Multiplicity Results for a Quasilinear Elliptic

System via Morse Theory

SILVIA CINGOLANIDipartimento di Matematica, Politecnico di Bari

Via Amendola 126/B, 70126 Bari, Italycingolan@poliba.it

MONICA LAZZODipartimento di Matematica, Universita degli Studi di Bari

Via Orabona 4, 70125 Bari, Italylazzo@dm.uniba.it

GIUSEPPINA VANNELLADipartimento di Matematica, Politecnico di Bari

Via Amendola 126/B, 70126 Bari, Italyvannella@poliba.it

Abstract

In this work we prove some multiplicity results for solutions of a system of ellipticquasilinear equations, involving the p-Laplace operator (p > 2). The proofs are basedon variational and topological arguments and make use of new perturbation results inMorse theory for the Banach space W 1,p

0 .

MSC 2000: 58E05, 35B20, 35J50

1 Introduction

Let us consider a system of linear Schrodinger equations

i~∂Ψ∂t

= − ~2

2m∆Ψ, (1)

where ~ is the Planck constant, Ψ : Ω → Rn+1, Ω is a domain of Rn, and n ≥ 3. It isknown that the linear Schrodinger equation is dispersive in nature, so that a wave packetdisperses in a short time. However, if a nonlinear term is added in (1), localized finiteenergy wave packets (the so-called solitons) may exist.

1

In this work, we investigate the existence of solitary wave solutions Ψ(x, t) = u(x)eiωt

(ω ∈ R) of the perturbed system of differential equations

i~∂Ψ∂t

= − ~2

2m∆Ψ + Mε(Ψ), (2)

where ε is a positive parameter and Mε(ψ) is the following nonlinear differential term:

Mε

(u(x)eiωt

)=

(−εp−2∆pu +

1ε2

W ′(u))

eiωt.

We are thus led to study the quasilinear elliptic problem −∆u + ωu− εp−2∆pu + ε−2W ′(u) = 0 in Ω

u = 0 on ∂Ω

(for simplicity, we set 2m = ~ = 1). Under the ansatz ω = ε−2, the problem abovebecomes

(Pε) −εp∆pu− ε2∆u + V ′ (u) = 0 in Ω

u = 0 on ∂Ω

where V ′(u) = u + W ′(u).Throughout the paper, we assume that Ω is a bounded domain of Rn, with smooth

boundary ∂Ω, and p > n ≥ 3. On the potential V we make the following hypotheses:

(V0) V ∈ C2(Rn+1 \ ξ,R), with ξ 6= 0;

(V1) V (ξ) ≥ V (0) = 0 for any ξ ∈ Rn+1 \ ξ;(V2) there exist c, ρ > 0 such that V (ξ + ξ) ≥ c |ξ|np/(n−p) for any |ξ| < ρ;

(V3) the Hessian matrix V ′′(0) is nondegenerate.

Due to the singularity of the potential V , finite energy solutions of (Pε) take value inRn+1 \ ξ, a topologically nontrivial space. This fact allows a classification of solutions,by means of the topological charge (see Section 2 and [1, 2], where the charge was firstintroduced). The existence of one solution of (Pε), with topological charge q, can beproved by a simple variational argument, for any positive ε and for any integer q (seeTheorem 2.1 below).

Let us notice that, under the change of variable x 7→ εx, (Pε) becomes −∆pu−∆u + V ′ (u) = 0 in Ωε

u = 0 on ∂Ωε

where Ωε =x ∈ Rn : εx ∈ Ω

. If Ω contains the origin, as we assume for simplicity,

then Ωε approaches Rn as ε → 0, and we obtain the limit problem

(L) −∆pu−∆u + V ′(u) = 0 in Rn

u → 0 as |x| → ∞.

2

As proved in [2], (L) has a weak solution U , corresponding to an energy ground state, withnonzero topological charge q∗ (see Section 3 for some details). The precise evaluation ofq∗ is still an open problem.

In this paper we prove that (Pε) has multiple solutions with topological charge q∗, forsmall values of the parameter ε. From a physical viewpoint, the corresponding solutionsof (2) can be interpreted as solitary waves with high frequencies.

It was proved by Bellazzini [3] that the number of solutions to (Pε), with topologicalcharge q∗, is at least cat(Ω) + 1, provided ε is sufficiently small (cat(Ω) is the Ljusternik–Schnirelman category of Ω). The results in [3] are based on Ljusternik–Schnirelman theory.

We aim to apply Morse theory to (Pε), in order to obtain better information aboutthe number of solutions, in the spirit of a celebrated paper by Benci and Cerami [4],in which a semilinear elliptic problem is studied. The relations between the topologicalproperties of the domain and the multiplicity of solutions to semilinear elliptic problemshave been extensively investigated, during the past years. On the contrary, as far as weknow, applications of Morse theory to estimating the number of solutions for quasilinearequations, involving the p-Laplace operator (p > 2), are not available in the literature.

From the variational point of view, solutions of (Pε) correspond to critical points ofthe energy functional fε : Λ(Ω) −→ R defined by

fε(u) =∫

Ω

(εp

p|∇u|p +

ε2

2|∇u|2 + V (u)

)dx,

whereΛ(Ω) =

u ∈ W 1,p

0 (Ω,Rn+1) : u(x) 6= ξ for any x ∈ Ω.

The global Morse relations, which apply to fε, and a topological argument (Proposi-tion 4.1) yield our first multiplicity result (Theorem 1.3).

Let us recall some basic definitions. Following common practice, for any k ∈ N wedenote by Hk(A,B) the k-th relative homology group of the topological pair (A,B), withcoefficient in some field K; we also set Hk(A) = Hk(A, ∅).

Definition 1.1 Let X be a Banach space, f ∈ C1(X,R), u ∈ X an isolated critical pointof f , and c = f(u).For k = 0, 1, 2, . . . , the k-th critical group of f in u is

Ck(f, u) = Hk(f c, f c \ u).The Morse polynomial of f in u is

Pt(f, u) =∞∑

k=0

dimCk(f, u) tk.

The multiplicity of u is the integer P1(f, u).

3

Definition 1.2 For any A ⊂ Rn, the Poincare polynomial of A is

Pt(A) =∞∑

k=0

dimHk(A) tk.

We are ready to state our first result:

Theorem 1.3 Assume (V0)–(V3). There exists ε0 > 0 such that, for any ε ∈ (0, ε0 ),(Pε) has at least P1(Ω) solutions, counted with their multiplicity, with topological chargeq∗.

