Morse index and critical groups for p-Laplace equations with critical exponents

MORSE INDEX AND CRITICAL GROUPS FOR p-LAPLACEEQUATIONS WITH CRITICAL EXPONENTS

SILVIA CINGOLANI AND GIUSEPPINA VANNELLA

Abstract. In this work we consider a class of Euler functionals defined in Banachspaces, associated to quasilinear elliptic problems involving the critical Sobolev expo-nent. We perform critical groups estimates via the Morse index.

1. Introduction

Morse theory is a very important tool in the study of nonlinear variational problems.Using Morse techniques one investigates the local behavior of the Euler functional asso-ciated to the problem near its critical points, by computing the so-called critical groups.

If H is a Hilbert space, f : H → R is a C2 functional and u is a nondegenerate criticalpoint of f (i.e. the second derivative f ′′(u) is invertible from H to its dual space H∗)then it is well known how to compute the critical groups of f in u. In fact, classicalresults, based on the Morse Lemma, guarantee that such groups are calculated using anHessian type notion like the Morse index of the critical point, which is the supremum ofthe dimensions of the subspaces on which f ′′(u) is negative definite.

These ideas were generalized by Gromoll and Meyer in order to compute the criticalgroups of an isolated critical point u, possibly degenerate, with finite Morse index, whenf ′′(u) is a Fredholm operator (see [2]).

Dealing with a variational problem set up in a Banach space, which is not isomorphicto its dual one, several difficulties arise in employing Morse techniques since the classicalnotion of nondegenerate is not reasonable and even the second derivatives are not ingeneral Fredholm operators (cf. [17]).

All the above quoted problems in extending Morse theory in Banach spaces occurin variational differential problems, involving the p-laplacian (p > 2), which naturallyarises in various physical contexts, for instance in the study of non-Newtonian fluids andelasticity problems (cf. [10, 16]).

In [23] Uhlenbeck cited an unpublished conjecture by Smale. He supposed that, inorder to develop a local Morse theory, the mere injectivity of the second derivative f ′′(u)could sometimes be the right notion of nondegeneracy in a Banach (not Hilbert) setting.

Recently in [5, 6] it has been proved that Smale’s conjecture is essentially true fora class of functionals, defined in the Banach space W 1,p

0 (Ω), (p > 2), associated toa class of quasilinear equations with subcritical growth. More precisely critical groupcomputations are performed via the Morse index just using the injectivity of the secondderivative in the critical point. Furthermore, in [8], Marino-Prodi perturbation type

1991 Mathematics Subject Classification. 58E05, 35B20, 35J60, 35J70.Key words and phrases. p-Laplace equations, critical exponents, critical groups, Morse index.The research of the authors was supported by the MIUR project “Variational and topological methods

in the study of nonlinear phenomena” (PRIN 2005).1

2 SILVIA CINGOLANI AND GIUSEPPINA VANNELLA

results are proved (cf. [17]) and existence results are obtained for p-Laplace equationsvia Morse techniques in [4, 7].

In this paper we focus more generally on a quasilinear elliptic equation with an addi-tional critical nonlinear term:

(Pα)

−div ((α + |∇u|2)(p−2)/2∇u) = g(u) + |u|p∗−2u in Ωu = 0 on ∂Ω

where Ω is a bounded domain in RN with smooth boundary, N > p, 2 ≤ p, p∗ =Np/(N − p), α > 0, and

(g) |g′(t)| ≤ c1|t|q + c2 with c1, c2 positive constants and 0 ≤ q < p∗ − 2 if N > p.

Solutions of (Pα) correspond to critical points of the C2 functional Jα : W 1,p0 (Ω) → R

defined by setting

(1.1) Jα(u) =1

p

∫

Ω

(α + |∇u|2)p/2 dx−∫

Ω

G(u) dx− 1

p∗

∫

Ω

|u|p∗ dx

where G(s) =∫ s

0g(s).

Since W 1,p0 (Ω) is not isomorphic to the dual space W−1,p′(Ω), with 1/p + 1/p′ = 1,

any critical point uα of Jα is degenerate, in the sense already given for Hilbert spaces.Furthermore, as J ′′α(u) is not a Fredholm operator, we can not apply the generalizedsplitting Morse lemma in order to describe the behavior of Jα near the critical points. Asin [5], we evaluate the critical groups of Jα near a critical point uα, if J ′′α(u) : W 1,p

0 (Ω) →W−1,p′(Ω) is an injective map.

Before stating the main results, we introduce some useful definitions (cf. [2]).

Definition 1.1. Let X be a Banach space and f be a C2 functional on X. Let K be afield. Let u be a critical point of f , c = f(u), and U be a neighborhood of u. We call

Cq(f, u) = Hq(f c ∩ U, (f c \ u) ∩ U)

the q-th critical group of f at u, q = 0, 1, 2, . . . , where f c = v ∈ X | f(v) ≤ c andHq(A,B) stands for the q-th Alexander-Spanier cohomology group of the pair (A,B)with coefficients in K.

Definition 1.2. Let X be a Banach space and f be a C2 functional on X. If u is acritical point of f , the Morse index of f in u is the supremum of the dimensions of thesubspaces of X on which f ′′(u) is negative definite. It is denoted by m(f, u). Moreover,the large Morse index of f in u is the sum of m(f, u) and the dimension of the kernel off ′′(u). It is denoted by m∗(f, u).

Now we state the main result of the work.

Theorem 1.3. Let uα be a critical point of the functional Jα such that J ′′α(uα) is injective.Then m(Jα, uα) is finite and

Cq(Jα, uα) ∼= K, if q = m(Jα, uα),

Cq(Jα, uα) = 0, if q 6= m(Jα, uα).

