A NONDEGENERACY CONDITION FOR A SEMILINEAR ELLIPTIC SYSTEM AND THE EXISTENCE OF 1- BUMP SOLUTIONS

. Combining situations originally considered in [7] - [8], a semilinear elliptic system is treated and a nondegeneracy condition leading to the existence of multibump solutions is considerably weakened.


1.
Introduction. Using variational methods, many papers such as [3], [6], [7], [9], [11], [16], [17], [23], [24], have studied the existence of homoclinic solutions for Hamiltonian systems. There are related studies for semilinear elliptic PDEs where one has solutions that decay to 0 as |x| → ∞. See e.g. [1], [3], [8], [12], [15], [22]. A common feature of these papers is that they begin by first determining what we will call basic solutions by means of arguments of "mountain pass theorem" type. These basic solutions decay to 0 as the temporal or spatial variables approach infinity. Thus these solutions are not near 0 only on a compact subset of R or R n and hence are often referred to as 1− bump solutions. Further variational methods are then used to find additional families of solutions, so-called multibump solutions, provided that the set of 1−bump solutions is not too degenerate. Indeed nondegeneracy conditions play an important role in the classical approaches to homoclinic solutions via the methods of analysis, generally in the form of a condition requiring the stable and unstable manifold at the corresponding homoclinic point to intersect transversally [10]. In the variational setting, the first nondegeneracy condition used was that the set of basic solutions is finite modulo a discrete set of phase shifts [23].
In this paper, our goal is to show how a much more general nondegeneracy condition can be employed to get the 1−bump solutions. This condition also leads to the existence of multibump solutions. The lengthy details of this latter application will appear elsewhere. Nondegeneracy conditions related to the one employed here have also been used in settings where the basic solutions are heteroclinics and where minimizers rather than mountain pass solutions are involved. See e.g. [3], [15] - [17].
The model case that will be studied here is a variant of that treated by [8]. Thus consider the semilinear elliptic system where L is an m by m matrix and F u denotes the gradient with respect to the variable u of a positive nonlinearity, F , which is 1− periodic in the components of the variable x = (x 1 , · · · , x n ) ∈ R n and superquadratic with respect to the variable u ∈ R m . More precisely suppose that L and F satisfy the following conditions: (L): L ∈ C 1 (R n , R m × R m ), is 1− periodic in x j for j = 1, . . . , n, and is positive definite for each x ∈ [0, 1] n , (F 1): F ∈ C 2 (R n × R m , R) and F (x, u) is 1− periodic in x j for j = 1, . . . , n, (F 2): there exists a constant, x ∈ R n . This system was studied in [19] when n = 1 (without (F 2) − (F 3)). It was shown that (PDE) then possess at least one classical solution, u, and u ∈ W 1,2 (R, R m ). Therefore u → 0 as x 1 → ∞. The solution, u, is obtained as a limit of k−periodic solutions as k → ∞. These subharmonic solutions are found via an application of the Mountain Pass Theorem.
To use variational arguments to obtain multibump solutions of (PDE) in this setting, it is necessary to have 1−bump solutions with some further properties. Unfortunately this is not the case for the simpler situation just described. However such solutions were obtained when n = 1 in [7] and when m = 1 in [8] provided that (F 1) − (F 4) hold and a nondegeneracy condition, (N D1), is satisfied. To introduce (N D1), for u ∈ E ≡ W 1,2 (R n , R m ), let Arguing as in [8], J ∈ C 1 (E, R). Solutions of (PDE) will be obtained as critical points of J on E. As was shown in [8], see also Section 2 here, J satisfies the geometrical hypotheses of the Mountain Pass Theorem. Define Then condition (N D1) is (N D1): There is an α > 0 such that K c+α /Z n is finite. Such a condition was introduced by Séré in [23] and used also in [7], [11], [1] for related classes of problems. Observe that condition (N D1) (as well as the other conditions mentioned below) fails to hold if F is independent of x. Subsequent to [23], milder conditions than (N D1) were introduced. For example, starting with [24], there are papers such as [6], [12], [13] that assume (N D2): There is an α > 0 such that K c+α is countable. Condition (N D2) has been further generalized leading to very weak global conditions sufficient to obtain chaotic dynamics for some classes of ordinary differential equations. An example of such a mild global condition is requiring the stable and unstable manifolds not to coincide for a class of Lagrangian equations. This was done in [2] for periodic perturbations of a nonlinear autonomous pendulum equation and in [14] for a periodic Duffing equation. A generalization of these results to Hamiltonian systems was given first in [4] in an analytic setting. A C ∞ case was treated in [5] where chaotic behavior was obtained whenever the set of homoclinic solutions has compact components with respect to the uniform metric on R. A related condition involving disconnectedness of the set of minimal homoclinic solutions was used in [21] (see also [20]) for singular Lagrangian systems on R 2 . Similar disconnectedness assumptions on the set of minimal solutions were also used for other settings for partial differential equations in [15] and [3]. For some Hamiltonian systems of double well potential type, related assumptions were made for a situation where minimal solutions were not involved in [16] and [17].
