Morse index estimates for quasilinear equations on Riemannian manifolds

MORSE INDEX ESTIMATES FOR QUASILINEAR EQUATIONS ONRIEMANNIAN MANIFOLDS

SILVIA CINGOLANI, GIUSEPPINA VANNELLA, AND DANIELA VISETTI

Abstract. The work deals with Morse index estimates for a solution u ∈ Hp1 (M)

of the quasilinear elliptic equation −divg

((α + |∇u|g2)(p−2)/2∇u

)= h(x, u), where

(M, g) is a compact, Riemannian manifold, 0 < α, 2 ≤ p < n. The right hand sidenonlinear term h(x, s) is allowed to be subcritical or critical.

2000 Mathematics Subject Classification: 58E05, 35B20, 35J60, 35J70.Key words: Quasilinear elliptic equations, Riemannian Manifold, Morse index

1. Introduction

In the recent years there was an increasing interest toward nonlinear differential prob-lems on manifolds. Several questions give rise to sophisticated analysis where the geome-try of the manifold plays a central role. We refer the reader to [13] for an overview on thetopic. In this work we consider (M, g) a compact, connected, orientable, boundarylessRiemannian manifold of class C∞ with Riemannian metric g and dim(M) = n ≥ 3. ByNash’s theorem [18], M can be isometrically embedded in RN , with N ≥ 2n, as a regularsub-manifold. Let Hp

1 (M) be the Sobolev space defined as the completion of C∞(M)with respect to the norm ‖u‖1,p = ‖∇u‖p +‖u‖p , where ‖·‖p is the usual norm of Lp(M)(see Section 2). We are concerned with Morse index estimates for the solutions of thefollowing quasilinear problem

(1.1)

−divg

((α + |∇u|g2)(p−2)/2∇u

)= h(x, u)

u ∈ Hp1 (M)

where 0 < α, 2 ≤ p < n. Here h : M × R→ R satisfies the assumption:

• (h) for any s ∈ R, h(·, s) is continuous on M , h(x, ·) is C1 on R and ∀(x, s) ∈M×R, |Dsh(x, s)| ≤ c1|s|p∗−2+c2, with c1, c2 positive constants, p∗ = pn/(n−p).

The above differential problem involves the p-Laplacian operator on (M, g) which isdefined as a natural extension of the Laplace-Beltrami operator corresponding to p = 2and arises in several physical contexts, for instance fluid dynamics, nonlinear elasticityand glaciology (see [1, 10]).

In a Euclidian context, some results in [5, 6] have been obtained concerning the com-putation of critical groups for solutions of quasilinear equations with Dirichlet boundaryvalue conditions with autonomous right hand side nonlinear term via differential notionslike the Morse index (see also [7, 8] for applications). The main difficulty is the fact thatthe variational setting is Banach (not Hilbert), so that every solution of problem (1.1) isdegenerate in the classical sense given in a Hilbert space and the classical Morse lemma

The research of the first and second author is supported by the MIUR project “Variational andtopological methods in the study of nonlinear phenomena” (PRIN 2007).

1

2 SILVIA CINGOLANI, GIUSEPPINA VANNELLA, AND DANIELA VISETTI

does not hold. Moreover one lacks Fredholm properties of the second derivatives of theEuler functional associated to (1.1), and also generalized versions of Morse lemma, dueto Gromoll and Meyer, fail (see [4]).

As far as we know, the critical group estimates for solutions of quasilinear problemson a Riemannian manifold (M, g) are an open problem (for p = 2 we refer to Theorem4.1 in [4]). Regularity results, developed by Druet in [11], have furnished us with thestarting point to perform a Morse Lemma for quasilinear equations on manifold. Thefirst step is to introduce a weighted Hilbert space, depending on the solution, in whichthe variational setting Hp

1 (M) can be embedded. In this way the second derivatives ofthe Euler functional associated to problem (1.1) can be extended to a Fredholm operator,which provides a natural splitting of the space with respect to its spectral decompositionin L2(M). By means of a priori C1-bounds of solutions along a suitable direction, one canregain a finite-dimensional reduction and the critical group estimates for each solutionto (1.1) via differential notions.

The main results of the work are Theorems 2.1 and 2.2, which cover non-autonomousright hand side nonlinear terms, having subcritical or critical growths. We emphasizethat when h(x, ·) has a critical growth, problem (1.1) is strictly related to the studyof generalized scalar curvature equations (see [9]). In such a critical case, in order toperform reduction arguments, we prove local weakly semicontinuous properties of theEuler functional associated to (1.1), by means of a generalization of the Lindqvist’sinequality [16]). Moreover a local Palais-Smale condition is established for each level insmall balls of Hp

1 (M).Finally we mention the papers [2, 21] where multiplicity results for semilinear elliptic

equations on a Riemannian manifold are obtained via Morse techniques.

