Amann-Zehnder type results for p-Laplace problems
aa r X i v : . [ m a t h . A P ] N ov AMANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS SILVIA CINGOLANI, MARCO DEGIOVANNI, AND GIUSEPPINA VANNELLA
Abstract.
The existence of a nontrivial solution is proved for a class of quasilinearelliptic equations involving, as principal part, either the p -Laplace operator or the oper-ator related to the p -area functional, and a nonlinearity with p -linear growth at infinity.To this aim, Morse theory techniques are combined with critical groups estimates. Introduction
In 1980, Amann and Zehnder [4] studied the asymptotically linear elliptic problem(1.1) ( − ∆ u = g ( u ) in Ω ,u = 0 on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary, g : R → R is a C -functionsuch that g (0) = 0 and there exists λ ∈ R such thatlim | s |→∞ g ′ ( s ) = λ . They proved that problem (1.1) admits a nontrivial solution u , supposing that λ is notan eigenvalue of − ∆, the so-called nonresonance condition at infinity, and that thereexists some eigenvalue of − ∆ between λ and g ′ (0). The same result was obtained byChang [9] in 1981, using Morse theory for manifolds with boundary, and by Lazer andSolimini [37] in 1988, combining mini-max characterization of the critical point and Morseindex estimates. More precisely, the basic idea in [37] is to recognize that the energyfunctional associated to the asymptotically linear problem (1.1) has a saddle geometry,which implies that a suitable Poincar´e polynomial is not trivial, and also to show that acertain critical group at zero is trivial, to ensure the existence of a solution u = 0 of (1.1).In the present work, we are interested in finding nontrivial solutions u for the quasilinearelliptic problem(1.2) − div h ( κ + |∇ u | ) p − ∇ u i = g ( u ) in Ω ,u = 0 on ∂ Ω , where Ω is a bounded open subset of R N , N ≥
1, with ∂ Ω of class C ,α for some α ∈ ]0 , κ ≥ p > g : R → R is a C -function such that: Mathematics Subject Classification.
Key words and phrases. p -Laplace operator, p -area functional, nontrivial solutions, Morse theory,critical groups, functionals with lack of smoothness.The research of the authors was partially supported by Gruppo Nazionale per l’Analisi Matematica,la Probabilit`a e le loro Applicazioni (INdAM). In particular S. Cingolani is supported by G.N.A.M.P.A.Project 2016 “Studio variazionale di fenomeni fisici non lineari”. ( a ) g (0) = 0 and there exists λ ∈ R such thatlim | s |→∞ g ( s ) | s | p − s = λ . About the principal part of the equation, the reference cases are κ = 0, which yields the p -Laplace operator , and κ = 1, which yields the operator related to the p -area functional .In the case p = 2 the value of κ is irrelevant.It is standard that weak solutions u of (1.2) correspond to critical points of the C -functional f : W ,p (Ω) → R defined as(1.3) f ( u ) = Z Ω Ψ p,κ ( ∇ u ) dx − Z Ω G ( u ) dx , where Ψ p,κ ( ξ ) = 1 p h(cid:0) κ + | ξ | (cid:1) p − κ p i , G ( s ) = Z s g ( t ) dt. With reference to the approach of [37], when p = 2 the new difficulties that one hasto face are related to both the main ingredients of the argument, namely to recognize asaddle structure, with a related information on a suitable Poincar´e polynomial, and toprovide an estimate of the critical groups at zero by some Hessian type notion.Concerning the first aspect, the spectral properties of − ∆ p are not yet well understood.We say that the real number λ is an eigenvalue of − ∆ p if the equation − ∆ p u = λ | u | p − u admits a nontrivial solution u ∈ W ,p (Ω) and we denote by σ ( − ∆ p ) the set of sucheigenvalues. It is known that there exists a first eigenvalue λ >
0, which is simple,and a second eigenvalue λ > λ , both possessing several equivalent characterizations(see [5, 6, 22, 26, 39]). Moreover, one can define in at least three different ways a divergingsequence ( λ m ) of eigenvalues of − ∆ p (see [13, 26, 41]), but it is not known if they agree for m ≥ σ ( − ∆ p ) is covered. Therefore it is not standard to recognizea saddle type geometry for the energy functional associated to the quasilinear problem.On the other hand, for functionals defined on Banach spaces, serious difficulties arise inextending Morse theory (see [48, 47, 10, 11, 12]). More precisely, by standard deformationresults, which hold also in general Banach spaces, one can prove the so-called Morserelations, which can be written as ∞ X m =0 C m t m = ∞ X m =0 β m t m + (1 + t ) Q ( t ) , where ( β m ) is the sequence of the Betti numbers of a pair of sublevels ( { f ≤ b } , { f < a } )and ( C m ) is a sequence related to the critical groups of the critical points u of f with a ≤ f ( u ) ≤ b (see e.g. the next Definition 2.1 and [11, Theorem I.4.3]). The problem,in the extension from Hilbert to Banach spaces, concerns the estimate of ( C m ), henceof critical groups, by the Hessian of f or some related concept. In a Hilbert setting,the classical Morse lemma and the generalized Morse lemma [30] provide a satisfactoryanswer. For Banach spaces, a similar general result is so far not known, also due to thelack of Fredholm properties of the second derivative of the functional.The first difficulty has been overcome by the first two authors in [13] for a problemquite similar to (1.2). By generalizing from [11] the notion of homological linking, in [13,Theorem 3.6] an abstract result has been proved which allows to produce a pair of sublevels MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 3 ( { f ≤ b } , { f < a } ) with a nontrivial homology group. In order to describe its dimensionin terms of λ in the setting of problem (1.2), it is then convenient to set, whenever m ≥ λ m = inf (cid:26) sup A E : A ⊆ M , A is symmetric and Index ( A ) ≥ m (cid:27) , where M = (cid:26) u ∈ W ,p (Ω) : Z Ω | u | p dx = 1 (cid:27) , E ( u ) = Z Ω |∇ u | p dx and Index denotes the Z -cohomological index of Fadell and Rabinowitz [27, 28]. For amatter of convenience, we also set λ = −∞ . It is well known that ( λ m ) is a nondecreasingdivergent sequence. The arguments of [13] apply for any p > p becomes relevant. In [16] the first and the lastauthor have proved, for p > κ >
0, an extension of the Morse Lemma and establisheda connection between the critical groups and the Morse index, taking advantage of thefact that, under suitable assumptions on g , the functional f is actually of class C on W ,p (Ω) and that Ψ ′′ p,κ ( η )[ ξ ] ≥ ν p,κ | ξ | with ν p,κ > . Related results in the line of Morse theory have been proved by the first and the lastauthor, in the case p >
2, in [17, 18, 19]. By means of the results of [19], an Amann-Zehnder type result has been proved in [13] for a problem quite similar to (1.2), providedthat p > < p < p ≥ g , we provide an Amann-Zehnder type result for any p > < p <
2, the functional f is not of class C on W ,p (Ω). For κ = 0, even the function Ψ p,κ is not of class C on R N .If κ > p ≥
2, let us denote by m ( f,
0) the supremum of the dimensions of thelinear subspaces where the quadratic form Q : W , (Ω) → R defined as Q ( u ) = κ p − Z Ω |∇ u | dx − g ′ (0) Z Ω u dx if κ > p > , Z Ω |∇ u | dx − g ′ (0) Z Ω u dx if κ = 0 and p = 2 , is negative definite. Let us also denote by m ∗ ( f,
0) the supremum of the dimensions ofthe linear subspaces where the quadratic form Q is negative semidefinite. If κ = 0 and1 < p <
2, we set m ( f,
0) = m ∗ ( f,
0) = 0.Our first result is the following:
Theorem 1.1.
Assume < p < ∞ , κ ≥ and hypothesis ( a ) on g . Suppose also that λ σ ( − ∆ p ) and denote by m ∞ the integer such that λ m ∞ < λ < λ m ∞ +1 .If m ∞ [ m ( f, , m ∗ ( f, , then there exists a nontrivial solution u of (1.2) . It is easily seen that, if p = 2, the assumption that there exists some eigenvalue of − ∆between λ and g ′ (0) is equivalent to m ∞ [ m ( f, , m ∗ ( f, SILVIA CINGOLANI, MARCO DEGIOVANNI, AND GIUSEPPINA VANNELLA
Differently from [13], we aim also to deal with the resonant case, namely λ ∈ σ ( − ∆ p ).This is not motivated by the pure wish of facing a more complicated situation. To ourknowledge, nobody has so far excluded the possibility that σ ( − ∆ p ) = { λ } ∪ [ λ , + ∞ [.In such a case, the restriction λ σ ( − ∆ p ) would be quite severe. Taking into accountTheorem 1.1, the next result has interest if λ ∈ σ ( − ∆ p ). Theorem 1.2.
Assume hypothesis ( a ) on g and one of the following: ( b − ) we have lim | s |→∞ [ pG ( s ) − g ( s ) s ] = −∞ ; then we denote by m ∞ the integer such that λ m ∞ < λ ≤ λ m ∞ +1 ;( b + ) we have lim | s |→∞ [ pG ( s ) − g ( s ) s ] = + ∞ and, moreover, either < p ≤ with κ ≥ or p > with κ = 0 ; then we denoteby m ∞ the integer such that λ m ∞ ≤ λ < λ m ∞ +1 . If m ∞ [ m ( f, , m ∗ ( f, , then there exists a nontrivial solution u of (1.2) . Remark 1.3.
Concerning the lower order term, examples of g satisfying ( a ) and ( b + ) or( b − ) are given by g ( s ) = λ (1 + s ) p − s + µ (1 + s ) q − s with µ = 0 and 0 < q < p ≤ ,g ( s ) = λ | s | p − s + µ | s | q − s with µ = 0 and 2 ≤ q < p , so that, respectively, G ( s ) = λp h(cid:0) s (cid:1) p − i + µq h(cid:0) s (cid:1) q − i ,G ( s ) = λp | s | p + µq | s | q . Remark 1.4.
Let p = 4, so thatΨ p,κ ( ξ ) = 14 | ξ | + 12 κ | ξ | , and let g ( s ) = λ m s + µs with m ≥ µ >
0, so that lim | s |→∞ [4 G ( s ) − g ( s ) s ] = + ∞ , while f ( u ) = 14 Z Ω (cid:2) |∇ u | − λ m | u | (cid:3) dx + 12 Z Ω (cid:2) κ |∇ u | − µ | u | (cid:3) dx . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 5 It is clear that we cannot describe the geometry of the functional f , if we have no infor-mation concerning κ and µ . For this reason in ( b + ) only the case κ = 0 is considered,when p > k > k = 0 and 1 < p < κ ≥ < p < ∞ (see Theorem 2.8).Sections 3, 4, 5 and 6 are devoted to the proof, by a finite dimensional reductionintroduced in a different setting in [36], of the results stated in Section 2, while Section 7contains the proof of Theorems 1.1 and 1.2.2. Critical groups estimates
In this section we consider a class of functionals including (1.3). More precisely, let Ωbe a bounded open subset of R N , N ≥
1, with ∂ Ω of class C ,α for some α ∈ ]0 , f : W ,p (Ω) → R be the functional defined as(2.1) f ( u ) = Z Ω Ψ( ∇ u ) dx − Z Ω G ( x, u ) dx where G ( x, s ) = R s g ( x, t ) dt . We assume that:(Ψ ) the function Ψ : R N → R is of class C with Ψ(0) = 0 and ∇ Ψ(0) = 0; moreover,there exist 1 < p < ∞ , κ ≥ < ν ≤ C such that the functions (Ψ − ν Ψ p,κ )and ( C Ψ p,κ − Ψ) are both convex; such a p is clearly unique;(Ψ ) if κ = 0 and 1 < p <
2, then Ψ is of class C on R N \ { } ; otherwise, Ψ is ofclass C on R N ;( g ) the function g : Ω × R → R is such that g ( · , s ) is measurable for every s ∈ R and g ( x, · ) is of class C for a.e. x ∈ Ω; moreover, we suppose that: – if p < N , there exist C, q > q ≤ p ∗ − NpN − p − | g ( x, s ) | ≤ C (1 + | s | q ) for a.e. x ∈ Ω and every s ∈ R ; – if p = N , there exist C, q > | g ( x, s ) | ≤ C (1 + | s | q ) for a.e. x ∈ Ω and every s ∈ R ; – if p > N , for every S > C S > | g ( x, s ) | ≤ C S for a.e. x ∈ Ω and every s ∈ R with | s | ≤ S ;( g ) for every S > b C S > | D s g ( x, s ) | ≤ b C S for a.e. x ∈ Ω and every s ∈ R with | s | ≤ S .