This theorem extends a classical result in Hilbert spaces for nondegenerate criticalpoints (cf. Theorem 4.1 in [2]), showing that the critical groups of Jα in u depend onlyon its Morse index.

We conclude with a quantitative result for a possibly degenerate critical point.

MORSE INDEX ESTIMATES 3

Theorem 1.4. Let uα be an isolated critical point of the functional Jα. Then m(Jα, uα)and m∗(Jα, uα) are finite and

Cq(Jα, uα) = 0for any q ≤ m(Jα, uα)− 1 and q ≥ m∗(Jα, uα) + 1.

In the proofs of Theorem 1.3 and 1.4 the critical group estimates of Jα in the criticalpoint uα is reduced to a finite dimensional problem by introducing an auxiliary Hilbertstructure, which depends on uα. Differently from [5, 6], the presence of the criticalterm in the right hand side of equation (Pα) produces a lack of compactness, so thatthe functional Jα does not satisfy the so-called Palais-Smale condition at all levels (cf.Definition 3.1). Nevertheless we prove a crucial lemma, in which we show that Jα satisfiesthe Palais-Smale condition in small balls around each critical point (see Lemma 3.2).

Concerning estimates of critical groups for continuous functionals associated to quasi-linear problems, we mention [14, 9].

Throughout the paper we use the following notations.

(1) (·|·) denotes the scalar product in RN

(2) ‖ · ‖ denotes the usual norm both in W 1,p0 (Ω) and in W−1,p(Ω)

(3) ‖ · ‖t denotes the usual norm in Lt(Ω)(4) ‖ · ‖∞ denotes the usual norm in L∞(Ω)(5) 〈·, ·〉 : W−1,p′(Ω)×W 1,p

0 (Ω) → R denotes the duality pairing(6) Br (u) = v ∈ W 1,p

0 (Ω) : ‖v − u‖ < r(7) f c = v ∈ W 1,p

0 (Ω) : f(v) ≤ c, f ba = v ∈ W 1,p

0 (Ω) : a ≤ f(v) ≤ b(8)

f b

a=v ∈ W 1,p

0 (Ω) : a < f(v) < b

(9) From time to time, we omit the symbol dx in integrals over Ω.

2. Auxiliary Hilbert structure

We begin to recall the following two theorems obtained in [12] by Guedda and Veron.

Theorem 2.1. Let G be a bounded open subset of RN with a C2 boundary ∂G, ε ∈ (0, 1],1 < p < N , f ∈ LN/p(G), K ∈ LN/p(G) and u ∈ W 1,p

0 (G) a solution of −div ((ε + |∇u|2)(p−2)/2∇u) + K(x)|u|p−2u = f in G

u = 0 on ∂G.

Then for any t ∈ [1, +∞) there exists Ct depending on G, K, ‖f‖LN/p(G), N and p,but not on ε ∈ (0, 1], such that

(2.1) ‖u‖Lt(G) ≤ Ct

Theorem 2.2. Let G be a bounded open subset of RN with a C2 boundary ∂G, 1 < p ≤N , ε ∈ (0, 1], f ∈ Ls(G) for some s > N/p and u ∈ W 1,p

0 (G) a solution of −div ((ε + |∇u|2)(p−2)/2∇u) = f in G

u = 0 on ∂G.


Then u ∈ L∞(G) and there exists C = C(N, p, |G|) such that

‖u‖L∞(G) ≤ C‖f‖1/(p−1)Ls(G) .

Consequently we can derive the following regularity result on the solutions of (P ).

Lemma 2.3. Let uα ∈ W 1,p0 (Ω) be a critical point of Jα. Then uα ∈ C1(Ω).

Proof. Set Kα(x) = sign(uα)(g(uα) + |uα|p∗−2uα)/(1 + |uα|p−1). We notice that uα

satisfies

(Q)

−div ((α + |∇u|2)(p−2)/2∇u) = Kα(x)|u|p−2u + sign(uα)Kα(x) on Ωu = 0 on ∂Ω.

From (g) we deduce that

|Kα(x)| ≤ C1|uα|p∗−p + C2

where C1, C2 are positive constants. As p∗−p = p2/(N−p), and uα ∈ LNp/(N−p)(Ω), thenKα ∈ LN/p(Ω). By Theorem 2.1, we have uα ∈ ∩1≤t<+∞Lt(Ω) and for any t ∈ [1, +∞[there exists a constant Ct depending also on Ω, ‖Kα‖LN/p , N , p such that

(2.2) ‖uα‖Lt(Ω) ≤ Ct.

Here Ct is independent of α ∈ (0, 1]. Moreover by Theorem 2.2, as Kα(x) ∈ Ls(Ω) withs > N/p, by a straightforward application of Moses’s iterative scheme [18], we can derivethat uα ∈ L∞(Ω) and there exists C = C(N, p, |Ω|) such that

(2.3) ‖uα‖L∞(Ω) ≤ C‖Kα‖1/(p−1)Ls(Ω) .

Finally, by the regularity results in [21, 22, 15], we derive that u ∈ C1(Ω). ¤

Now we fix a critical point uα of Jα. By Lemma 2.3, it follows that uα ∈ C1(Ω).As in [5, 6], we introduce a Hilbert space, depending on the critical point uα, in which

W 1,p0 (Ω) is embedded, so that a suitable splitting can be obtained.Precisely, set b(x) = ∇uα(x), let Hα be the closure of C∞

0 (Ω) under the scalar product

(v, w)α =∫Ω[(α + |b(x)|2)(p−2)/2(∇v|∇w)

+(p− 2)(α + |b(x)|2)(p−4)/2(b(x)|∇v)(b(x)|∇w)] dx.