In order to obtain a more meaningful multiplicity result, we need a deeper insight intothe notion of multiplicity for critical points of fε. In other words, we need to correlatetopological objects, like the critical groups of fε in each critical point, with a Hessian–typenotion, namely the Morse index:

Definition 1.4 Let X be a Banach space and f ∈ C2(X,R). Let u ∈ X be a criticalpoint of f . The Morse index of f in u is the supremum of the dimensions of the subspacesof X on which f ′′(u) is negative definite; it is denoted by m(f, u). Moreover, the largeMorse index of f in u is the sum of m(f, u) and the dimension of the kernel of f ′′(u); itis denoted by m∗(f, u).

In developing a local Morse theory for fε, we are faced with many difficulties, due tothe lack of a Hilbert structure in the Banach space W 1,p

0 (Ω,Rn+1). We refer to [5, 6] foran extensive discussion on the problems that arise in studying the quantitative aspectsof a Morse theory in a Banach space. Here, we limit ourselves to underline the followingfeatures (all of which occur in the variational framework associated with (Pε)):

• The Morse Splitting Lemma does not hold.

• If X is a Banach space and f : X → R is a C2 functional, giving a reasonabledefinition of nondegenerate critical point is not a trivial issue. In fact, it makes nosense to require that the second derivative of f in a critical point is invertible, since,in general, a Banach space is not isomorphic to its dual space.

• If X is a Banach space and f : X → R is a C2 functional, the existence of a nondegen-erate critical point u ∈ X of f with finite Morse index, which is the most interestingcase in Morse theory, implies the existence of an equivalent Hilbert structure (see [5]for details).

• The second derivative of f in a critical point is not even a Fredholm operator, sothat Gromoll and Meyer–type extensions of the Morse Lemma do not apply either.

4

Recently, Cingolani and Vannella introduced in [6, 7] a new definition of nondegeneratecritical point for functionals associated to a class of quasilinear equations, and found aconnection between the critical groups and the Morse index in a nondegenerate criticalpoint. The results in [6] were generalized by Carmona, Cingolani and Vannella [8] tofunctionals associated to quasilinear elliptic systems.

In the setting of (Pε), the above-mentioned definition of nondegenerate critical pointreads as follows:

Definition 1.5 A critical point u of fε is nondegenerate if the map

f ′′ε (u) : W 1,p0 (Ω,Rn+1) −→ W−1,p′(Ω,Rn+1)

is injective.

In a nondegenerate critical point, we are able to compute all the critical groups of fε:

Theorem 1.6 Assume (V0)–(V2) and fix ε > 0. Let u be a nondegenerate critical pointof fε. Then m(fε, u) is finite and

Ck(fε, u) ' K if k = m(fε, u),Ck(fε, u) = 0 if k 6= m(fε, u).

In the case in which fε has only nondegenerate critical points, we have the followingresult:

Theorem 1.7 Assume (V0)–(V3). There exists ε0 > 0 such that, for any ε ∈ (0, ε0), if fε

has only nondegenerate critical points, then (Pε) has at least P1(Ω) distinct solutions withtopological charge q∗.

For an isolated (possibly degenerate) critical point u of fε, we prove that the numberof nontrivial critical groups in u is finite (see Theorem 5.5 below). We also obtain aMarino–Prodi perturbation–type result (Theorem 6.1): in a chosen neighborhood of anisolated critical point u, fε can be approximated in the C2 norm with functionals whosecritical points are finitely many and nondegenerate (in the sense of Definition 1.5).

Theorem 6.1 yields an interpretation of the notion of multiplicity. In this context, acrucial role is played by an abstract result (Theorem 7.1), that can be roughly described asfollows: if u is an isolated critical point of a C1 functional f in a Banach space, and g is aC1-small perturbation of f around u, with a finite number of critical points, then the Morsepolynomial of f in u can be computed in terms of the sum of the Morse polynomials of gin each critical point and a partially controlled remainder term. In view of Theorem 7.1and Theorem 1.6, up to a local perturbation of fε every isolated critical point u can beresolved into a finite number of nondegenerate critical points, this number being at leastP1(f, u). We refer to Section 8 for further details.

5

We conclude this section by observing that, in general, in a topologically rich domainthe application of Morse theory yields better results than the application of Ljusternik–Schnirelman theory. As an example, we consider a domain with holes (cf. Corollary 1.4in [4]):

Corollary 1.8 Assume (V0)–(V3). Let A and Ci (i = 1, 2, . . . , k) be contractible, open,smooth and bounded non-empty sets in Rn. Suppose that Ci ⊂ A for any i = 1, 2, . . . , kand Ci ∩ Cj = ∅ for any i 6= j, and set

Ω = A \k⋃

i=1

Ci. (3)

Then there exists ε0 > 0 such that, for any ε ∈ (0, ε0), the problem (Pε) has at least k + 1solutions with topological charge q∗, counted with their multiplicity. Moreover, if fε hasnondegenerate critical points for ε ∈ (0, ε0), then (Pε) has at least k + 1 distinct solutionswith topological charge q∗.

Remark that, independently of the number of holes in Ω ⊂ Rn, the Ljusternik–Schnirelman category of Ω is 2. As a consequence, the results in [3], based on Ljusternik–Schnirelman theory, provide at most 3 as a lower bound for the number of solutions to(Pε).

Notations

(·|·) and | · | denote the inner product and the Euclidean norm in Rk (k ∈ N).

Br(x) = z ∈ Rk : |z − x| < r for any x ∈ Rk and r > 0.

∆pu =(div (|∇u|p−2∇ui)

)1≤i≤n+1

for any u = (u1, . . . , un+1) and p > 2.

‖ · ‖ denotes the usual norm in W 1,p0 (Ω,Rn+1).

Br(u) = v ∈ W 1,p0 (Ω) : ‖v − u‖ < r for any u ∈ W 1,p

0 (Ω,Rn+1) and r > 0.

〈·, ·〉 : W−1,p′(Ω,Rn+1)×W 1,p0 (Ω,Rn+1) → R denotes the duality pairing.

f c = v ∈ X : f(v) ≤ c for any f : X −→ R and c ∈ R.

2 The Variational Setting

We define the set

Λ(Ω) =u ∈ W 1,p

0 (Ω,Rn+1) : u(x) 6= ξ for any x ∈ Ω

6

and the energy functional fε : Λ(Ω) −→ R

fε(u) =∫

Ω

(εp

p|∇u|p +

ε2

2|∇u|2 + V (u)

)dx.

The space W 1,p0 (Ω,Rn+1) is embedded in the space of continuous and bounded functions

in Ω, hence the functional fε is well defined and of class C2 in Λ(Ω). Furthermore, thecritical points of fε on Λ(Ω) are weak solutions to (Pε).

With no loss of generality, we assume ξ = (1, 0) ∈ R × Rn. For any u ∈ Λ(Ω), letu(x) = (u0(x), u(x)), with u0(x) ∈ R and u(x) ∈ Rn. We define the support of u as theopen, bounded set

Ku =x ∈ Ω : u0(x) > 1

.