From (Ψ ) it follows that Ψ is strictly convex. Moreover, under these assumptions, it iseasily seen that f : W ,p (Ω) → R is of class C , while it is of class C if p > max { N, } .Finally, even in the case g = 0, f is never of class C for 1 < p < C inthe case p = 2 iff Ψ is a quadratic form on R N (see [1, Proposition 3.2]). SILVIA CINGOLANI, MARCO DEGIOVANNI, AND GIUSEPPINA VANNELLA
Now, let u ∈ W ,p (Ω) be a critical point of the functional f , namely a weak solutionof ( − div [ ∇ Ψ( ∇ u )] = g ( x, u ) in Ω ,u = 0 on ∂ Ω . According to [31, 25, 38, 45, 46], u ∈ C ,β (Ω) for some β ∈ ]0 ,
1] (see also the nextTheorems 3.1 and 3.2).Let us recall the first ingredient we need from [11, 23, 40].
Definition 2.1.
Let G be an abelian group, c = f ( u ) and f c = (cid:8) u ∈ W ,p (Ω) : f ( u ) ≤ c (cid:9) .The m -th critical group of f at u with coefficients in G is defined by C m ( f, u ; G ) = H m ( f c , f c \ { u } ; G ) , where H ∗ stands for Alexander-Spanier cohomology [44]. We will simply write C m ( f, u ),if no confusion can arise.In general, it may happen that C m ( f, u ) is not finitely generated for some m and that C m ( f, u ) = { } for infinitely many m ’s. If however u is an isolated critical point, underassumptions (Ψ ) and ( g ) it follows from [14, Theorem 1.1] and [3, Theorem 3.4] that C ∗ ( f, u ) is of finite type.The other ingredient is a notion of Morse index, which is not standard, as the func-tional f is not in general of class C .In the case κ > < p < ∞ , observe that ν min { ( p − , } ( κ + | η | ) p − | ξ | ≤ Ψ ′′ ( η )[ ξ ] ≤ C max { ( p − , } ( κ + | η | ) p − | ξ | for any η, ξ ∈ R N , as (Ψ − ν Ψ p,κ ) and ( C Ψ p,κ − Ψ) are both convex. Therefore, there exists ˜ ν > ν | ξ | ≤ Ψ ′′ ( ∇ u ( x )) [ ξ ] ≤ ν | ξ | for any x ∈ Ω and ξ ∈ R N , as ∇ u is bounded. Moreover, D s g ( x, u ) ∈ L ∞ (Ω), as u is bounded. Thus, we can definea smooth quadratic form Q u : W , (Ω) → R by Q u ( v ) = Z Ω Ψ ′′ ( ∇ u )[ ∇ v ] dx − Z Ω D s g ( x, u ) v dx and define the Morse index of f at u (denoted by m ( f, u )) as the supremum of thedimensions of the linear subspaces of W , (Ω) where Q u is negative definite and the large Morse index of f at u (denoted by m ∗ ( f, u )) as the supremum of the dimensionsof the linear subspaces of W , (Ω) where Q u is negative semidefinite. We clearly have m ( f, u ) ≤ m ∗ ( f, u ) < + ∞ . Let us point out that Q u is well behaved on W , (Ω),while f is naturally defined on W ,p (Ω).In the case κ = 0 and p >
2, we still haveΨ ′′ ( ∇ u ( x )) [ ξ ] ≤ ν | ξ | for any x ∈ Ω and ξ ∈ R N , so that Q u : W , (Ω) → R , m ( f, u ) and m ∗ ( f, u ) can be defined as before and m ( f, u ) ≤ m ∗ ( f, u ). However, m ( f, u ) and m ∗ ( f, u ) might take the value + ∞ . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 7 Finally, in the case κ = 0 and 1 < p < p − ν | η | − p | ξ | ≤ Ψ ′′ ( η )[ ξ ] ≤ C | η | − p | ξ | for any η, ξ ∈ R N with η = 0 , whence Ψ ′′ ( ∇ u ( x )) [ ξ ] ≥ ˜ ν | ξ | for any x ∈ Ω with ∇ u ( x ) = 0 and ξ ∈ R N . Set Z u = { x ∈ Ω : ∇ u ( x ) = 0 } ,X u = (cid:26) v ∈ W , (Ω) : ∇ v ( x ) = 0 a.e. in Z u and |∇ v | |∇ u | − p ∈ L (Ω \ Z u ) (cid:27) . Then ( v | w ) u = Z Ω \ Z u Ψ ′′ ( ∇ u ) [ ∇ v, ∇ w ] dx is a scalar product on X u which makes X u a Hilbert space continuously embeddedin W , (Ω). Moreover, we can define a smooth quadratic form Q u : X u → R by Q u ( v ) = Z Ω \ Z u Ψ ′′ ( ∇ u )[ ∇ v ] dx − Z Ω D s g ( x, u ) v dx and denote again by m ( f, u ) the supremum of the dimensions of the linear subspaces of X u where Q u is negative definite and by m ∗ ( f, u ) the supremum of the dimensions ofthe linear subspaces of X u where Q u is negative semidefinite. Since the derivative of Q u is a compact perturbation of the Riesz isomorphism, we have m ( f, u ) ≤ m ∗ ( f, u ) < + ∞ .For a sake of uniformity, let us set X u = W , (Ω) when κ > < p < ∞ .Now we can state the results concerning the critical groups estimates for the func-tional (2.1). Theorem 2.2.
Let κ > and < p < ∞ . Let u ∈ W ,p (Ω) be a critical point of thefunctional f defined in (2.1) .Then we have C m ( f, u ) = { } whenever m < m ( f, u ) or m > m ∗ ( f, u ) . When the quadratic form Q u has no kernel, we can provide a complete description ofthe critical groups. Theorem 2.3.
Let κ > and < p < ∞ . Let u ∈ W ,p (Ω) be a critical point of thefunctional f defined in (2.1) with m ( f, u ) = m ∗ ( f, u ) .Then u is an isolated critical point of f and we have ( C m ( f, u ) ≈ G if m = m ( f, u ) ,C m ( f, u ) = { } if m = m ( f, u ) . If u is an isolated critical point of f , then a sharper form of Theorem 2.2 can be proved.Taking into account Theorem 2.3, only the case m ( f, u ) < m ∗ ( f, u ) is interesting. Theorem 2.4.
Let κ > and < p < ∞ . Let u ∈ W ,p (Ω) be an isolated critical pointof the functional f defined in (2.1) with m ( f, u ) < m ∗ ( f, u ) .Then one and only one of the following facts holds: SILVIA CINGOLANI, MARCO DEGIOVANNI, AND GIUSEPPINA VANNELLA ( a ) we have ( C m ( f, u ) ≈ G if m = m ( f, u ) ,C m ( f, u ) = { } if m = m ( f, u ) ;( b ) we have ( C m ( f, u ) ≈ G if m = m ∗ ( f, u ) ,C m ( f, u ) = { } if m = m ∗ ( f, u ) ;( c ) we have C m ( f, u ) = { } whenever m ≤ m ( f, u ) or m ≥ m ∗ ( f, u ) . Remark 2.5.
Since the value of κ is irrelevant in the case p = 2, Theorems 2.2, 2.3and 2.4 cover also the case κ = 0 with p = 2.In the case κ = 0 and p = 2, we cannot provide such a complete description. Let usmention, however, that critical groups estimates have been obtained in [2] when Ω is aball centered at the origin, and the critical point u is a positive and radial function suchthat |∇ u ( x ) | 6 = 0 for x = 0.Apart from the radial case, if p > g is subjected to assumptions that imply f tobe of class C on W ,p (Ω), it has been proved in [36, Theorem 3.1] that C m ( f, u ) = { } whenever m < m ( f, u ). On the contrary, there is no information, in general, when p > m > m ∗ ( f, u ).In the case 1 < p <
2, the situation turns out to be in some sense reversed. Wewill prove a result when m > m ∗ ( f, u ), while we have no information, in general, when m < m ( f, u ). Theorem 2.6.
Let κ = 0 and < p < . Let u ∈ W ,p (Ω) be a critical point of thefunctional f defined in (2.1) .Then we have C m ( f, u ) = { } whenever m > m ∗ ( f, u ) . However, in the case u = 0, we can provide an optimal result in the line of Theorem 2.3. Theorem 2.7.
Let κ = 0 and < p < . Let be a critical point of the functional f defined in (2.1) .Then we have m ( f,
0) = m ∗ ( f,
0) = 0 and is a strict local minimum and an isolatedcritical point of f with ( C m ( f, ≈ G if m = 0 ,C m ( f,
0) = { } if m = 0 . Finally, under more specific assumptions we can extend Theorem 2.2 to any κ and p .This will be enough for the results stated in the Introduction. Theorem 2.8.
Let κ ≥ and < p < ∞ . Let be an isolated critical point of thefunctional f defined in (2.1) and suppose that g is independent of x and satisfies assump-tion ( g ) with q < p ∗ − in the case p < N .Then we have C m ( f,
0) = { } whenever m < m ( f, or m > m ∗ ( f, . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 9 Some auxiliary results
In the following, for any q ∈ [1 , ∞ ] we will denote by k k q the usual norm in L q (Ω). Wealso set, for any u ∈ C ,α (Ω), k u k C ,α = sup Ω | u | + sup Ω |∇ u | + sup x,y ∈ Ω x = y |∇ u ( x ) − ∇ u ( y ) || x − y | α . Throughout this section, we assume that Ω is a bounded open subset of R N and we supposethat Ψ and g satisfy assumptions (Ψ ), (Ψ ) and ( g ), without any further restriction on p and κ .In the first part, we adapt to our setting some regularity results from [31, 25, 38, 45, 46]. Theorem 3.1.
For every u ∈ W ,p (Ω) , there exists r > such that, for any u ∈ W ,p (Ω) and w ∈ W − , ∞ (Ω) satisfying Z Ω [ ∇ Ψ( ∇ u ) · ∇ v − g ( x, u ) v ] dx ≤ h w, v i for any v ∈ W ,p (Ω) with vu ≥ a.e. in Ω , k∇ u − ∇ u k p ≤ r , we have u ∈ L ∞ (Ω) and k u k ∞ ≤ C ( k w k W − , ∞ ) . Proof.
We only sketch the proof the case 1 < p < N . The case p ≥ N is similar and evensimpler. Since (Ψ − ν Ψ p,κ ) is convex, we have ∇ Ψ( ξ ) · ξ = ( ∇ Ψ( ξ ) − ∇ Ψ(0)) · ξ ≥ ν (cid:0) κ + | ξ | (cid:1) p − | ξ | for every ξ ∈ R N . Then the argument is the same of [31, Corollary 1.1]. We only have to remark that, forevery V ∈ L N/p (Ω) and q < ∞ , there exists r > u ∈ W ,p (Ω), V ∈ L N/p (Ω) and w ∈ W − , ∞ (Ω) satisfying Z Ω (cid:2) ∇ Ψ( ∇ u ) · ∇ v − V | u | p − uv (cid:3) dx ≤ h w, v i for any v ∈ W ,p (Ω) with vu ≥ , k V − V k N/p ≤ r , we have u ∈ L q (Ω) and k u k q ≤ C ( q, k u k p ∗ , k w k W − , ∞ )(see, in particular, [31, Proposition 1.2 and Remark 1.1]). The key point is that, for any ε >
0, there exist r, k such that k ≥ k = ⇒ Z {| V | >k } | V | N/p dx ≤ ε whenever k V − V k N/p ≤ r . After removing the dependence on V in [31, Proposition 1.2],hence on u in [31, Corollary 1.1], the argument is the same of [31]. (cid:3) Theorem 3.2.
