Multiple positive solutions for a critical quasilinear equation via Morse theory
✩Existence de plusieurs solutions positives d’une équation quasi-linéaire avec exposant critique par la théorie de Morse
Silvia Cingolani
∗, Giuseppina Vannella
Dipartimento di Matematica, Politecnico di Bari, Via Orabona 4, 70125 Bari, Italy Received 6 June 2007; accepted 24 September 2007
Available online 1 November 2007
Abstract
We deal with the existence of solutions for the quasilinear problem
(Pλ)
⎧⎨
⎩
−pu=λuq−1+up∗−1 inΩ,
u >0 inΩ,
u=0 on∂Ω,
whereΩis a bounded domain inRNwith smooth boundary,Np2, 1< pq < p∗,p∗=Np/(N−p),λ >0 is a parameter.
Using Morse techniques in a Banach setting, we prove that there existsλ∗>0 such that, for anyλ∈(0, λ∗),(Pλ)has at least P1(Ω)solutions, possibly counted with their multiplicities, wherePt(Ω)is the Poincaré polynomial ofΩ. Moreover forp2 we prove that, for eachλ∈(0, λ∗), there exists a sequence of quasilinear problems, approximating(Pλ), each of them having at leastP1(Ω)distinct positive solutions.
©2007 Elsevier Masson SAS. All rights reserved.
Résumé
On s’interesse à l’existence de solutions pour l’équation quasi-linéaire
(Pλ)
⎧⎨
⎩
−pu=λuq−1+up∗−1 inΩ,
u >0 inΩ,
u=0 on∂Ω,
oùΩ est un domaine deRN avec frontière régulière,Np2, 1< pq < p∗,p∗=Np/(N−p),λ >0 est un paramètre.
Par des techniques de la théorie de Morse dans le cadre des espaces de Banach, un démontre l’existence de λ∗>0 tel que, pour toutλ∈(0, λ∗),(Pλ)possède au moinsP1(Ω)solutions, considerées avec leur multiplicité, oùP1(Ω)est le polynôme de
✩ The research of the authors was supported by the MIUR project “Variational and topological methods in the study of nonlinear phenomena”
(PRIN 2005).
* Corresponding author.
E-mail addresses:[email protected] (S. Cingolani), [email protected] (G. Vannella).
0294-1449/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2007.09.003
Poincaré deΩ. En outre, pourp2, on démontre que, pour toutλ∈(0, λ∗), il existe une suite de problèmes quasi-linéaires qui approchent(Pλ), chacun desquels a au moinsP1(Ω)solutions positives différentes.
©2007 Elsevier Masson SAS. All rights reserved.
MSC:58E05; 35B20; 35B33; 35J60
Keywords:p-Laplace equations; Critical Sobolev exponent; Perturbation results; Morse theory; Critical groups
1. Introduction
Let us consider the quasilinear elliptic problem
(Pλ)
⎧⎨
⎩
−pu=λuq−1+up∗−1 inΩ,
u >0 inΩ,
u=0 on∂Ω,
whereΩ is a bounded domain inRN with smooth boundary,Np2, 1< pq < p∗,p∗=Np/(N−p),λ >0 is a parameter.
The first striking result, in the casep=2, is due to Pohozaev [30] who proved that ifΩ is starshaped with respect to some point andλ=0, then(Pλ)has no solution. Some years later, in the celebrated paper [9], Brezis and Nirenberg showed that ifN4 andp=q=2, problem(Pλ)has a solution for anyλ∈(0, λ1)whereλ1is the first eigenvalue of−with Dirichlet boundary condition onΩ (cf. [2]) and no solution whenλλ1orλ0 andΩ is starshaped.
Moreover they proved that whenN=3,p=q=2,Ωis the ball inRN, then problem(Pλ)has a solution if and only ifλ∈(λ41, λ1). The paper [9] stimulated a vast amount of research on this subject.
The quasilinear critical problem(Pλ)was considered by Azorero and Peral in [3]. They showed that ifN p2, andp=q, then(Pλ)has a solution for anyλ∈(0, λ1), whereλ1denotes the first eigenvalue ofp-Laplace operator
−p with Dirichlet boundary condition onΩ, no solutions whenλλ1 orλ0 andΩ is starshaped. The same results are obtained by Guedda and Veron in [23] with a different approach.
Moreover in [3] (see also [4]) Azorero and Peral proved that whenq∈(p, p∗),Np2,(Pλ)has a solution for anyλ >0.