Notice that for any x ∈ ∂Ku we have u0(x) = 1, which together with u(x) 6= ξ impliesu(x) 6= 0. Thus, it makes sense to define the topological charge of u as the integer

ch(u) =

deg(u,Ku, 0) if Ku 6= ∅0 if Ku = ∅.

By means of the topological charge, Λ(Ω) can be split into the union of the sets

Λq(Ω) = u ∈ Λ(Ω) : ch(u) = q, q ∈ Z.

By arguing as in Section 5.1 in [2], it is possible to see that Λq(Ω) is not empty. More-over, since the topological charge is continuous with respect to the uniform convergence,Λq(Ω) is closed in Λ(Ω); Λq(Ω) is also open in Λ(Ω) (see [1, 2] for details). Thus, for anyq ∈ Z, Λq(Ω) is a connected component of Λ(Ω).

It is not difficult to prove that, for any q, fε has a critical point in Λq(Ω). Indeed, asproved in [3], the infimum of fε in Λq(Ω) is attained. For the reader’s convenience, wegive a sketch of the proof.

Theorem 2.1 For any q ∈ Z, there exists uq ∈ Λq(Ω) such that

fε(uq) = infv∈Λq(Ω)

fε(v).

Proof. If vn ⊂ Λ(Ω) is a bounded sequence in W 1,p0 (Ω,Rn+1) and vn weakly con-

verges to v ∈ ∂Λ(Ω), then by (V2) we have∫Ω V (vn)dx →∞ as n →∞. This implies that

the sublevel sets of fε in Λ(Ω) are weakly closed.Moreover, by (V1) we have fε(vk) →∞ for every vk ⊂ Λ(Ω) such that ‖vk‖ → ∞, thatis, fε is coercive. As a consequence, the sublevel sets of fε are weakly compact.Since Ω is bounded, each connected component Λq(Ω) is weakly closed in Λ(Ω), so thatf c

ε ∩ Λq(Ω) is weakly compact.Therefore, the weak lower semicontinuity of fε implies the existence of a minimum pointin each Λq(Ω). ¤

In order to apply Morse theory to (Pε), we need the following property:

7

Lemma 2.2 The functional fε satisfies the Palais–Smale condition in Λq(Ω), for anyq ∈ Z.

Proof. Let vk ⊂ Λq(Ω) be a Palais–Smale sequence for fε, namely fε(vk) → c andf ′ε(vk) → 0 in W−1,p′(Ω,Rn+1) (1/p + 1/p′ = 1) as k →∞.By (V1)− (V3), ‖vk‖ is bounded and vk weakly converges to some v ∈ Λq(Ω). This impliesthat V ′(vk) → V ′(v) uniformly and in W−1,p′(Ω,Rn+1).Since p > 2, it is easy to prove that the operator A, defined by A(u) = ∆u + ∆pu, ismonotone (cf. [2]). As a consequence, vk strongly converges to v. ¤

3 The Limit Problem

The quasilinear system

(L) −∆pu−∆u + V ′(u) = 0 in Rn

u → 0 as |x| → ∞was recently proposed by Benci, Fortunato, and Pisani as a multidimensional Lorentzinvariant model having soliton-like solutions (see [1] for the case n = 3 and [2] for arbitrarydimension n ≥ 3; see also the related papers [9, 10, 11]).

Weak solutions of (L) are critical points of the energy functional

I(u) =∫

Rn

(1p|∇u|p +

12|∇u|2 + V (u)

)dx

on the set

Λ(Rn) =

u ∈ W 1,p(Rn,Rn+1) ∩W 1,2(Rn,Rn+1) : u(x) 6= ξ for any x ∈ Rn

.

For any u = (u0, u) ∈ Λ(Rn), the support of u is the set Ku =x ∈ Rn : u0(x) > 1

;

the topological charge of u is the integer

ch(u) =

deg(u,Ku, 0) if Ku 6= ∅0 if Ku = ∅.

Since u(x) → 0 as |x| → ∞, Ku is a bounded open set; furthermore, u(x) 6= 0 for anyu ∈ ∂Ku, so that ch(u) is well defined.

It is proved in [2] (Theorem 1) that I has a global minimizer in the class

Λ∗(Rn) =u ∈ Λ(Rn) : ch(u) 6= 0

;

in other words, there exists U ∈ Λ(Rn) such that

I∗ := infΛ∗(Rn)

I = I(U), ch(U) = q∗ 6= 0.

8

Due to the invariance under translations, we can assume that |U | has a global maximumpoint at x = 0.

Remark 3.1 The set Λ(Rn) can be decomposed into infinitely many connected compo-nents Λq(Rn) :=

u ∈ Λ(Rn) : ch(u) = q

, q ∈ Z. Since these connected components are

not weakly closed in Λ(Rn), one cannot use a simple argument as in the proof of Theo-rem 2.1 to prove the existence of a minimizer for I in each Λq(Rn). In fact, the variationalapproach in [2] relies on a concentration–compactness argument, the so-called SplittingLemma (see Proposition 3 in [2]).

As an easy corollary to the Splitting Lemma, we get the following compactness prop-erty: if vk ⊂ Λ∗(Rn) is a minimizing sequence for I, there exists xk ⊂ Rn such that,up to a subsequence, vk(· + xk) converges weakly in W 1,p(Rn,Rn+1) ∩W 1,2(Rn,Rn+1) tosome v satisfying I(v) = I∗.

Remark 3.2 In [2], the regularity assumption on V is weaker than (V0). Indeed, theauthors assume that V is of class C1 in Rn+1 \ ξ and twice differentiable at the origin.

4 A Topological Argument

In this section we correlate the topology of a suitable energy sublevel of fε with thetopology of Ω, by means of their Poincare polynomials.

Proposition 4.1 There exist ε0 > 0 and γ > 0 such that, for any ε ∈ (0, ε0), we have

Pt

(f εn(I∗+γ)

ε ∩ Λq∗(Ω))

= Pt(Ω) + Zε(t), (4)

where Zε(t) is a series with non-negative coefficients.

The proof of Proposition 4.1, inspired by some ideas introduced in [12], is based on theconstruction of two suitable maps. The ground state solution U and the ground energyI∗, defined in the previous section, play an important role.The map Φε. Let δ > 0 be such that the sets

Ω−δ = x ∈ Ω : dist(x, ∂Ω) > 2δ, Ω+δ = x ∈ Rn : dist(x,Ω) < 2δ

are homotopically equivalent to Ω. Let η : [0, +∞) → [0, 1] be a smooth, nonincreasingfunction such that η(t) = 1 if 0 ≤ t ≤ δ, η(t) = 0 if t ≥ 2δ, |η′(t)| ≤ c for some c > 0.For ε > 0 and for x0 ∈ Ω−δ , define the function ϕε,x0 : Ω → Rn+1 by setting

ϕε,x0(x) = η(|x− x0|) U(x− x0

ε

).