Assume that ∂ Ω is of class C ,α for some α ∈ ]0 , . Then there exists β ∈ ]0 , such that any solution u of ( u ∈ W ,p (Ω) , − div [ ∇ Ψ( ∇ u )] = w − div w in W − ,p ′ (Ω) , with w ∈ L ∞ (Ω) and w ∈ C ,α (Ω; R N ) , belongs also to C ,β (Ω) and we have k u k C ,β ≤ C ( k w k ∞ , k w k C ,α ) . Proof.
Since Ψ is strictly convex and ∇ Ψ( ξ ) · ξ ≥ ν (cid:0) κ + | ξ | (cid:1) p − | ξ | for every ξ ∈ R N ,it is standard that, for every w ∈ L ∞ (Ω) and w ∈ C ,α (Ω; R N ), there exists one andonly one u ∈ W ,p (Ω) ∩ L ∞ (Ω) such that − div [ ∇ Ψ( ∇ u )] = w − div w . Moreover, wehave k u k ∞ ≤ C ( k w k ∞ , k w k C ,α )(see e.g. [35]).Now, for every N ≥
1, fix a nonnegative smooth function ̺ with compact support inthe unit ball of R N and unit integral. Then define, for every Φ ∈ L loc ( R N ) and ε > R ε Φ)( ξ ) = Z ̺ ( y )Φ( ξ − εy ) dy . It is easily seen that there exist 0 < ˇ ν ( N, p ) ≤ ˇ C ( N, p ) such thatˇ ν (1 + | ξ | ) p − ≤ Z ̺ ( y )(1 + | ξ − ty | ) p − dy ≤ ˇ C (1 + | ξ | ) p − , ˇ ν (1 + | ξ | ) p − ≤ Z ̺ ( y )( t + | ξ − y | ) p − dy ≤ ˇ C (1 + | ξ | ) p − , for every t ∈ [0 ,
1] and ξ ∈ R N . Then there exist 0 < ˆ ν ( N, p ) ≤ b C ( N, p ) such thatˆ ν ( ε + κ + | ξ | ) p − ≤ Z ̺ ( y )( κ + | ξ − εy | ) p − dy ≤ b C ( ε + κ + | ξ | ) p − , for every κ ≥ ε > ξ ∈ R N .Observe that Ψ p,κ ∈ W , loc ( R N ) and p −
12 ( κ + | η | ) p − | ξ | ≤ Ψ ′′ p,κ ( η )[ ξ ] ≤ ( κ + | η | ) p − | ξ | for every η, ξ ∈ R N with η = 0.It follows that there exist 0 < ˜ ν ( N, p ) ≤ e C ( N, p ) such that˜ ν ( ε + κ + | η | ) p − | ξ | ≤ ( R ε Ψ p,κ ) ′′ ( η )[ ξ ] ≤ e C ( ε + κ + | η | ) p − | ξ | , for every κ ≥ ε > ξ, η ∈ R N . Since (Ψ − ν Ψ p,κ ) and ( C Ψ p,κ − Ψ) are bothconvex, we infer that R ε Ψ : R N → R is a smooth function satisfying(3.1) ν ˜ ν ( ε + κ + | η | ) p − | ξ | ≤ ( R ε Ψ) ′′ ( η )[ ξ ] ≤ C e C ( ε + κ + | η | ) p − | ξ | , for every ε > ξ, η ∈ R N . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 11 Again, from the results of [35], it follows that there exists one and only one u ε ∈ W ,p (Ω) ∩ L ∞ (Ω) such that − div [ ∇ ( R ε Ψ)( ∇ u ε )] = w − div w . Moreover, we have k u ε k ∞ ≤ C ( k w k ∞ , k w k C ,α )and the estimate is independent of ε for, say, 0 < ε ≤ u ε ∈ C ,β (Ω) and k u ε k C ,β ≤ C ( k w k ∞ , k w k C ,α )for some β ∈ ]0 , ε ∈ ]0 , u ε ) is convergent, as ε →
0, to u in C (Ω) and the assertion follows. (cid:3) From (3.1) we also infer the next result.
Proposition 3.3.
We have Ψ ∈ W ,qloc ( R N ) for some q > N , so that the map ∇ Ψ : R N → R N is locally H¨older continuous. Now let X be a reflexive Banach space. The next concept is taken from [8, 43]. Definition 3.4.
Let D ⊆ X . A map F : D → X ′ is said to be of class ( S ) + if, for everysequence ( u k ) in D weakly convergent to u in X withlim sup k h F ( u k ) , u k − u i ≤ , we have k u k − u k → Proposition 3.5.
Let f : X → R be a function of class C and let C be a closed andconvex subset of X . Assume that f ′ is of class ( S ) + on C .Then the following facts hold: ( a ) f is sequentially lower semicontinuos on C with respect to the weak topology; ( b ) if ( u k ) is a sequence in C weakly convergent to u with lim sup k f ( u k ) ≤ f ( u ) , we have k u k − u k → ; ( c ) any bounded sequence ( u k ) in C , with k f ′ ( u k ) k → , admits a convergent subse-quence.Proof. Let ( u k ) be a sequence in C weakly convergent to u . To prove ( a ) we may assume,without loss of generality, that lim sup k f ( u k ) ≤ f ( u ) . Let t k ∈ ]0 ,
1[ be such that f ( u k ) = f ( u ) + h f ′ ( v k ) , u k − u i , v k = u + t k ( u k − u ) . Then ( v k ) also is a sequence in C weakly convergent to u andlim sup k h f ′ ( v k ) , v k − u i = lim sup k t k h f ′ ( v k ) , u k − u i = lim sup k t k ( f ( u k ) − f ( u )) ≤ . Since f ′ is of class ( S ) + on C , we infer that k v k − u k →
0, hence thatlim k f ( u k ) = lim k [ f ( u ) + h f ′ ( v k ) , u k − u i ] = f ( u )and assertion ( a ) follows. To prove ( b ), let τ k ∈ (cid:3) , (cid:2) be such that f ( u k ) − f (cid:18) u k + 12 u (cid:19) = 12 h f ′ ( w k ) , u k − u i , w k = u + τ k ( u k − u ) . Observe that (cid:0) u k + u (cid:1) also is a sequence in C weakly convergent to u , whencelim inf k f (cid:18) u k + 12 u (cid:19) ≥ f ( u ) . It follows lim sup k h f ′ ( w k ) , u k − u i = lim sup k (cid:20) f ( u k ) − f (cid:18) u k + 12 u (cid:19)(cid:21) ≤ k w k − u k →
0. Since ( τ k ) is bounded away from 0, we conclude that k u k − u k → c ) we may assume that ( u k ) is weakly convergent to some u , whencelim k h f ′ ( u k ) , u k − u i = 0 . Since f ′ is of class ( S ) + on C , assertion ( c ) also follows. (cid:3) We end the section with a result relating the minimality in the C -topology and thatin the W ,p -topology. When W = W ,p (Ω) and Ψ( ξ ) = p | ξ | p , the next theorem has beenproved in [29], which extends to the p -Laplacian the well-known result by Brezis andNirenberg [7] for the case p = 2 (see also [32] for p > Theorem 3.6.
Assume that ∂ Ω of class C ,α and that u ∈ W ,p (Ω) ∩ C ,α (Ω) for some α ∈ ]0 , . Suppose also that W ,p (Ω) = V ⊕ W , where V is a finite dimensional subspaceof W ,p (Ω) , W is closed in W ,p (Ω) and the projection P V : W ,p (Ω) → V , associatedwith the direct sum decomposition, is continuous from the topology of L (Ω) to that of V .If u is a strict local minimum for the functional f defined in (2.1) along u + ( W ∩ C (Ω)) for the C (Ω) -topology, then u is a strict local minimum of f along u + W forthe W ,p (Ω) -topology.Proof. Define a convex and coercive functional h : W ,p (Ω) → R by h ( u ) = Z Ω (cid:2) Ψ( ∇ u ) − Ψ( ∇ u ) − ∇ Ψ( ∇ u ) · ( ∇ u − ∇ u ) (cid:3) dx and observe that v k → u in W ,p (Ω) if and only if h ( v k ) →
0. Actually, if v k → u in W ,p (Ω), it is clear that h ( v k ) →
0. Conversely, assume that h ( v k ) →
0. SinceΨ( ∇ v k ) − Ψ( ∇ u ) − ∇ Ψ( ∇ u ) · ( ∇ v k − ∇ u ) → L (Ω) , up to a subsequence we haveΨ( ∇ v k ) − Ψ( ∇ u ) − ∇ Ψ( ∇ u ) · ( ∇ v k − ∇ u ) → , hence ∇ v k → ∇ u a.e. in Ω by the strict convexity of Ψ. On the other hand,Ψ( ∇ v k ) − Ψ( ∇ u ) − ∇ Ψ( ∇ u ) · ( ∇ v k − ∇ u ) ≥ νp |∇ v k | p − ∇ Ψ( ∇ u ) · ∇ v k − z a.e. in Ω MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 13 for some z ∈ L (Ω). Therefore, ( ∇ v k ) is convergent to ∇ u also weakly in L p (Ω). If weapply Fatou’s Lemma to the sequence[Ψ( ∇ v k ) − Ψ( ∇ u ) − ∇ Ψ( ∇ u ) · ( ∇ v k − ∇ u )] − νp |∇ v k | p + ∇ Ψ( ∇ u ) · ∇ v k + z ≥ , we find that lim sup k Z Ω |∇ v k | p dx ≤ Z Ω |∇ u | p dx , whence the convergence of ( v k ) to u in W ,p (Ω).Since h is of class C with h h ′ ( u ) , u − u i = Z Ω (cid:0) ∇ Ψ( ∇ u ) − ∇ Ψ( ∇ u ) (cid:1) · ( ∇ u − ∇ u ) dx > u = u , for every r > { w ∈ W : h ( u + w ) = r } is a C -hypersurface in W . Moreover, if r is small enough, the map f ′ is of class ( S ) + on (cid:8) u ∈ W ,p (Ω) : h ( u + u ) ≤ r (cid:9) . If Ψ = Ψ p, with 1 < p < N , this is proved in [14, Theorem 1.2], while the general casefollows from [3, Theorem 3.4]. From Proposition 3.5 we infer that { u f ( u + u ) } isweakly lower semicontinuous on (cid:8) u ∈ W ,p (Ω) : h ( u + u ) ≤ r (cid:9) , hence on { w ∈ W : h ( u + w ) ≤ r } , which is weakly compact.If we argue by contradiction, we find a sequence ( w k ) in W such that w k is a minimumof { w f ( u + w ) } on { w ∈ W : h ( u + w ) ≤ r k } with r k → w k = 0, in particular f ( u + w k ) ≤ f ( u ). Therefore, there exists λ k ≥ h f ′ ( u + w k ) + λ k h ′ ( u + w k ) , u i = 0 for any u ∈ W , namely Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ u dx − Z Ω g ( x, u + w k ) u dx + λ k Z Ω (cid:0) ∇ Ψ( ∇ ( u + w k )) − ∇ Ψ( ∇ u ) (cid:1) · ∇ u dx = 0 for any u ∈ W , which is equivalent to Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ u dx −
11 + λ k Z Ω g ( x, u + w k ) u dx = λ k λ k Z Ω ∇ Ψ( ∇ u ) · ∇ u dx for any u ∈ W .