The first multiplicity result for(Pλ)has been achieved by Rey in [31] in the semilinear case. Precisely Rey proved that ifN5,p=q=2, forλsmall enough(Pλ)has at leastcat(Ω)solutions, wherecat(Ω)denotes the Ljusternik–
Schnirelmann category ofΩ in itself. This result, as Rey wrote in the introduction of the paper [31], was suggested by the fact that the number of solutions to(Pλ)is related to the properties of the Green’s function ofΩ. Precisely in [32], he has showed that ifN4,p=q=2, anduλis a solution of(Pλ), which concentrates around a pointx0 as λ→0, thenx0is a critical point of the Robin’s function, the regular part of the Green’s function. Conversely if N5,p=q=2, any nondegenerate critical pointx0of the Robin’s function generates a family of solutions of(Pλ), concentrating aroundx0asλ→0.
Throughout a different approach, based on some ideas introduced by Benci and Cerami in [6], Lazzo obtained in [24] the same result of Rey [31], under the weaker condition N4. Really, for p=q =2, in [28], Passaseo improved the results in [31,24] proving that ifΩis not contractible,(Pλ)has at leastcat(Ω)+1 solutions. Furthermore Passaseo showed that the number of solutions of(Pλ)is not related to the topology ofΩ, but to the topology of a domainΩ˜ which differs fromΩ by a set of small capacity. For instance ifΩ is obtained byΩ˜ cutting off a set of small capacity, then problem(Pλ)has at leastcat(Ω)˜ +1 distinct solutions, for λsmall, even if the domainΩ is contractible in itself.
Recently in [1], Alves and Ding have proved a multiplicity result for the quasilinear problem(Pλ), in the spirit of the papers [31,24]. They showed that ifNp2and 2pq < p∗, then there existsλ∗>0 such that for each λ∈(0, λ∗)problem(Pλ)has at leastcat(Ω)solutions.
In this work we aim to obtain a better information on the number of solutions of problem(Pλ), for small value of parameterλ, via the Morse theory and the domain topology. We recall the following definitions.
Definition 1.1.LetKbe a field. For anyB⊂A⊂Rn, we denotePt(A, B)the Poincaré polynomial of the topological pair(A, B), defined by
Pt(A, B)=+∞
k=0
dimHk(A, B)tk,
whereHk(A, B)stands for thekth Alexander–Spanier relative cohomology group of(A, B), with coefficient in K;
we also setHk(A)=Hk(A,∅)andPt(A)=Pt(A,∅)is called the Poincaré polynomial ofA.
Definition 1.2.LetXbe a Banach space andf be aC1functional onX. LetKbe a field. Letube a critical point off,c=f (u), andUbe a neighborhood ofu. We call
Cq(f, u)=Hq
fc∩U,
fc\ {u}
∩U
the qth critical group of f at u,q =0,1,2, . . ., where fc= {v∈X: f (v)c}, Hq(A, B) stands for the qth Alexander–Spanier cohomology group of the pair (A, B)with coefficients in K. By the excision property of the singular cohomology theory the critical groups do not depend on a special choice of the neighborhoodU.
Definition 1.3.We denotePt(f, u)the Morse polynomial off inu, defined by Pt(f, u)=
+∞
k=0
dimCk(f, u)tk.
We call themultiplicityofuthe numberP1(f, u)∈N∪ {+∞}.
In the past years the relations between the topological properties of the domain and the multiplicity of solutions to semilinear elliptic problems have been largely investigated. We mention the celebrated paper [7] where Benci and Cerami estimated the number of solutions of the semilinear elliptic problem
(Sλ)
−λu+u=f (u) inΩ,
u >0 inΩ,
u=0 on∂Ω,
whereΩ is a bounded domain inRN with smooth boundary,N 3, λ >0 is a parameter, f:R+→Ris aC1,1 function withf (0)=f(0)=0, having a subcritical growth at infinity. By means of Morse techniques, they showed that the number of positive solutions of problem(Sλ), counted with their multiplicities, depends on the topology ofΩ, actually onPt(Ω), the Poincaré polynomial ofΩ.
The functional analytic setting, when(Sλ)is set up, is a Hilbert space. The multiplicity is exactly one, if the solution is nondegenerate in the classical sense given in a Hilbert space, namely the second derivative of the associated Euler functional in the solution is an isomorphism between the Hilbert space and its dual. The nondegeneracy condition is generally verified via the perturbation results in [26], which guarantee that each isolated critical point can be resolved in a finite number of nondegenerate critical points of aC1locally approximating functional. Let us emphasize that the perturbation results in [26] rely on a infinite dimensional version of Sard’s Theorem, due to Smale [34] so that they need the Fredholm properties of the second derivates in the critical points.