9

If ε is sufficiently small, ϕε,x0 is in Λq∗(Ω) for any x0 ∈ Ω−δ . Indeed, by construction ϕε,x0

belongs to W 1,p0 (Ω,Rn+1) for any ε and x0. Moreover, since U(x) vanishes at infinity, it is

easy to see that ϕε,x0(x)−U(

x−x0ε

)goes to 0 as ε goes to 0, uniformly in x and x0. This

implies that, for small ε, |ϕε,x0(x) − ξ| is bounded from below away from zero. Finally,the continuity of the topological charge with respect to the uniform convergence yieldsch(ϕε,x0) = ch(U) = q∗.

For ε > 0 sufficiently small, we define the map

Φε : Ω−δ −→ Λq∗(Ω), Φε(x0) = ϕε,x0 .

Lemma 4.2 Uniformly in x0 ∈ Ω−δ , we have

limε→0

ε−nfε

(Φε(x0)

)= I∗. (5)

Proof. Let us denote Ωε = x : x0 + εx ∈ Ω; a simple change of variable gives∫

Ω|∇ϕε,x0(x)|pdx = εn−p

∫

Ωε

∣∣∣∣εη′(ε|x|)x

|x|U(x) + η(ε|x|)∇U(x)∣∣∣∣p

dx;

since ∣∣∣∣εη′(ε|x|)U(x)x

|x| + η(ε|x|)∇U(x)∣∣∣∣p

≤ C(|U(x)|p + |∇U(x)|p)

and both U and ∇U are Lp-summable, the Lebesgue Theorem applies and gives∫

Ω|∇ϕε,x0(x)|pdx = εn−p

[∫

Rn

|∇U(x)|pdx + o(1)]

as ε → 0. (6)

In a similar way, because of the L2-summability of U and ∇U , we obtain∫

Ω|∇ϕε,x0(x)|2dx = εn−2

[∫

Rn

|∇U(x)|2dx + o(1)]

as ε → 0. (7)

Now, due to (V0)–(V1) there exist c1, ρ1 > 0 such that V (ξ) ≤ c1|ξ|2 for any |ξ| < ρ1.Since U vanishes at infinity, if ε is small and |x| > δ

ε we have |η(ε|x|)U(x)| ≤ ρ1, whenceV

(η(ε|x|)U(x)

) ≤ c1|η(ε|x|)U(x)|2 ≤ c1|U(x)|2 and∫

Ωε\B δε(0)

V(η(ε|x|)U(x)

)dx ≤ c1

∫

Ωε\B δε(0)|U(x)|2dx = o(1) as ε → 0. (8)

On the other hand, we obviously have∫

B δε(0)

V(U(x)

)dx =

∫

Rn

V(U(x)

)dx + o(1) as ε → 0. (9)

10

From (8) and (9) we deduce∫

ΩV

(ϕε,x0(x)

)dx = εn

[∫

Rn

V(U(x)

)dx + o(1)

]. (10)

Finally, from (6), (7) and(10) we get fε

(Φε(x0)

)= εn [I(U) + o(1)], which in turn gives

(5). Let us note that the limit in (5) is uniform in x0, because Ω−δ is a compact set. ¤

The map βε. We define the barycenter map

βε : W 1,p0 (Ω,Rn+1) \ 0 −→ Rn, βε(u) =

∫Ω µε,u(x) x dx∫Ω µε,u(x) dx

,

where µε,u(x) =εp

p|∇u(x)|p +

ε2

2|∇u(x)|2.

Lemma 4.3 There exist γ, ε > 0 such that βε(u) ∈ Ω+δ for any u ∈ Λq∗(Ω) ∩ f

εn(I∗+γ)ε ,

for any ε ∈ (0, ε).

Proof. By way of contradiction, let εk → 0 as k →∞ and let uk ∈ Λq∗(Ω) be such thatfεk

(uk) ≤ εnk

(I∗ + 1

k

)and βεk

(uk) 6∈ Ω+δ .

Let vk(x) = uk(εkx) if εkx ∈ Ω, vk(x) = 0 otherwise. Obviously, vk is in Λ∗(Rn) and, byconstruction, it is a minimizing sequence for I. By Remark 3.1, there exists xk ⊂ Rn

such that, up to a subsequence, wk(x) := vk(x+xk) converges weakly in W 1,p(Rn,Rn+1)∩W 1,2(Rn,Rn+1) to some W satisfying I(W ) = I∗.It is easy to see that

∫Rn V (wk(x)) dx approaches

∫Rn V (W (x)) dx as k →∞, therefore

∫

Rn

(1p|∇wk(x)|p +

12|∇wk(x)|2

)dx −→

∫

Rn

(1p|∇W (x)|p +

12|∇W (x)|2

)dx (11)

as k →∞. Obviously, for any 1− δdiam(Ω)

< θ < 1 there exists R > 0 such that∫

BR(0)

(1p|∇W (x)|p +

12|∇W (x)|2

)dx > θ

∫

Rn

(1p|∇W (x)|p +

12|∇W (x)|2

)dx;

by taking (11) into account, it is easy to see that the same inequality holds for wk, for klarge. As a consequence,

∫

BRεk(εkxk)∩Ω

µεk,uk(x)dx > θ

∫

Ωµεk,uk

(x)dx. (12)

This easily implies dist(εkxk, Ω) → 0 as k →∞, thus we can assume that εkxk convergesto some x ∈ Ω. If k is large enough, (12) gives

∫

Bδ(x)∩Ωµεk,uk

(x)dx > θ

∫

Ωµεk,uk

(x)dx,

11

thus

|βεk(uk)− x| =

∣∣∣∣∫Ω µεk,uk

(x) x dx∫Ω µεk,uk

(x) dx− x

∣∣∣∣ =

∣∣∫Ω µεk,uk

(x) (x− x) dx∣∣

∫Ω µεk,uk

(x) dx

≤∫Bδ(x)∩Ω µεk,uk

(x) |x− x| dx∫Ω µεk,uk

(x) dx+

∫Ω\Bδ(x) µεk,uk

(x) |x− x| dx∫Ω µεk,uk

(x) dx

≤ δ

∫Bδ(x)∩Ω µεk,uk

(x) dx∫Ω µεk,uk

(x) dx+ diam(Ω)

∫Ω\Bδ(x) µεk,uk

(x) dx∫Ω µεk,uk

(x) dx

≤ δ + (1− θ)diam(Ω) < 2δ,

a contradiction. ¤

Proof of Proposition 4.1. Let γ be chosen according to Lemma 4.3 and let

Σε = fεn(I∗+γ)ε ∩ Λq∗(Ω).