As b ∈ C(Ω), the norm ‖·‖α induced by (·, ·)α is equivalent to the usual norm of W 1,20 (Ω).

Hence Hα is isomorphic to W 1,20 (Ω) and the embedding W 1,p

0 (Ω) → Hα is continuous.

Denoting by 〈·, ·〉 : H∗α ×Hα → R the duality pairing in Hα, J ′′α(uα) can be extended to

the operator Lα : Hα → H∗α defined by setting

〈Lαv, w〉 = (v, w)α + 〈Kv,w〉where 〈Kv, w〉 = − ∫

Ωg′(uα)vw − (p∗ − 1)

∫Ω(uα)p∗−2vw, for any v, w ∈ Hα.

Lemma 2.4. Lα is a compact perturbation of the Riesz isomorphism from Hα to H∗α.

In particular, Lα is a Fredholm operator with index zero.


Proof. In order to prove the assertion it is sufficient to show that K is a compact operatorfrom Hα to H∗

α. Let vn be a bounded sequence in Hα. Then there exists v ∈ Hα suchthat, up to a subsequence, vn converges weakly to v in Hα and strongly in L2(Ω).There is a constant C > 0 such that, for any w ∈ Hα, ‖w‖α = 1 we have

∫

Ω

g′(uα)|vn − v||w|+ (p∗ − 1)

∫

Ω

|uα|p∗−2|vn − v||w| ≤ C(∫

Ω

|vn − v|2 dx)1/2

which tends to zero as n → +∞, uniformly with respect to w. This implies that Kα isa compact operator. ¤

Now let us denote by m(Lα) the maximal dimension of a subspace of Hα on which Lα isnegative definite. Obviously m(Jα, uα) ≤ m(Lα). Furthermore let us denote by m∗(Lα)the sum of m(Lα) and the dimension of the kernel of Lα. By Lemma 2.4 we concludethat m(Lα) and m∗(Lα) are finite.Since Lα is a Fredholm operator in Hα, we can consider the natural splitting

Hα = H− ⊕H0 ⊕H+

where H−, H0, H+ are, respectively, the negative, null and positive spaces, according tothe spectral decomposition of Lα in L2(Ω). Therefore one can easily show that thereexists a γα > 0 such that

〈Lαv, v〉 ≥ γα‖v‖2α ∀v ∈ H+

where m(Lα), m∗(Lα) are respectively the dimensions of H− and H− ⊕H0.Since uα ∈ C1(Ω), we can deduce from standard regularity theory (see Theorem 8.15,8.24 and 8.29 in [11]) that H− ⊕H0 ⊂ W 1,p

0 (Ω) ∩ L∞(Ω).The second order differential of Jα in uα is given by

(2.4) 〈J ′′α(uα)v, w〉 =

∫

Ω

(α + |∇uα|2)(p−2)/2(∇v|∇w) dx

+

∫

Ω

(p− 2)(α + |∇uα|2)(p−4)/2(∇uα|∇v)(∇uα|∇w) dx

−∫

Ω

g′(uα)vw dx− (p∗ − 1)

∫

Ω

|uα|p∗−2vw dx ∀v, w ∈ W 1,p0 (Ω).

Consequently, denoted by W = H+ ∩W 1,p0 (Ω) and V = H− ⊕H0, we get the splitting

W 1,p0 (Ω) = V ⊕W and,

(2.5) 〈J ′′α(uα)v, v〉 ≥ γα‖v‖2α ∀v ∈ W,

so that m(Lα) = m(Jα, uα) and m∗(Lα) = m∗(Jα, uα).

3. Palais-Smale condition

We recall the following definition (see [20]).

Definition 3.1. Let X be a Banach space and f be a C1 real function on X. LetC be a closed subset of X. A sequence (un) in C is a Palais-Smale sequence for f if‖f(un)‖ ≤ M uniformly in n, while ‖f ′(un)‖ → 0 as n → ∞. We say that f satisfies(P.S.) on C if any Palais-Smale sequence in C has a strongly convergent subsequence.


We prove that the functionals Jα and, in some way, Jα|W with α ≥ 0 satisfy a localPalais Smale condition on each level.

In what follows, as usually, S denotes the best Sobolev constant of the embeddingW 1,p

0 (Ω) → Lp∗(Ω) given by S = inf ‖u‖p : u ∈ W 1,p0 (Ω), ‖u‖p∗ = 1.

Lemma 3.2. There exists a R > 0 such that, for any fixed α ≥ 0 and any u ∈ W 1,p0 (Ω),

the functional Jα satisfies (P.S.) condition on BR (u).

Proof. For convenience we fix α ≥ 0 and denote Jα = J . Fixing R ∈ (0, SN/p2

2), if (um) ⊂

BR (u) is a sequence such that J ′(um) → 0, then (um) is bounded, thus converges to

some u ∈ BR (u), weakly in W 1,p0 (Ω) and strongly in each Lr(Ω), with r < p∗. Moreover,

arguing as in Lemma 3.1 in [19], one can prove that (α + |∇um|2) p−22 ∇um converges to

(α + |∇u|2) p−22 ∇u weakly in Lp/(p−1)(Ω) and a.e. in Ω.

Therefore, for any z ∈ W 1,p0 (Ω),

〈J ′(u), z〉 = limm→∞

〈J ′(um), z〉 = 0

so that u is a critical point and, in particular,

(3.1) 〈J ′(um), um〉 − 〈J ′(u), u〉 = o(1).

