THE GENERALIZED CONLEY INDEX AND MULTIPLE SOLUTIONS OF SEMILINEAR ELLIPTIC PROBLEMS

We establish some framework so that the generalized Conley 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.


Introduction
In this paper, we continue our efforts to show how the generalized Conley index developed by Rybakowski can be applied to multiple solution problems of semilinear 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 of the elliptic problems. A simple example with superlinearity is given to illustrate some advantages of our approach compared with other well-established methods. We also study in some detail a more general version of a 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 approach gives better results and when it does not. Here we also improve some results of A. Ambrosetti et al [1] and answer affirmatively a question asked therein.
In [26], various applications of the generalized Conley index to asymptotically linear parabolic and elliptic problems can be found. Recently, in [10] and [12], we used the generalized Conley index to study the multiplicity and related problems of some sublinear and asymptotically linear elliptic equations, where we made use of the Morse inequalities of the Morse decompositions together with the order-preserving property of the parabolic flow. The results obtained there seem difficult to get by using the standard topological and variational methods. In this paper, we extend some of the ideas in [10] and [12] to superlinear problems. As before, we use Rybakowski's version of the Conley index. We remark that Benci [4] developed a different version of the Conley index and he applied it also to some superlinear problems [4], [5]. He made use of his Morse index of critical 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 of the Conley index.
We need to establish proper compactness settings in the generalized Conley index for superlinear nonlinearities. This is done by making use of the energy functionals, a priori estimates (for the parabolic flow) and P.S. condition (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 of the 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 of the gradient flow are compact in C 1 (D), and the fact that the flow has a certain invariance property.
If the 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 condition) in the generalized Conley index theory, we make use of its energy functional J : W 1,2 B (D) → R, where F (u) = u 0 f (s)ds, and a priori estimates for parabolic equations. Here and throughout this paper, we understand that W 1,2 B = W 1,2 0 if Bu = u (i.e., the Dirichlet case), and W 1,2 B = 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 (H 1 ), and • (H 2 ) For some M > 0 and q > 1, 0 < (1 + q)F (u) ≤ uf (u) for all |u| ≥ M , then the set of points 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 for π for superlinear problems. Note that condition (H 2 ) implies that F (u) ≥ C 1 (|u| q+1 −1) for some C 1 > 0 and all u. Therefore we necessarily have q ≤ γ if f satisfies both (H 1 ) and (H 2 ). It then follows from γ < 1 + (4q + 8)/(3N ) that γ < (3N + 8)/(3N − 4) if N > 1.
It is well known that conditions (H 1 ) and (H 2 ) guarantee that the energy functional J satisfies the P.S. condition on W 1,2 B , i.e., {u n } has a convergent subsequence in W 1,2 B 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 of the negative gradient flow of J, that is, the global flow η : 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 (H 1 ) and (H 2 ) only. In fact, we will show that, under condition (H 1 ), J satisfies the P.S. condition if and only if {u ∈ W 1,2 B (D) : a ≤ J(u) ≤ b} is strongly η-admissible for any finite numbers a < 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 of η compared with π is that it does not have the order preserving property that the parabolic flow enjoys. Moreover, unlike the parabolic flow, solutions along η 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 of η used by Hofer [21], and an observation that any entire orbit lying between two energy levels is precompact in C 1 (D). This makes it possible to use upper and lower solution arguments in our generalized Conley index setting. These properties of η come from the fact that J (u) = u − Au where A is 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: where f is C 1 and satisfies (H 1 ), (H 2 ) and • (H 3 ) There exist real numbers α and β with α < β such that is superlinear near infinity. It follows from condition (H 3 ) 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 of the point (f (α), f (β)) in R 2 and the number of nonconstant solutions of (1.4) which are outside (α, β). Here we say u = u(x) is outside (α, β) if for some x 0 ∈ D, u(x 0 ) ∈ (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 C 1 , by varying f in (α, β) one can produce many constant and nonconstant solutions of (1.4) which lie inside this interval. The point of our method is that it does not depend on how f varies in (α, β).
.. 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 of the problem under consideration.
Define the sets G 1 , G 2 and G 3 in R 2 as follows: Note that the union of the closure of G 1 , G 2 and G 3 is the entire first quadrant in R 2 (where the Neumann boundary condition is used).
We have the following result.
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 of a problem studied by [1] and [2] recently, i.e., the following Dirichlet problem: where λ > 0 and r ∈ (0, 1) are constants, g : We will show that if g is asymptotically linear, then our generalized Conley index approach as developed in [12] gives better results than other wellknown methods; however, if g is superlinear, the degree method used in [11] seems to give the best result.
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 (H 4 ) and (H 6 ) for 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−1 u, where p > 1 and p < (N + 2)/(N − 2) when N > 2, then the above result follows directly from [2]. In fact, in this case, using the oddness of the 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 proof at the end of [2]. 4. Conditions (H 5 ) and (H 6 ) can be relaxed considerably. For example, (H 5 ) can be replaced by and (H 6 ) can be replaced by where γ ± satisfies the same condition as γ. See also Remark 4.4 at the end of section 4.
Our results on the parabolic and gradient flows, especially that on the parabolic 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 generalized 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 flow η 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 of the flow π 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 of the less natural gradient flow) and for possible future applications.
Most of our results on the flows π and η hold true for much more general flows, where they are induced by more general uniformly elliptic operators, and the nonlinearity f can also be x dependent.
The rest of this 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 of Proposition 1.1 is proved using the flow π. In section 3, we study the gradient flow η 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.