2. Definitions, recalls and main results

Any solution u ∈ Hp1 (M) corresponds to a critical point of the Euler functional Jα :

Hp1 (M) → R, defined by

(2.1) Jα(u) =

∫

M

1

p

(α + |∇u(x)|2g

) p2 dµg −

∫

M

H(x, u(x)) dµg

where H(x, s) =∫ s

0h(x, t) dt .

In what follows, the Morse index of Jα in u, denoted by m(Jα, u), is defined as thesupremum of the dimensions of the subspaces of Hp

1 (M) on which J ′′α(u) is negativedefinite. Moreover, the large Morse index of Jα in u, denoted by m∗(Jα, u), is defined asthe sum of m(Jα, u) and the dimension of the kernel of J ′′α(u).

We need to introduce the following definitions (cf. [4]). Let K be a field. We call

(2.2) Cq(Jα, u) = Hq(J cα ∩ U, (J c

α \ u) ∩ U)

the q-th critical group of Jα at u, q = 0, 1, 2, . . . , where U is a neighborhood of u,Jα(u) = c, J c

α = v ∈ Hp1 (M) | Jα(v) ≤ c and Hq(A,B) stands for the q-th Alexander-

Spanier cohomology group of the pair (A,B) with coefficients in K (cf. [4]).

Now we state the main results of the work.

QUASILINEAR EQUATIONS ON RIEMANNIAN MANIFOLDS 3

Theorem 2.1. Let u0 be a critical point of the functional Jα such that J ′′α(u0) is injective.Then m(Jα, u0) is finite and

Cq(Jα, u0) ∼= K, if q = m(Jα, u0),

Cq(Jα, u0) = 0, if q 6= m(Jα, u0).

This theorem extends a classical result in Hilbert spaces for nondegenerate criticalpoints (cf. Theorem 4.1 in [4]), showing that the critical groups of Jα in u0 depend onlyon its Morse index. We conclude with a quantitative result for a possibly degeneratecritical point.

Theorem 2.2. Let u0 be an isolated critical point of the functional Jα. Then m(Jα, u0)and m∗(Jα, u0) are finite and

Cq(Jα, u0) = 0for any q ≤ m(Jα, u0)− 1 and q ≥ m∗(Jα, u0) + 1.

We recall some definitions and results about compact connected Riemannian manifoldsof class C∞ (see for example [13]).

We denote by B(0, R) the ball in Rn of center 0 and radius R and by Bg(x,R) theball in M of center x and radius R.

On the tangent bundle TM of M the exponential map exp : TM → M is defined.This map has the following properties:

(i) exp is of class C∞;(ii) there exists a constant R > 0 such that

expx |B(0,R) : B(0, R) → Bg(x,R)

is a diffeomorphism for all x ∈ M .

It is possible to choose an atlas C on M , whose charts are given by the exponentialmap (normal coordinates). We denote by ψCC∈C a partition of unity subordinate tothe atlas C. Let gx0 be the Riemannian metric in the normal coordinates of the mapexpx0

.For any u ∈ Hp

1 (M) we have that

‖∇u‖pp =

∫

M

|∇u(x)|pgdµg =∑C∈C

∫

C

ψC(x)|∇u(x)|pgdµg.

In particular, if u has support inside one chart C = Bg(x0, R), then setting u(z) =u(expx0

(z)) we have

‖∇u‖pp =

∫

B(0,R)

n∑i,j=1

(gij

x0(z)

∂u(z))

∂zi

∂u(z))

∂zj

) p2

|gx0(z)| 12 dz ,

where |gx0(z)| = det(gx0 ij(z)) and (g ijx0

(z)) is the inverse matrix of gx0(z). In particularwe have that gx0(0) = Id.

A similar relation holds for the integration of (α + |∇u(x)|2) p2 .


We also recall that, for any u ∈ Hp1 (M)

‖u‖1,p =

(∫

M

|∇u(x)|pgdµg

) 1p

+

(∫

M

|u(x)|pdµg

) 1p

.

Moreover, by Theorem 4.1 in [13], there exists A >> 0 such that for any u ∈ Hp1 (M)

(2.3) (

∫

M

|u|p∗dµg)1/p∗ ≤ A(

∫

M

|∇u|pgdµg)1/p + V ol

−1/n(M,g)(

∫

M

|u|pdµg)1/p.

3. Properties of Jα

For any α ∈ [0, 1], the functional Jα is C2 and for any u, v, w ∈ Hp1 (M) we have

〈J ′α(u), v〉 =

∫

M

(α + |∇u(x)|2g

) p−22 (∇u(x)|∇v(x))g dµg

−∫

M

h (x, u(x)) v(x) dµg;

〈J ′′α(u)v, w〉 =

∫

M

(α + |∇u(x)|2g

) p−22 (∇v(x)|∇w(x))g dµg

+ (p− 2)

∫

M

(α + |∇u(x)|2g

) p−42 (∇u(x)|∇v(x))g(∇u(x)|∇w(x))g dµg

−∫

M

Dsh(x, u(x))v(x)w(x) dµg.

We now prove that, in spite of the possible critical growth of h, Jα is locally weaklylower semicontinuous.