It follows Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ u dx −
11 + λ k Z Ω g ( x, u + w k ) u dx = λ k λ k Z Ω ∇ Ψ( ∇ u ) · ∇ u dx + Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ P V u dx −
11 + λ k Z Ω g ( x, u + w k ) P V u dx − λ k λ k Z Ω ∇ Ψ( ∇ u ) · ∇ P V u dx for any u ∈ W ,p (Ω) . Since P V is continuous from the topology of L (Ω) to that of W ,p (Ω), we have Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ P V u dx −
11 + λ k Z Ω g ( x, u + w k ) P V u dx − λ k λ k Z Ω ∇ Ψ( ∇ u ) · ∇ P V u dx = Z Ω z k u dx for any u ∈ W ,p (Ω)with ( z k ) bounded in L ∞ (Ω). It follows(3.2) − div [ ∇ Ψ( ∇ ( u + w k ))] −
11 + λ k g ( x, u + w k ) = z k − div (cid:20) λ k λ k ∇ Ψ( ∇ u ) (cid:21) and ∇ Ψ( ∇ u ) ∈ C ,β (Ω; R N ) for some β ∈ ]0 , p < N , from ( g ) we infer that11 + λ k [ g ( x, u + w k ) u − g ( x, u ≤ | g ( x, u + w k ) − g ( x, | | u | = | g ( x, u + w k ) − g ( x, || u + w k | ( u + w k ) u ≤ C (1 + | u + w k | p ∗ − ) | u + w k | ( u + w k ) u = C u + w k | u + w k | u + C | u + w k | p ∗ − ( u + w k ) u , whenever u ( u + w k ) ≥ Z Ω ∇ Ψ( ∇ ( u + w k )) · ∇ u dx − Z Ω C | u + w k | p ∗ − ( u + w k ) u dx ≤ Z Ω (cid:20)
11 + λ k g ( x,
0) + ˆ z k + z k (cid:21) u dx + λ k λ k Z Ω ∇ Ψ( ∇ u ) · ∇ u dx for any u ∈ W ,p (Ω) with u ( u + w k ) ≥ , where ˆ z k = C u + w k | u + w k | where u + w k = 0 , u + w k = 0 . From Theorem 3.1 it follows that ( u + w k ) is bounded in L ∞ (Ω). Coming back to (3.2),from Theorem 3.2 we conclude that ( u + w k ) is bounded in C ,β (Ω) for some β ∈ ]0 , u + w k ) is convergent to u in C (Ω) and a contradiction follows.If p ≥ N , the argument is similar and even simpler. (cid:3) MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 15 Parametric minimization
Throughout this section, we assume that Ω is a bounded open subset of R N with ∂ Ωof class C ,α for some α ∈ ]0 ,
1] and that Ψ and g satisfy assumptions (Ψ ), (Ψ ), ( g )and ( g ) with either κ > < p < ∞ or κ = 0 and 1 < p < u denote a critical point of the functional f defined in (2.1). According to Theo-rems 3.1 and 3.2, we have u ∈ C ,β (Ω) for some β ∈ ]0 , R N → R , for any x, v ∈ R N we setΦ ′′ ( x )[ v ] = lim inf y → xt → w → v Φ( y + tw ) + Φ( y − tw ) − y ) t . Then the function { ( x, v ) Φ ′′ ( x )[ v ] } is lower semicontinuous. If Φ is convex, it is alsoclear that Φ ′′ ( x )[ v ] ∈ [0 , + ∞ ] and that Φ ′′ ( x )[0] = 0. In particular, it is easily seen that κ = 0 and 1 < p < ⇒ Ψ ′′ p,κ (0)[ ξ ] = ( ξ = 0 , + ∞ if ξ = 0 . Since (Ψ − ν Ψ p,κ ) is convex, we also have κ = 0 and 1 < p < ⇒ Ψ ′′ (0)[ ξ ] = ( ξ = 0 , + ∞ if ξ = 0 , while Ψ ′′ ( η )[ ξ ] = Ψ ′′ ( η )[ ξ ] in the other cases. In particular, the function { ξ Ψ ′′ ( η )[ ξ ] } is convex for any η ∈ R N : Proposition 4.1.
For every u, v ∈ W ,p (Ω) , the function (cid:26) ( x, t ) (1 − t )Ψ ′′ (cid:0) ∇ u ( x ) + t ( ∇ v ( x ) − ∇ u ( x )) (cid:1)(cid:2) ∇ v ( x ) − ∇ u ( x ) (cid:3) (cid:27) belongs to L (Ω × ]0 , and one has Z Ω Ψ( ∇ v ) dx − Z Ω Ψ( ∇ u ) dx − Z Ω ∇ Ψ( ∇ u ) · ( ∇ v − ∇ u ) dx = Z (1 − t ) (cid:26)Z Ω Ψ ′′ (cid:0) ∇ u ( x ) + t ( ∇ v ( x ) − ∇ u ( x )) (cid:1)(cid:2) ∇ v ( x ) − ∇ u ( x ) (cid:3) dx (cid:27) dt . Proof.
Let us treat the case κ = 0 and 1 < p <
2. The case κ > < p < ∞ issimilar and even simpler. First of all, { ( η, ξ ) Ψ ′′ ( η )[ ξ ] } is a Borel function, being lowersemicontinuous. Moreover, we haveΨ ′′ ( η )[ ξ ] = Ψ ′′ ( η )[ ξ ] ≤ C | η | − p | ξ | for any η, ξ ∈ R N with η = 0 . Therefore, for every η, ξ ∈ R N , the function { t Ψ( η + t ( ξ − η )) } belongs to W , loc ( R )and we haveΨ( ξ ) − Ψ( η ) − ∇ Ψ( η ) · ( ξ − η ) = Z (1 − t )Ψ ′′ (cid:0) η + t ( ξ − η ) (cid:1)(cid:2) ξ − η (cid:3) dt . Then, given u, v ∈ W ,p (Ω), we have a.e. in ΩΨ( ∇ v ( x )) − Ψ( ∇ u ( x )) − ∇ Ψ( ∇ u ( x )) · ( ∇ v ( x ) − ∇ u ( x ))= Z (1 − t )Ψ ′′ (cid:0) ∇ u ( x ) + t ( ∇ v ( x ) − ∇ u ( x )) (cid:1)(cid:2) ∇ v ( x ) − ∇ u ( x ) (cid:3) dt . By integrating over Ω and applying Fubini’s theorem, the assertion follows. (cid:3)
Theorem 4.2.
Let ( u k ) be a sequence in W ,p (Ω) ∩ L ∞ (Ω) and ( v k ) a sequence in W , (Ω) such that ( u k ) is bounded in L ∞ (Ω) and convergent to u in W ,p (Ω) , while ( v k ) is weaklyconvergent to v in W , (Ω) .Then we have Z Ω Ψ ′′ ( ∇ u )[ ∇ v ] dx − Z Ω D s g ( x, u ) v dx ≤ lim inf k (cid:18)Z Ω Ψ ′′ ( ∇ u k )[ ∇ v k ] dx − Z Ω D s g ( x, u k ) v k dx (cid:19) . Proof.
Since ( v k ) is convergent to v in L (Ω), we clearly have Z Ω D s g ( x, u ) v dx = lim k Z Ω D s g ( x, u k ) v k dx . Then the assertion follows from the Theorem in [33]. (cid:3)
Proposition 4.3.
There exists a direct sum decomposition L (Ω) = V ⊕ f W such that: ( a ) V ⊆ X u ∩ W ,p (Ω) ∩ L ∞ (Ω) with dim V = m ∗ ( f, u ) < + ∞ , while f W is closedin L (Ω) ; ( b ) we have Z Ω Ψ ′′ ( ∇ u )[ ∇ ( v + w )] dx − Z Ω D s g ( x, u )( v + w ) dx = Z Ω Ψ ′′ ( ∇ u )[ ∇ v ] dx − Z Ω D s g ( x, u ) v dx + Z Ω Ψ ′′ ( ∇ u )[ ∇ w ] dx − Z Ω D s g ( x, u ) w dx for any v ∈ V and w ∈ f W ∩ W , (Ω) , Z Ω Ψ ′′ ( ∇ u )[ ∇ v ] dx − Z Ω D s g ( x, u ) v dx ≤ for any v ∈ V , Z Ω Ψ ′′ ( ∇ u )[ ∇ w ] dx − Z Ω D s g ( x, u ) w dx > for any w ∈ ( f W ∩ W , (Ω)) \ { } . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 17 Proof.
Let us treat in detail the case κ = 0 and 1 < p <
2. Since the derivative of thesmooth quadratic form Q u : X u → R is a compact perturbation of the Riesz isomor-phism, it is standard that there exists a direct sum decomposition X u = V ⊕ c W such that dim V = m ∗ ( f, u ) < + ∞ , c W = (cid:26) w ∈ X u : Z Ω vw dx = 0 for any v ∈ V (cid:27) ,Q u ( v + w ) = Q u ( v ) + Q u ( w ) for any v ∈ V and w ∈ c W ,Q u ( v ) ≤ v ∈ V ,Q u ( w ) > w ∈ c W \ { } . Moreover, either V = { } or V = span { e , . . . , e m ∗ } and each e j ∈ X u \ { } is a solutionof Z Ω \ Z u Ψ ′′ ( ∇ u )[ ∇ e j , ∇ u ] dx − Z Ω D s g ( x, u ) e j u dx = λ j Z Ω e j u dx for any u ∈ X u for some λ j ≤ k∇ u k ∞ > ϕ : R → R is a nondecreasing Lipschitz function with ϕ (0) = 0, then ϕ ( e j ) ∈ X u ,whence Z Ω \ Z u ϕ ′ ( e j )Ψ ′′ ( ∇ u )[ ∇ e j ] dx − Z Ω D s g ( x, u ) e j ϕ ( e j ) dx = λ j Z Ω e j ϕ ( e j ) dx ≤ . On the other hand, we haveΨ ′′ ( ∇ u ( x ))[ ξ ] ≥ ( p − ν |∇ u ( x ) | − p | ξ | ≥ ( p − ν k∇ u k − p ∞ | ξ | for any x ∈ Ω \ Z u and ξ ∈ R N , whence ( p − ν k∇ u k − p ∞ Z Ω ϕ ′ ( e j ) |∇ e j | dx − Z Ω D s g ( x, u ) e j ϕ ( e j ) dx ≤ . Since D s g ( x, u ) ∈ L ∞ (Ω), it is standard (see e.g. [35]) that e j ∈ L ∞ (Ω), whence V ⊆ X u ∩ L ∞ (Ω) ⊆ W ,p (Ω), as p < f W = (cid:26) w ∈ L (Ω) : Z Ω vw dx = 0 for any v ∈ V (cid:27) , then f W is a closed linear subspace of L (Ω) and L (Ω) = V ⊕ f W .
Since Z Ω Ψ ′′ ( ∇ u )[ ∇ u ] dx − Z Ω D s g ( x, u ) u dx = Q u ( u ) if u ∈ X u , Z Ω Ψ ′′ ( ∇ u )[ ∇ u ] dx − Z Ω D s g ( x, u ) u dx = + ∞ if u ∈ W , (Ω) \ X u , the other assertions easily follow. In the case κ >
0, one has X u = W , (Ω) and the adaptation of the previous argumentis very simple if 1 < p ≤
2. If p >
2, one has to remark that Ψ ′′ ( ∇ u ) is continuous.By standard regularity results (see e.g. [42, Theorem 7.6]) it follows that e j ∈ W ,p (Ω),whence V ⊆ W ,p (Ω). (cid:3) In the following, we consider a direct sum decomposition as in the previous proposition.In particular, the projection e P V : L (Ω) → V , associated with the direct sum decomposi-tion, is continuous with respect to the L -topology. Since V ⊆ W , (Ω) ∩ L ∞ (Ω) is finitedimensional, it is equivalent to consider the norm of W , (Ω) ∩ L ∞ (Ω) on V .Then we set W = f W ∩ W ,p (Ω), which is a closed linear subspace of W ,p (Ω), so that W ,p (Ω) = V ⊕ W and P V = e P V (cid:12)(cid:12) W ,p is L -continuous as well.We also set, for any r > B r = (cid:8) u ∈ W ,p (Ω) : k∇ u k p < r (cid:9) ,D r = (cid:8) u ∈ W ,p (Ω) : k∇ u k p ≤ r (cid:9) . Lemma 4.4.
For any
M > , there exist r, δ > such that, for every u ∈ ( u + D r ) ∩ W , ∞ (Ω) with k u k ∞ + k∇ u k ∞ ≤ M and every w ∈ f W ∩ W , (Ω) , one has Z Ω Ψ ′′ ( ∇ u )[ ∇ w ] dx − Z Ω D s g ( x, u ) w dx ≥ δ Z Ω |∇ w | dx . Proof.