The Parabolic Flow
In this section, we study (1.1) by making use of its 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 J 1 = J| X θ and for any interval Λ in R, define The following result plays an important role in this section. is strongly π-admissible. Recall that a set N ∈ X θ is called strongly π-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 any u n ∈ X θ , The proof of Theorem 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. . Then for any r ∈ (1, (2q + 4)/3), there exists M = M r independent of u and T 1 , T 2 such that sup Lemma 2.1 follows from some well known energy estimates and interpolation inequalities, which were used, for example, in [7] and [19]. For the convenience of the readers, we will give the proof of Lemma 2.1 in the Appendix.
The proof of Lemma 2.2 is based on some well known methods of F. Rothe [25] and is rather technical. Since the ideas needed in proving Lemma 2.2 are scattered in [25], we will give a complete proof of Lemma 2.2 in the Appendix.
(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 t n > 1 for all n. Then using Lemma 2.2 and Theorem 4.3 of [26] as in (ii), we deduce that π(t, u n ) X σ ≤ M 1 for all n and t ∈ [1, t n ], where M 1 is a positive number and σ ∈ (θ, 1). Since X σ imbeds compactly into X θ , {π(t n , u n )} is precompact in X θ and hence has a convergent subsequence.
The proof of Theorem 2.1 is complete.

then the generalized Morse index h(π, K) is well-defined and
A detailed proof of these facts can be found in [27] where the Dirichlet boundary condition is considered, but the proof for the Neumann problem is the same. As in [27], one finds easily from these two facts that, for a ≤ a 0 , By Palais' theorem (see, Theorem 3.2 in [8]), ). Thus, using (2.2), we have is strongly πadmissible. We show that N is an isolating neighborhood of K. It suffices to show that K is the maximal invariant set of π in N and that K ⊂ J −1 1 (a 0 , b 0 ). Let K 1 be the maximal invariant set of π in N and u ∈ K 1 (If K 1 is empty, then the conclusion of the theorem holds trivially). Then u = u(t, x) is a full 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 set α(u) are both nonempty. Since J is a Lyapunov functional for (1.2), ω(u) and α(u) consist of equilibria 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 u 1 ∈ α(u) and u 2 ∈ ω(u), for any t ∈ (−∞, ∞). This shows that u ∈ J −1 1 (a 0 , b 0 ). Thus K = K 1 and N is an isolating neighborhood of K. Therefore h(π, K) is well-defined.
Since J 1 is a quasi-potential of π, by Corollary 4.7 of [26], A simple variant of the argument on page 59 of [8] shows that, if a < a 1 ≤ a 0 The proof is 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 of Proposition 1.1. We remark that though we give a unified generalized Conley index approach, a number of subcases in Proposition 2.1 can be proved by the standard topological and variational methods.