In next preliminary lemma, if B is an n × n real, symmetric and positive definitematrix, let us denote by (·|·)B the inner product in Rn defined by

(x|y)B = x>By =n∑

i,j=1

bij xiyj ∀x, y ∈ Rn.

We begin to prove the following inequality (the case α = 0 and B = I has been provedby Lindqvist in Lemma 4.2 of [16]).

Lemma 3.1. Let B be an n× n real, symmetric and positive definite matrix, (·|·)B and| · |B the corresponding inner product and norm in Rn. For any α ≥ 0, p ≥ 2 andx, y ∈ Rn

(α + |y|B2)p/2 ≥ (α + |x|B2)p/2 + p(α + |x|B2)(p−2)/2(x|y − x)B +|y − x|Bp

2p−1 − 1.

Proof. Fix x, y ∈ Rn, x 6= y. We define the function f(t) = (α + |x + t(y − x)|B2)p/2,t ≥ 0. Since the map f is strictly convex, we have f(1) > f(0) + f ′(0), that is

(3.1) (α + |y|B2)p/2 > (α + |x|B2)p/2 + p(α + |x|B2)(p−2)/2(x|y − x)B.

Replacing y by x+y2

in (3.1), we have

(3.2)

(α +

∣∣∣∣x + y

2

∣∣∣∣B

2

)p/2

> (α + |x|B2)p/2 +p

2(α + |x|B2)(p−2)/2(x|y − x)B.


Now we derive a generalized Clarkson’s inequality (see, for instance, Theorem IV.10[3]) when α = 0). Recalling that ap/2 + bp/2 ≤ (a + b)p/2 for any a, b ≥ 0, we have

(3.3)

(α +

∣∣∣∣x + y

2

∣∣∣∣B

2

)p/2

+

(∣∣∣∣x− y

2

∣∣∣∣B

2

)p/2

≤(

α +

∣∣∣∣x + y

2

∣∣∣∣B

2 +

∣∣∣∣x− y

2

∣∣∣∣B

2

)p/2

= (α +1

2(|x|B2 + |y|B2))p/2 ≤ 1

2

((α + |x|B2)p/2 + (α + |y|B2)p/2

)

where the last inequality follows from the fact that t ∈ [0, +∞[7→ (α + t)p/2 is convex.Now putting together (3.2), (3.3), we derive

(3.4) (α + |y|B2)p/2 > −(α + |x|B2)p/2 + 2

(α +

∣∣∣∣x + y

2

∣∣∣∣B

2

)p/2

+ 2

∣∣∣∣x− y

2

∣∣∣∣B

p

≥ (α + |x|B2)p/2 + p (α + |x|B2)(p−2)/2 (x|y − x)B + 2

∣∣∣∣y − x

2

∣∣∣∣B

p

.

This is the desired inequality with the constant 21−p in place of 12p−1−1

.

Replacing y by x+y2

in (3.4) and combining again with (3.3), we get the constantimproved to 21−p+41−p. By iteration we finally find the constant 21−p+41−p+81−p+... =

12p−1−1

. ¤

Proposition 3.2. There exists R > 0 such that, for any fixed α ≥ 0 and any u ∈ Hp1 (M),

the functional Jα is weakly lower semicontinuous in BR(u) = v ∈ Hp1 (M) | ‖v− u‖1,p ≤

R.Proof. Let R > 0, u ∈ Hp

1 (M). Let uk a sequence in Hp1 (M) such that ‖uk − u‖1,p ≤ R.

Assume that uk weakly converges to u.We will show that Jα(u) ≤ lim infk Jα(uk), if R is chosen small enough.Since uk → u weakly in Hp

1 (M), then uk → u strongly in Lr(M) with r < p∗ anduk(x) → u(x) a.e. in M .

By Lemma 3.1, we have

(3.5)

∫M

(α + |∇uk|g2)p/2 dµg −∫

M(α + |∇u|g2)p/2 dµg

≥ p∫

M(α + |∇u|g2)(p−2)/2 (∇u|∇uk −∇u)g dµg + 1

2p−1−1

∫M|∇uk −∇u|pg dµg.

Moreover, since uk → u strongly in Lr(M) with r < p∗ and uk(x) → u(x) a.e. in M ,we have

(3.6)

∫

M

H(x, uk) dµg −∫

M

H(x, uk − u) dµg =

∫

M

H(x, u) dµg + o(1).

Indeed, using Vitali’s theorem, it results

(3.7)

∫M

H(x, uk) dµg −∫

MH(x, uk − u) dµg

=∫

M

∫01 d

dtH(x, uk + (t− 1)u) dt dµg =

∫01∫

Mh(x, uk + (t− 1)u)u dµg dt

tends to

(3.8)

∫

0

1

∫

M

h(x, tu)u dµg dt =

∫

M

∫

0

1d

dtH(x, tu) dt dµg =

∫

M

H(x, u) dµg.