Next our new nondegeneracy condition, (N D3) will be introduced. It is related to the one employed in [16] and [17] and is also the analogue for the current setting of the conditions of mountain pass type that have been used in [15] and [3]. Let the family of restrictions to T 0 of critical points of J in D such that J(U ) ≤ d. As will be shown in section 2, S d = S d ∪ {0} is a compact metric space under the metric obtained from · W 1,2 (T0,R m ) and if C d (0) denotes the component containing 0 in S d and R d = {u ∈ S d | u W 1,2 (T0,R m ) ≥ 2ρ}, either A major consequence of Theorem 1.1 is that it serves as a stepping stone to constructing more complex multibump solutions of (PDE). Carrying out this construction is a rather lengthy and technical process and will be carried out in a sequel to this paper. However as an illustration, in section 4, we will state a precise such result. For now, we simply note that roughly stated, such multibump existence theorems say that either (PDE) is degenerate in having a very large number of solutions near the collection of 1-bump solutions given by Theorem 1.1 or for any k ∈ N, there is a solution of (PDE) near the sum of k sufficiently separated 1−bump solutions.
In section two, to properly formulate the variational problem for (PDE), some preliminary results from [8] and [16] will be recalled. In addition, the properties of 6998 PIERO MONTECCHIARI AND PAUL H. RABINOWITZ (N D3) will be developed. Then in section 3, the existence of the 1−bump solutions will be proved with the aid of an existence theorem from [16]. Lastly in section 4, as was mentioned above, a multibump result will be described.
2. The variational setting for (PDE) and the new nondegeneracy condition (ND3). This section consists of two types of results. The first type includes the presentation of some properties of (PDE) and its variational formulation. It also includes some properties of Palais -Smale sequences. Many of these results are well known in simpler contexts like that of [8]. Therefore when their proofs are small modifications of those in [8], we omit them here. The second class of results involve the formation of the new nondegeneracy condition, (ND3) and the development of its properties.
To begin, by (F 1), (F 3), We seek solutions of (PDE) which belong to E = W 1,2 (R n , R m ) as critical points of the functional J. As norm in E, we can take Now for u ∈ E, J can be written more simply as For what follows, for given Ω ⊂ R n and u, v ∈ E, set As was noted above, The functional J satisfies the geometrical hypotheses of the Mountain Pass Theorem: With H and c as in the Introduction, by (2.5), c ≥ 1 4 ρ 2 and by (2.6) and the argument of the Mountain Pass Theorem (see e.g. [18]), there exists a Palais-Smale, or (PS) for short, sequence for J at level c. Note that by (F 4), In particular, by (2.7), (PS) sequences for J are bounded in E and so converge weakly up to subsequences. Moreover, by (2.7), Hence J has no negative critical values. Note also that by Some properties of the (PS) sequences of J will be recalled next. Set We claim there exists aρ To verify (2.10), observe that there exists a constant, κ > 0 such that u L p+1 (T k ) ≤ κ u T k for all u ∈ E, k ∈ Z n . For future reference, κ will be used as the common constant of the embeddings W 1,2 (Ω) → L s+1 (Ω) when Ω is a measurable set of R n which satisfies the cone property with respect to e.g. the right-spherical cone 4 , let C 1/4 be given by (2.2). Then, for any u, v ∈ E, k ∈ Z n , Condition (2.11) applies to the critical points of J. Setting An important property of (PS) sequences -see e.g. [8] -is that any weak limit u of a (PS) sequence (u q ) is a critical point of J and moreover the new sequence u q − u is a new (PS) sequence. More explicitly: 7000 PIERO MONTECCHIARI AND PAUL H. RABINOWITZ Proposition 2.14. If u q → u weakly in E and is such that Another useful property of (PS) sequences is: Setting v q = u q −u, by Proposition 2.14, v q is a (PS) sequence for J and it weakly converges to 0 in E. Note that As in [8], using (2.12), (2.13) and Propositions 2.14 and 2.15 gives more precise information on (P S) sequences: . . , U β ∈ D ∩ {J ≤ b}, a subsequence of (u q ), again denoted by (u q ), and sequences (ξ 1 q ), . . . , (ξ β q ) ∈ Z n such that, as q → ∞, |ξ j q | → ∞ for any j ∈ {1, . . . , β} and Remark 2.21. When (2.20) holds, we say u q converges to the formal (β + 1)-chain, U 0 , · · · , U β .