Proof. It follows from (H 3 ) that (1.4) has no constant solution outside [α, β].
Moreover, there exist unique α 1 > α and β 1 < β such that α 1 ≤ β 1 and 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 Second, for any α ∈ (α, α 1 ) and β ∈ (β 1 , β), u ≡ α and v ≡ β are strict lower and upper solutions of (1.4) respectively. Fix such a pair and define As in [10], one can easily show that N 0 is a strongly π-admissible isolating neighborhood of K 0 , where K 0 is the maximal invariant set of π in N 0 . Moreover (2.6) can be proved by either a simple homotopy argument (see Remark 3.2 later) or by calculating h(π, K 0 ) directly (observing that N 0 is in fact an isolating block of K 0 ).
We have three cases to consider: 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)) ∈ G 2 . We may assume that f (α) ∈ (λ 1 , λ 2 ) and f (β) ∈ (λ 3 , ∞) as the remaining case can be proved analogously.
A simple variant of the proof of Proposition 2 in [12] (where the Dirichlet boundary condition is considered) shows that u = β 1 is a strict local minimum of J restricted to the set S = {u ∈ W 1,2 (D) : u ≥ β 1 }, and (1.4) has a solution u 1 > β 1 which is of mountain pass type. We may assume that (1.4) has only finitely many solutions in S. Then it follows as in [12] that Here, and in what follows, C q (J, u) denotes the critical groups of the critical point u of J. By Proposition 2.1 in [10] (applied to the Neumann boundary condition case), Since f (β) ∈ (λ 3 , ∞), we can use the shifting theorem (see, e.g., [22]) to calculate the critical groups of the critical point u 2 = β, and obtain Now use this and Proposition 2.1 in [10] as above, we obtain This implies in particular that u 2 = u 1 . As u 3 = α ∈ S, we also have Now clearly u 1 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, if K is the maximal invariant set of π in J −1 1 [a, b], then by Theorem 2.2, h(π, K) is well-defined and By changing the subscripts in u 1 , u 2 and u 3 we can suppose that J( 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 of solutions of (1.4). Since both α(u) and ω(u) are connected, each of them is contained in one and only one K i . Choose w ∈ α(u). If w ∈ K j , j > 0, then we necessarily have α(u) = K j , and J(ω(u)) < J(α(u)). Hence we must have ω(u) ⊂ K i , i < j. It remains to check the case that w ∈ K 0 . In this case, w ∈ [α 1 , β 1 ]. It follows from the parabolic maximum principle and the fact that u = 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 ∈ K 0 . But this contradicts our initial assumption. Therefore the last case never occurs. This proves that {K 0 , K 1 , K 2 , K 3 } is a Morse decomposition of K.
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 , and obtain by comparing the coefficients: This The proof is complete.

Remark 2.2.
As in [26], one can use irreducibility properties of the generalized Conley index to discuss the existence of connecting orbits of (1.2). For example, under the conditions of Proposition 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 conditions) having this solution of (1.4) as its alpha or omega limit set.

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 that σ(η)J (η) is locally Lipschitz in η and σ(η)J (η) W 1,2 B ≤ 1. Therefore η is defined on the whole R × W 1,2 B . Fix any k > 0 and define A : where (−∆ + k) −1 denotes the inverse of −∆ + k under the homogeneous boundary conditions Bu = 0. It is well known that, under condition (H 1 ), A is compact, and if we use the equivalent norm For any u 0 ∈ W 1,2 B , let . Then it is well known (see, e.g., [8] or [21]) that there exist a finite sequence of Banach spaces X 1 , ..., X k such that Using this and (3.1), one sees that if η(t, u 0 ) is a solution of (1.3), then Since A is a bounded operator from X k to X k−1 , we can find Repeating the above argument k − i times, we obtain where K k−i depends only on i and M 1 , M 2 . Now use the argument once more, and the proof is complete. Proof. Let U be any bounded closed set in W 1,2 B . Since η is a global flow, it does not explode in U . Therefore, it suffices to show that {η(t n , u n )} has a convergent subsequence whenever η([0, t n ], u n ) ⊂ U and t n → ∞. Since U is bounded, it is easy to see that δ = min{σ(u) : u ∈ U } is positive. It follows that By (3.1) and Lemma 3.1, there exists C > 0 such that Thus {η(t n , u n )−e −w(tn,un) u n } has a convergent subsequence in W 1,2 B . Since e −w(tn,un) u n → 0 in W 1,2 B , we conclude that {η(t n , u n )} has a convergent subsequence. The proof is complete.
The following result establishes the relationship between the P.S. condition and η-admissibility.