Assume, for a contradiction, that there exist a sequence ( v k ) in W ,p (Ω) ∩ W , ∞ (Ω),strongly convergent to u in W ,p (Ω) and bounded in W , ∞ (Ω), and a sequence ( w k ) in f W ∩ W , (Ω) such that(4.1) Z Ω Ψ ′′ ( ∇ v k )[ ∇ w k ] dx − Z Ω D s g ( x, v k ) w k dx < k Z Ω |∇ w k | dx . Without loss of generality, we may assume that k∇ w k k = 1. Then, up to a subsequence,( w k ) is weakly convergent to some w in W , (Ω). In particular, w ∈ f W . From Theorem 4.2we infer that Z Ω Ψ ′′ ( ∇ u )[ ∇ w ] dx − Z Ω D s g ( x, u ) w dx ≤ , whence w = 0.Coming back to (4.1), now we deduce thatlim k Z Ω Ψ ′′ ( ∇ v k )[ ∇ w k ] dx = 0 . Since ( ∇ v k ) is bounded in L ∞ (Ω), in both cases κ = 0 with 1 < p < κ > < p < ∞ we infer that ∇ w k → L (Ω). Since k∇ w k k = 1, a contradictionfollows. (cid:3) Theorem 4.5.
There exist
M, r > and β ∈ ]0 , such that: ( a ) the map f ′ is of class ( S ) + on u + D r ; MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 19 ( b ) for every v ∈ V ∩ D r , the derivative of the functional W → R w f ( u + v + w ) is of class ( S ) + on W ∩ D r ; moreover, if w is a critical point of such a functionalwith w ∈ D r , then v + w ∈ C ,β (Ω) and k v + w k C ,β < M ; finally, the functional { w f ( u + v + w ) } is strictly convex on (cid:8) w ∈ W ∩ D r : ( v + w ) ∈ W , ∞ (Ω) and k v + w k ∞ + k∇ ( v + w ) k ∞ ≤ M (cid:9) ;( c ) u is a strict local minimum of f along u + W for the W ,p (Ω) -topology.Proof. As already observed in the proof of Theorem 3.6, there exists r > f ′ is of class ( S ) + on u + D r . It follows that { w f ( u + v + w ) } is of class ( S ) + on W ∩ D r for any v ∈ V ∩ D r . Moreover, if w is a critical point, we have h f ′ ( u + v + w ) , u − P V u i = 0 for any u ∈ W ,p (Ω) , whence h f ′ ( u + v + w ) , u i = h f ′ ( u + v + w ) , P V u i for any u ∈ W ,p (Ω) . Since P V is continuous from the topology of L (Ω) to that of W ,p (Ω) and f ′ is boundedon bounded sets, it follows that h f ′ ( u + v + w ) , P V u i = Z Ω zu dx for any u ∈ W ,p (Ω)with z uniformly bounded in L ∞ (Ω) with respect to v and w , whence h f ′ ( u + v + w ) , u i = Z Ω zu dx for any u ∈ W ,p (Ω) . From Theorems 3.1 and 3.2, possibly by decreasing r , we conclude that u + v + w , hence v + w , is uniformly bounded in C ,β (Ω).Finally, again by decreasing r , we infer by Lemma 4.4 that(4.2) Z Ω Ψ ′′ ( ∇ ( u + u ))[ ∇ w ] dx − Z Ω D s g ( x, u + u ) w dx ≥ δ Z Ω |∇ w | dx for every u ∈ D r ∩ W , ∞ (Ω) with k u k ∞ + k∇ u k ∞ ≤ M and every w ∈ f W ∩ W , (Ω).If v ∈ V ∩ D r , t ∈ [0 ,
1] and w , w ∈ W ∩ D r with ( v + w j ) ∈ W , ∞ (Ω) and k v + w j k ∞ + k∇ ( v + w j ) k ∞ ≤ M , we have w j ∈ W ,p (Ω) ∩ W , (Ω), hence w j ∈ W , (Ω), as ∂ Ω is smooth enough. ByProposition 4.1 and (4.2) we easily deduce that(1 − t ) f ( u + v + w ) + tf ( u + v + w ) ≥ f ( u + v + (1 − t ) w + tw ) + δ t (1 − t ) Z Ω |∇ w − ∇ w | dx . Therefore { w f ( u + v + w ) } is strictly convex. In particular, the critical point u is a strict local minimum of f along u + ( W ∩ C (Ω))for the C (Ω)-topology. From Theorem 3.6 we infer that u is a strict local minimum of f along u + W for the W ,p (Ω)-topology. (cid:3) Theorem 4.6.
There exist
M, r > , β ∈ ]0 , and ̺ ∈ ]0 , r ] such that, for every v ∈ V ∩ D ̺ , there exists one and only one w ∈ W ∩ D r such that f ( u + v + w ) ≤ f ( u + v + w ) for any w ∈ W ∩ D r . Moreover, v + w ∈ C ,β (Ω) with k v + w k C ,β ≤ M , w ∈ B r and w is the unique criticalpoint of { w f ( u + v + w ) } in W ∩ D r .Finally, if we set ψ ( v ) = w , the map { v v + ψ ( v ) } is continuous from V ∩ D ̺ into C (Ω) , while the map ψ is continuous from V ∩ D ̺ into W ,p (Ω) ∩ L ∞ (Ω) . Moreover, ψ (0) = 0 .In the case κ > , the map ψ is also of class C from V ∩ B ̺ into W , (Ω) and, forevery z ∈ V ∩ B ̺ and v ∈ V , we have that ψ ′ ( z ) v is the minimum point of the functional (cid:26) w Z Ω (cid:8) Ψ ′′ ( ∇ u )[ ∇ w ] − D s g ( x, u ) w (cid:9) dx + Z Ω { Ψ ′′ ( ∇ u )[ ∇ v, ∇ w ] − D s g ( x, u ) vw } dx (cid:27) on f W ∩ W , (Ω) , where u = u + z + ψ ( z ) . Moreover, ψ ′ (0) = 0 .Proof. Let
M, r > β ∈ ]0 ,
1] be as in Theorem 4.5. In particular, we may supposethat f ( u ) < f ( u + w ) for every w ∈ W ∩ D r with w = 0. By Lemma 4.4 we may alsoassume that there exists δ > Z Ω Ψ ′′ ( ∇ ( u + u ))[ ∇ w ] dx − Z Ω D s g ( x, u + u ) w dx ≥ δ Z Ω |∇ w | dx for every u ∈ D r ∩ C ,β (Ω) with k u k C ,β ≤ M and every w ∈ f W ∩ W , (Ω).We claim that there exists ̺ ∈ ]0 , r ] such that f ( u + v ) < f ( u + v + w ) for any v ∈ V ∩ D ̺ and any w ∈ W with k∇ w k p = r .By contradiction, let ( v k ) be a sequence in V with v k → w k ) be a sequence in W with k∇ w k k p = r and f ( u + v k ) ≥ f ( u + v k + w k ). Up to a subsequence, ( w k ) isweakly convergent to some w ∈ W ∩ D r . Then ( u + v k + w k ) is weakly convergent to u + w with lim sup k f ( u + v k + w k ) ≤ lim k f ( u + v k ) = f ( u ) ≤ f ( u + w ) . Combining Proposition 3.5 with Theorem 4.5, we deduce that ( u + v k + w k ) is stronglyconvergent to u + w , whence f ( u + w ) = f ( u ) with k∇ w k p = r , and a contradictionfollows.Again from Proposition 3.5 and Theorem 4.5 we know that { w f ( u + v + w ) } isweakly lower semicontinuous on W ∩ D r for any v ∈ V ∩ D ̺ . Therefore there exists aminimum point w ∈ W ∩ D r and in fact w ∈ B r . In particular, we have h f ′ ( u + v + w ) , w i = 0 for any w ∈ W .
MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 21 From Theorem 4.5 we infer that v + w ∈ C ,β (Ω) with k v + w k C ,β ≤ M . Since { w f ( u + v + w ) } is strictly convex on (cid:8) w ∈ W ∩ D r : ( v + w ) ∈ W , ∞ (Ω) and k v + w k ∞ + k∇ ( v + w ) k ∞ ≤ M (cid:9) , the minimum is unique. If v = 0, then w = 0.Finally, if we set ψ ( v ) = w , the map { v v + ψ ( v ) } is defined from V ∩ D ̺ into (cid:8) u ∈ C ,β (Ω) : k u k C ,β ≤ M (cid:9) , which is a compact subset of C (Ω), and has closed graph, as f is continuous. Thereforeit is a continuous map. The continuity of ψ follows.In the case κ >
0, the function Ψ is of class C on R N . Therefore, there exists C > (cid:12)(cid:12)(cid:12)(cid:12)Z Ω Ψ ′′ ( ∇ ( u + u ))[ ∇ u , ∇ u ] dx − Z Ω D s g ( x, u + u ) u u dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ u k k∇ u k for every u ∈ D r ∩ C ,β (Ω) with k u k C ,β ≤ M and every u , u ∈ W , (Ω). Moreover, wehave(4.5) h f ′ ( u + v + ψ ( v )) , u i − h f ′ ( u + v + ψ ( v )) , u i = Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( v − v + ψ ( v ) − ψ ( v )) , ∇ u ] − D s g ( x, γ t )( v − v + ψ ( v ) − ψ ( v )) u (cid:27) dx dt for any v , v ∈ V ∩ D ̺ and u ∈ W , (Ω), where γ t = u + v + ψ ( v ) + t ( v − v + ψ ( v ) − ψ ( v )) . Now let z ∈ V ∩ B ̺ and let u = u + z + ψ ( z ). From (4.3) and (4.4) it follows that, forevery v ∈ V , the functional (cid:26) w Z Ω (cid:8) Ψ ′′ ( ∇ u )[ ∇ w ] − D s g ( x, u ) w (cid:9) dx + Z Ω { Ψ ′′ ( ∇ u )[ ∇ v, ∇ w ] − D s g ( x, u ) vw } dx (cid:27) admits one and only one minimum point L z v in f W ∩ W , (Ω), which satisfies(4.6) Z Ω { Ψ ′′ ( ∇ u )[ ∇ ( L z v ) , ∇ w ] − D s g ( x, u )( L z v ) w } dx = − Z Ω { Ψ ′′ ( ∇ u )[ ∇ v, ∇ w ] − D s g ( x, u ) vw } dx for any w ∈ f W ∩ W , (Ω) . Moreover, the map L z : V → W , (Ω) is linear and continuous, as V is finite dimensional.Since Q u ( v + w ) = Q u ( v ) + Q u ( w ) for any v ∈ V and w ∈ f W ∩ W , (Ω), we also have L = 0. By (4.5), for every v , v ∈ V ∩ B ̺ and w ∈ f W ∩ W , (Ω), it holds0 = h f ′ ( u + v + ψ ( v )) , w i − h f ′ ( u + v + ψ ( v )) , w i = Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( v − v + ψ ( v ) − ψ ( v )) , ∇ w ] − D s g ( x, γ t )( v − v + ψ ( v ) − ψ ( v )) w (cid:27) dx dt . Taking into account (4.6), we deduce that Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( ψ ( v ) − ψ ( v )) , ∇ w ] − D s g ( x, γ t )( ψ ( v ) − ψ ( v )) w (cid:27) dx dt − Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( L z ( v − v )) , ∇ w ] − D s g ( x, u )( L z ( v − v )) w (cid:27) dx = − Z Z Ω (cid:26) [Ψ ′′ ( ∇ γ t ) − Ψ ′′ ( ∇ u )] [ ∇ ( v − v ) , ∇ w ] − [ D s g ( x, γ t ) − D s g ( x, u )] ( v − v ) w (cid:27) dx dt . It follows Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( ψ ( v ) − ψ ( v ) − L z ( v − v )) , ∇ w ] − D s g ( x, γ t )( ψ ( v ) − ψ ( v ) − L z ( v − v )) w (cid:27) dx dt = − Z Z Ω (cid:26) [Ψ ′′ ( ∇ γ t ) − Ψ ′′ ( ∇ u )] [ ∇ ( L z ( v − v )) , ∇ w ] − [ D s g ( x, γ t ) − D s g ( x, u )] ( L z ( v − v )) w (cid:27) dx dt − Z Z Ω (cid:26) [Ψ ′′ ( ∇ γ t ) − Ψ ′′ ( ∇ u )] [ ∇ ( v − v ) , ∇ w ] − [ D s g ( x, γ t ) − D s g ( x, u )] ( v − v ) w (cid:27) dx dt . Since the map { v v + ψ ( v ) } is continuous from V ∩ B ̺ into C (Ω), from (4.3) we inferthat lim ( v ,v ) → ( z,z ) v = v k∇ ( ψ ( v ) − ψ ( v ) − L z ( v − v )) k k∇ ( v − v ) k = 0 . Therefore ψ is of class C from V ∩ B ̺ into W , (Ω) and ψ ′ ( z ) = L z . (cid:3) The finite dimensional reduction
Throughout this section we keep the assumptions and the notations of Section 4. Wealso define the reduced functional ϕ : V ∩ B ̺ → R as ϕ ( v ) = f ( u + v + ψ ( v )) = min { f ( u + v + w ) : w ∈ W ∩ D r } . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 23 Theorem 5.1.