By (3.5) and (3.6) we have


(3.9)

Jα(uk)− Jα(u)≥ 1

p

∫M|∇u|p−2

g (∇u|(∇uk −∇u))g dµg + 1(2p−1−1) p

∫M|∇uk −∇u|pg

− ∫M

H(x, uk − u) dµg + o(1)≥ 1

(2p−1−1) p

∫M|∇uk −∇u|pg − C

∫M|uk − u|p∗ dµg + o(1)

≥ ‖uk − u‖p1,p

(1

(2p−1−1) p− C Kp‖uk − u‖p∗−p

1,p

)+ o(1).

where K = maxA, V ol−1/n(M,g) (see (2.3)). By choosing R > 0 small enough, we derive

that ‖uk − u‖p∗−p1,p < 1/(C Kp), which implies

lim infk→+∞

Jα(uk) ≥ Jα(u).

¤

We now prove that a local Palais-Smale condition holds for Jα.

Proposition 3.3. There exists R > 0 such that, for any fixed α ≥ 0 and any u ∈ Hp1 (M),

the functional Jα satisfies (P.S.) condition on BR(u) = v ∈ Hp1 (M) | ‖v − u‖1,p ≤ R.

Proof. Let R > 0. For convenience, fixed α ≥ 0, we denote Jα = J . Let (um) ⊂ BR(u)be a sequence such that J ′(um) → 0, then (um) is bounded, thus converges to someu ∈ BR(u), weakly in Hp

1 (M) and strongly in each Lr(M), 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))N norm and a.e. in M .

Therefore, for any z ∈ Hp1 (M),

〈J ′(u), z〉 = limm→∞

〈J ′(um), z〉 = 0

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

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

We can show that

(3.11)

∫M

(α + |∇um −∇u|g2)p−22 |∇um −∇u|g2 dµg

=∫

M(α + |∇um|g2)

p−22 |∇um|g2 dµg −

∫M

(α + |∇u|g2)p−22 |∇u|g2 dµg + o(1).

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

∫M

(α + |∇um|g2)p−22 |∇um|g2 dµg −

∫M

(α + |∇um −∇u|g2)p−22 |∇um −∇u|g2 dµg

=∫

M

∫01 d

dt

[(α + |∇um + (t− 1)∇u|g2)

p−22 |∇um + (t− 1)∇u|g2

]dt dµg

=∫01∫

M(p− 2) (α + |∇um,t|g2)(p−4)/2 |∇um,t|g2 (∇um,t|∇u)g

+2 (α + |∇um,t|g2)(p−2)/2 (∇um,t|∇u)g dµg dt

tends to∫

01∫

M

((p− 2) (α + t2|∇u|g2)(p−4)/2 |∇u|g4 t3 + 2 (α + t2|∇u|g2)(p−2)/2 |∇u|g2 t

)dµgdt

=∫

M

∫01 d

dt[(α + t2|∇u|g2)

p−22 |∇u|g2t2] dt dµg

=∫

M(α + |∇u|g2)

p−22 |∇u|g2 dµg.


In a similar way, using Vitali’s Convergence Theorem one can prove that

(3.12)

∫M

h(x, um)umdµg −∫

Mh(x, um − u)(um − u)dµg

=∫

M

∫01 d

dt

[h(x, um + (t− 1)u)(um + (t− 1)u)]dt dµg

=∫

01∫

M

[Dsh(x, um + (t− 1)u)(um + (t− 1)u)u + h(x, um + (t− 1)u)u

]dµg

which tends to∫

0

1

∫

M

[Dsh(x, tu)(tu)u+h(x, tu)u

]dµg dt =

∫

M

∫

0

1d

dt

[h(x, tu)(tu)

]dt dµg =

∫

M

h(x, u)u dµg.

Since |Dsh(x, t)| ≤ c1|t|p∗−2+c2, with c1 and c2 positive constants, there exist C,D > 0such that h(x, t)t ≤ C|t|p∗ + Dt2 for any t ∈ R. From (3.10), (3.11), (3.12), we deduce(3.13)∫

M|∇um −∇u|pg dµg − C

∫M|um − u|p∗ dµg

≤ ∫M

(α + |∇um −∇u|g2)p−22 |∇um −∇u|g2 dµg −

∫M

h(x, um − u)(um − u) dµg + o(1)

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

where C is a positive constant. Denoting

(3.14) a = lim supm→+∞ ‖um − u‖p1,p ≤ 2p lim supm→+∞

∫M|∇um −∇u|pg dµg

by (3.13), (2.3) we have

a ≤ 2p C lim supm→+∞

∫

M

|um − u|p∗dµg ≤ 2p CKp∗ap∗/p

where K = maxA, V ol−1/n(M,g).

Therefore, if a > 0, this implies a ≥ (2pCKp∗)−(N−p)/p, hence

(3.15) (2pCKp∗)−(N−p)/p ≤ a ≤ 2p C lim supm→+∞(‖um − u‖1,p + ‖u− u‖1,p

)p∗

≤ 2p C (2R)p∗

where K > 0 is a positive constant. If we take R > 0 small enough, such that2p C (2R)p∗ < (2pCKp∗)−(N−p)/p, we derive a contradiction. Therefore it must be a = 0and thus um strongly converges to u in Hp

1 (M). ¤

4. Critical Group estimates

We recall this regularity result, which can be proved using Moser’s iterative scheme[17] and following the arguments of Druet in [11], taking account of [12] and [20].

