i, M1 and M2 such that η(t, u_n)−e^−w(t,un)u_nXi−1 ≤C, ∀t∈[0, T), n≥1.

Proof. By (3.1),

η(t, u_n)−e^−w(t,un)u_nXk−1 =e^{−w(t,un) t}

0 e^wA(η)σ(η)ds_Xk−1

≤e^−w(t,un) _t

0 e^wA(η)_Xk−1σ(η)ds.

Since A is a bounded operator fromX_k toX_k−1, we can findC₁ =C₁(M₁) such that

A(u)_Xk−1 ≤C₁, ∀u∈X_k,u_Xk ≤M₁. Using this we obtain

e^{−w(t,un) t}

0 e^wA(η)_Xk−1σ(η)ds

≤e^−w(t,un) _t

0 e^w(s,un)C1σ(η(s, un))ds

=e^{−w(t,un) w(t,un)}

0 C₁e^ξdξ=C₁(1−e^−w(t,un))

≤C₁, ∀t∈[0, T), n≥1.

Thus η(t, un)−e^−w(t,un)unXk−1 ≤C1, ∀t∈[0, T), and η(t, u_n)_Xk−1 ≤C₁+u_nXk−1 ≤K₁, ∀t∈[0, T), n≥1.

Repeating the above argument k−itimes, we obtain η(t, u_n)_Xi ≤K_k−i, ∀t∈[0, T), n≥1

where K_k−i depends only on i and M₁, M₂. Now use the argument once more, and the proofis complete.

Theorem 3.1. Let (H₁) be satisfied. Then any bounded closed set in W_B^1,2 is stronglyη-admissible.

Proof. LetU be any bounded closed set inW_B^1,2. Sinceη is a global flow, it does not explode inU. Therefore, it suffices to show that {η(t_n, u_n)} has a convergent subsequence wheneverη([0, tn], un)⊂U and tn→ ∞. SinceU is bounded, it is easy to see that δ = min{σ(u) :u ∈U} is positive. It follows that

w(t_n, u_n) = ^tn

0 σ(η(s, u_n))ds≥δt_n→ ∞.

By (3.1) and Lemma 3.1, there exists C >0 such that

η(tn, un)−e^−w(tn,un)unXk−1 ≤C for all n.

ButX_k−1imbeds compactly intoX_k=W_B^1,2. Thus{η(t_n, u_n)−e^−w(tn,un)u_n} has a convergent subsequence in W_B^1,2. Since e^−w(tn,un)u_n → 0 in W_B^1,2, we conclude that {η(t_n, u_n)} has a convergent subsequence. The proofis complete.

The following result establishes the relationship between the P.S. condition and η-admissibility.

Theorem 3.2. Suppose that (H1) is satisfied. Then J satisfies the P.S.

condition if and only if, for any −∞ < a < b < ∞, J⁻¹[a, b] is strongly η-admissible.

Proof. We show first that strongly η-admissibility implies the P.S. condi- tion. Suppose that J⁻¹[a, b] is strongly η-admissible for any finite inter- val [a, b]. Let {u_n} ⊂ W_B^1,2 be a P.S. sequence: {J(u_n)} is bounded and J(un) → 0. It follows easily that we can find tn > 0, tn → ∞ and a fi- nite interval [a, b] such that η([−t_n,0], u_n)⊂J⁻¹[a, b]. Let v_n=η(−t_n, u_n).

Then η([0, tn], vn) ⊂ J⁻¹[a, b]. Since tn → ∞, and J⁻¹[a, b] is strongly η- admissible, u_n =η(t_n, v_n) has a convergent subsequence. This proves that J satisfies the P.S. condition.

Next we prove the converse. Suppose that J satisfies the P.S. condition.

We need to show that {η(t_n, u_n)} has a convergent subsequence whenever η([0, tn], un)⊂J⁻¹[a, b] and tn→ ∞. The proofofTheorem 3.1 shows that it suffices to prove that {η(t, u_n) : 0 ≤ t ≤t_n, n ≥ 1} is bounded in W_B^1,2. Suppose that this set is not bounded. Then, by passing to a subsequence, we can find sn∈[0, tn] such that

Rn≡ η(sn, un)_W^1,2

B → ∞.

Since J satisfies the P.S. condition, its critical points in J⁻¹[a, b] form a bounded setK, sayK ⊂BR0−1 ={u∈W_B^1,2 :u_W^1,2

B ≤R0−1}. Moreover, the P.S. condition implies that, for some δ ∈ (0,1), J(u)_W^1,2

B ≥ δ for all u∈J⁻¹[a, b]\BR0. By definition,

min{1,J(u)_W^1,2

B } ≤σ(u)J(u)_W^1,2

B ≤1.

Thus, ifwe denotevn=η(sn, un), then η(t, v_n)−v_nW^1,2

B ≤ |t|, ∀t.

It follows that η(t, v_n)_W^1,2

B ≥ v_nW^1,2

B − |t|> R₀ whenever |t|< R_n−R₀. Since t_n→ ∞, we have, subject to a subsequence, either (i) s_n→ ∞, or (ii) tn−sn→ ∞.

In case (i), choose T_n = −min{R_n −R₀, s_n}. Then T_n → −∞ and η(t, vn)∈J⁻¹[a, b]\BR0 for all t∈[Tn,0]. Therefore,

a−b≤J(vn)−J(η(Tn, vn)) = ₀

Tn(d/dt)J(η(t, vn))dt

= ⁰

Tn

−σ(η(t, v_n))J(η(t, v_n))²_W1,2 B dt

≤δ²Tn→ −∞, a contradiction.

In case (ii), we define Tn= min{Rn−R0, tn−sn} and similarly derive b−a≥J(vn)−J(η(Tn, vn))≥δ²Tn→ ∞,

again a contradiction. This finishes the proof.

Remark 3.1. Under the condition (H1), in general, J satisfies the P.S.

condition (or equivalently, by Theorem 3.2,J⁻¹[a, b] is stronglyη-admissible) does not imply that J⁻¹[a, b]∩ Xi is strongly η|R×Xi-admissible for any i ≤k−1. To see this, we choose f such that (H₁) and the P.S. condition are satisfied and that (1.1) has a solution u₀ in J⁻¹[a+ 1, b−1] for some a < b−2. By elliptic regularity,u₀ ∈X₀. Now choose u_n ∈ X_i satisfying that

un−u0Xi+1 →0, unXi → ∞.

Since η(t, u0) ≡ u0, we can find an increasing sequence {tn} with tn → ∞ such that η([0, t_n], u_n) ⊂ B₁(u₀)∩J⁻¹[a, b], where B₁(u₀) = {u ∈ X_i+1 : u_Xi+1 ≤1}. By Lemma 3.1 there existsC >0 independent ofnsuch that

η(t, u_n)−e^−w(t,un)u_nXi ≤C, ∀t∈[0, t_n], n≥1.

Since u_nXi → ∞and u_n→u₀ inX_k, for any fixed j, we can find n_j ≥j such that unjXie^−w(tj,u0) > j and |w(tj, u0) −w(tj, unj)| < 1. Thus, unj_Xie^−w(tj,unj)→ ∞, and

η(t_j, u_nj)_Xi ≥ u_njXie^−w(tj,unj)−C→ ∞.

Sinceη([0, t_j], u_nj)⊂η([0, t_nj], u_nj)⊂J⁻¹[a, b]∩X_i, andt_nj → ∞, our above discussion means that J⁻¹[a, b]∩X_i is not stronglyη|_R×Xi-admissible.

The following is a useful result.