Using [1] (cf. [20]), we have that

(3.2) ‖um − u‖p∗p∗ = ‖um‖p∗

p∗ − ‖u‖p∗p∗ + o(1).

Analogously we can deduce that

(3.3)

∫Ω(α + |∇um −∇u|2) p−2

2 |∇um −∇u|2 dx

=∫Ω(α + |∇um|2) p−2

2 |∇um|2 dx− ∫Ω(α + |∇u|2) p−2

2 |∇u|2 dx + o(1).

Indeed, by Vitali’s Convergence Theorem, denoting by um,t = um + (t − 1)u, we inferthat∫

Ω(α + |∇um|2)

p−22 |∇um|2 dx− ∫

Ω(α + |∇um −∇u|2) p−2

2 |∇um −∇u|2 dx

=∫Ω

∫ 1

0ddt

[(α + |∇um + (t− 1)∇u|2) p−2

2 |∇um + (t− 1)∇u|2]dt dx

=∫ 1

0

∫Ω

((p− 2) (α + |∇um,t|2)(p−4)/2 |∇um,t|2∇um,t∇u + 2 (α + |∇um,t|2)(p−2)/2∇um,t∇u

)dx dt

tends to∫ 1

0

∫Ω

((p− 2) (α + t2|∇u|2)(p−4)/2 |∇u|4 t3 + 2 (α + t2|∇u|2)(p−2)/2 |∇u|2 t

)dxdt

=∫Ω

∫ 1

0ddt

[(α + t2|∇u|2) p−22 |∇u|2t2] dt dx

=∫Ω

(α + |∇u|2) p−22 |∇u|2 dx.

From (3.1), (3.2) and (3.3) we deduce

(3.4)

∫Ω|∇um −∇u|p dx− ∫

Ω|um − u|p∗ dx

≤ ∫Ω

(α + |∇um −∇u|2) p−22 |∇um −∇u|2 dx− ∫

Ω|um − u|p∗ dx

= 〈J ′(um), um〉 − 〈J ′(u), u〉+ o(1) = o(1).

Denoting a = lim supm→+∞ ‖um − u‖p, by (3.4) and the definition of S we have


a ≤ lim supm→+∞

∫

Ω

|um − u|p∗ ≤ S−p∗/pap∗/p.

Therefore, if a > 0, this implies a ≥ SN/p, hence

SN/p ≤ a ≤ lim supm→+∞

(‖um − u‖+ ‖u− u‖)p ≤ (2R)p < SN/p

which is absurd. Therefore it must be a = 0 and thus um strongly converges to u inW 1,p

0 (Ω). ¤

The next proposition states a sort of (P.S.) condition for Jα in the direction of W .

Proposition 3.3. There exists R > 0 such that, for any fixed α ≥ 0 and u ∈ W 1,p0 (Ω),

if (um) ⊂ BR (u) and supw∈W\0

〈Jα′(um), w〉/‖w‖ → 0, then (um) has a convergent

subsequence.

Proof. Reasoning as in the previous proof, let us fix R ∈ (0, SN/p2

2), α ≥ 0, and denote

Jα = J .

With the same arguments, we infer that (um) weakly converges to some u ∈ BR (u)and, for any z ∈ W 1,p

0 (Ω),

(3.5) 〈J ′(um), z〉 → 〈J ′(u), z〉.Now let e1, . . . em∗ be an L2-orthogonal basis of V , where m∗ = m∗(Jα, uα). Denoting

by PV (z) =∑m∗

i=1

(∫Ω

eiz dx)ei, it is clear that z − PV (z) ∈ W , for any z ∈ W 1,p

0 (Ω).Moreover PV (um) strongly converges to PV (u). Exploiting (3.5) we get

(3.6)〈J ′(um), um〉 − 〈J ′(u), u〉= 〈J ′(um), PV (um)〉 − 〈J ′(u), PV (u)〉+ o(1)= 〈J ′(um), PV (um)− PV (u)〉+ 〈J ′(um), PV (u)〉 − 〈J ′(u), PV (u)〉+ o(1) = o(1)

Consequently, as (3.2) and (3.3) hold too, we can complete this proof in the same wayas the previous one, using (3.6) instead of (3.1). ¤

4. A finite dimensional reduction

We recall an abstract result, contained in [13].

Theorem 4.1. Let Θ : Lp(Ω,Rk)×Lq(Ω,Rm) →]−∞, +∞] be a functional of the form

Θ(u, v) =

∫

Ω

θ(x, u, v) dx

where θ(x, u, v) is a nonnegative, continuous function and θ(x, u, ·) is convex. Then Θis lower discontinuous with respect to the strong convergence of the component u in Lp

and with respect to the weak convergence of the component v in Lq.

First we show that uα is a strict minimum point in the direction of W .


Lemma 4.2. There exists δ > 0 such that for any w ∈ W \ 0 with ‖w‖ ≤ δ, we have

Jα(uα + w) > Jα(uα).

More precisely, there exists µ′ > 0 such that, for any w ∈ W with ‖w‖ ≤ δ, we have

Jα(uα + w)− Jα(uα) ≥ µ′‖w‖p.

Proof. For simplicity, set Jα = J and define h(s) = −g(s)− |s|p∗−2s.There exist a constant c(uα) > 0, depending on the critical point uα, and a constant

d > 0 such that for any x ∈ Ω and for any s ∈ R we have

|h′(s)| ≤ c(uα) + d|s− uα(x)|p∗−2.