Theorem 4.1. Let (M, g) be a compact Riemannian n-manifold, n ≥ 2, 1 < p < n,α ∈ [0, 1], C > 0 and let h ∈ C0(M × R) be such that

∀(x, r) ∈ M × R, |h(x, r)| ≤ C(|r|p∗−1 + 1

).

If u ∈ Hp1 (M) is a solution of divg

((α + |∇u|2g

)(p−2)/2∇u)

+ h(x, u) = 0, then u ∈C1,β(M), with β ∈]0, 1[. Moreover, if u belongs to a bounded set A ⊂ Hp

1 (M), then‖u‖C1,β(M) ≤ K, where K depends only on n, p , g, A, α, C.

Now we fix a critical point u0 of Jα. It is standard that u0 solves the quasilinearproblem (1.1). By Theorem 4.1, then u0 ∈ C1,β(M). As in the Euclidean case [5], wecan introduce a Hilbert space, depending on the critical point u0, in which Hp

1 (M) is


embedded, so that a suitable splitting can be obtained. Precisely, let H0 be the closureof C∞

0 (M) under the scalar product

(v, w)0 =

∫

M

((α + |∇u0|g2)(p−2)/2(∇v|∇w)g

+ (p− 2)(α + |∇u0|g2)(p−4)/2(∇u0|∇v)g(∇u0|∇w)g

)dµg +

∫

M

vw dµg.

As u0 ∈ C1,β(M), the norm ‖ · ‖0 induced by (·, ·)0 is equivalent to the usual normof H12(M). Hence H0 is isomorphic to H12(M) and the embedding Hp

1 (M) → H0 iscontinuous.

Denoting by 〈·, ·〉 : H∗0 ×H0 → R the duality pairing in H0, J ′′α(u0) can be extended to

the operator L0 : H0 → H∗0 defined by setting

〈L0v, w〉 = (v, w)0 − 〈Kv, w〉where

〈Kv, w〉 =

∫

M

(Dsh(x, u0) + 1

)vw dµg,

for any v, w ∈ H0.

Lemma 4.2. L0 is a compact perturbation of the Riesz isomorphism from H0 to H∗0 . In

particular, L0 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 H0 to H∗

0 . Let vn be a bounded sequence in H0. Then there exists v ∈ H0 suchthat, up to a subsequence, vn converges weakly to v in H0 and strongly in L2(M).There is a constant C > 0 such that, for any w ∈ H0, ‖w‖0 = 1 we have

|∫

M

(Dsh(x, u0) + 1

)(vn − v)w dµg| ≤ C

(∫

M

|vn − v|2 dµg

)1/2

which tends to zero as n → +∞, uniformly with respect to w. This implies that K is acompact operator. ¤

Now let us denote by m(L0) the maximal dimension of a subspace of H0 on which L0 isnegative definite. Obviously m(Jα, u0) ≤ m(L0). Furthermore let us denote by m∗(L0)the sum of m(L0) and the dimension of the kernel of L0. By Lemma 4.2 we concludethat m(L0) and m∗(L0) are finite.Since L0 is a Fredholm operator in H0, we can consider the natural splitting

H0 = H− ⊕H0⊕H+

where H−, H0, H+ are, respectively, the negative, null and positive spaces, accordingto the spectral decomposition of L0 in L2(M). Therefore one can easily show that thereexists a γ0 > 0 such that

〈L0v, v〉 ≥ γ0‖v‖02 ∀v ∈ H+

moreover m(L0), m∗(L0) are respectively the dimensions of H− and H− ⊕H0.Since u0 ∈ C1,β(M), following the same arguments in Lemma 2.2 and Theorem 2.3 in[11], we derive that H− ⊕H0 ⊂ Hp

1 (M) ∩ C1(M).


Consequently, denoted by W = H+ ∩Hp1 (M) and V = H− ⊕H0, we get the splitting

Hp1 (M) = V ⊕W and,

(4.1) 〈J ′′α(u0)w, w〉 ≥ γ0‖w‖20 ∀w ∈ W,

so that m(L0) = m(Jα, u0) and m∗(L0) = m∗(Jα, u0).

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

Proposition 4.3. There exists R > 0 such that, for any fixed α ≥ 0 and u ∈ Hp1(M),if (um) ⊂ BR(u) and sup

w∈W\0〈Jα

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

subsequence.

Proof. Reasoning as in the proof of Proposition 3.3, let us fix R > 0, α ≥ 0, and denoteJα = J .

With the same arguments, we infer that (um) weakly converges to some u ∈ BR(u)and, for any z ∈ Hp1(M),

(4.2) 〈J ′(um), z〉 → 〈J ′(u), z〉.Now let e1, . . . em∗ be an L2-orthogonal basis of V , where m∗ = m∗(Jα, u0). Denoting

by PV (z) =∑m∗

i=1

(∫M

eiz dx)ei, it is clear that z − PV (z) ∈ W , for any z ∈ Hp1(M).