Theorem 3.2. Suppose that (H 1 ) is satisfied. Then J satisfies the P.S. condition if and only if, for any
Proof. We show first that strongly η-admissibility implies the P.S. condition. Suppose that J −1 [a, b] is strongly η-admissible for any finite interval [a, b]. Let {u n } ⊂ W 1,2 B be a P.S. sequence: {J(u n )} is bounded and J (u n ) → 0. It follows easily that we can find t n > 0, t n → ∞ and a fi- Since t n → ∞, and J −1 [a, b] is strongly ηadmissible, 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, t n ], u n ) ⊂ J −1 [a, b] and t n → ∞. The proof of Theorem 3.1 shows that it suffices to prove that {η(t, u n ) : 0 ≤ t ≤ t n , n ≥ 1} is bounded in W 1,2 B . Suppose that this set is not bounded. Then, by passing to a subsequence, we can find s n ∈ [0, t n ] such that Since J satisfies the P.S. condition, its critical points in Moreover, the P.S. condition implies that, for some δ ∈ (0, 1), J (u) W 1,2 Thus, if we denote v n = η(s n , u n ), then It follows that Since t n → ∞, we have, subject to a subsequence, either (i) s n → ∞, or (ii) t n − s n → ∞. In In case (ii), we define T n = min{R n − R 0 , t n − s n } and similarly derive again a contradiction. This finishes the proof. By elliptic regularity, u 0 ∈ X 0 . Now choose u n ∈ X i satisfying that Since η(t, u 0 ) ≡ u 0 , we can find an increasing sequence Since u n X i → ∞ and u n → u 0 in X k , for any fixed j, we can find n j ≥ j such that u n j X i e −w(t j ,u 0 ) > j and |w(t j , u 0 ) − w(t j , u n j )| < 1. Thus, u n j X i e −w(t j ,un j ) → ∞, and The following is a useful result.
Since B is precompact in X k , we can find t n → −∞ and some v 0 ∈ X k such that η(t n , v) → v 0 in X k . Using w(t n , v) → −∞, and passing to the limit in Since B is precompact and hence bounded in X k , by the properties of A, But v is an arbitrary point in B. Therefore, Now, by the properties of A, we can find C 2 such that sup u∈B A(u) X k−2 ≤ C 2 , and by (3.3), we deduce Clearly, Lemma 3.2 follows by repeatedly using the above argument. The proof is complete.
The following result follows from Theorems 3.1, 3.2 above and Corollary 4.7 and Theorem 4.8 in Chapter III of Rybakowski [26].   any a ≤ a 0 and b ≥ Proof. By standard argument, all the solutions of (1.1) contained in Thus both α(η) and ω(η) are non-empty. Since J is a quasi-potential of η, α(η) and ω(η) consist of equilibria of η, that is, solutions of (1.1). By our choice of a 0 and b 0 , α(η)∪ω(η) ⊂ J −1 (a 0 , b 0 ) and thus η( does not depend on a, b. Denote this set by K 0 and the proof for (i) is finished.
We may assume that the constant k in the definition of A is chosen large enough so that u → f (u) + ku is strictly increasing for u ∈ [min D u 0 (x) − 1, max D v 0 (x) + 1]. Then it follows from standard argument (see, e.g., [3]) that u n = A n u 0 and v n = A n v 0 satisfy: and u * , v * are the minimal and maximal solutions of (1. B . We show that, for all small δ > 0, any solution of (1.1) in N δ belongs to S 0 . In fact, if w is any solution of (1.1) Fix such a small δ > 0 and let K 1 be the maximal invariant set of η in is an arbitrary orbit in K 1 , then it is precompact in W 1,2 B and α(η) ∪ ω(η) ⊂ S 0 by what we have just proved. Let w ∈ α(η). By Lemma 3.2, η((−∞, ∞), u 0 ) is precompact in C 1 . We may assume that η(t n , u 0 ) → w in C 1 , where t n → −∞. Since w ∈ [u * , v * ] C 1 , for any fixed positive integer m, η(t n , is A-invariant, as in Hofer [21], it follows from a well-known result in Banach space ordinary differential for all t ∈ (−∞, ∞). Since this is true for any m and u m → . This shows in particular that K 1 = K 0 .
The above argument also shows that N δ ∩ J −1 [a 0 , b 0 ] is an isolating neighborhood of K 0 . Since it is strongly η-admissible by Theorem 3.2, we conclude that h(η, K 0 ) is well-defined.
Next we calculate h(η, K 0 ) by a homotopy argument. First we may assume that u * = 0 and f (0) = 0. Otherwise, we can make a change of variables w = u − u * and replace f bỹ from the very beginning. One easily sees that under this change of variables, the new f and new J satisfy all the conditions of the Lemmas and Theorems in this section so far. Therefore, this change of variables does not affect the previous results in this section.
We make a remark here for our later purpose. Since in generalf is x dependent even if f is not, we need to be a little careful here if we want to use this theorem to prove Proposition 1.1, where condition (H 3 ) is needed. We observe that the change of variables here does not produce trouble for the proof of Proposition 1.1, as we will use u 0 = α , v 0 = β and u * = α 1 is a constant in the proof of Proposition 1.1, where α , β and α 1 are defined in the proof of Proposition 2.1. Now go back to our calculation of h(η, K 0 ) under the assumptions that u * = 0 and f (0) = 0. We consider the following homotopy: Let J µ , η µ and A µ denote the corresponding J, η and A, where f is replaced by f µ . It is easy to check that u 0 < 0 and v 0 > 0 are strict lower and upper solutions of (3.4) for any fixed µ ∈ [0, 1] (here we use f (u 0 ) + ku 0 < 0 < f (v 0 ) + kv 0 ). Moreover, by standard regularity argument, the set B . Thus we can find real numbers a 1 < b 1 such that . It is easy to see that u * µ ≥ u * 0 = u * and v * µ ≤ v * 0 = v * for any µ ∈ [0, 1]. Now we use a similar argument to that used in showing that N δ ∩ J −1 [a 0 , b 0 ] is a strongly η-admissible isolating neighborhood of K 0 , and obtain that, for δ > 0 small, ] is a strongly η µ -admissible isolating neighborhood of K µ for each µ ∈ [0, 1], where K µ denotes the maximal invariant set of η µ in N , and N δ is the closed It is easily seen that if µ n → µ 0 with µ n , µ 0 ∈ [0, 1], then η n → η 0 . Moreover, a simple variant of the proof of Proposition 2.1 shows that N is {η µn }-admissible. Therefore, we can use the continuation property of the generalized Conley index to conclude that h(η µ , K µ ) is independent of µ ∈ [0, 1]. In particular, But clearly K 0 = {0} and u = 0 is a global minimum of J 0 . Thus, by Lemma 3.3, Remark 3.2. The homotopy (3.4), together with the change of variables trick, also works for the parabolic flow. In this case the argument is much more simpler because we can simply use [u 0 , v 0 ] X θ as the common isolating neighborhood. This observation can be used to simplify the arguments in a number of places in section 3 of [10].