Now let us define for any x ∈ Ω and for any s ∈ Rh(x, s) = h(s) +

d

p∗ − 1|s− uα(x)|p∗−2(s− uα(x))

and

H(x, s) = H(s) +d

p∗(p∗ − 1)|s− uα(x)|p∗ .

It is immediate to check that

(4.1) Dsh(x, s) ≥ −c(uα).

Set k(ξ) = 1p|ξ|p for any ξ ∈ RN . Obviously there exist C > 0 such that |k′′(ξ1 + ξ2)| ≤

C(|ξ1|p−2 + |ξ2|p−2) for any ξ1, ξ2 ∈ RN .From (2.5) we can say that there is γ′α > 0 such that

〈J ′′(uα)v, v〉 ≥ γ′α‖v‖2W 1,2

0∀v ∈ W .

Now let us fix µ > 0 such that

(4.2)α

p−22

2− Cµ‖∇uα‖p−2

∞ − Cµαp−22 > 0, γ′α − 2Cµ‖∇uα‖p−2

∞ > 0.

Moreover let us define the functional tµ : W 1,p0 (RN) → R by setting

tµ(z) =µ

p

∫

Ω

|∇z −∇uα|p dx− d

p∗(p∗ − 1)

∫

Ω

|z − uα|p∗ dx, z ∈ W 1,p0 (Ω)

and let us setJ(z) = J(z)− tµ(z), z ∈ W 1,p

0 (Ω).

First, we observe that there exist γ > 0 and µ′ > 0 such that

(4.3) tµ(z) ≥ µ′∫

Ω

|∇z −∇uα|p dx

for any z ∈ W 1,p0 (Ω) such that ‖z − uα‖ ≤ γ.

We shall prove that there exist σ > 0, C > 0 such that for any z ∈ W 1,p0 (Ω) with

‖z − uα‖ ≤ σ we have

(4.4) 〈J ′′(z)v, v〉 ≥ C‖v‖2α ∀v ∈ W.

At this point, we define δ = minγ, σ.Moreover, from the previous relation, for any w ∈ W with ‖w‖ ≤ δ there exists

ϑ ∈ (0, 1) such that

(4.5) J(uα + w)− J(uα) =1

2〈J ′′(uα + ϑw)w, w〉 ≥ C‖w‖2

α.


Hence, by (4.3) and (4.5), for any w ∈ W with ‖w‖ ≤ δ, we have

J(uα + w)− J(uα) = J(uα + w)− J(uα) + tµ(uα + w)− tµ(uα)

≥ C‖w‖2α + µ′‖w‖p.

It remains to prove (4.4). By way of contradiction, we assume that there exist twosequences zn ∈ W 1,p

0 (Ω) and vn ∈ W , such that ‖vn‖W 1,20

= 1, ‖zn − uα‖ → 0 and

(4.6) lim infn→∞

〈J ′′(zn)vn, vn〉 ≤ 0.

As vn is bounded in Hα, there exists v ∈ W such that vn converges to v weakly inHα and strongly in L2(Ω), up to subsequences. First we prove that v 6= 0. In fact, if weassume that v = 0, then by (4.1) we have

(4.7)〈J ′′(zn)vn, vn〉 =

∫Ω(α + |∇zn|2)(p−2)/2|∇vn|2 +

∫Ω(p− 2)(α + |∇zn|2)(p−4)/2(∇zn|∇vn)2

−µ∫

Ω(k′′(∇zn −∇uα)∇vn|∇vn) +

∫Ω

h′(x, zn)v2n

≥ αp−22 /2 + (1/2− Cµ)

∫Ω|∇zn|p−2|∇vn|2 dx − Cµ‖∇uα‖p−2

∞∫Ω|∇vn|2 dx

−c(uα)∫Ω

v2n ≥ α

p−22 /2− Cµ‖∇uα‖p−2

∞ − c(uα)∫Ω

v2n dx.

Since vn → 0 in L2(Ω) and (4.2) holds, we derive that (4.7) contradicts (4.6) as n → +∞,so that v 6= 0.Furthermore by (4.1), we also infer that

(4.8)

∫

Ω

h′(uα)v2 dx ≤ lim infn→∞

∫

Ω

h′(x, zn)v2n.

Applying Theorem 4.1, by (4.8) and (4.2) we have

0 ≥ lim infn→∞〈J ′′(zn)vn, vn〉≥ lim infn→∞(

∫Ω

((α + |∇zn|2)(p−2)/2 − |∇zn|p−2/2

) |∇vn|2+

∫Ω(p− 2)(α + |∇zn|2)(p−4)/2(∇zn|∇vn)2 + (1/2− Cµ)

∫Ω|∇zn|p−2|∇vn|2

−Cµ∫Ω|∇uα|p−2|∇vn|2 +

∫Ω

h′(x, zn)v2n )

≥ ∫Ω(α + |∇uα|2)(p−2)/2|∇v|2 +

∫Ω(p− 2)(α + |∇uα|2)(p−4)/2(∇uα|∇v)2

−2Cµ‖∇uα‖p−2∞

∫Ω|∇v|2 +

∫Ω

h′(uα)v2

= 〈J ′′(uα)v, v〉 − 2Cµ‖∇uα‖p−2∞

∫Ω|∇v|2 ≥ (γ′α − 2Cµ‖∇uα‖p−2

∞ )‖v‖2W 1,2

0

which is a contradiction. ¤

Now in order to carry out a finite dimensional reduction, we need some preliminaryresults.