Moreover PV (um) strongly converges to PV (u). Exploiting (4.2) we get

(4.3)〈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 (3.11) and (3.12) hold too, we can complete this proof in the sameway as the previous one, using (4.3) instead of (3.10). ¤

Now we prove that u0 is a strict minimum point in the direction of W .In order to do that we need some preliminary results. The first one is found in [14].

Lemma 4.4. Let I : Lp(M,Rk)× Lq(M,Rm) →]−∞, +∞] be a functional of the form

I(u, v) =

∫

M

φ(x, u, v) dµg

where φ(x, u, v) is a nonnegative, continuous function and φ(x, u, ·) is convex. Then Iis lower semicontinuous 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.

Lemma 4.5. For any K > 0 there exist r0 > 0 and C > 0 such that, for any z ∈ C1(M),with ‖z‖C1(M) ≤ K, ‖z − u0‖ < r0, we have

(4.4) 〈J ′′α(z)w, w〉 ≥ C‖w‖20 ∀w ∈ W.

Proof. By contradiction, there exist K > 0 and two sequences zn ∈ C1(M) and wn ∈ Wsuch that ‖wn‖0 = 1, ‖zn‖C1(M) ≤ K, ‖zn − u0‖1,p → 0 and

(4.5) 〈J ′′α(zn)wn, wn〉 <1

n.


Firstly, it is immediate to see that, for a suitable positive constant C0, we have

(4.6) 〈J ′′α(zn)wn, wn〉 ≥ C0‖wn‖02−∫

M

(Dsh(x, zn) + 1)wn2 dµg.

As wn is bounded in H0, it converges to some w weakly in H0 and strongly in L2(M).Moreover, as h(x, ·) is C1, we derive

(4.7) limn→∞

∫

M

(Dsh(x, zn) + 1)wn2 dµg =

∫

M

(Dsh(x, u0) + 1)w2 dµg

We see that w 6= 0, in fact if not, by (4.5), (4.6) and (4.7) we should have C0 ≤ 0.By Lemma 4.4 we have

(4.8)∫

M

((α + |∇u0|g2)(p−2)/2|∇w|2g + (p− 2)(α + |∇u0|g2)(p−4)/2(∇u0|∇w)g2

)dµg

≤ lim infn→∞

∫

M

((α + |∇zn|g2)(p−2)/2|∇wn|2g + (p− 2)(α + |∇zn|g2)(p−4)/2(∇zn|∇wn)g2

)dµg .

So, by (4.1), (4.5) and (4.8) we have

γ0‖w‖02 ≤ 0

which is an absurd as w 6= 0. ¤

Following the same arguments of Lemma 4.5 in [5] and Lemma 4.2 in [6], one canrecognize that u0 is a strict minimum point in the direction of W .

Lemma 4.6. There exist δ > 0 and µ′ > 0 such that, for any w ∈ W with ‖w‖ ≤ δ, wehave

Jα(u0 + w)− Jα(u0) ≥ µ′‖w‖p.

We can now obtain the following result which is crucial for developing a finite dimen-sional reduction.

Proposition 4.7. There exist r > 0 and ρ ∈ ]0, r[ such that for each v in V ∩ Bρ(0),there exists one and only one w ∈ W ∩Br(0) ∩ L∞(Ω) such that for any z ∈ W ∩Br(0)we have

Jα(v + w + u0) ≤ Jα(v + z + u0).

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

(4.9) 〈J ′α(u0 + v + w), z〉 = 0 ∀z ∈ W

and

(4.10) S =v + z + u0 | v ∈ V ∩Bρ(0), w ∈ W ∩Br(0)

⊂ BR′(u0)

where R′ = minR, R and R, R are respectively defined by Proposition 4.3 and Propo-sition 3.2.


Proof. Now let m∗ = m∗(Jα, u0), e1, . . . em∗ be an L2-orthogonal basis of V ⊂ C1(M)and δ be defined by Lemma 4.6. There exists M depending just on δ such that ifz ∈ Bδ(u0) is a solution of 〈J ′α(z), w〉 = 0 for any w ∈ W , denoting by fz(x) =∑m∗

i=1

(∫M

eiz dx)ei(x), we have fz ∈ C1(M) and ‖fz‖C1(M) ≤ M . So z solves the

equation

divg

((α + |∇u|2g

)(p−2)/2∇u)

+ h(x, u) + fz(x) = 0.

and there exist D > 0 such that

∀(x, r) ∈ M × R, |h(x, r) + fz(x)| ≤ D(|r|p∗−1 + 1

).

Thus, by Theorem 4.1, z ∈ C1(M) and ‖z‖C1(M) ≤ K, where K > 0. Now byLemma 4.5 in correspondence of 2K there exist r0 ∈ (0, δ) and C > 0 such that (4.4)holds.