Let κ > with < p < ∞ or κ = 0 with < p < . Then the functional ϕ is of class C and (5.1) h ϕ ′ ( z ) , v i = h f ′ ( u + z + ψ ( z )) , v i for any z ∈ V ∩ B ̺ and v ∈ V .
In particular, is a critical point of ϕ . Moreover, we have C m ( ϕ, ≈ C m ( f, u ) for any m ≥ . Finally, is an isolated critical point of ϕ if and only if u is an isolated critical pointof f .Proof. For any v , v ∈ V ∩ B ̺ , we have ϕ ( v ) = f ( u + v + ψ ( v ))= f ( u + v + ψ ( v )) + h f ′ ( u + v + t ( v − v ) + ψ ( v )) , v − v i≥ f ( u + v + ψ ( v )) + h f ′ ( u + v + t ( v − v ) + ψ ( v )) , v − v i = ϕ ( v ) + h f ′ ( u + v + t ( v − v ) + ψ ( v )) , v − v i for some t ∈ ]0 , ψ is continuous from V ∩ B ̺ into W ,p (Ω), it follows thatlim inf ( v ,v ) → ( z,z ) v = v ϕ ( v ) − ϕ ( v ) − h f ′ ( u + z + ψ ( z )) , v − v ik v − v k ≥ . We also have ϕ ( v ) = f ( u + v + ψ ( v )) ≤ f ( u + v + ψ ( v ))= f ( u + v + ψ ( v )) + h f ′ ( u + v + t ( v − v ) + ψ ( v )) , v − v i = ϕ ( v ) + h f ′ ( u + v + t ( v − v ) + ψ ( v )) , v − v i for some t ∈ ]0 , ( v ,v ) → ( z,z ) v = v ϕ ( v ) − ϕ ( v ) − h f ′ ( u + z + ψ ( z )) , v − v ik v − v k ≤ . Therefore ϕ is of class C with h ϕ ′ ( z ) , v i = h f ′ ( u + z + ψ ( z )) , v i . Since ψ (0) = 0, we also have ϕ ′ (0) = 0.Now consider Y = { u + z + ψ ( z ) : z ∈ V ∩ B ̺ } endowed with the W ,p (Ω)-topology. Since { z u + z + ψ ( z ) } is a homeomorphismfrom V ∩ B ̺ onto Y which sends 0 into u , it is clear that C m ( ϕ, ≈ C m ( f (cid:12)(cid:12) Y , u ) for any m ≥ . On the other hand, from Proposition 3.5 and Theorem 4.5 we see that the functional { w f ( u + v + w ) } satisfies the Palais-Smale condition over W ∩ D r for any v ∈ V ∩ B ̺ .Moreover, ψ ( v ) is the unique critical point, in fact the minimum, of such a functionalin W ∩ D r . Arguing as in the Second Deformation Lemma, it is possible to define adeformation H : ( u + ( V ∩ B ̺ ) + ( W ∩ D r )) × [0 , → ( u + ( V ∩ B ̺ ) + ( W ∩ D r )) such that H ( u, t ) − u ∈ W , f ( H ( u, t )) ≤ f ( u ) , H ( u, ∈ Y , H ( u, t ) = u if u ∈ Y , whence H m ( f c , f c \ { u } ) ≈ H m ( f c ∩ Y, ( f c ∩ Y ) \ { u } ) . This is proved in [16, Theorem 5.4] in the case p >
2, but the argument works also for1 < p ≤
2. See also [36, Theorem 4.7] in a nonsmooth setting.Therefore we have C m ( ϕ, ≈ C m ( f (cid:12)(cid:12) Y , u ) ≈ C m ( f, u ) for any m ≥ . Since any critical point u of f in u + ( V ∩ B ̺ ) + ( W ∩ D r ) must be of the form u = u + z + ψ ( z ) with z ∈ V ∩ B ̺ , from (5.1) we infer that 0 is isolated for ϕ if and only if u is isolated for f . (cid:3) Theorem 5.2.
Let κ > with < p < ∞ . Then ϕ is of class C and ϕ ′′ ( z )[ v ] = Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( v + ψ ′ ( z ) v )] − D s g ( x, u )( v + ψ ′ ( z ) v ) (cid:27) dx (5.2) = Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ v ] − D s g ( x, u ) v (cid:27) dx − Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( ψ ′ ( z ) v )] − D s g ( x, u )( ψ ′ ( z ) v ) (cid:27) dx for any z ∈ V ∩ B ̺ and v ∈ V , where u = u + z + ψ ( z ) . In particular, we have ϕ ′′ (0)[ v ] = Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ v ] − D s g ( x, u ) v (cid:27) dx for any v ∈ V .
Proof.
By Theorem 4.6, the map ψ is of class C from V ∩ B ̺ into W , (Ω) with ψ (0) = 0and ψ ′ (0) = 0. For any z ∈ V ∩ B ̺ , let L z : V → V ′ be the linear map defined by(5.3) h L z v , v i = Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ v , ∇ v ] − D s g ( x, u ) v v (cid:27) dx + Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( ψ ′ ( z ) v ) , ∇ v ] − D s g ( x, u )( ψ ′ ( z ) v ) v (cid:27) dx , where u = u + z + ψ ( z ). By (4.5), for every v , v ∈ V ∩ B ̺ and v ∈ V , we have h ϕ ′ ( v ) , v i − h ϕ ′ ( v ) , v i = h f ′ ( u + v + ψ ( v )) , v i − h f ′ ( u + v + ψ ( v )) , v i = Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( v − v + ψ ( v ) − ψ ( v )) , ∇ v ] − D s g ( x, γ t )( v − v + ψ ( v ) − ψ ( v )) v (cid:27) dx dt , MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 25 where γ t = u + v + ψ ( v ) + t ( v − v + ψ ( v ) − ψ ( v )). Taking into account (5.3), wededuce that h ϕ ′ ( v ) , v i − h ϕ ′ ( v ) , v i − h L z ( v − v ) , v i = Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( v − v + ψ ( v ) − ψ ( v )) , ∇ v ] − D s g ( x, γ t )( v − v + ψ ( v ) − ψ ( v )) v (cid:27) dx dt − Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( v − v ) , ∇ v ] − D s g ( x, u )( v − v ) v (cid:27) dx − Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( ψ ′ ( z )( v − v )) , ∇ v ] − D s g ( x, u )( ψ ′ ( z )( v − v )) v (cid:27) dx . It follows h ϕ ′ ( v ) , v i − h ϕ ′ ( v ) , v i − h L z ( v − v ) , v i = Z Z Ω (cid:26) [Ψ ′′ ( ∇ γ t ) − Ψ ′′ ( u )] [ ∇ ( v − v ) , ∇ v ] − [ D s g ( x, γ t ) − D s g ( x, u )] ( v − v ) v (cid:27) dx dt + Z Z Ω (cid:26) Ψ ′′ ( ∇ γ t )[ ∇ ( ψ ( v ) − ψ ( v ) − ψ ′ ( z )( v − v )) , ∇ v ] − D s g ( x, γ t )( ψ ( v ) − ψ ( v ) − ψ ′ ( z )( v − v )) v (cid:27) dx dt + Z Z Ω (cid:26) [Ψ ′′ ( ∇ γ t ) − Ψ ′′ ( u )] [ ∇ ( ψ ′ ( z )( v − v )) , ∇ v ] − [ D s g ( x, γ t ) − D s g ( x, u )] ( ψ ′ ( z )( v − v )) v (cid:27) dx dt . Since the map { v v + ψ ( v ) } is continuous from V ∩ B ̺ into C (Ω), we infer thatlim ( v ,v ) → ( z,z ) v = v h ϕ ′ ( v ) , v i − h ϕ ′ ( v ) , v i − h L z ( v − v ) , v ik∇ ( v − v ) k = 0 for any v ∈ V .
Therefore ϕ is of class C and ϕ ′′ ( z )[ v , v ] = Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ v , ∇ v ] − D s g ( x, u ) v v (cid:27) dx + Z Ω (cid:26) Ψ ′′ ( ∇ u )[ ∇ ( ψ ′ ( z ) v ) , ∇ v )] − D s g ( x, u )( ψ ′ ( z ) v ) v (cid:27) dx . By Theorem 4.6, we also have Z Ω { Ψ ′′ ( ∇ u )[ ∇ ( ψ ′ ( z ) v ) , ∇ w ] − D s g ( x, u )( ψ ′ ( z ) v ) w } dx = − Z Ω { Ψ ′′ ( ∇ u )[ ∇ v, ∇ w ] − D s g ( x, u ) vw } dx for any v ∈ V and w ∈ f W ∩ W , (Ω) , whence (5.2).Since ψ (0) = 0 and ψ ′ (0) = 0, the formula for ϕ ′′ (0) follows. (cid:3) Proof of the results of Section 2
Proof of Theorems 2.6, 2.7, 2.2, 2.3 and 2.4.
From Theorem 5.1 we know that C m ( f, u ) ≈ C m ( ϕ,
0) for any m ≥ . Since the critical groups are defined using Alexander-Spanier cohomology, it is clear that C m ( ϕ,
0) = { } whenever m > dim V = m ∗ ( f, u ), both in the case κ > < p < ∞ and in the case κ = 0 with 1 < p < u = 0 with κ = 0 and 1 < p <
2, we clearly have Z u = Ωand X u = { } , whence m ( f,
0) = m ∗ ( f,
0) = 0, V = { } and W = W ,p (Ω). FromTheorem 4.6 it follows that 0 is a strict local minimum and an isolated critical point of f .By the excision property, it follows C m ( f, ≈ H m ( { } , ∅ ) , whence ( C m ( f, ≈ G if m = 0 ,C m ( f,
0) = { } if m = 0 . Now assume that κ > < p < ∞ . From Theorem 5.2 and Proposition 4.3 we inferthat ϕ is of class C with ϕ ′′ (0)[ v ] = Q u ( v ) ≤ v ∈ V .
Let V − be a subspace of X u = W , (Ω) of dimension m ( f, u ) such that Q u is negativedefinite on V − . Then it is easily seen that Q u is negative definite also on P V ( V − ), whichhas the same dimension of V − . Therefore we may assume, without loss of generality, that V − ⊆ V and we have ϕ ′′ (0)[ v ] = Q u ( v ) < v ∈ V − \ { } . It follows (see e.g. [36, Theorem 3.1]) that C m ( ϕ,
0) = { } whenever m < dim V − = m ( f, u ). The proof of Theorems 2.6, 2.7 and 2.2 is complete.If m ( f, u ) = m ∗ ( f, u ), we have V − = V . Then 0 is a nondegenerate critical point of ϕ with Morse index dim V = m ( f, u ). It follows that 0 is an isolated critical point of ϕ and C m ( f, u ) ≈ C m ( ϕ, ≈ δ m,m ( f,u ) G . Moreover, u is an isolated critical point of f by Theorem 5.1 and Theorem 2.3 follows.Finally, assume that u is an isolated critical point of f with m ( f, u ) < m ∗ ( f, u ). ByTheorem 5.1 we infer that 0 is an isolated critical point of ϕ and Theorem 2.4 followsfrom [40, Corollary 8.4]. (cid:3) MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 27 Proof of Theorem 2.8.