In our work, we obtain a first result, which correlates the topological properties of the domain and the number of solutions of(Pλ), counted with their multiplicities.
Theorem 1.4.Assume that Np2,1< pq < p∗,p∗=Np/(N−p). There existsλ∗>0 such that, for any λ∈(0, λ∗),(Pλ)has at leastP1(Ω)solutions, possibly counted with their multiplicities.
As showed in [7], the application of the Morse theory yields better results than the application of Ljusternik–
Schnirelman theory for topologically rich domain. For example if Ω is obtained by an open contractible domain cutting offkholes, we derive that the number of solutions of(Pλ)is affected byk, even if the category ofΩ is 2 (see Remark 3.11 and also [7]).
Theorem 1.4 assuresP1(Ω)distinct solutions, if one is able to interpret themultiplicityfor a solution of(Pλ). For this reason, we need a deep insight into this notion. In the present work we consider the casep2. In this context, the variational setting of the quasilinear problem(Pλ)is the Banach spaceW01,p(Ω), which is not Hilbert forp=2. The Euler functionals associated to problem(Pλ)areC2, but several conceptual difficulties arise in order to perform a local Morse theory and perturbation results like in [26]. Firstly all the solutions of(Pλ)are degenerate in the classic sense, as W01,p(Ω)is not isomorphic to its dual space (cf. [19]). Moreover, denoting byIλ the Euler functional associated to(Pλ), we lack the Fredholm properties ofIλin its critical points, so that the perturbation results in [26] cannot be applied. As far as we know, there are no kind of results in the spirit of the paper by Rey [32].
As in [15], we introduce the followingweakernotion of nondegeneracy.
Definition 1.5.LetAbe an open subset ofW01,p(Ω)andg:A→Rbe aC2functional. We say that a critical pointu ofgis nondegenerate ifg(u)is injective fromW01,p(Ω)to its dualW−1,p(Ω).
We remark that the above notion of nondegeneracy coincides with the usual one if the space is Hilbert and the operator is Fredholm. This is not our case, whenp >2. We emphasize that in 1969 Smale, as written by Uhlenbeck in [38], conjectured that injectivity is enough for developing Morse theory in some Banach settings.
Using the above notion of nondegeneracy, we give a sharp interpretation of the multiplicity of a critical point of(Pλ) in terms of approximating elliptic problems. This result is contained in Theorem 5.1. We remark that this approach is new also for the casep=2. Indeed the perturbation results by Marino and Prodi [26] furnish an interpretation of the multiplicity in terms ofC1locally approximating functional, which cannot be, in general, the Euler functional of some semilinear problem.
Using the result in Theorem 5.1, we prove the following multiplicity results. In what follows, we say that ∂Ω satisfies the interior sphere condition if for eachx0∈∂Ω there exists a ballBR(x1)⊂Ω such thatBR(x1)∩∂Ω= {x0}.
Theorem 1.6. Assume that ∂Ω satisfies the interior sphere condition and that N p2, 2< pq < p∗, p∗= Np/(N−p). There existsλ∗>0such that, for anyλ∈(0, λ∗), either(Pλ)has at leastP1(Ω)distinct solutions or, if not, for any sequence(αn), withαn>0,αn→0, there exists a sequence(fn)withfn∈C1(Ω),fnC1 →0such that problem
(Pn)
⎧⎨
⎩
−div
|∇u|2+αn(p−2)/2
∇u
=λuq−1+up∗−1+fn inΩ,
u >0 inΩ,
u=0 on∂Ω
has at leastP1(Ω)distinct solutions, fornlarge enough.
Theorem 1.7.Assume that∂Ωsatisfies the interior sphere condition and thatN4,2q <2∗,2∗=2N/(N−2).
Then there existsλ∗>0 such that, for anyλ∈(0, λ∗), either (Pλ)has at leastP1(Ω)distinct solutions or, if not, there exists a sequence(fn)withfn∈C1(Ω),fnC1→0such that problem
(Ln)
⎧⎨
⎩
−u=λuq−1+up∗−1+fn inΩ,
u >0 inΩ,
u=0 on∂Ω
has at leastP1(Ω)distinct solutions, fornlarge enough.