Lemma 4.3. For any M > 0 there exist r0 > 0 and C > 0 such that for any z ∈W 1,p

0 (Ω) ∩ L∞(Ω), with ‖z‖∞ ≤ M , ‖z − uα‖ < r0, we have

(4.9) 〈J ′′α(z)w, w〉 ≥ C‖w‖2α ∀w ∈ W.

Proof. The thesis follows arguing similarly to the proof of (4.4). ¤


Lemma 4.4. Let a > 0. If z ∈ Ba (uα) is a solution of 〈J ′α(z), w〉 = 0 for any w ∈ W ,then z ∈ L∞(Ω). Moreover there exists M > 0 such that ‖z‖∞ ≤ M , M depending juston a.

Proof. Let z ∈ Ba (uα) be a solution of 〈J ′α(z), w〉 = 0 for any w ∈ W . Let e1, . . . em∗be an L2-orthogonal basis of V , where m∗ = m∗(Jα, uα). For any v ∈ W 1,p

0 (Ω) we can

choose v − ∑m∗i=1

(∫Ω

eiv dx)ei, as test function in 〈J ′α(z), w〉 = 0. So we derive that z

solves the equation

〈J ′α(z), v〉 =

∫

Ω

r(x)v dx

where r(x) =∑m∗

i=1

(∫Ω(α+ |∇z|2)(p−2)/2(∇z|∇ei)−g(z)ei−|z|p∗−2zei

)ei. As V ⊂ L∞(Ω)

we have r(x) ∈ L∞(Ω) and ‖r‖∞ ≤ C, where C is a positive constant depending on a.Now, arguing as in the proof of Lemma 2.3, we infer that z ∈ L∞(Ω) and the analogousof (2.3) holds, so that ‖z‖∞ ≤ M for a suitable positive constant M depending only ona. ¤

Lemma 4.5. For any δ′ ∈ (0, δ) there exists % = %(δ′) ∈ (0, δ′) such that, for each

v ∈ V ∩ B% (0), we have

(4.10) infJα(uα + v + w) : w ∈ W, ‖w‖ ∈ [δ′, δ] > Jα(uα + v).

Proof. Let us fix δ′ ∈ (0, δ). Arguing by way of contradiction, we assume that thereexist two sequences vn ∈ V and wn ∈ W such that vn → 0, ‖wn‖ ∈ [δ′, δ] and

(4.11) Jα(uα + vn + wn) ≤ Jα(uα + vn).

By Lemma 4.2, we have that, for any n ∈ N(4.12) µ′δ′p ≤ µ′‖wn‖p ≤ Jα(uα + wn)− Jα(uα).

Moreover, as vn → 0 strongly in W 1,p0 (Ω) and Jα is continuous, we have

(4.13) Jα(uα + vn) = Jα(uα) + o(1).

Furthermore there exists a sequence βn ∈ (0, 1) such that

(4.14) |Jα(uα + vn + wn)− Jα(uα + wn)| ≤ ‖J ′α(uα + vn + βnwn)‖‖vn‖ = o(1).

By (4.11), (4.12), (4.13), (4.14), we have

µ′δ′p ≤ Jα(uα + wn + vn)− Jα(uα + vn) + o(1) ≤ o(1)

which is a contradiction. ¤Now we can apply Ekeland Principle to derive the existence of a local minimum along

the direction of W .

Lemma 4.6. There exist r ∈ (0, δ) and % ∈ (0, r) such that for any v ∈ V ∩B% (0) there

exists one and only one w ∈ W ∩ Br (0) ∩ L∞(Ω) such that for any z ∈ W ∩ Br (0) wehave

Jα(uα + v + w) ≤ Jα(uα + v + z).

Moreover w is the only element of W ∩ Br (0) such that

〈J ′α(uα + v + w), z〉 = 0 ∀z ∈ W.


Proof. For simplicity set Jα = J . Let R > 0 be defined by Proposition 3.3.By Lemma 4.4 there is a positive constant M > 0, depending just by R, such that,if z ∈ BR (uα) is a solution of 〈J ′(z), w〉 = 0, for all w ∈ W , then z ∈ L∞(Ω) and‖z‖∞ ≤ M .By Lemma 4.3, in correspondence of 2M , there exists r0 such that (4.9) holds. Nowlet r = minδ/2, R/2, r0/3 and % = %(r), where δ and %(r) are respectively defined byLemma 4.2 and Lemma 4.5. Observe that

(uα +

(V ∩ B% (0)

)+

(W ∩ Br (0)

))⊂ BR(uα).

By Ekeland Principle and Lemma 4.5, for any v ∈ V ∩ B% (0) there exists a sequencewn ∈ W ∩Br(0) such that

J(uα + v + wn) → infw∈W∩Br(0)

J(uα + v + w)

and

supw∈W\0

〈Jα′(uα + v + wn), w〉

‖w‖ → 0 n → +∞.

Proposition 3.3 assures that wn strongly converges, up to subsequences. Moreover, de-noting with w(v) the limit of wn, 〈J ′(uα + v + w(v)), z〉 = 0 for any z ∈ W , so that

‖uα + v + w(v)‖∞ ≤ M , independently of v ∈ V ∩ B% (0) . We conclude that, for any

v ∈ V ∩ B% (0), there exists a minimum point w ∈ W ∩ Br (0) ∩ L∞(Ω) of the function

w ∈ W ∩ Bδ (0) 7→ J(uα + v + w). We can also show that w is the unique minimum

point. Actually, we even prove that w is the only element of W ∩ Br (0) such that

(4.15) 〈J ′α(uα + v + w), z〉 = 0 ∀z ∈ W.