By Theorem 2.2, Remark 2.5 and Theorem 2.7, we have only to treat the case κ = 0 with p >
2, so that Q u ( v ) = Q ( v ) = − Z Ω g ′ (0) v dx ∀ v ∈ W , (Ω) . If g ′ (0) = 0, we have m ( f,
0) = 0, m ∗ ( f,
0) = + ∞ and the assertion is obvious.If g ′ (0) <
0, we have m ( f,
0) = m ∗ ( f,
0) = 0. On the other hand, it is easily seen that f : W ,p (Ω) ∩ C (Ω) → R is strictly convex in a neighborhood of 0 for the C (Ω)-topology. In particular, 0 is a strictlocal minimum for the C (Ω)-topology. From Theorem 3.6 we infer that 0 is a strict localminimum of f : W ,p (Ω) → R for the W ,p (Ω)-topology. By the excision property we have C m ( f, ≈ H m ( { } , ∅ )and the assertion follows.If g ′ (0) >
0, we have m ( f,
0) = m ∗ ( f,
0) = + ∞ . If p > N , the functional f is of class C on W ,p (Ω) with f ′′ (0)( v ) = − Z Ω g ′ (0) v dx ∀ v ∈ W ,p (Ω) . From [36, Theorem 3.1] we infer that C m ( f,
0) = { } for any m and the assertion follows.If p ≤ N , recall that | g ( s ) | ≤ C (1 + | s | q )with q < p ∗ − p < N , and consider a C ∞ -function ϑ : R → [0 ,
1] with ϑ ( s ) = 1for | s | ≤ ϑ ( s ) = 0 for | s | ≥
2. Then define, for every t ∈ [0 , C -functional f t : W ,p (Ω) → R as f t ( u ) = Z Ω Ψ( ∇ u ) dx − Z Ω G t ( u ) dx , where g t ( s ) = g ( ϑ ( ts ) s ) , G t ( s ) = Z s g t ( σ ) dσ . For any t ∈ ]0 ,
1] the functional f t is of class C with f ′′ t (0)( v ) = − Z Ω g ′ (0) v dx ∀ v ∈ W ,p (Ω) . Again from [36, Theorem 3.1] we infer that C m ( f t ,
0) = { } for any t ∈ ]0 ,
1] and any m .Let r > f = f in D r = (cid:8) u ∈ W ,p (Ω) : k∇ u k p ≤ r (cid:9) and such that the assertion of Theorem 3.1 holds forˆ g ( s ) = C | s | q − s . Then the map { t f t } is continuous from [0 ,
1] into C ( D r ). Moreover from [3, Theo-rem 3.5] we infer that f ′ t is of class ( S ) + , so that f t satisfies the Palais-Smale conditionover D r , for any t ∈ [0 , We claim that there exists t ∈ ]0 ,
1] such that 0 is the unique critical point of f t in D r whenever 0 ≤ t ≤ t . Assume, for a contradiction, that t k → u k ∈ D r \ { } is acritical point of f t k . Then, for every v ∈ W ,p (Ω) with vu k ≥
0, we have Z Ω ∇ Ψ( ∇ u k ) · ∇ v dx = Z Ω g t k ( u k ) v dx ≤ Z { u k =0 } | g ( ϑ ( t k u k ) u k ) | | v | dx = Z { u k =0 } | g ( ϑ ( t k u k ) u k ) || u k | u k v dx ≤ Z { u k =0 } C (1 + | u k | q ) | u k | u k v dx = Z { u k =0 } C u k | u k | v dx + Z Ω C | u k | q − u k v dx . It follows Z Ω [ ∇ Ψ( ∇ u k ) · ∇ v − ˆ g ( u k ) v ] dx ≤ h ˆ w k , v i , where ˆ w k = ( C u k | u k | where u k = 0 , u k = 0 . From Theorem 3.1 we infer that ( u k ) is bounded in L ∞ (Ω), so that ϑ ( t k u k ) = 1 eventuallyas k → ∞ . Then u k is a critical point of f and a contradiction follows.From [21, Theorem 5.2] we deduce that C m ( f, ≈ C m ( f t ,
0) (for related results, seealso [11, Theorem I.5.6], [14, Theorem 3.1] and [40, Theorem 8.8]) and the assertionfollows. (cid:3) Proof of the main results
In this last section we prove the main results stated in the Introduction. Let us recallsome variants of the results of [13] suited for our purposes. We start with a saddle theorem,where linear subspaces are substituted by symmetric cones.
Theorem 7.1.
Let X be a real Banach space and let X − , X + be two symmetric conesin X such that X + is closed in X , X − ∩ X + = { } and such that Index( X − \ { } ) = Index( X \ X + ) < + ∞ . Let r > and let D − = { u ∈ X − : k u k ≤ r } , S − = { u ∈ X − : k u k = r } . Let f : X → R be a function of class C such that inf X + f > −∞ , sup D − f < + ∞ , if X − = { } , we have f ( u ) < inf X + f whenever u ∈ S − . Set a = inf X + f , b = sup D − f , m = Index( X − \ { } ) MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 29 and assume that every sequence ( u n ) in X , with f ( u n ) → c ∈ [ a, b ] and (1 + k u n k ) k f ′ ( u n ) k → , admits a convergent subsequence ( Cerami-Palais-Smale condition ) and that f − ([ a, b ]) contains a finite number of critical points.Then there exists a critical point u of f with a ≤ f ( u ) ≤ b and C m ( f, u ) = { } .Proof. From [24, Theorems 2.7 and 2.8] we infer that ( D − , S − ) links X + cohomologically indimension m over Z . According to [20, Remark 4.4], the Cerami-Palais-Smale conditionis just the usual Palais-Smale condition with respect to an auxiliary distance function.Then the assertion follows from [23, Theorem 5.2, Remark 5.3 and Theorem 7.5]. (cid:3) Theorem 7.2.
Let ( λ m ) be defined as in the Introduction and let m ≥ be such that λ m < λ m +1 . If we set X − = (cid:26) u ∈ W ,p (Ω) : Z Ω |∇ u | p dx ≤ λ m Z Ω | u | p dx (cid:27) X + = (cid:26) u ∈ W ,p (Ω) : Z Ω |∇ u | p dx ≥ λ m +1 Z Ω | u | p dx (cid:27) if m ≥ , ( X − = { } X + = W ,p (Ω) if m = 0 , then X − , X + are two closed symmetric cones in W ,p (Ω) such that X − ∩ X + = { } andsuch that Index( X − \ { } ) = Index( W ,p (Ω) \ X + ) = m . Proof. If m ≥
1, the result is contained in [24, Theorem 3.2]. The case m = 0 is obvious. (cid:3) Now let f : W ,p (Ω) → R be the C -functional defined in (1.3) by setting f ( u ) = Z Ω Ψ p,κ ( ∇ u ) dx − Z Ω G ( u ) dx . Proof of Theorem 1.1.
Let us show that f satisfies the Cerami-Palais-Smale condition. Let ( u n ) be a sequencein W ,p (Ω) with f ( u n ) bounded and (1 + k u n k ) k f ′ ( u n ) k →
0, so that(7.1) lim n h f ′ ( u n ) , v − u n i = 0 ∀ v ∈ W ,p (Ω) . First of all, let us show that ( u n ) is bounded in W ,p (Ω). By contradiction, assume that k u n k → ∞ and set z n = u n k u n k . Up to a subsequence, z n is convergent to some z weakly in W ,p (Ω), strongly in L p (Ω) and a.e. in Ω. Since h f ′ ( u n ) , z − z n i →
0, dividing by k u n k p − and taking into account ( a ), we getlim n Z Ω (cid:18) κ k u n k + |∇ z n | (cid:19) p − ∇ z n · ∇ ( z − z n ) dx = 0 . By the convexity of Ψ p,κ , it followslim sup n Z Ω |∇ z n | p dx ≤ lim sup n Z Ω (cid:18) κ k u n k + |∇ z n | (cid:19) p dx ≤ lim n Z Ω (cid:18) κ k u n k + |∇ z | (cid:19) p dx = Z Ω |∇ z | p dx , so that z n → z strongly in W ,p (Ω) and z = 0.Given v ∈ W ,p (Ω), we also have h f ′ ( u n ) , v i → k u n k p − ,lim n Z Ω "(cid:18) κ k u n k + |∇ z n | (cid:19) p − ∇ z n · ∇ v − g ( k u n k z n ) k u n k p − v dx = 0 . Taking again into account ( a ), we get Z Ω |∇ z | p − ∇ z · ∇ v dx = λ Z Ω | z | p − zv dx ∀ v ∈ W ,p (Ω) , which contradicts the assumption that λ σ ( − ∆ p ). Therefore ( u n ) is bounded in W ,p (Ω), hence convergent, up to a subsequence, to some u weakly in W ,p (Ω).According to [3, Theorem 3.5], the operator f ′ is of class ( S ) + . From (7.1) we inferthat ( u n ) is strongly convergent to u in W ,p (Ω).Now define X − , X + according to Theorem 7.2 with m = m ∞ , so that X − , X + are twosymmetric cones in W ,p (Ω) satisfying the assumptions of Theorem 7.1 with Index( X − \{ } ) = m ∞ . Let us treat the case m ∞ ≥
1. The case m ∞ = 0 is similar and simpler. If λ m ∞ < α ′ < α ′′ < λ < β ′ < β ′′ < λ m ∞ +1 , taking into account assumption ( a ) we infer that there exists C > β ′′ pλ m ∞ +1 | ξ | p − C ≤ Ψ p,κ ( ξ ) ≤ α ′ pλ m ∞ | ξ | p + C ∀ ξ ∈ R N ,α ′′ p | s | p − C ≤ G ( s ) ≤ β ′ p | s | p + C ∀ s ∈ R . It easily follows that inf X + f > −∞ , lim k u k→∞ u ∈ X − f ( u ) = −∞ . In particular, there exists r > ∀ u ∈ S − : f ( u ) < inf X + f and, since f is bounded on bounded subsets, we also have sup D − f < + ∞ .If f has infinitely many critical points, we are done. Otherwise, from Theorem 7.1 weinfer that there exists a critical point u of f with C m ∞ ( f, u ) = { } .Since m ∞ [ m ( f, , m ∗ ( f, C m ∞ ( f,
0) = { } .Therefore u = 0 and the assertion follows. (cid:3) In order to prove Theorem 1.2, we need an auxiliary result.
MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 31 Proposition 7.3.
Let γ ∈ R and Γ : R → R be a function of class C such that lim | s |→∞ Γ( s ) | s | p = γ , lim | s |→∞ [ p Γ( s ) − s Γ ′ ( s )] = + ∞ . Then we have lim | s |→∞ [Γ( s ) − γ | s | p ] = + ∞ . Proof.
Let H ( s ) = Γ( s ) − γ | s | p , so thatlim | s |→∞ H ( s ) | s | p = 0 , lim | s |→∞ [ pH ( s ) − sH ′ ( s )] = + ∞ . For every
M >
0, there exists s > pH ( s ) − sH ′ ( s ) ≥ pM for any s ≥ s . Itfollows (cid:18) H ( s ) − Ms p (cid:19) ′ = sH ′ ( s ) − pH ( s ) + pMs p +1 ≤ ∀ s ≥ s , which implies that H ( t ) t p − Mt p ≤ H ( s ) s p − Ms p whenever t ≥ s ≥ s . Passing to the limit as t → + ∞ , we get0 ≤ H ( s ) s p − Ms p ∀ s ≥ s , namely H ( s ) ≥ M ∀ s ≥ s . Therefore lim s → + ∞ H ( s ) = + ∞ . The limit as s → −∞ can be treated in a similar way. (cid:3) Proof of Theorem 1.2.