For ε sufficiently small, Lemma 4.2 and Lemma 4.3 yield Φε(Ω−δ ) ⊂ Σε and βε(Σε) ⊂ Ω+δ ,

respectively. By arguing as in the proof of Lemma 4.2, we can prove that βε

(Φε(x0)

) → x0

as ε → 0, uniformly in x0 ∈ Ω−δ . As a consequence, it is easy to prove that, for small ε,the map H(t, x) = tx + (1− t)β

(Φε(x)

)is an omotopy between βε Φε and the inclusion

map j : Ω−δ → Ω+δ .

Let Ψε,k and βε,k be the homomorphisms induced by the maps Φε and βε between thek-th homology groups

Hk(Ω−δ )Ψε,k−→ Hk(Σε)

βε,k−→ Hk(Ω+δ ).

Since βε,k Ψε,k is the identity map, the homology group Hk(Ω−) is homotopic to asubspace of Hk(Σε), whence

dimHk(Ω−δ ) ≤ dimHk(Σε). (13)

By our choice of δ, Ω and Ω−δ are homotopically equivalent, so (13) yields

dimHk(Ω) = dimHk(Ω−δ ) ≤ dimHk(Σε),

and (4) follows from the definition of Poincare polynomial. ¤

5 Critical Group Estimates

Let H be a Hilbert space, f a C2 functional on H, and u a nondegenerate critical point off (that is, f ′′(u) is an isomorphism between H and the dual space H∗). Classical resultsallow to estimate the critical groups of f in u via its Morse index, by using the MorseLemma (see, for example, Theorem 4.1 in [13]). On the contrary, in a Banach space not

12

isomorphic to its dual space, the (generalized) Morse Lemma does not hold, and estimateson the critical groups are not available in general.

In this section we develop a local Morse theory and compute the critical groups for asuitable class of functionals, associated with the Euler functionals fε.

Since the parameter ε plays no role here, in order to simplify the notation we write finstead of fε; moreover, q ∈ Z \ 0 is fixed.

Definition 5.1 Let A be an open subset of Λq(Ω). We define F(A) as the set of func-tionals g : A → R such that there exist a finite-dimensional subspace V ofW 1,p

0 (Ω,Rn+1) ∩ L∞(Ω,Rn+1), a C2 function h : V → R, and a continuous projectionP

V: W 1,p

0 (Ω,Rn+1) → V , such that g(v) = f(v) + h(P

Vv).

Definition 5.2 Let A be an open subset of Λq(Ω). We define M(A) as the set of func-tionals g : A → R such that g ∈ C2(A), the critical points of g are nondegenerate (in thesense of Definition 1.5) and, for any critical point u of g, there exists an open neighborhoodU of u such that g|U ∈ F(U).

The main result in this section is the following theorem:

Theorem 5.3 Assume (V0)–(V2). Let A be an open subset of Λq(Ω) and g ∈ M(A).Then, for each critical point u of g, the Morse index m(g, u) is finite and

Cj(g, u) ∼= K if j = m(g, u), (14)Cj(g, u) = 0 if j 6= m(g, u). (15)

Proof. Theorem 5.3 can be proved by arguing as in [6, 7]. For convenience, we give asketch of the proof, which we divide into four steps.Step 1. Let g ∈ F(

Λq(Ω))

be fixed, let u ∈ Λq(Ω) be a critical point of g and c = g(u).The second-order differential of g in u is given by

〈g′′(u) v, w〉 =∫

Ω(1 + |∇u|p−2)(∇v|∇w) dx

+ (p− 2)∫

Ω|∇u|p−4(∇u|∇v)(∇u|∇w) dx +

∫

Ω(V ′′(u)v|w) dx

+ 〈h′′(PV u) PV v, PV w〉for any v, w ∈ TuΛq(Ω) = W 1,p

0 (Ω,Rn+1).Since the critical point u belongs to C1(Ω,Rn+1) (see the proof of Theorem 2.3 in [8]),the vector function ri(x) = |∇u(x)|(p−4)/2∇ui(x) belongs to C0(Ω,Rn+1), for each i =1, ..., n + 1. Set r = (r1, ..., rn+1). Let Hr be the completion of C∞

0 (Ω,Rn+1) under theinner product

(v, w)r =∫

Ω(1 + |r|2)(∇v|∇w) dx + (p− 2)(r|∇v)(r|∇w) dx;

13

let ‖ · ‖r be the norm induced by (·, ·)r and let 〈·, ·〉r : H∗r ×Hr → R denote the duality

pairing in Hr.We emphasize that the space Hr, whose Hilbert structure depends on the critical pointu and is closely related with g′′(u), is equivalent to the Hilbert space W 1,2

0 (Ω,Rn+1). Byconstruction, the embedding W 1,p

0 (Ω,Rn+1) ⊂ Hr is continuous. Furthermore, g′′(u) canbe extended to the operator Lr : Hr → H∗

r by setting

〈Lrv, w〉r = (v, w)r +∫

ΩV ′′(u)vw dx + 〈h′′(PV u) PV v, PV w〉 for any v, w ∈ Hr.

The operator Lr is a compact perturbation of the Riesz isomorphism from Hr to the dualspace H∗

r . In particular, Lr is a Fredholm operator with zero index.Step 2. The Sobolev space W 1,p

0 (Ω,Rn+1) can be split into the direct sum V ⊕W , whereV is a finite dimensional space, with dimV = m∗(g, u), and

〈g′′(u)w, w〉 ≥ µ1‖w‖2r for any w ∈ W ,

for a suitable constant µ1 > 0. Furthermore, there exist r0, C > 0 such that Br0(u) ⊂Λq(Ω) and

〈g′′(z)w, w〉 ≥ C‖w‖2r for any w ∈ W , for any z ∈ Br0(u). (16)

As a consequence, both m(g, u) and m∗(g, u) are finite.Since 0 is a local minimum of g in the direction of W , we are able to perform a reductionargument. Indeed, there exist R > 0 and ρ ∈ (0, R) such that

U := u +(V ∩ Bρ(0)

)+

(W ∩ BR(0)

) ⊂ Λq(Ω)

and, for any v ∈ V ∩ Bρ(0), there exists a unique w ∈ W ∩ BR(0) such that

g(u + v + w) ≤ g(u + v + z) for any z ∈ W ∩ BR(0).

Moreover, w is the only element in W ∩ BR(0) such that

〈g′(u + v + w), z〉 = 0 for any z ∈ W

(see Lemma 4.6 in [6]). It is then natural to define the map

ψ : V ∩ Bρ(0) → W ∩ BR(0)

by setting ψ(v) = w, the unique minimum point of the function w ∈ W ∩ BR(0) 7→g(u + v + w). We also define the function

φ : V ∩ Bρ(0) → R, φ(v) = g(u + v + ψ(v)

)(17)

14

and we introduce the set

Y =

u + v + ψ(v) : v ∈ V ∩ Bρ(0)

.