Remark 3.3.
All the above results in this section also hold true for much more general problems than (1.1). For example, they are true for the cases indicated in Remark 2.1. Therefore, they should have many other applications.
Proof of Proposition 1.1. This is just a simple variant of that for Proposition 2.1. Therefore we just give an outline of the proof by pointing out the points where care is needed.
We make use of the previous results in this section. First, we observe that Theorem 2.2 is still true with π replaced by η. Then we follow the proof of Proposition 2.1 with π replaced by η. Note that (2.6) is now replaced by Theorem 3.3, and (2.10) by Theorem 2.2 with π replaced by η.
Second, we notice that the only other point we need to be careful with is in checking the Morse decompositions. Since by Lemma 3.2 any entire orbit is compact in C 1 , we see that any orbit approaches its alpha limit set in the norm of C 1 (actually C 0 is enough for our present purpose because we are dealing with Neumann boundary conditions). Moreover, as in the proof of
Then the conclusion follows from a simple variant of the proof of Lemma 3.1 in [2].
(ii) The same upper and lower solution argument as in [2] shows that (1.5) has a minimal positive solution u λ for 0 < λ < Λ + , and a maximal negative solution v λ for 0 < λ < Λ − .
(iii) This follows from a variant of the proof of Theorem 1 in [11] based on Hofer [21]. We just sketch the main steps. By truncating g outside [− v λ ∞ , u λ ∞ ], we may assume that J λ , defined by Note first that J λ is C 1 and the monotonicity condition for I − J λ in [21] can be easily met as λ|u| r−1 u is increasing with u and g is locally Lipschitz. Then as in [11], v λ and u λ are strict local minima of J λ on C. It then follows from the mountain pass theorem on C (see [21]) that J λ has a critical point w of type −I in C( see [21] for the definition of type −I critical points). Note that C 1 smoothness for J is enough here. Since u λ is the minimal positive solution and v λ is the maximal negative solution, w must be a sign-changing solution if we can show that w = 0. Since the righthand side of (1.5) is greater than au for all small positive u and some a > λ 2 , a variant of the argument in Step 2 of the proof of Theorem 1 in [11] shows that 0 is a critical point of type X ( see [21] for the definition of type X critical points). Hence w = 0.
The proof is complete.