Assume ( b − ). Let us show that f satisfies the Cerami-Palais-Smale condition. Let ( u n )be a sequence in W ,p (Ω) with f ( u n ) bounded and (1 + k u n k ) k f ′ ( u n ) k →
0. First of all,let us show that ( u n ) is bounded in W ,p (Ω). By contradiction, assume that k u n k → ∞ and set z n = u n k u n k . Up to a subsequence, z n is convergent to some z weakly in W ,p (Ω),strongly in L p (Ω) and a.e. in Ω. As in the proof of Theorem 1.1, we infer that z n → z strongly in W ,p (Ω) and z = 0.We also have lim sup n | pf ( u n ) − h f ′ ( u n ) , u n i| < + ∞ . Since p Ψ p,κ ( ξ ) − ∇ Ψ p,κ ( ξ ) · ξ = κ (cid:0) κ + | ξ | (cid:1) p − − κ p is bounded from below, we infer thatlim inf n Z Ω [ pG ( u n ) − g ( u n ) u n ] dx > −∞ . On the other hand, there exists
C > pG ( s ) − g ( s ) s ≤ C ∀ s ∈ R whence, by Fatou’s lemma, Z Ω (cid:26) lim sup n [ pG ( u n ) − g ( u n ) u n ] (cid:27) dx > −∞ . Since we havelim n [ pG ( u n ( x )) − g ( u n ( x )) u n ( x )] = −∞ for a.e. x ∈ Ω with z ( x ) = 0 , we infer that z = 0 a.e. in Ω and a contradiction follows.We conclude that the sequence ( u n ) is bounded, hence convergent, up to a subsequence,to some u weakly in W ,p (Ω). As in the proof of Theorem 1.1, we get that ( u n ) is stronglyconvergent to u in W ,p (Ω).Now let λ m ∞ < λ ≤ λ m ∞ +1 and define X − , X + as in the proof of Theorem 1.1.We have Ψ p,κ ( ξ ) ≥ p | ξ | p − p κ p ∀ ξ ∈ R N and, by Proposition 7.3, lim | s |→∞ (cid:20) G ( s ) − λp | s | p (cid:21) = −∞ . Therefore, there exists
C > G ( s ) ≤ λp | s | p + C ∀ s ∈ R . It easily follows that inf X + f > −∞ and we conclude as in the proof of Theorem 1.1.Now assume ( b + ), so that λ m ∞ ≤ λ < λ m ∞ +1 and either 1 < p ≤ κ ≥ p > κ = 0. It follows that p Ψ p,κ ( ξ ) − ∇ Ψ p,κ ( ξ ) · ξ is even bounded and the Cerami-Palais-Smale condition can be proved as in the previouscase.Now let us show that(7.2) lim k u k→∞ u ∈ X − f ( u ) = −∞ . MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 33 Let u n ∈ X − with k u n k → ∞ and let z n = u n k u n k . Up to a subsequence, ( z n ) is convergentto some z weakly in W ,p (Ω), strongly in L p (Ω) and a.e. in Ω. Since z n ∈ X − , we alsohave z = 0. From Proposition 7.3 we infer thatlim | s |→∞ (cid:20) G ( s ) − λp | s | p (cid:21) = + ∞ . In particular, there exists
C > G ( s ) ≥ λp | s | p − C ∀ s ∈ R . From Fatou’s lemma we infer thatlim n Z Ω (cid:20) G ( u n ) − λp | u n | p (cid:21) dx = + ∞ . Since Ψ p,κ ( ξ ) ≤ p | ξ | p ∀ ξ ∈ R N , it follows that f ( u n ) ≤ p Z Ω [ |∇ u n | p − λ | u n | p ] dx − Z Ω (cid:20) G ( u n ) − λp | u n | p (cid:21) dx , whence (7.2). Now we conclude as in the proof of Theorem 1.1. (cid:3) References [1]
A. Abbondandolo and M. Schwarz , A smooth pseudo-gradient for the Lagrangian action func-tional,
Adv. Nonlinear Stud. (2009), no.
4, 597–623.[2]
A. Aftalion and F. Pacella , Morse index and uniqueness for positive solutions of radial p − Laplace equations,
Trans. Amer. Math. Soc. (2004), no.
11, 4255-4272.[3]
S. Almi and M. Degiovanni , On degree theory for quasilinear elliptic equations with naturalgrowth conditions, in
Recent Trends in Nonlinear Partial Differential Equations II: Stationary Prob-lems (Perugia, 2012), J.B. Serrin, E.L. Mitidieri and V.D. R˘adulescu eds., 1–20,
ContemporaryMathematics , , Amer. Math. Soc., Providence, R.I., 2013.[4] H. Amann and E. Zehnder , Nontrivial solutions for a class of nonresonance problems and appli-cations to nonlinear differential equations,
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1980), no.
4, 539–603.[5]
A. Anane , Simplicit´e et isolation de la premi`ere valeur propre du p -laplacien avec poids, C. R.Acad. Sci. Paris S´er. I Math. (1987), no.
16, 725–728.[6]
A. Anane and N. Tsouli , On the second eigenvalue of the p -Laplacian, in Nonlinear PartialDifferential Equations (F`es, 1994), A. Benkirane, J.-P. Gossez eds., 1–9,
Pitman Res. Notes Math.Ser. , , Longman, Harlow, 1996.[7] H. Brezis and L. Nirenberg , H versus C local minimizers, C. R. Acad. Sci. Paris S´er. I Math. (1993), no.
5, 465–472.[8]
F.E. Browder , Fixed point theory and nonlinear problems,
Bull. Amer. Math. Soc. (N.S.) (1983), no.
1, 1–39.[9]
K.C. Chang , Solutions of asymptotically linear operator equations via Morse theory,
Comm. PureAppl. Math. (1981), no.
5, 693–712.[10]
K.C. Chang , Morse theory on Banach space and its applications to partial differential equations,
Chinese Ann. Math. Ser. B (1983), no.
3, 381–399.[11]
K.C. Chang , “Infinite-dimensional Morse theory and multiple solution problems”,
Progress in Non-linear Differential Equations and their Applications , , Birkh¨auser, Boston, 1993. [12] K.C. Chang , Morse theory in nonlinear analysis, in
Nonlinear Functional Analysis and Applicationsto Differential Equations (Trieste, 1997), A. Ambrosetti, K.C. Chang, I. Ekeland eds., 60–101, WorldSci. Publishing, River Edge, NJ, 1998.[13]
S. Cingolani and M. Degiovanni , Nontrivial solutions for p -Laplace equations with right handside having p -linear growth at infinity, Comm. Partial Differential Equations (2005), no.
8, 1191–1203.[14]
S. Cingolani and M. Degiovanni , On the Poincar´e-Hopf Theorem for functionals defined onBanach spaces,
Adv. Nonlinear Stud. (2009), no.
4, 679-699.[15]
S. Cingolani, M. Degiovanni and G. Vannella , On the critical polynomial of functionalsrelated to p -area (1 < p < ∞ ) and p -Laplace (1 < p ≤
2) type operators,
Atti Accad. Naz. LinceiRend. Lincei Mat. Appl. , (2015), no.
1, 49-56.[16]
S. Cingolani and G. Vannella , Critical groups computations on a class of Sobolev Banach spacesvia Morse index,
Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2003), no.
2, 271–292.[17]
S. Cingolani and G. Vannella , Morse index computations for a class of functionals definedin Banach spaces, in
Nonlinear Equations: Methods, Models and Applications (Bergamo, 2001),D. Lupo, C. Pagani and B. Ruf, eds., 107–116,
Progr. Nonlinear Differential Equations Appl. , ,Birkh¨auser, Basel, 2003.[18] S. Cingolani and G. Vannella , Morse index and critical groups for p -Laplace equations withcritical exponents, Mediterr. J. Math. (2006), no. S. Cingolani and G. Vannella , Marino-Prodi perturbation type results and Morse indices ofminimax critical points for a class of functionals in Banach spaces,
Ann. Mat. Pura Appl. (4) (2007), no.
1, 157–185.[20]
J.-N. Corvellec , Quantitative deformation theorems and critical point theory,
Pacific J. Math. (1999), no.
2, 263–279.[21]
J.-N. Corvellec and A. Hantoute , Homotopical stability of isolated critical points of continuousfunctionals,
Set-Valued Anal. (2002), no. M. Cuesta , Eigenvalue problems for the p -Laplacian with indefinite weights, Electron. J. DifferentialEquations , No. 33, 9 pp.[23]
M. Degiovanni , On topological Morse theory,
J. Fixed Point Theory Appl. (2011), no.
2, 197-218.[24]
M. Degiovanni and S. Lancelotti , Linking over cones and nontrivial solutions for p -Laplaceequations with p -superlinear nonlinearity, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2007), no.
6, 907–919.[25]
E. DiBenedetto , C α local regularity of weak solutions of degenerate elliptic equations, NonlinearAnal. (1983), no.
8, 827–850.[26]
P. Dr´abek and S.B. Robinson , Resonance problems for the p -Laplacian, J. Funct. Anal. (1999), no.
1, 189–200.[27]
E.R. Fadell, P.H. Rabinowitz , Bifurcations for odd potential operators and an alternative topo-logical index,
J. Funct. Anal. (1977), no.
1, 48–67.[28]
E.R. Fadell, P.H. Rabinowitz , Generalized cohomological index theories for Lie group actionswith an application to bifurcation questions for Hamiltonian systems,
Invent. Math. (1978), no.
2, 139–174.[29]
J.P. Garc´ıa Azorero, I. Peral Alonso and J.J. Manfredi , Sobolev versus H¨older localminimizers and global multiplicity for some quasilinear elliptic equations,
Commun. Contemp. Math. (2000), no.
3, 385–404.[30]
D. Gromoll and W. Meyer , On differentiable functions with isolated critical points,
Topology (1969), 361–369.[31] M. Guedda and L. V´eron , Quasilinear elliptic equations involving critical Sobolev exponents,
Nonlinear Anal. (1989), no.
8, 879–902.[32]
Z. Guo and Z. Zhang , W ,p versus C local minimizers and multiplicity results for quasilinearelliptic equations, J. Math. Anal. Appl. (2003), no.
1, 32–50.[33]
A.D. Ioffe , On lower semicontinuity of integral functionals. II,
SIAM. J. Control Optimization (1977), no.
6, 991–1000.
MANN-ZEHNDER TYPE RESULTS FOR p -LAPLACE PROBLEMS 35 [34] S.T. Kyritsi and N. Papageorgiou , Minimizers of nonsmooth functionals on manifolds andnonlinear eigenvalue problems with constraints,
Publ. Math. Debrecen (2005), no. O.A. Ladyzhenskaya and N.N. Ural’tseva , “Linear and quasilinear elliptic equations”, NaukaPress, Moscow, 1964. Academic Press, New York-London, 1968.[36]
S. Lancelotti , Morse index estimates for continuous functionals associated with quasilinear ellipticequations,
Adv. Differential Equations (2002), no.
1, 99–128.[37]
A.C. Lazer, S. Solimini , Nontrivial solutions of operator equations and Morse indices of criticalpoints of min-max type,
Nonlinear Anal. (1998), no.
8, 761–775.[38]
G.M. Lieberman , Boundary regularity for solutions of degenerate elliptic equations,
NonlinearAnal. (1988), no.
11, 1203–1219.[39]
P. Lindqvist , On the equation div ( |∇ u | p − ∇ u ) + λ | u | p − u = 0, Proc. Amer. Math. Soc. (1990), no.
1, 157–164 and (1992), no.
2, 583–584.[40]
J. Mawhin and M. Willem , “Critical point theory and Hamiltonian systems”,
Applied Mathe-matical Sciences , , Springer-Verlag, New York, 1989.[41] K. Perera , Nontrivial critical groups in p -Laplacian problems via the Yang index, Topol. MethodsNonlinear Anal. (2003), no.
2, 301–309.[42]
C.G. Simader , “On Dirichlet’s boundary value problem”,
Lecture Notes in Mathematics , ,Springer-Verlag, Berlin-New York, 1972.[43] I.V. Skrypnik , “Methods for analysis of nonlinear elliptic boundary value problems”,
Translationsof Mathematical Monographs , , American Mathematical Society, Providence, RI, 1994.[44] E.H. Spanier , “Algebraic topology”, McGraw-Hill Book Co., New York, 1966.[45]
P. Tolksdorf , On the Dirichlet problem for quasilinear equations in domains with conical boundarypoints,
Comm. Partial Differential Equations (1983), no.
7, 773–817.[46]
P. Tolksdorf , Regularity for a more general class of quasilinear elliptic equations,
J. DifferentialEquations (1984), no.
1, 126–150.[47]
A.J. Tromba , A general approach to Morse theory,
J. Differential Geometry (1977), no. K. Uhlenbeck , Morse theory on Banach manifolds,
J. Funct. Anal. (1972), 430–445. Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari, Via Ora-bona 4, 70125 Bari, Italy
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