We fix 0 < r < min

R2, R

2, r0

3

where R and R are introduced respectively in Proposi-

tion 4.3 and Proposition 3.2, so that (4.10) holds and Jα is sequentially lower semicontin-uous in Br(u) with respect the weak topology of Hp

1 (M) for any u ∈ Hp1 (M). Therefore,

for any fixed v ∈ V ∩Br(0), there exists a minimum point w ∈ W ∩Br(0) of the functionw ∈ W ∩Br(0) 7→ Jα(u0 + v + w).

We shall prove that there exists 0 < ρ < r such that for any v ∈ V ∩Bρ(0)

(4.11) infJα(u0 + v + w) |w ∈ W, ‖w‖ = r > Jα(u0 + v).

Arguing by contradiction, we assume that there exist two sequences wn in W and vnin V such that ‖wn‖ = r, ‖vn‖ → 0 and

(4.12) Jα(u0 + vn + wn) ≤ Jα(u0 + vn) + o(1).

Moreover, Jα(u0 + vn + wn)− Jα(u0 + wn) = 〈J ′α(u0 + βnvn + wn), vn〉, where βn ∈ (0, 1),so that

Jα(u0 + vn + wn) = Jα(u0 + wn) + o(1)

which combined with (4.12) and Lemma 4.6 gives the absurd

µ′rp ≤ Jα(u0 + wn)− Jα(u0) = Jα(u0 + vn + wn)− Jα(u0) + o(1)

≤ Jα(u0 + vn)− Jα(u0) + o(1) = o(1).

Consequently, by (4.11), we have that for any v ∈ V ∩ Bρ(0) the minimum point wbelongs to W ∩Br(0) and then solves

(4.13) 〈J ′α(u0 + v + w), z〉 = 0 ∀ z ∈ W.

Therefore w ∈ C1(M) and ‖u0 + v + w‖C1(M) ≤ K. We can also recognize that w isunique. By contradiction we suppose that there exist w1 6= w2 ∈ W which solve (4.13).Then for any t ∈ [0, 1] we have ‖v +w1 + t(w2−w1)‖1,p ≤ 3r < r0, ‖u0 + v +w1 + t(w2−w1)‖C1(M) ≤ 2K and applying Lemma 4.5,

0 = 〈J ′α(u0 + v + w2)− J ′α(u0 + v + w1), w2 − w1〉

=

∫

0

1〈J ′′α(u0 + v + w1 + t(w2 − w1))(w2 − w1), w2 − w1〉 dt > 0.

¤


So we can introduce the map

(4.14) Ψ : V ∩Bρ(0) → W ∩Br(0)

where Ψ(v) = w is the unique minimum point of the function w ∈ W ∩ Br(0) 7→Jα(u0 + v + w), and the function

(4.15) Φ : V ∩Bρ(0) → Rdefined by Φ(v) = Jα(u0 + v + Ψ(v)).

Now we will show that

(4.16) Cj(Jα, u0) ' Cj(Φ, 0).

and, from this, we will prove Theorem 2.1 and Theorem 2.2.We first need the regularity of Ψ introduced in (4.14).

Proposition 4.8. Ψ is a continuous function.

Proof. Let vn be a convergent sequence in V ∩ Bρ(0), and let v be its limit. ByProposition 4.3, Ψ(vn) → w ∈ W ∩ Br(0). Moreover 〈J ′α(u0 + v + w), z〉 = 0 for anyz ∈ W , thus Proposition 4.7 assures that w = Ψ(v).

¤Moreover we need the following technical Lemma.

Lemma 4.9. For any σ ∈ (0, ρ], let us introduce the set

Mσ = u0+v+(1−t)w+tΨ(v) : v ∈ V ∩Bσ(0), w ∈ W∩Br(0), Jα(u0+v+w) ≤ c, t ∈ [0, 1].There exists ρ′ ∈ (0, ρ] such that, if u0 + v + w ∈ Mρ′, where (v, w) ∈ V ×W , then we

have, for any z ∈ W with ‖z‖ = r

Jα(u0 + v + w) < Jα(u0 + v + z).

Proof. Arguing by way of contradiction, let vn in V , wn in W ∩ Br(0), zn in W ,tn in [0, 1] be sequences such that vn → 0, Jα(u0 + vn + wn) ≤ c, ‖zn‖ = r and

(4.17) Jα (u0 + vn + (1− tn)wn + tnΨ(vn)) ≥ Jα(u0 + vn + zn).

By Lemma 4.6, there exists µ′ such that

c + µ′‖wn‖p ≤ Jα(u0 + wn) = Jα(u0 + vn + wn) + o(1) ≤ c + o(1)

hence wn → 0. Analogously

(4.18) c + µ′rp ≤ Jα(u0 + vn + zn) + o(1).