Since ψ is a continuous map, the set

φc =

v ∈ V ∩ Bρ(0) : φ(v) ≤ c

is homeomorphic to

(g|Y )c =

u + v + ψ(v) : v ∈ V ∩ Bρ(0), φ(v) ≤ c

.

Furthermore, since ψ(0) = 0, we deduce that φc \ 0 is homeomorphic to (g|Y )c \ u.Therefore, for any integer k, we have

Ck(φ, 0) ' Ck(g|Y , u). (18)

Now, we claim that the topological pair((g|Y )c, (g|Y )c \ u)

is a deformation retract of (gc ∩ U, gc ∩ U \ u).

Indeed, set X = gc ∩ U and X ′ = gc ∩ U \ u, and define

η(t, u + v + w) = u + v + w + t(ψ(v)− w

),

where t ∈ [0, 1] and x = u + v + w ∈ X. By (16), g(η(t, x)) ≤ c for any t ∈ [0, 1] andx ∈ X, so that η : [0, 1] × X → X is well posed. Moreover it is easy to see that η iscontinuous, η(0, ·) = IX (where IX is the identity map in X), η(1, X) ⊂ (g|Y )c and for anyt ∈ [0, 1], η(t, ·)|(g|Y )c = I(g|Y )c . Furthermore, since 0 is the minimum point of the functionw 7→ g(u+w), we deduce that η(t,X ′) ⊂ X ′ for any t ∈ [0, 1], and η(1, X ′) ⊂ (g|Y )c \ u.Our claim is thus proved.As a consequence, for any k ∈ Z we have

Hk

(gc|Y , gc

|Y \ u)' Hk

(gc ∩ U, gc ∩ U \ u

),

thus the definition of critical groups, the excision property and (18) yield

Ck(g, u) ' Ck(g|Y , u) ' Ck(φ, 0). (19)

Step 3. Assume now that u is a nondegenerate critical point of g (in the sense of Defini-tion 1.5). The following inequality holds:

〈g′′(u)v, v〉 ≤ −µ2‖v‖2 for any v ∈ V

15

for a suitable constant µ2 > 0. As a consequence, u is a local isolated maximum of g alongV , thus 0 is a local maximum of φ in V ∩Bρ(0). Taking (19) into account yields (14) and(15).Step 4. Let A be an open subset of Λq(Ω), and let g ∈ M(A). Plainly, Step 1–3 aboveapply to each critical point of g. Theorem 5.3 is thus proved. ¤

Proof of Theorem 1.6. If u ∈ Λq(Ω) is a nondegenerate critical point of f , thenf ∈M(A) for some open subset A of Λq(Ω), and Theorem 5.3 applies to f . ¤

Remark 5.4 If H is a Hilbert space, the Morse polynomial of g ∈ C2(H,R) in a nonde-generate critical point u is Pt(g, u) = tm(g,u) and the multiplicity of u is P1(g, u) = 1 (see[14], for example). In view of Step 1–3 in the proof of Theorem 5.3, the same propertyholds in every nondegenerate (in the sense of Definition 1.5) critical point of every functionin the class F(

Λq(Ω)).

We conclude this section with a quantitative result for an isolated, possibly degenerate,critical point:

Theorem 5.5 Assume (V0)–(V2) and fix ε > 0. Let u be an isolated critical point of fε.Then m(fε, u) and m∗(fε, u) are finite and

Ck(fε, u) = 0 for k ≤ m(fε, u)− 1 and k ≥ m∗(fε, u) + 1.

Moreover, dimCk(fε, u) < ∞ for any k ∈ N.

Proof. Step 1 and 2 in the proof of Theorem 5.3 apply to any isolated critical point uof f , so that

Ck(f, u) ' Ck(φ, 0). (20)

Plainly, (20) implies Ck(f, u) = 0 for any k > dimV = m∗(f, u). On the other hand,Corollary 6.4 in [15] gives Ck(f, u) = 0 for any k ≤ m(f, u)− 1. ¤

6 A Perturbation Result of Marino–Prodi Type

Let f be a C2 functional on a Hilbert space H and let u be an isolated critical point forf , such that the second derivative f ′′(u) is a Fredholm operator. A celebrated paper byMarino and Prodi [16] guarantees that f can be locally approximated by a C2 functionalg, having a finite number of nondegenerate critical points, where nondegenerate meansthat the second derivative of g in a critical point is an isomorphism between H and thedual space.

16

Let us emphasize that the perturbation results by Marino and Prodi rely on an infinite-dimensional version of Sard’s Theorem, due to Smale [17]. Therefore, in their approach itis crucial to assume that the second derivative f ′′(u) is a Fredholm operator.

In the variational setting of (Pε), the second derivative of the Euler functional fε inthe critical points is not a Fredholm operator, thus the perturbation results mentionedabove cannot be applied. Nonetheless, for an isolated critical point u of fε in Λq(Ω), wecan prove that fε can be locally approximated in the C2 norm by functionals in the classM(A), for a suitable neighborhood A of u. Note that, by Theorem 5.3, all the criticalgroups of the approximating functionals can be computed.

As in the previous section, q ∈ Z \ 0 is fixed. Moreover, we write f instead of fε inorder to simplify the notation.

Theorem 6.1 Let u ∈ Λq(Ω) be an isolated critical point of f . Let A ⊂ Λq(Ω) be anopen neighborhood of u such that f has no critical points other than u in A. Then for anyµ > 0 and for any γ > 0 such that Bγ(u) ⊂ A, there exists a functional g ∈ M(A) withthe following properties:

(1) f(v) = g(v) if v ∈ A \ Bγ(u);

(2) ‖f − g‖C2(A) < µ;

(3) the critical points of g, if any, are in Bγ(u) and finitely many;

(4) g satisfies the Palais–Smale condition in Bγ(u).

Proof. We show only how to construct the approximating functional g; for furtherdetails, we refer to the proof of Theorem 4.3 in [18].First of all, we remark that φ, defined in (17), is a C2 function (see Lemma 3.6 in [18]).Therefore, well-known perturbation results, based on Sard’s Lemma (see [19, 20]), yieldthe following property:

Lemma 6.2 In correspondence with every pair (µ, δ), with µ > 0 and δ ∈ (0, ρ), thereexists a C2 function α : V ∩ Bρ(0) → R such that

(a) α(v) = 0 if ‖v‖ ≥ δ;

(b) ‖α‖C2(V ∩Bρ(0)) < µ;

(c) the critical points of φ + α, if any, are nondegenerate, finitely many, and they are inBδ(0).