These results lead to: Suppose there exists an R > 0 independent of q such that whenever k ∈ Z n with max 1≤i≤n |k i | ≥ R. Then there is U 0 ∈ {0} ∪ D ∩ {J ≤ b} such that, up to a subsequence, u q → U 0 in E as q → ∞.
Next in preparation to formulating the new nondegeneracy condition, let S d be as in the Introduction. An important property of S d follows from a unique continuation property for (PDE) (see [22]) : the map u ∈ S d → U ∈ D d is invertible. (2.25) Due to the precompactness of (PS) sequences in W 1,2 loc (R n , R m ), the set S d possesses the following property. Letting (e 1 , . . . , e n ) denote the canonical orthonormal base of R n , for ∈ {1, . . . , n}, consider the shift map g : S d → S d , g (U | T0 ) = U (· + e )| T0 and more generally g k ≡ g k1 1 • . . . • g kn n for k = (k 1 , . . . , k n ) ∈ Z n Then g k is a homeomorphism on S d and g k (0) = 0 for any k ∈ Z n . For u ∈ S d , (2.27) Moreover by (2.12), for any u ∈ S d , there exists q(u) = (q 1 (u), · · · , q n (u)) ∈ Z n such that The set R d is a compact subset of S d , not containing 0. Moreover whenever u ∈ S d , R d , any "trajectory" {g k (u) | k ∈ Z n } intersects R d via (2.28): Proof. Note that for any k ∈ Z n , since g k is a homeomorphism on S d , g k (C d (0)) is compact, connected and contains 0. Hence In particular Then (2.31) follows by (2.29) and (2.32).
By Proposition 2.30, either case (2 o ) occurs and C d (0) is a "large" continuum which intersects R d , or C d (0) reduces to the singleton {0} in the case (1 o ). The above alternative can be viewed as the analogue of the All or Nothing Lemma used in [15] and [3], and generalized in [16]. Indeed, as in [16], in the present work we use the Lemma at the mountain pass critical level, c. In both [15] and [3] the use of the Lemma is made at a minimization level of the functional where the situation simplifies considerably due to related compactness properties.
Recall our new nondegeneracy condition is Our goal is to show that when (N D3) is satisfied for some d > c, then Theorem 1.1 holds, i.e. J has a critical value arbitrarily close to c even though c may not be a critical value. Before proving this result in §3, some implications of (N D3) will be developed next. In particular it will be shown that it leads to some important separation properties of the set S d and related rather technical concentration properties of the elements of D d that will be needed in the proof of Theorem 1.1 in section 3. Suppose that (N D3) holds. By classical topological separation theorems, see e.g. [25], S d can then be divided into the disjoint union of two compact sets K 1 and K 2 , one containing 0 and the other R d . More precisely The decomposition is not unique and it can be assumed that iv) u W 1,2 (T0,R m ) ≤ρ/2 for any u ∈ K 1 .
For any U ∈ D d and k ∈ Z n , the function g k (U | T0 ) is either in K 2 or K 1 . It is useful to quantify this behaviour. If U ∈ D d and ∈ {1, . . . n}, U {x∈R n ||x |>j} → 0 as j → +∞. (2.33) Then, since by (v) inf u∈K2 u W 1,2 (T0,R m ) ≥ 5r 0 , it follows that Remark 2.35. Suppose n = 1 so (PDE) is a Hamiltonian system of ordinary differential equations. Let Ψ s (0) and Ψ u (0) denote respectively the stable and unstable manifolds of (PDE) at 0. Then from a dynamical systems point of view, k s (U ) can be interpreted roughly as the smallest value of x 1 above which the homoclinic orbit, U , of (PDE) remains in the portion of Ψ s (0) associated with K 1 . Likewise k u (U ) can be interpreted roughly as the largest value of x 1 below which U remains in the portion of Ψ u (0) associated with K 1 .
Remark 3.1. It is natural to ask whether c is a critical value of J. Since J does not verify the (PS) condition, the answer is unclear. However, if c is not a critical value of J, Theorem 1.1 guarantees the existence of an infinitude of distinct solutions of (PDE). Indeed taking a sequence ε i → 0 as i → ∞, by Theorem 1.1, there is a corresponding sequence of solutions, V i , of (PDE) with J(V i ) ∈ (c, c + ε i ). Applying Proposition 2.19 to the (PS) sequence, (V i ), shows that along a subsequence, it converges to a formal (β + 1)-chain, (U 0 , · · · , U β ). Since c is not a critical value, β = 0; otherwise by (2.20) J(V i ) → c = J(U 0 ). Hence β > 0 and each of the functions, U q , q = 0, · · · , β, are solutions of (P DE) with Σ β 0 J(U q ) = c so 0 < J(U q ) < c for each q.