Consequently u0 + vn + (1− tn)wn + tnΨ(vn) → u0 and, applying (4.17) and (4.18),

c + µ′rp ≤ c + o(1)

which is absurd.¤

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

M = Mρ′

Φc = v ∈ V ∩Bρ′(0) : Jα(u0 + v + Ψ(v)) ≤ c


Y = u0 + v + Ψ(v) : v ∈ V ∩Bρ′(0)

Y c = u0 + v + Ψ(v) : v ∈ V ∩Bρ′(0), Jα(u0 + v + Ψ(v)) ≤ c

U =(u0 +

(V ∩Bρ′(0)

)+

(W ∩Br(0)

)).

Since Ψ is a continuous map and Ψ(0) = 0, the topological pair (Φc, Φc \ 0) is homeo-morphic to (Y c, Y c \ u0). Therefore, according to (2.2), for any integer j we have

Cj(Φ, 0) ∼= Hj (Y c, Y c \ u0) .

Relation (4.16) will be proved if we show that the topological pair (Y c, Y c \ u0) isa deformation retract of ((Jα)c ∩ U, (Jα)c ∩ U \ u0).

This will be done in Corollary 4.11, whose proof relies on the following result.

Proposition 4.10. 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 : Hp1 (M) → R, β(z) = sup

w∈W\0

〈J ′α(z), w〉‖w‖

and the set

Z∗ = z ∈ Hp1 (M) : β(z) 6= 0 .

From standard arguments concerning the construction of a pseudogradient vector field(see e.g. [4]) 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

(4.19)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 4.9 and (4.19) 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 for


the initial data z0, we have limt→T− Jα(σ(t, z0)) > c. By Proposition 4.3 there is ε > 0such that, for any t,

β(σ(t, z0)) > ε,

hence by (4.19)

(4.20) c− Jα(z0) ≤∫ t

0

d

dsJα(σ(s, z0))ds < −ε2t

so that T < Jα(z0)−cε2

.

Moreover for any t1 < t2 we get

(4.21)

‖σ(t2)− σ(t1)‖ ≤∫ t2

t1‖σ(t)‖ dt ≤ 2

∫ t2t1

β(σ(t)) dt

≤ 2√

(t2 − t1)(Jα(σ(t1))− Jα(σ(t2))

) ≤ 2√

(t2 − t1)(Jα(z0)− c

).

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 (4.19),

∂

∂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. Wewill prove now that in this case β(r(zn)) → 0, so that the assert comes from Proposition4.3.

Indeed, arguing by contradiction, suppose that

(4.22) β(r(zn)) is not infinitesimal,

then, there exist T > 0 and ε0 > 0 such that

(4.23) ∀n ∈ N, ∀t ∈ [Tzn − T , Tzn ] β(σ(t, zn)) > ε0.

In fact, if (4.23) is not true, there is a subsequence znk such that, for any k ∈ N

β(σ(tk, znk)) ≤ 1

k


at least for a suitable tk ∈ [Tznk− 1

k, Tznk

]. Hence, by Proposition 4.3,

σ(tk, znk) → z0,

while, by (4.21),

‖r(znk)− σ(tk, znk

)‖ ≤ 2√

1/k · (Jα(znk)− c) → 0,

so that also r(znk) → z0, and in particular β(r(znk

)) → 0, in contradiction with (4.22).Therefore (4.23) holds and, reasoning as in (4.20),

c− Jα

(σ(Tzn − T , zn)

) ≤ −ε02T

which gives the absurd

c + ε02 T ≤ Jα(σ(Tzn − T , zn)) ≤ Jα(zn) = c + o(1).

¤In order to prove (4.16) and finally Theorems 2.1 and 2.2, we need the following step.

Corollary 4.11. (Y c, Y c \u0) is a deformation retract of((Jα)c∩ U, (Jα)c∩ U \u0

).

Proof. It immediately follows from the previous proposition defining

η : [0, 1]× (Jα)c∩U → (Jα)c∩U, η(t, u0 + v +w) = r (u0 + v + (1− t)w + tΨ(v)) .

¤

Proof of Theorem 2.1. As J ′′α(u0) is injective, we have H0 = 0 and there is asuitable constant µ > 0 such that

〈J ′′α(u0)v, v〉 ≤ −µ‖v‖2 for any v ∈ V.

As a consequence, u0 is a local isolated maximum of Jα along V , thus 0 is a local isolatedmaximum of Φ in V ∩Bρ′ and by (4.16) the assert comes (see [4, Example 1, page 33]).¤

Proof of Theorem 2.2. By (4.16), it is clear that Cj(Jα, u0) = 0 if j > dim V .Moreover, by Theorem 3.1 proved by Lancelotti in [15], we have Cj(Jα, u0) = 0 forany j < m(Jα, u0).

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Dipartimento di Matematica, Politecnico di Bari, Via Orabona 4, 70125 Bari, ItalyE-mail address: [email protected]

Dipartimento di Matematica, Politecnico di Bari, Via Orabona 4, 70125 Bari, ItalyE-mail address: [email protected]

Dipartimento di Matematica, Universita di Trento, via Sommarive 14 I-38123 Povo(TN), Italy

E-mail address: [email protected]

Morse index estimates for quasilinear equations on Riemannian manifolds

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