Theorems 1.6 and 1.7 are quantitative results which give an interpretation to the number of solutions of(Pλ). The proofs of these theorems rely on the construction of an approximating functional toIλ, having onlynondegenerate critical points in the sense introduced in Definition 1.5. For nondegenerate critical points of the approximating func- tional, we are able to compute the critical groups, which are topological objects, in terms of differential notions, like the Morse index (see Theorem 4.2), so that the multiplicity of a nondegenerate critical point is exactly one. By The- orem 2.4 it follows that the Morse polynomialPt(Iλ, u0)(see Definition 1.3) can be computed in terms of the sum of the Morse polynomials of the approximating functional in each critical point and a partially controlled remainder term.
We remark that the idea of combining the Splitting Theorem and Sard’s Lemma, in the finite dimensional case, can be traced back to Chang [10] in the special case of aC2functional, defined on a Hilbert space, having an isolated critical point.
Perturbations results in Morse theory for quasilinear problem having a right-hand side subcritically at infinity are obtained in [17,14] (see also [13,20]).
Concerning multiplicity results of nontrivial solutions (not necessarily positive) for some critical quasilinear prob- lem, we quote [33,29]. Finally we mention a recent result by Degiovanni and Lancelotti [21], where the existence of a nontrivial solution (not necessarily positive) for the critical problem(Pλ)is proved for anyλ > λ1,λ=λm, where (λm)is a suitable sequence of eigenvalues of−p.
Throughout the paper we use the following notations:
(1) · denotes the usual norm both inW01,p(Ω)and inW−1,p(Ω);
(2) · 1,2denotes the usual norm inW01,2(Ω);
(3) · ∞denotes the usual norm inL∞(Ω);
(4) | · |r denotes the usual norm inLr(Ω);
(5) · C1 and · C2 denote the usual norms inC1(W01,p(Ω))andC2(W01,p(Ω));
(6) ·,·:W−1,p(Ω)×W01,p(Ω)→Rdenotes the duality pairing;
(7) d(·,·)denotes the distance function in each metric space;
(8) Mr denotes{v∈W01,p(Ω): d(v, M) < r}, whereM⊂W01,p(Ω)andr >0;
(9) fc= {v∈W01,p(Ω): f (v)c},fab= {v∈W01,p(Ω): af (v)b}, int(fab)= {v∈W01,p(Ω): a < f (v) < b}.
2. Some abstract recalls in Morse theory
We need to recall some useful definitions and results (cf. [11,12,35]).
Definition 2.1.LetXbe a Banach space andf be aC1functional onX. LetC be a closed subset ofX. A sequence (un)inCis a Palais–Smale sequence forf iff (un)Muniformly inn, whilef(un)→0 asn→ +∞.
We say thatf satisfies(P .S.)onCif any Palais–Smale sequence inChas a strongly convergent subsequence.
Letc∈R. We say thatf satisfies(P .S.)c if any sequence(un)inX, such thatf (un)→candf(un)→0 as n→ +∞, has a strongly convergent subsequence.
Definition 2.2.LetXbe a Banach space andf be aC2functional onX. Ifuis a critical point off, the Morse index off inuis the supremum of the dimensions of the subspaces ofXon whichf(u)is negative definite. It is denoted bym(f, u). Moreover, the large Morse index off inuis the sum ofm(f, u)and the dimension of the kernel off(u).
It is denoted bym∗(f, u).
Next theorem is a topological version of the classical Morse relation (cf. Theorem 4.3 in [11]).
Theorem 2.3.LetXbe a Banach space andf be aC1functional onX. Leta, b∈Rbe two regular values forf, with a < b. Iff satisfies the(P .S.)ccondition for allc∈(a, b)andu1, . . . , ul are the critical points off inf−1(a, b), then
+∞
q=0
l j=1
dimCq(f, uj)
tq=Pt(fb, fa)+(1+t )Q(t ), (2.1)
whereQ(t )is a formal series with coefficients inN∪ {+∞}.
We point out that the above series are formal, as (2.1) means that the coefficients (possibly+∞) of eachtqare the same on both sides of the equality.
In order to obtain a multiplicity result of solutions to problem (Pλ)via Morse relations, we recall an abstract theorem, proved in [14] (see also [5] and [11]).
Theorem 2.4.LetAbe a open subset of a Banach spaceX. Letf be aC1functional onAandu∈Abe an isolated critical point off. Assume that there exists an open neighborhoodUofusuch thatU⊂A,uis the only critical point off inUandf satisfies the Palais–Smale condition inU.