Remark 4.1.
Note that in Theorem 4.1, there is no growth restriction on g from above. One easily sees that our method used to prove (iii) above can be used to answer affirmatively a question in Remark 2.9 of [1], where they conjecture that their growth restriction on g is not necessary for the existence of w (in their setting, one easily shows that J λ (w) < 0). (ii) for 0 < λ < Λ + (resp Λ − ), (1.5) has at least two positive (resp. negative) solutions; (iii) for 0 < λ ≤ max{Λ + , Λ − }, (1.5) has at least two sign-changing solutions.
(ii) Again we consider the case 0 < λ < Λ + only. The other case is similar. Choose λ ∈ (λ, Λ + ) and define u = u λ . Then u is an upper solution to (1.5). Let u * be the maximal solution of (1.5) between w λ and u. Then the proof of Proposition 2 in [12] shows that (1.5) has a mountain pass solution u > u * (some details are recalled in the proof of (iii) below).
(iii) We break the proof into three steps.
For λ = min{Λ + , Λ − }, the proof is the same as in (iii) Theorem 4.1, since we have proved in (i) that (1.5) has a minimal positive solution and a maximal negative solution. Suppose next that min{Λ + , We may assume that Λ − < Λ + ; the other case is similar. Now for Λ − < λ ≤ Λ + , (1.5) has a minimal positive solution u λ and no negative solution. Let C = {u ∈ W 1,2 0 (D) : u ≤ u λ }. We first use a trick in the proof of Proposition 2 in [12]. Denote f (u) = λ|u| r−1 u + g(u) and definẽ Thenf is continuous, and for any given finite interval in R 1 , we can find a positive constant k such thatf (x, u) + ku is strictly increasing in u for u in that interval and all x in D. Moreover, it is easily seen that w λ and u λ + εφ 1 is a pair of lower and upper solutions to −∆u =f (x, u), u| ∂D = 0.
Here ε > 0 is a constant and φ 1 is given by As in [12], it follows from [6] that the functional restricted to E = C 1 0 (D) has a local minimizer in the order interval [w λ , u λ + εφ 1 ] E , and it is also a local minimizer ofJ λ in H = W 1,2 0 (D). By the definition off , clearly u λ is the only critical point ofJ λ in the above interval. Therefore, u λ is an isolated critical point ofJ λ in H. One easily checks that J λ (−tφ 1 ) → −∞ as t → +∞. The argument in [12] shows thatJ λ satisfies the P.S. condition. Since u λ is an isolated critical point ofJ λ , as in [12], inf{J λ (u) : u − u λ H = ε} >J λ (u λ ). Thus we can use the well-known mountain pass theorem to obtain a critical point u 0 ofJ λ . It follows from the definition off that u 0 ∈ C, hence it is a solution to (1.5). Clearly u 0 = u λ . At this stage, we are not sure if u 0 = 0. If we can show that (1.5) has a solution in C \ {0, u λ }, then this solution must be a sign-changing solution as there is no negative solution and u λ is the minimal positive solution.
Next we use an idea of [21]. Arguing indirectly, we assume that 0 and u λ are the only solutions of (1.5) in C. Hence we necessarily have u 0 = 0 and thusJ λ (u λ ) <J λ (u 0 ) = 0. Using a suitable equivalent norm of H, we can assume that I −J λ is increasing in the order interval [−1, u λ + 1] H . From the behaviour of the nonlinearity near 0, as before, 0 is a critical point of J λ of type X. This would contradicts the well-known characterization of a mountain pass solution if our functional is smooth enough near 0. To get around this smoothness problem, we use the proof of Lemma 3 in [21] to find a continuous map γ : [0, 1] → H such that γ(0) = u λ ,J λ (γ(1)) <J λ (u λ ) andJ λ (γ(t)) <J λ (0) = 0 for all t ∈ [0, 1]. This implies that the mountain pass solution satisfiesJ λ (u 0 ) < 0, a contradiction. This finishes the proof of Step 1.
Step 2. Let 0 < λ ≤ min{Λ + , Λ − }. Then (1.5) has a sign-changing solution outside the order interval [v λ , u λ ] W 1,2 0 . We use some results and techniques from [12]. Since the righthand side of (1.5) is not C 1 at u = 0, we cannot use [12] directly. We modify |u| r−1 u near u = 0 as follows. For δ > 0 small, let Then consider −∆u = λh δ (u) + g(u), u| ∂D = 0. With care, it is easy to see that we can construct h δ as above and such that h δ (u) ≤ |u| r−1 u for u > 0 and the reverse inequality holds for u < 0. It follows that u λ is a strict upper solution of (4.1) and v λ is a strict lower solution of (4.1). Since δ is small, we can find ε 0 > 0 small such that εφ 1 ≤ u λ and εφ 1 is a lower solution of (4.1) for all ε ∈ (0, ε 0 ). It follows that (4.1) has a minimal positive solution u δ . Similarly, it has a maximal negative solution v δ . Now the modified problem (4.1) satisfies all the conditions of Proposition 2 in [12]. By Propositions 2, 2 , 2 and their proofs in [12], we conclude that (4.1) has a sign-changing solution w δ outside the order interval [v δ , u δ ]. We want to show that, for some δ n → 0, w δn converges to a signchanging solution of (1.5) outside [v λ , u λ ]; this would finish our proof of Step 2.
We show first that v δ → v λ and u δ → u λ as δ → 0. Since h δ 1 (u) ≥ h δ 2 (u) if u ≥ 0 and δ 1 ≤ δ 2 , an easy upper and lower solution argument shows that u δ 1 ≥ u δ 2 . By a simple regularity argument, {u δ : 0 < δ ≤ δ 0 } is precompact in C 1 (D) for any fixed small positive δ 0 . Thus for any sequence δ n → 0 there is a subsequence still denoted by δ n such that u δn → u in C 1 . One easily sees that u is a solution to (1.5). Moreover, u δn ≤ u ≤ u λ . Hence we must have u = u λ as u λ is the minimal positive solution of (1.5). This implies that u δ → u λ in C 1 as δ → 0. The proof that v δ → v λ is similar.
We show next that w δ ∞ ≤ M for all small δ and some fixed positive constant M . We argue indirectly. Suppose to the contrary that there exists a sequence of positive numbers δ n → 0 such that w δn ∞ → ∞. Then let w n = w δn / w δn ∞ , and divide (4.1) with (δ, u) = (δ n , w δn ) by w δn ∞ , and use lim |u|→∞ g(u)/u = a and |h δn (w δn )| ≤ |w δn | r . It results From the L ∞ boundedness of the righthand side of (4.2) we infer that {w n } is precompact in C 1 . By passing to a subsequence we may assume that w n → w in C 1 . Now let n → ∞ in (4.2) we obtain −∆w = aw, w| ∂D = 0.
Thus we have proved that w δ ∞ ≤ M for all δ ∈ (0, δ 0 ], where δ 0 > 0 is small. Using this fact, we find that the righthand side of (4.1) with u = w δ is L ∞ bounded uniformly for all small positive δ. Hence it follows from elliptic regularity that {w δ : 0 < δ ≤ δ 0 } is precompact in C 1 . Thus we can find a sequence of positive numbers δ n → 0 such that w δn → w in C 1 . One easily sees that w is a solution of (1.5). Since w δn is outside [v δn , u δn ] C 1 and v δn → v λ , u δn → u λ in C 1 , it follows that w is not in the interior of [v λ , u λ ] C 1 considered as a subset of C 1 (D).
We prove that w changes sign. Otherwise w is a positive solution or negative solution. If it is a positive solution, then, by the maximum principle, we deduce that w > 0 in D and ∂w/∂n < 0 on ∂D. This implies that w δn > 0 in D for all large n, which contradicts the fact that w δn changes sign. Similarly w cannot be negative. Hence w must change sign.
It remains to show that w is outside [v λ , u λ ] W 1,2 0 . It suffices to show that w is outside [v λ , u λ ] C 1 . If this is not true, then choosing constant k > 0 such that λ|u| r−1 u + g(u) + ku is increasing for u in the range − v λ ∞ ≤ u ≤ u λ ∞ , we obtain that It follows then from the maximum principle that u λ − w > 0 in D and ∂(u λ − w)/∂n < 0 on ∂D. The same is true also for w − v λ . Thus w is in the interior of [v λ , u λ ] C 1 , which contradicts our earlier observation. This finishes the proof of Step 2.
The proof of Step 3 is a variant of that of Step 2. We will just sketch the proof. Again we consider the modified problem (4.1) first and then pass to the limit δ → 0. As before, for all small positive ε and any λ ∈ (Λ − , Λ + ], εφ 1 and u λ is a pair of strict lower and upper solutions to (4.1), and it follows that (4.1) has a minimal positive solution u δ . Using the Lemma in the proof of Proposition 2 of [12], we can find a strict upper solutionũ δ of (4.1) satisfying u δ <ũ δ ≤ u λ such that ω(ũ δ ) = u * δ is a positive solution of (4.1) which is comparable with any positive solution of (4.1). Here ω(v) denotes the omega limit set of the solution of the corresponding parabolic problem of (4.1) which passes through v. Now using Proposition 2 of [12] to (4.1) with u = εφ 1 and u =ũ δ we conclude that (4.1) has a solution w δ not comparable with at least one of u δ = ω(εφ 1 ) and u * δ = ω(ũ δ ). It follows that w δ must be a sign-changing solution as any positive solution of (4.1) is comparable with both u * δ and the minimal positive solution u δ . Now as in the proof of Step 2, we can show by a compactness argument that for some δ n → 0, w δn → w in C 1 , and w is a solution of (1.5). Since w δn is not comparable with u δn which was shown in Step 2 to converge to u λ in C 1 , we conclude that w = 0. Since w δn changes sign, we conclude that w = u λ . Now as in the last part of the proof of Step 2, we find that if w and u λ are comparable, then either w − u λ or u λ − w is in the interior of the natural positive cone of C 1 0 (D). But since w δn → w, u δn → u λ and u * δn → u λ in C 1 , it follows that either both w δn − u δn and w δn − u * δn , or both u δn − w δn and u * δn −w δn are in the interior of the positive cone in C 1 0 if w is comparable with u λ . But this contradicts the fact that w δn is not comparable with at least one of u δn and u * δn . This shows that w is a sign-changing solution which is not comparable with u λ , as required. The case Λ − > Λ + can be considered in a similar way.
The proof of Theorem 4.2 is complete.