The next Corollary shows that the stronger assumption (ND1) made in [8] always implies that c is a critical value for J. Proof. If (ND1) holds, so does 1 o of (2.31) with d = c + α. If (i) of Remark 3.1 fails, by (ii) of Remark 3.1, J has infinitely many critical values in (c, c + α) contrary to (ND1). Theorem 1.1 will be proved with the aid of the following abstract result from [16]. It is a variant of the Mountain Pass Theorem that is suitable for some classes of functionals for which the underlying Hilbert space involves an unbounded spatial or temporal domain. : (J 3 ) There are constants, b * > b, ν > 0, r * > 0 and a sequence, is a sequence in A j with J(u k ) bounded and J (u k ) → 0, then u k has a convergent subsequence in A j . Then for any ε > 0, J possesses a critical value, b ε ∈ [b, b + ε) and a critical point, u ε with J(u ε ) = b ε . Moreover u ε is not a local minimum of J.
Remark 3.4. For each ε, u ε lies in one of the sets, A j , but the set may vary with ε.
It remains to verify (J 3 ). Towards that end, let d > c and k − , k + ∈ Z n with k − ≺ k + . Define Remark 3.5. Note that if U ∈ A d k − ,k + , property (v) of K 1 and K 2 , (A2) and (A3) imply , . . . , n}. Indeed, suppose that for some pair i, ∈ {1, . . . , n} we have k −, But then the triangle inequality shows K 1 − K 2 T0 ≤ 2r 0 contrary to property (v). The remaining cases of k −, Remark 3.6. If U ∈ A d k − ,k + for some k − ≺ k + ∈ Z n , then U ≥ 4r 0 . To see this note by property (v) again, inf u∈K2 u T0 ≥ 5r 0 and by (A3) and the triangular inequality, Observe that by Proposition 2.39, if U ∈ D d , then U ∈ A d k u (U ),k s (U ) . Hence, setting

PIERO MONTECCHIARI AND PAUL H. RABINOWITZ
With these observations, the main properties of A d can be given. In what follows, B r (z) always denotes an open ball of radius r about z. It will be clear from the context whether the underlying space is E or R n . Proposition 3.7. A d possesses the following properties: Before proving Proposition 3.7, the proof of Theorem 1.1 can be completed by using Proposition 3.7 to verify (J 3 ). Indeed take b * = d, ν = ε 0 , r * = r 0 /3, A = A d , and after an isomorphism between Z and Z n × Z n , A j = A d k − ,k + . Then by (P 1) − (P 3), J satisfies (J 3 ) and Theorem 1.1 is proved.
Lastly we give the Proof of Proposition 3.7. To verify (P1), let . . , n}. Thus property (iv) of K 1 gives u T0 ≤ r 0 +ρ/2 <ρ when k ≥ k + or k ≤ k − for an ∈ {1, . . . , n}. By Proposition 2.22, there is a U ∈ D d such that U p − U → 0 along a subsequence (still denoted by U p ). Thus J(U ) ≤ d and (A1) holds for U . To show that U ∈ A d k − ,k + , it remains to prove that U satisfies (A2) and (A3). Property (A2) is immediate since it is satisfied by any function U p and U p − U → 0 as p → ∞. For property (A3), observe that via Proposition 2.39, for any fixed p ∈ N, U p satisfies (A3) for n pairs of members of Z n ; k −, ,p , k +, ,p ∈ Z n . By Remark 1.1, , k +, ,p i ≤ k + i for any i, ∈ {1, . . . , n} and p ∈ N. Thus passing to a further subsequence, it can be assumed that all the elements, U p , satisfy (A3), with respect to some fixed n pairs of indices, say k −, , k +, ∈ Z n , 1 ≤ ≤ n, independently of p. Hence, by the strong convergence, U satisfies (A3) with respect to k −, , k +, (1 ≤ ≤ n) and (P1) follows.
In either of the above cases, J has infinitely many distinct critical points.
Thus Theorem 4.1 tells us that either the set of critical values of J nearc is highly degenerate in the sense of (i) or infinitely many multibump solutions of (PDE) as in (ii) can be constructed.
We suspect that Theorem 4.1 is not an optimal result. Note thatL depends on k, the number of "bumps". A better result would be to find solutions with L independent of k. Indeed some such results are known and can be found in dynamical systems settings in [24], [9], [11] and [6]. The only case we know of involving PDEs is [12]. The existence of multibump solutions of (PDE) will be proved in a future paper. Proofs of such results are lengthy and technical and involve the construction of appropriate pseudogradient vector fields and deformation mappings. We have a proof of Theorem 4.1 as stated but hope to obtain the stronger k−independent result just mentioned.