17

Let us fix µ > 0 and γ > 0 such that Bγ(u) ⊂ A. For any fixed δ ∈ (0, γ), there existsν > 0 such that ‖f ′(z)‖ ≥ ν for any z ∈ Bγ(u) \ Bδ(u) (we use the notation ‖ · ‖ also forthe norm in W−1,p(Ω,Rn+1)).For µ > 0, let α be the function which corresponds with the pair (µ, ρ/2) via Lemma 6.2,and let β : V → R be the natural extension of α to V , that is,

β(v) =

α(v) if ‖v‖ ≤ ρ0 otherwise.

Finally, let η : Λq(Ω) → R be any C2 function such that

η(z) = 0 if z /∈ Bγ(u), η(z) = 1 if z ∈ Bδ(u).

We define the C2 perturbed functional g : A → R by setting

g(z) = f(z) + η(z) β(PV (z − u)

).

By construction, (1) holds. Furthermore, since η and its derivatives are uniformly bounded,a suitable choice of µ yields

‖f − g‖C2(A) ≤ minµ, ν/2and (2) follows. See [18] for the proof of (3) and (4). ¤

7 An Abstract Result on Morse Polynomials

This section is devoted to a general result, concerning a C1 functional in a Banach space.

Theorem 7.1 Let A be an open subset of a Banach space X. Let f ∈ C1(A,R) and u ∈ Abe an isolated critical point of f . Assume that there exists an open neighborhood U of usuch that U ⊂ A, u is the only critical point of f in U and f satisfies the Palais–Smalecondition in U .Then there exists µ > 0 such that, for any g ∈ C1(A,R) such that

(1) ‖f − g‖C1(A) < µ,

(2) g satisfies the Palais–Smale condition in U ,

(3) g has a finite number u1, u2, . . . , um of critical points in U , contained in U ,

we havem∑

j=1

Pt(g, uj) = Pt(f, u) + (1 + t)Q(t),

where Q(t) is a formal series with coefficients in N ∪ ∞.

18

Proof. We prove Theorem 7.1 under the restrictive assumption that the Banach spaceX is equipped with a norm ‖ ·‖ such that ‖ ·‖2 is C1 (this assumption is clearly satisfied inthe variational framework of problem (Pε)). We adapt to our setting some ideas developedby Gromoll and Meyer [21] for an isolated critical point at an isolated critical level in aHilbert space; see also Theorems 5.2 and 5.3 in [14]. In the general case, Theorem 7.1 canbe proved by adapting several results in [22].Let A, f and u be as above. For simplicity, we assume u = 0 and f(0) = 0. Let U bean open neighborhood of 0 in A such that U ⊂ A, u is the only critical point of f in U ,and f satisfies the Palais–Smale condition in U . For U∗ := U \ 0, there exists a locallyLipschitz continuous vector field F : U∗ → X such that

‖F (z)‖ ≤ 2‖f ′(z)‖ and 〈f ′(z), F (z)〉 ≥ ‖f ′(z)‖2

for any z ∈ U∗ (throughout the proof, we use the notation ‖ · ‖ also for the norm in thedual space).Let δ > 0 be such that Bδ(0) ⊂ U ; the Palais–Smale condition implies

β := inf‖f ′(x)‖ : x ∈ Bδ(0) \Bδ/2(0)

> 0.

Choose λ > 4δβ , 0 < γ < 3δ2

8λ , and δ2

4 + λγ ≤ µ ≤ δ2 − λγ. Let us define the C1 functional

h(z) = λf(z) + ‖z‖2

and the setsW = f−1[−γ, γ] ∩ hµ, W− = f−1(−γ) ∩W.

It is easy to see that

Bδ/2(0) ∩ f−1[−γ, γ] ⊂ W ⊂ Bδ(0) ∩ f−1[−γ, γ], (21)

f−1[−γ, γ] ∩ h−1(µ) ⊂ Bδ(0) \Bδ/2(0), (22)

〈h′(z), F (z)〉 > 0 for any z ∈ Bδ(0) \Bδ/2(0). (23)

Now, for any x ∈ U∗ we consider the negative flow for F which solves the Cauchy problem

(Px)

η(t, x) = − F (η(t,x))

‖F (η(t,x))‖2η(0, x) = x.

We haved

dtf (η(t, x)) ≤ −‖f

′ (η(t, x)) ‖2

‖F (η(t, x))‖2≤ − 1

4(24)

and, for any t2 > t1 > 0 in which η(t, x) is defined,

f (η(t2, x))− f (η(t1, x)) =∫ t2

t1

d

dsf (η(s, x)) ds ≤ − 1

4(t2 − t1). (25)

The following properties can be easily deduced from (21)–(24):

19

• If η(t1, x), η(t2, x) ∈ W , then η(t, x) ∈ W for any t ∈ [t1, t2];

• W− = x ∈ W : η(t, x) /∈ W for any t > 0 .

Claim 1: W− is a strong deformation retract of f0 ∩W \ 0.Let x0 ∈ f0 ∩W \ 0 =: W ′. First of all, we notice that ‖f ′(η(t, x0))‖ ≥ εx0 > 0, for anyt ≥ 0 such that η(t, x0) ∈ W .Now we can prove that the flow η(t, x0) reaches W−. Indeed, if the maximal existenceinterval for the initial data x0 is [ 0, Tx0) and limt→T−x0

f(η(t, x0)) > −γ, then (25) impliesTx0 ≤ 4γ. Moreover, for any t1 < t2,

‖η(t2, x0)− η(t1, x0)‖ ≤∫ t2

t1

‖η(t, x0)‖ dt ≤ 1εx0

(t2 − t1).

This implies that limt→T−x0η(t, x0) exists and it is not a critical point. Therefore, the flow

can be extended beyond Tx0 , against the maximality.As a consequence, for any x ∈ W ′ there exists Tx ≥ 0 such that η(Tx, x) ∈ W−; notice thatTx = 0 when x ∈ W−. Finally, the function x ∈ W ′ 7→ Tx is continuous by the ImplicitFunction Theorem.The function σ : W ′ × [0, 1] → W ′, defined by σ(x, t) = η(tTx, x), is the required strongdeformation retraction.Claim 2: f0 ∩W is a strong deformation retract of W .Let x0 ∈ W \ f0, and let η(t, x0) be the maximal solution of (Px0). We distinguish twocases:

(a) inft>0

‖η(t, x0)‖ > 0 (b) inft>0

‖η(t, x0)‖ = 0.

In case (a), by arguing as in the proof of Claim 1, it is possible to show that there existsTx0 ∈ R+ such that η(t, x0) is defined at least in [0, Tx0) and limt→T−x0

f (η(t, x0)) = 0.Moreover, the map x 7→ Tx is continuous in x0.In case (b) we have a sequence tn of positive numbers such that lim

n→∞ η(tn, x0) = 0. First

we prove that limt→T−x0

η(t, x0) = 0, where Tx0 := lim supn→∞

tn.