Remark 4.2.
Note that our proof of (iii) gives information on the location of the two sign-changing solutions, i.e., one is inside the order interval [v λ , u λ ], another is outside this interval. Here we use the convention for the notation of this order interval as in the statement of Step 1. Note also that the argument used in the proof of Step 3 also works for the case of Step 2 but it gives less information on the location of the second sign-changing solution in this case. If we assume further that g satisfies (H 6 ) (which implies (H 1 ) and (H 2 )), then (iv) for any 0 < λ < max{Λ + , Λ − }, (1.5) has at least two sign-changing solutions.
(ii) and (iii) These follow from simple variations of the proofs of (ii) and Step 1 in (iii) of Theorem 4.2. Note that (H 1 ) and (H 2 ) guarantee that we still have the P.S. condition and the mountain pass theorem applies as before.
(iv) We may assume that Λ − < Λ + . The proofs for the other cases are similar or simpler. By (iii) and its proof we need only show that (1.5) has a sign-changing solution not comparable with u λ for any 0 < λ < Λ ≡ max{Λ + , Λ − }. We use a degree argument.
Then, as in [11], A = KG is completely continuous as a mapping of both H = W 1,2 0 (D) to itself and E = C 1 0 (D) to itself. Moreover, A is strongly increasing in [u λ − u λ , u λ − u λ ] E under the order induced by the natural positive cone P in E, and maps both P and −P to themselves. Finally, u is a solution of (1.5) if and only if u − u λ is a fixed point of A. Therefore, it suffices to show that A has a sign-changing fixed point.
As in [11], by using a priori estimates from [14] or [18] for positive solutions of the corresponding elliptic equation of Au = u, we obtain via a simple homotopy that Since u λ is a strict upper solution of (1.5), A maps the order interval [0, u λ − u λ ] E to its relative interior in P , which we denote by C + . Since C + is a bounded convex set in P , it follows from the basic properties of the degree (see e.g., [3] and [23]) that Hence, by the additivity property of the degree, Let C − denote the relative interior of [u λ − u λ , 0] E in −P , and enlarge R when necessary, we obtain similarly Then A maps C into C and we obtain as before that Notice that now the degree is on the whole space E.
Suppose that (1.5) has only one sign-changing solution obtained in (iii) . Then necessarily, A has no sign-changing fixed point, and thus we can classify all the solutions of Au = u into three types: (a) Solutions contained in C; (b) Positive solutions outside C -they form a compact set K + contained in P ∩B R \C + ; (c) Negative solutions outside C -they form a compact set K − contained in −P ∩ B R \ C − . By the maximum principle, and compactness of K + , one easily deduces that K + lies in the interior of P . Similarly, K − is in the interior of −P . Hence we can find small neighborhoods N + and N − of K + and K − in E respectively such that they are in the interior of P and −P respectively. Now by the excision property of the degree, Since N + and N − are in the interior of P and −P respectively, we also have, by the properties of the degree, that Finally, as in [11] section 3, we use the assumptions (H 1 ), (H 2 ), the fact that the solution set of (1.5) is bounded (since we assume that there is only one sign-changing solution), and obtain where M > 0 is large enough such that B M contains all the three types of solutions listed above. By the additivity of the degree, we obtain This contradiction finishes our proof.