Then there existsμ >¯ 0such that, for anyg∈C1(A,R)such that
• f −gC1(A)<μ,¯
• gsatisfies the Palais–Smale condition inU,
• ghas a finite number{u1, u2, . . . , um}of critical points inU, we have
m j=1
Pt(g, uj)=Pt(f, u)+(1+t )Q(t ),
whereQ(t )is a formal series with coefficients inN∪ {+∞}. 3. The topological result
Assume thatNp2and 1< pq < p∗,p∗=pN/(N−p). Standard arguments prove that the solutions of(Pλ) correspond to the critical points of theC1functionalIλ:W01,p(Ω)→Rdefined by setting
Iλ(u)= 1 p
Ω
|∇u|pdx−λ q
Ω
(u+)qdx− 1 p∗
Ω
(u+)p∗dx. (3.1)
We introduce the Nehari manifolds Σλ=
u∈W01,p(Ω)\ {0}: Iλ(u), u =0 .
Suitably modifying the proof of [7, Lemma 2.2], it is easy to show that, for any λ >0, Σλ is a 1-codimensional submanifold ofW01,p(Ω), as it isC1-diffeomorphic to
u∈W01,p(Ω): u =1
\
u∈W01,p(Ω): u0 a.e.
.
Moreover, each nontrivial critical point ofIλis a nonnegative function which belongs toΣλ. We state some results, which are proved in [3,4,1].
As usually, we denote by S the best Sobolev constant of the embedding W01,p(Ω) → Lp∗(Ω) given by S=inf{up: u∈W01,p(Ω),|u|p∗=1}.
Lemma 3.1.LetNp2. ThenIλsatisfies the(P .S.)ccondition for allc∈(0,SN/pN ).
Theorem 3.2.LetNp2.Iλpossesses the mountain-pass geometry(M.P., for short), is bounded from below onΣλ andinfIλ(Σλ), which we denote bycλ, is the M.P. level, i.e.
cλ
def= infIλ(Σλ)=inf
v=0max
t0Iλ(t v).
Moreovercλis decreasing inλandlimλ→0cλ=SN/pN .
Up to translations, we may assume that 0 ∈ Ω. Moreover, in what follows, we fix r > 0 such that Br(0)= {x∈Rn: d(x,0) < r} ⊂Ω and the sets
Ωr+=
x∈Rn: d(x, Ω) < r
, Ωr−=
x∈Ω: d(x, ∂Ω) > r are both homotopically equivalent toΩ.
Further, we consider the space W0,rad1,p (Br)=
u∈W01,p Br(0)
: u(x)=u
|x|
and set
Iλ,rad(u)= 1 p
Br(0)
|∇u|pdx−λ q
Br(0)
(u+)qdx− 1 p∗
Br(0)
(u+)p∗dx ∀u∈W0,rad1,p (Br), Σrad=
u∈W0,rad1,p (Br)\ {0}:
Irad (u), u
=0 , mr(λ)=infIrad(Σrad).
Theorem 3.3.Using the previous notations,mr(λ)is the M.P. level ofIrad, i.e.
mr(λ)=inf
maxt0Irad(t u): u∈W0,rad1,p (Br), u=0
.
Moreovermr(λ)is decreasing inλandlimλ→0mr(λ)=SN/pN .
Remark 3.4.Once fixedr >0, we can repeat the same construction for any ∈(0, r), defining the levelsm (λ). If m (λ)is a critical level for any ∈(0, r), then Theorem 1.4 is proved. So we can suppose thatmr(λ)is not a critical level forIλ.
We define the continuous mapβ:Σλ→RNby setting β(u)=
Ωx(u+(x))p∗dx
Ω(u+(x))p∗dx .
Lemma 3.5.There existsλ∗>0such that ifλ∈(0, λ∗),u∈ΣλandIλ(u)mr(λ), thenβ(u)∈Ωr+.
Proof. By way of contradiction, let{λn}and{un}be such thatλn→0,un∈Σλn,Iλn(un)mr(λn)andβ(un) /∈Ωr+. Since limλ→0mr(λ)=limλ→0cλ=SN/pN ,
n→+∞lim Iλn(un)=SN/p
N . (3.2)
Fromun∈ΣλnandIλn(un)mr(λn), we have thatunis bounded and henceλn|u+n|q→0.
Consequently, asun∈Σλnand (3.2) holds, we get
n→+∞lim unp= lim
n→+∞|u+n|pp∗∗=SN/p. (3.3)
Definingwn=un/|u+n|p∗, we see that|wn+|p∗=1 and, by (3.3), limn→+∞wnp=S.
Furthermore, the functionsw˜n(x)=w+n(x)satisfy
| ˜wn|p∗=1 and ˜wnp→S.
