A Generalized Mountain Pass Lemma with a Closed Subset for Locally Lipschitz Functionals
aa r X i v : . [ m a t h . C A ] F e b A Generalized Mountain Pass Lemma with a Closed Subsetfor Locally Lipschitz Functionals
In memory of Professor Shi Shuzhong for his 80th birthday
Fengying Li ∗ Bingyu Li Shiqing Zhang School of Economic and Mathematics, Southwestern University of Finance andEconomics, Chengdu, Sichuan, 611130, P.R.China. College of Information Science and Technology, Chengdu University of Technology,Chengdu, Sichuan, 610059, P.R.China. Department of Mathematics, Sichuan University,Chengdu, Sichuan, 610064, P.R.China.
Abstract
The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been stud-ied, extended and modified in several directions. Notable examples would cer-tainly include the generalization to locally Lipschitz functionals by K.C. Chang,analyzing the structure of the critical set in the mountain pass theorem in theworks of Hofer, Pucci-Serrin and Tian, and the extension by Ghoussoub-Preiss toclosed subsets in a Banach space with recent variations. In this paper, we utilizethe generalized gradient of Clarke and Ekeland’s variatonal principle to generalizethe Ghoussoub-Preiss’s Theorem in the setting of locally Lipschitz functionals.We give an application to periodic solutions of Hamiltonian systems.
Keywords : Mountain Pass Lemma of Ambrosetti-Rabinowitz, Ekeland’s varia-tional principle, locally Lipschitz functionals, Clarke’s generalized gradient, gen-eralized Mountain Pass Lemma. : 34A34, 34C25, 35A15. ∗ Corresponding Author, Email: [email protected] Introduction and Main Results
Saddle points in the Mountain pass Lemma ( [1] - [28]) are different from maximumpoints and minimum points. Maximum and Minimum problems in infinite dimensionalspace have a very long and prominent history ( [25]) with ”isoperimetric problems” andthe ”problem of the brachistochrone” as two notable examples. In the 19th century”Dirichlet principle” we essentially encountered the problem of minimizing a func-tional; however, complete rigor was mostly lacking and we had to wait for Hilbert forsatisfactory completion of the Dirichlet principle. Continuing in the 20th century, Ital-ian mathematician Tonelli introduced the concept of a weakly lower semi-continuous(w.l.s.c) functional and proved that a w.l.s.c functional defined on a weakly closed sub-set of a reflexive Banach space can attain its infimum if it is coercive ( [25]). At times,the existence of a saddle point, which is neither a maximum nor minimum point, isof considerable importance. Minimax methods in the finite dimensional case ( [25],[28]) can be traced back to Birkhoff in 1917 and von Neumann’s minimax theorem in1928. We can also observe that the Mountain Pass Lemma of Ambrosseti-Rabinowitz( [1]) in 1973 is a type of minimax theorem, which can be traced back to Courantin 1950 for the finite dimensional case ( [25]). The Palais-Smale condition first ap-peared in connection with infinite dimensional problems, but it is necessary even infinite dimensional situations. Reference ( [4]) gives an example of a polynomial in twovariables that has exactly two non-degenerate critical points, both of which are globalminimizers, and points out that the given polynomial does not satisfy the Palais-Smalecondition. From the finite dimensional case to the infinite dimensional case, the keystep in the proof of the Mountain Pass Lemma is the use of a Palais-Smale type com-pactness condition(
P S ) to drive Palais’s Deformation Lemma. We should note theoriginal proof of the Ambrosseti-Rabinowitz’s Mountain Pass Lemma used Palais’s De-formation Lemma ( [25]). In the 1970’s, Ekeland discovered a very important principlefor lower semi-continuous functions on a complete metric space. Until the middle ofthe 1980’s, Aubin-Ekeland ( [3]), Shi ( [24]) discovered the relationship between theMountain Pass Lemma of Ambrosseti-Rabinowitz and Ekeland’s variational priciple.The Mountain Pass Lemma of Ambrosseti-Rabinowitiz has been intensively studiedand has found numerous applications ( [1] - [28]). Of special note, it was generalized2o the case of locally Lipschitz functionals by K.C. Chang ( [5]) where he also obtainedmore minimax theorems by using a deformation lemma.In this paper, we use Ekeland’s variational principle to prove a generalized MountainPass Lemma for locally Lipschitz functionals related with a closed subset, and we alsofound an applications to Hamiltonian systems with local Lispschtiz potential and afixed energy.In 1973, Ambrosetti and Rabinowitz [1] published the famous Mountain-Pass The-orem:
Theorem 1.1. ([1])
Let f be a C − real functional defined on a Banach space X satisfying the following ( P S ) condition:Every sequence { x n } ⊂ X such that { f ( x n ) } is bounded and k f ′ ( x n ) k → in X ∗ has a strongly convergent subsequence.Suppose there is an open neighborhood Ω of x and a point x / ∈ ¯Ω such that f ( x ) , f ( x ) < c ≤ inf ∂ Ω f, and let Γ := { g ∈ C ([0 , X ) : g (0) = x , g (1) = x } . Then c := inf g ∈ Γ max t ∈ [0 , f ( g ( t )) ≥ c is a critical value of f : that is, there is ¯ x ∈ X such that f (¯ x ) = c and f ′ (¯ x ) = 0 , where f ′ (¯ x ) denotes the Fr´echet derivative of f at ¯ x . Let C − ( X ; R ) be the space of locally Lipschitz mappings from X to R . ForΦ ∈ C − ( X ; R ) set (Clarke[6]) ∂ Φ( x ) := { x ∗ ∈ X ∗ : < x ∗ , v > ≤ Φ ( x, v ) , ∀ v ∈ X } , where Φ ( x, v ) := lim sup w → x t ↓ Φ( w + tv ) − Φ( w ) t denotes the generalized directional derivative ofΦ at the point x along the direction v . We should note that if Φ ∈ C , then Φ ( x, v )reduces to the Gˆateaux directional derivative and ∂ Φ reduces to the classical derivative.