Remark 4.3.
The result of Theorem 4.3 (iv) seems difficult to obtain by Morse theory or our generalized Conley index approach. The difficulty comes from the fact that |u| r−1 u is not locally Lipschitz near 0. Note that the modification trick used in the proof of Theorem 4.2 is difficult to use for the superlinear case because a priori bounds for sign-changing solutions of the modified equations are at least difficult to obtain (if obtainable at all) and such bounds are essential for the limiting process to work.

Remark 4.4.
From the proof, it is clear that condition (H 6 ) can be replaced by any other conditions which guarantee that the positive and negative solutions of (1.5) are a priori bounded.

Remark 4.5.
It is possible to show that the conclusion of (iv) in Theorem 4.3 is also true for the case that λ = max{Λ + , Λ − }. We did not include this case in Theorem 4.3 because its proof is rather different from that of Theorem 4.3 above and is very technical. Remark 4.6. Under the condition (H 4 ), if 0 is an isolated solution of (1.5), then it can be shown that the critical groups of 0 as a critical point of the corresponding functional are all trivial. It can also be proved that the fixed point index of 0 for the corresponding abstract operator is zero. Note that this latter conclusion does not follow directly from the well-known relation between the fixed point index and the critical groups because the right side of (1.5) is not smooth enough. Note also that it follows easily from this fixed point index result that 0 is never isolated if g is an odd function. This implies a partial answer to a recent question of Bartsch

Appendix
In this section, we prove Lemmas 2.1 and 2.2.
Since such M and δ are independent of u and T 1 , T 2 , our proof of Lemma 2.2 is complete.