Let us introduce the following notation D1,p(Rn)=
u∈Lp∗(Rn): ∂u
∂xi ∈Lp(Rn)fori=1, . . . , N
.
By Lemma 3.1 in [1], there is {εn} in R+ and {yn} in RN, such that εn→0, yn →y ∈ Ω and vn(x)= ε(Nn −p)/pw˜n(εnx+yn)→vinD1,p(RN), withv(x) >0.
Consideringφ∈C0∞(RN)such thatφ(x)=x inΩ, we infer β(un)=
Ωx(u+n(x))p∗dx
Ω(u+n(x))p∗dx =
RN
φ(x)
˜ wn(x)p∗
dx=
RN
φ(εnz+yn) vn(z)p∗
dz.
Moreover, by Lebesgue Theorem,
RN
φ(εnx+yn)
vn(x)p∗
dx→y∈Ω,
so that limn→∞β(un)=y∈Ω, in contradiction withβ(un) /∈Ωr+. 2
By Theorem 3.3, for each λ >0 we can consider vλ∈Σλ,rad, such that Iλ,rad(vλ)=mr(λ). Let us introduce γ:Ωr−→Iλmr(λ)∩Σλdefined by
γ (y)(x)=vλ(x−y) ifx∈Br(y),
0 otherwise.
Note thatγ is continuous and, asvλis radial,
β◦γ (y)=y ∀y∈Ωr−. (3.4)
So we get the following result.
Lemma 3.6.There existsλ∗>0such that ifλ∈(0, λ∗), then dimHk(Iλmr(λ)∩Σλ)dimHk(Ω).
Proof. Letλ∗be as in Lemma 3.5. We denote byγk andβk the homomorphisms induced byγ andβ respectively between thekth cohomology groups, i.e.
Hk(Ωr+) β
−→k Hk(Iλmr(λ)∩Σλ) γ
−→k Hk(Ωr−).
Since, from (3.4),γk◦βk=idk andΩr+is homotopically equivalent toΩ, the assert follows. 2 From now on, for anyba, we will denote(Iλ)basimply byIab.
Lemma 3.7.Iλ−1{a}is a deformation retract ofIab\Σλ, for anya∈(0, cλ)andba. Proof. LetC=W01,p(Ω)\(Σλ∪ {0}). For anyu∈C, the function
t∈ [0,+∞)−→Iλ(t u)
has one maximum pointθu, andθu=1 sincet u∈Σλif and only ift=θu.
Adapting the proof of [7, Lemma 2.2], we infer that u→θu is continuous, so that A= {u∈C: θu<1}and B= {u∈C: θu>1}are open sets andIab\Σλ⊂A∪B.
Letu∈Iab\Σλ. Ifu∈A, letδ(u)be the only valuet1 such thatIλ(t u)=a.
The functionδ:Iab∩A→Ris continuous. In fact, letF:(0,+∞)×A→Rbe defined byF (t, u)=Iλ(t u)−a andu0∈Iab∩A. Lett01 be such thatF (t0, u0)=0. Sinceθu0<1 whilet01, we get thatt0u0∈/Σλand
∂F
∂t (t0, u0)=
Iλ(t0u0), u0
=0,
so, by the Implicit Function Theorem,δis continuous.
Analogously, if u∈B, let δ(u) be defined as the only t ∈(0,1] such that Iλ(t u)=a, so that the function δ:Iab∩B→Ris continuous too.
Now letH:[0,1] ×(Iab\Σλ)→W01,p(Ω)be defined byH (t, u)=(t δ(u)+1−t )u. The proof is completed, as we see immediately that:
• His continuous;
• H (0, u)=u∀u;
• Iλ(H (1, u))=a∀u;
• H (t, u)∈Iab\Σλ∀t, ∀u;
• H (t, u)=u∀u∈Iλ−1{a} ∀t. 2
We now give a technical lemma (see [7, Lemma 5.3] for the proof).
Lemma 3.8.LetMbe a manifold andN⊂Mbe a closed oriented submanifold of codimensiond. IfW is a subset ofN closed inN, then
Pt(M,M\W )=tdPt(N,N\W ).
Proposition 3.9.Ifa∈(0, cλ)andbais a noncritical level forIλ, then Pt(Iλb, Iλa)=tPt(Iab∩Σλ).