In fact, if w1 and w2 ∈ W ∩ Br (0) verify (4.15), then they belong to L∞(Ω) and‖w1‖∞ ≤ M , ‖w2‖∞ ≤ M . We notice that, for all t ∈ [0, 1],

‖v + w2 + t(w1 − w2)‖ < 3r ≤ r0 and ‖uα + v + w2 + t(w1 − w2)‖∞ ≤ 2M.

Therefore by (4.9) we have

0 = 〈J ′α(uα + v + w1)− J ′α(uα + v + w2), w1 − w2〉=

∫ 1

0〈J ′′α(uα + v + w2 + t(w1 − w2))(w1 − w2), w1 − w2〉 dt ≥ C‖w1 − w2‖2

α

so that w1 = w2. ¤

Now we can introduce the map

(4.16) Ψα : V ∩ B% (0) → W ∩ Br (0)

where Ψα(v) is such element w. We also define the function

(4.17) Φα : V ∩ B% (0) → R, Φα(v) = Jα(uα + v + Ψα(v)).

In Appendix, we will prove that

(4.18) Cj(Jα, uα) = Cj(Φα, 0)


Proof of Theorem 1.3. As uα is nondegenerate, we have H0 = 0 and there is asuitable constant µ > 0 such that

〈J ′′α(uα)v, v〉 ≤ −µ‖v‖2 for any v ∈ V.

As a consequence, uα is a local isolated maximum of Jα along V , thus 0 is a local isolatedmaximum of Φα in V ∩ B% (0) and by (4.18) the assert comes (see [3, Example 2]). ¤

In order to prove Theorem 1.4, we recall Corollary 6.4 proved by Lancelotti in [14].

Theorem 4.7. Let X be a Banach space, f : X → R a continuous function and V asubspace of X of finite dimension m. We assume that:

i) for every u ∈ X, the function f is of class C2 on u + V and for every v ∈ V thefunctions u 7→ 〈f ′(u), v〉 and u 7→ 〈f ′′(u)v, v〉 are continuous on X;

ii) 〈f ′′(u)v, v〉 < 0 for every v ∈ V \ 0.Then we have Cq(f, u) = 0 for every q ≤ m− 1.

Proof of Theorem 1.4. By (4.18), it is clear that Cj(Jα, uα) = 0 if j > dim V .Moreover, by Theorem 4.7, we have Cj(Jα, uα) = 0 for any j < m(Jα, uα).

5. Appendix: proof of (4.18)

In this appendix we will denote by c = Jα(uα).We first need the regularity of Ψα introduced in (4.16).

Proposition 5.1. Ψα is a continuous function.

Proof. Let (vn) ⊂ V ∩ B% (0) be a convergent sequence, and let v be its limit. ByProposition 3.3 Ψα(vn) → w ∈ W ∩ Br (0). Moreover 〈J ′(uα + v + w), z〉 = 0 for anyz ∈ W , thus Lemma 4.6 assures that w = Ψ(v).

¤Moreover we need the following technical Lemma.

Lemma 5.2. For any %′ ∈ (0, %], let us introduce the set

M%′ = uα+v+(1−t)w+tΨα(v) : v ∈ V ∩B%′ (0), w ∈ W∩Br (0), Jα(uα+v+w) ≤ c, t ∈ [0, 1].There exists %′ ∈ (0, %] such that, for any uα + v + w ∈ M%′ and for any z ∈ W with

‖z‖ = r, we have thatJα(uα + v + w) < Jα(uα + v + z).

Proof. Arguing by way of contradiction, let (vn) ⊂ V , (wn) ⊂ W ∩ Br (0), (zn) ⊂ W ,(tn) ⊂ [0, 1] be such that vn → 0, Jα(uα + vn + wn) ≤ c, ‖zn‖ = r and

(5.1) Jα (uα + vn + (1− tn)wn + tnΨα(vn)) ≥ Jα(uα + vn + zn).

By Lemma 4.2 wn converges to zero. In fact, there exists µ′ such that, for any n ∈ N,Jα(uα + wn) ≥ Jα(uα) + µ′‖wn‖p, moreover |Jα(uα + vn + wn)− Jα(uα + wn)| = o(1), sothat

c + µ′‖wn‖p + o(1) ≤ Jα(uα + wn) + o(1) = Jα(uα + vn + wn) ≤ c

hence wn → 0. Analogously

(5.2) c + µ′rp + o(n) ≤ Jα(uα + vn + zn).


Consequently uα + vn + (1− tn)wn + tnΨα(vn) → uα and, applying (5.1) and (5.2),

c + µ′rp + o(1) ≤ Jα(uα + vn + zn)≤ Jα (uα + vn + (1− tn)wn + tnΨα(vn)) = c + o(1)

which is absurd.¤

First, fixing %′ ∈ (0, %] such that the previous Lemma is satisfied, we introduce thefollowing sets

Y = uα + v + Ψα(v) : v ∈ V ∩ B%′ (0)

M = M%′

(Φα)c = v ∈ V ∩ B%′ (0) : Jα(uα + v + Ψα(v)) ≤ c

(Jα|Y )c = uα + v + Ψα(v) : v ∈ V ∩ B%′ (0), Jα(uα + v + Ψα(v)) ≤ c

U =(uα +

(V ∩ B%′ (0)

)+

(W ∩ Br (0)

)).

Since Ψα is a continuous map and Ψα(0) = 0, the topological pair ((Φα)c, (Φα)c \ 0)is homeomorphic to ((Jα|Y )c, (Jα|Y )c \ uα). Therefore, according to Definition 1.1, forany integer j we have

Cj(Φα, 0) = Cj(Jα|Y , uα).