By way of contradiction, assume that there is a sequence sn → T−x0such that ‖η(sn, x0)‖ ≥

ε0 > 0. Thus we have two sequences, hn < kn, which both converge to Tx0 , such that

‖η(hn, x0)‖ =ε0

2, ‖η(kn, x0)‖ = ε0

andε0

2≤ ‖η(t, x0)‖ ≤ ε0 for all t ∈ [hn, kn].

As a result,inf

t∈ [ hn,kn]‖f ′(η(t, x0))‖ > α > 0

20

andε0

2≤ ‖η(kn, x0)− η(hn, x0)‖ ≤

∫ kn

hn

‖η(t, x0)‖ dt ≤ 1α

(kn − hn) → 0,

a contradiction.From (25) it easily follows that the map x ∈ W \ f0 7→ Tx ∈ R+ is continuous in x0.The map σ : [0, 1]×W → W , defined as

σ(t, x) =

x if x ∈ f0

η(tTx, x) if (t, x) ∈ [0, 1)×W \ f0

limτ→T−x

η(τ, x) if (t, x) ∈ 1 ×W \ f0,

is the required strong deformation retraction. Claim 2 is thus proved.Now, Claim 1, Claim 2, Definition 1.1 and the excision property give

Hk(W,W−) ' Hk(f0 ∩W, f0 ∩W \ 0) ' Ck(f, 0) (26)

for every k ∈ N.Let 0 < µ < minγ/3, β, (λβ − 4δ)/(3λ) and choose g ∈ C1(A,R), satisfying (1)–(3). It

is easy to see that g has no critical points in U \ W . Plainly,

W− = f−γ ∩W ⊂ g−2γ/3 ∩W ⊂ f−γ/3 ∩W ⊂ fγ/3 ∩W ⊂ g2γ/3 ∩W ⊂ fγ ∩W = W ;

moreover, with the same arguments used in the proof of Claim 1 and 2, we infer thatf−γ∩W is a strong deformation retract of f−γ/3∩W and fγ/3∩W is a strong deformationretract of fγ ∩W . As a consequence:

Hk(W,W−) ' Hk(g2γ/3 ∩W, g−2γ/3 ∩W ) (27)

which, together with (26), yields

Hk(g2γ/3 ∩W, g−2γ/3 ∩W ) ' Ck(f, 0). (28)

Since g has a finite number u1, u2, ...um of critical points in W, contained in

W , we have

v ∈ W | g′(v) = 0 ∩ ∂W = ∅.

In view of (1), there exist a pseudo–gradient vector field Y of g and a relative negativepseudo–gradient flow ηg such that

〈h′(z), Y (z)〉 > 0 for all z ∈ Bδ(0) \Bδ/2(0), (29)

thus ηg(t, x) ∈ W for any x ∈ ∂W \ g−1(−2γ/3) and t > 0 small.

21

At this point, Theorem 4.3 in [14] applies to the function g on W (see also Remark 4.1 in[14]), so that

m∑

j=1

Pt(g, uj) =∞∑

k=0

dimHk(g2γ/3 ∩W, g−2γ/3 ∩W ) tq + (1 + t)Q(t), (30)

where Q(t) is a formal series with coefficients in N ∪ ∞. The proof is concluded bytaking into account (30), (28) and the definition of Morse polynomial. ¤

8 On the Multiplicity of Isolated Critical Points

Throughout this section, ε > 0 is fixed. Our goal is to provide an interpretation of thenotion of multiplicity.

Let u ∈ Λq(Ω) be an isolated critical point of fε. Let A ⊂ Λq(Ω) be an open neighbor-hood of u such that f has no critical points other than u in A.

Choose γ > 0 such that Bγ(u) ⊂ A and let µ > 0 be chosen in correspondence withU = Bγ(u), via Theorem 7.1.

By Theorem 6.1, for any µ ∈ (0, µ) there exists gµ ∈M(A) such that ‖fε− gµ‖C2(A) <

µ, gµ satisfies the Palais–Smale condition in Bγ(u) and has finitely many critical pointsu1, . . . , um, all contained in Bγ(u). Theorem 7.1 implies

m∑

j=1

Pt(gµ, uj) = Pt(f, u) + (1 + t)Q(t), (31)

where Q(t) is a formal series with coefficients in N∪∞. Since gµ ∈M(A), Theorem 5.3applies to all the critical points of gµ, thus (31) gives

m∑

j=1

tm(gµ,uj) = Pt(f, u) + (1 + t)Q(t).

For t = 1, we get m ≥ P1(f, u).We have thus proved that, arbitrarily closed (in the C2 norm) to fε, there is a functional

that has at least P1(f, u) nondegenerate critical points around u.

9 Proof of the Multiplicity Results

Proof of Theorem 1.3. Let ε0 be chosen in accordance with Proposition 4.1 and let0 < ε < ε0.

22

Let uj (1 ≤ j ≤ m) be the critical points of fε in Σε. Since fε is bounded from below andsatisfies the Palais–Smale condition (Lemma 2.2), the global Morse relations give

∞∑

k=0

aktk =

∞∑

k=0

dimHk(Σε) tk + (1 + t)Qε(t), (32)

where ak =∑m

j=1 dimCk(fε, uj) and Qε(t) is a formal series with coefficients in N∪ ∞.Proposition 4.1 implies

∞∑

k=0

aktk = Pt(Ω) + Zε(t) + (1 + t)Qε(t)

whence, for t = 1, we get

m∑

j=1

P1(fε, uj) = P1(Ω) + Zε(1) + 2Qε(1). (33)

Since both Zε(1) and Qε(1) have nonnegative coefficients, (33) implies that the numberof critical points of fε in Σε, each counted with its own multiplicity, is at least P1(Ω). ¤

Remark 9.1 Let ε ∈ (0, ε0). From the proof of Theorem 1.3 and the remarks in Section 8,it follows that, for any µ > 0, there exists a functional g, with at least P1(Ω) critical points,such that ‖fε − g‖C2(Λq∗ (Ω)) < µ.

Proof of Corollary 1.7. By Remark 5.4, if uj is a nondegenerate critical point of fε,then P1(fε, uj) = 1, and (33) implies that the number of critical points of fε in Σε is atleast P1(Ω). ¤

Proof of Corollary 1.8. If Ω is as in (3), then

Pt(Ω) = 1 + k tn−1

(see the proof of Corollary 1.4 in [4]). Plainly, P1(Ω) = 1 + k, and the result follows fromTheorem 1.3.In particular, if all the critical points of fε are nondegenerate, the thesis follows fromCorollary 1.7. ¤

Acknowledgements

The authors would like to thank Prof. Vieri Benci and Prof. Marco Degiovanni for severalcomments and helpful suggestions.

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