Proof. If we set M=int(Iab),N=M∩ΣλandW=N, from Lemma 3.8 we get Pt
int(Iab),int(Iab)\Σλ
=tPt
int(Iab)∩Σλ
. (3.5)
Furthermore,aandbbeing not critical values forIλ, we have Pt(Iab, Iab\Σλ)=Pt
int(Iab),int(Iab)\Σλ
and Pt(Iab∩Σλ)=Pt
int(Iab)∩Σλ
. (3.6)
So, since
Pt(Iλb, Iλa)=Pt
Iab, Iλ−1(a) ,
the assert comes by (3.5), (3.6) and Lemma 3.7. 2
Corollary 3.10.There existsλ∗>0such that ifλ∈(0, λ∗)anda∈(0, cλ), then Pt(Iλmr(λ), Iλa)=t
Pt(Ω)+Zλ(t ) ,
whereZλ(t )is a polynomial with nonnegative integer coefficients.
Proof. Letλ∗be as in Lemma 3.6 and let us fixλ∈(0, λ∗)anda∈(0, cλ).
By Remark 3.4, we can assume thatmr(λ)is a noncritical value forIλ, so the assert derives from Lemma 3.6 and Proposition 3.9. 2
Proof of Theorem 1.4. Letλ∗be chosen in accordance with Corollary 3.10 andλ∈(0, λ∗). Letuj(1j m) be the critical points ofI in the strip(I )mar(λ), wherea∈(0, cλ). SinceI satisfies(P .S.)ccondition for allc∈(0, SN/p/N ) (see Lemma 3.1), the global Morse relation (2.1) gives
+∞
k=0
aktk=
+∞
k=0
dimHk(Imr(λ), Ia)tk+(1+t )Qλ(t ), (3.7) whereak=m
j=1dimCk(fλ, uj)andQλ(t )is a formal series with coefficients inN∪ {+∞}. Corollary 3.10 implies
+∞
k=0
aktk=t
Pt(Ω)+Zλ(t )
+(1+t )Qλ(t ) whence, fort=1, we get
m j=1
P1(I, uj)=P1(Ω)+Zλ(1)+2Qλ(1). (3.8)
Since bothZλ(1)andQλ(1)have nonnegative coefficients, problem(Pλ)has at leastP1(Ω)positive solutions, each counted with its own multiplicity. 2
Remark 3.11.If we considerΩ=A\k
i=1Ci, whereAandCi (i=1,2, . . . , k)are contractible, open, smooth and bounded nonempty sets inRn,Ci⊂Afor anyi=1,2, . . . , kandCi∩Cj= ∅for anyi=j, Theorem 1.4 guarantees that(Pλ)has at leastk+1 solutions, each counted with its own multiplicity.
4. Nondegeneracy and local Morse theory
Theorem 1.4 assures that problem(Pλ)has at leastP1(Ω)solutions, which can be distinct or, if not, counted with their multiplicities. However the evaluation of the multiplicity of a critical point is not easy, in general.
In a Hilbert space, the local behavior of the functional near a critical point is quite clear if the critical point is nondegenerate and computing the critical groups of a nondegenerate critical point is possible via its Morse index.
Successively, Gromoll and Meyer generalized these ideas in order to compute the critical groups of an isolated critical pointu, possibly degenerate, having finite Morse index, if the second derivative of the functional inuis a Fredholm operator. The generalized Morse lemma is a basic tool for computing the critical groups and the theory of Fredholm operators provides a natural setting for this lemma. Moreover we emphasize that such critical groups estimates seem to require a Hilbert space structure.
We remark that whenp=2, several conceptual difficulties arise for developing a local Morse theory forIλ. Firstly, all the critical points ofIλare degenerate in the classical sense given in Hilbert spaces, as the Banach spaceW01,p(Ω) is not isomorphic to its dual space. Moreover the second derivative ofIλin each critical point is not Fredholm, so that the (generalized) Morse Lemma does not hold, and relations between differentiable notions, like Morse index, and critical groups are not available in general. Our idea is to perform an approximation ofIλ in terms of functionals on W01,p(Ω), for which we are able to develop a local Morse theory.
In what follows, we assume thatp2. In this case it is standard to check thatIλis aC2functional. Ifp=q=2 the functionalIλis notC2, nevertheless this case can be easily covered just by replacingIλwith the functional
I˜λ(u)=1 2
Ω
|∇u|2dx−λ 2
Ω
u2dx− 1 2∗
Ω
(u+)2∗dx.