The aim of this section will be achieved if we show that the topological pair

((Jα|Y )c, (Jα|Y )c \ uα)is a deformation retract of

((Jα)c ∩ U, (Jα)c ∩ U \ uα.)The crucial point is to prove the following result.

Lemma 5.3. There exists a continuous function r : M → (Jα)c ∩ U such that

(a) for any z ∈ M : r(z)− z ∈ W(b) for any z ∈ (Jα)c ∩ U : r(z) = z.

Proof. Let us introduce the continuous function

β : W 1,p0 (Ω) → R, β(z) = inf

w∈W\0〈J ′α(z), w〉‖w‖

and the set

Z∗ = z ∈ W 1,p0 (Ω) : β(z) 6= 0 .

From standard arguments concerning the construction of a pseudogradient vector field(see e.g. [2]) we infer that there exists a continuous vector field X : Z∗ → W such that,for all z ∈ Z∗,

(1) ‖X(z)‖ ≤ 2β(z),


(2) 〈J ′α(z), X(z)〉 ≥ β2(z).

The existence of X gives a decreasing flow for Jα in the direction of W . In fact thefollowing Cauchy problem

(Pz)

σ(t) = −X(σ(t))

σ(0) = z

is locally solvable for any z ∈ Z∗ and the function t 7→ Jα (σ(t)) is decreasing as

(5.3)d

dtJα(σ(t)) = 〈J ′α(σ(t)), σ(t)〉 = −〈J ′α(σ(t)), X(σ(t))〉 < −β2(σ(t)) .

Now let z0 be an element of M ∩ J−1α [c, +∞) \ Y , so that the Cauchy problem (Pz0)

is locally solvable. We first note that Lemma 5.2 and (5.3) assure that σ(t, z0) ∈ U , forany t in which the solution σ(t, z0) to (Pz0) is defined.

It can be proved that there exists Tz0 ≥ 0 such that σ(t, z0) is defined at least in [0, Tz0 ]and Jα(σ(Tz0 , z0)) = c. In fact, if not, denoting by T the maximal existence interval forthe initial data z0, we have limt→T− Jα(σ(t, z0)) > c. By Proposition 3.3 there is ε > 0such that, for any t,

β(σ(t, z0)) > ε,

hence by (5.3)

(5.4) c− Jα(z0) ≤∫ t

0

d

dsJα(σ(t, z0))ds < −ε2t

so that T < Jα(z0)−cε2 .

Moreover for any t1 < t2 we get

(5.5) ‖σ(t2)− σ(t1)‖ ≤∫ t2

t1

‖σ(t)‖ dt ≤ 2

∫ t2

t1

β(σ(t)) dt ≤√

(t2 − t1)(c− Jα(z0)

).

This implies that σ(T ) exists and is not a critical point, hence the flow can be extendedbeyond T , contradicting the maximality.

Now, denoting by A the set of all (t, z) ∈ R × Z∗ such that the solution to (Pz) isdefined in t, we see that the function (t, z) ∈ A 7→ Jα(σ(t, z)) ∈ R is C1 and, by (5.3),

∂

∂tJα(σ(Tz0 , z0)) < −β2(σ(Tz0 , z0)) < 0,

so continuity of z 7→ Tz in z0 is assured by Implicit Function Theorem.Hence we are ready to define the function r : M → (Jα)c ∩ U given by

r(z) =

z if z ∈ (Jα)c

σ(Tz, z) if z /∈ (Jα)c

which verifies (a) and (b).


It is clear that r is continuous in the interior of (Jα)c and, by ODE theory, also inM ∩ J−1

α [c, +∞) \ Y , thus it remains only to verify continuity of r in J−1α c ∩ Y .

To this aim, let us fix z0 ∈ J−1α c∩Y and let (zn) ⊂ M be a sequence converging to z0.

If zn belongs to (Jα)c, then the assert easily comes, so let us suppose zn /∈ (Jα)c. We willprove now that in this case β(r(zn)) → 0, so that the assert comes from Proposition 3.3.

Indeed, arguing by contradiction, suppose that

(5.6) β(r(zn)) is not infinitesimal,

then, again by Proposition 3.3, there exists ε0 > 0 such that β(σ(t, zn)) > ε0 for anyn ∈ N and t ∈ [0, Tzn ]. Reasoning as in (5.4) and (5.5) we have that

c− Jα(zn) < −ε20 Tzn and ‖r(zn)− zn‖ ≤ 2

√Tzn (Jα(zn)− c) ,

thus

‖r(zn)− zn‖ ≤ 2Jα(zn)− c

ε0

which gives that r(zn) → z0, in contradiction with (5.6). ¤Corollary 5.4. ((Jα|Y )c, (Jα|Y )c \ uα) is a deformation retract of ((Jα)c ∩ U, (Jα)c ∩ U \ uα) .

Proof. It immediately follows from the previous Lemma defining

η : [0, 1]×(Jα)c∩U → (Jα)c∩U, η(t, uα+v+w) = r (uα + v + (1− t)w + tΨα(v)) .

¤

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Acknowledgment. We would like to thank Prof. Marco Degiovanni for useful discus-sions.

Dipartimento di Matematica, Politecnico di Bari, Via Amendola 126/B, 70126 Bari,Italy

E-mail address: [email protected]

Dipartimento di Matematica, Politecnico di Bari, Via Amendola 126/B, 70126 Bari,Italy

E-mail address: [email protected]

Morse index and critical groups for p-Laplace equations with critical exponents

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