In the sequel of the work we will simply refer toIλ, as all the arguments analogously work forI˜λ. For any 2pq < p∗,α0 we consider the followingC2functionals
Tα:W01,p(Ω)→R, Tα(u)= 1 p
Ω
α+ |∇u|2p/2
−λ q
Ω
(u+)qdx− 1 p∗
Ω
(u+)p∗ (4.1)
which approximateIλinC1(A)forα→0, ifAis a bounded subset ofW01,p(Ω). Forp=q=2 we replaceTαwith T˜α:W01,p(Ω)→R, T˜α(u)=1
2
Ω
α+ |∇u|2
−λ 2
Ω
u2dx− 1 2∗
Ω
(u+)2∗. (4.2)
Moreover we consider a functionalJα:W01,p(Ω)→Rof the type Jα(u)=Tα(u)−
Ω
f u
withf ∈C1(Ω). Forp=q=2 we setJα= ˜Tα−
Ωf u.
We begin to establish that, for anyα0,Jα satisfy a local Palais–Smale condition on each level. It can be proved reasoning as in Lemma 3.2 of [18]. For reader’s convenience, we sketch the proof.
Lemma 4.1.Assumep2. There existsR >0such that, for any fixedα0,f ∈C1(Ω)and anyu∈W01,p(Ω), the functionalJαsatisfies(P .S.)condition onBR(u)= {v∈W01,p(Ω): v−uR}.
Proof. For convenience we fixα0 and denoteJα=J. FixingR∈(0,SN/p
2
2 ), if(um)⊂BR(u)is a sequence such thatJ(um)→0, then(um)is bounded, thus converges to someu¯∈BR(u), weakly inW01,p(Ω)and strongly in each Lr(Ω), withr < p∗. Moreover, arguing as in Lemma 3.1 in [27], one can prove that(α+ |∇um|2)p−22∇umconverges to(α+ |∇ ¯u|2)p−22∇ ¯uweakly inLp/(p−1)(Ω)and a.e. inΩ.
Therefore, for anyz∈W01,p(Ω), J(u), z¯
= lim
m→+∞
J(um), z
=0 so thatu¯is a critical point and, in particular,
J(um), um
−
J(u),¯ u¯
=o(1). (4.3)
Using [8] (cf. [35]), we have that (um− ¯u)+p∗
p∗=(um)+p∗
p∗−(u)¯ +p∗
p∗+o(1). (4.4)
Moreover arguing as in Lemma 3.2 in [18] we can infer that
Ω
α+ |∇um− ∇ ¯u|2p−2
2 |∇um− ∇ ¯u|2dx
=
Ω
α+ |∇um|2p−22
|∇um|2dx−
Ω
α+ |∇ ¯u|2p−22
|∇ ¯u|2dx+o(1). (4.5)
From (4.3), (4.4) and (4.5) we deduce
Ω
|∇um− ∇ ¯u|pdx−
Ω
|um−u|p∗dx
Ω
α+ |∇um− ∇ ¯u|2p−2
2 |∇um− ∇ ¯u|2dx−
Ω
(um− ¯u)+p∗
dx
=
J(um), um
−
J(u),¯ u¯
+o(1)=o(1). (4.6)
Denotinga=lim supm→+∞um− ¯up, by (4.6) and the definition ofSwe have alim sup
m→+∞
Ω
|um− ¯u|p∗S−p∗/pap∗/p.
Therefore, ifa >0, this impliesaSN/p, hence SN/palim sup
m→+∞
um−u + u− ¯up
(2R)p< SN/p
which is absurd. Therefore it must bea=0 and thusumstrongly converges tou¯inW01,p(Ω). 2
Now we state two results concerning critical group computations via Morse index. For the proofs, we refer the reader to Theorems 1.3 and 1.4 of [18] (see also [16]).
Theorem 4.2.Letp >2 andα >0. Letu∈W01,p(Ω)be a nondegenerate(in the sense of Definition 1.5)critical point ofJα. Then the Morse indexm(Jα, u)is finite and
Cj(Jα, u)∼=K ifj =m(Jα, u), (4.7)
Cj(Jα, u)= {0} ifj =m(Jα, u). (4.8)
Theorem 4.3.Letp >2andα >0. Letu∈W01,p(Ω)be an isolated critical point ofJα. Thenm(Jα, u)andm∗(Jα, u) are finite and
Cj(Jα, u)= {0} for anyjm(Jα, u)−1andjm∗(Jα, u)+1.
Moreover,dimCj(Jα, u) <∞for anyj∈N.
Remark 4.4.Forp=2 Theorems 4.2 and 4.3 hold for the functionalJα, as consequence of classical results in Morse theory, based on Morse Lemma. We refer to Theorem 4.1 and Corollary 5.1 in [11].