Proposition 1.2. Φ ( x, v ) have some useful properties ( [6]):(1) The function v → Φ ( x, v ) is subadditive and positively homogeneous, and thenis convex,(2) | Φ ( x, v ) | ≤ K k v k ,(3) The function v → Φ ( x, v ) is continuous, Φ ( x, − v ) = ( − Φ) ( x, v ) . K.C. Chang [5] generalized the classical (
P S ) condition and the Mountain PassTheorem to local Lipschitz functions. Ribarska-Tsachev- Krastanov [22] gave a gener-alization of a result of Chang for the case when ”the separating mountain range has zeroaltitude” which is a version of the general mountain pass principle of Ghoussoub-Preissfor locally Lipschitz functions.The generalization of the Mountain Pass Lemma of Ghoussoub-Preiss [10] involvesthe modification of the classical Palais-Smale condition:
Definition 1.3.
Let X be a Banach space, F a closed subset of X and ϕ a Gˆateaux-differentiable functional on X . The ( P S ) F,c condition is the following: if { x n } ⊂ X isa sequence satisfying the three conditions(i) d ( x n , F ) →
0, where d ( x, F ) := inf y ∈ F || x − y || denotes the distance between thepoint x and the set F ,(ii) ϕ ( x n ) → c ,(iii) ϕ ′ ( x n ) → { x n } has a strongly convergent subsequence. Definition 1.4.
Let X be a Banach space, and F ⊂ X a closed subset. We sayΦ ∈ C − ( X ; R ) meets the ( CP S ) F,c condition when the following is true: if { x n } ⊂ X satisfies(1) d ( x n , F ) → x n ) → c ,(3) (1 + k x n k ) · min y ∗ ∈ ∂ Φ( x n ) k y ∗ k → { x n } has a convergent subsequence.It should be noticed that in Definition 1.4, the ( P SC ) F,c condition reduces to theCerami Condition when F = X and Φ ∈ C ( X ; R ).We can define the δ -distance (geodesic distance) ( [8]): δ ( x , x ) := inf { l ( c ) : c ∈ C ([0 , , X ) , c (0) = x , c (1) = x } , (11)4here l ( c ) := Z k ˙ c ( t ) k k c ( t ) k dt. we let dist δ ( x, F ) = inf { δ ( x, y ) : y ∈ F } . Proposition 1.5.
The geodesic distance δ has the following properties ( [8]):(1). δ ( x , x ) ≤ k x − x k ,(2). For every norm-bounded set B in X , there is γ > such that ∀ x , x ∈ B , thereholds δ ( x , x ) ≥ γ k x − x k . Definition 1.6.
Let X be a Banach space, and F ⊂ X a closed subset. We sayΦ ∈ C − ( X ; R ) meets the ( CP S ) F,c ; δ condition when the following is true: if { x n } ⊂ X satisfies(1) dist δ ( x n , F ) → x n ) → c ,(3) (1 + k x n k ) · min y ∗ ∈ ∂ Φ( x n ) k y ∗ k → { x n } has a convergent subsequence.We can now state the Mountain-Pass Theorem generalized by Ghoussoub-Preiss[10] for a continuous and Gˆateaux-differentiable functional statisfying the ( P S ) F,c con-dition:
Theorem 1.7. ( [10])
Let ϕ : X → R be a continuous and Gˆateaux-differentiablefunctional on a Banach space X such that ϕ ′ : X → X ∗ is continuous from the normtopology of X to the w ∗ − topology of X ∗ . Take u, v ∈ X , and let c := inf g ∈ Γ max t ∈ [0 , ϕ ( g ( t )) where Γ = Γ vu is the set of all continuous paths joining u and v . Suppose F is a closedsubset of X such that F ∩ { x ∈ X : ϕ ( x ) ≥ c } separates u and v and ϕ satisfies the ( P S ) F,c condition, then there exists a critical point ¯ x ∈ F for ϕ on F with critical value c : ϕ (¯ x ) = c, ϕ ′ (¯ x ) = 0 . Theorem 1.8. ( [7])
Let ( X, d ) be a complete metric space with metric d and f : X → R ∪ { + ∞} a lower semi-continuous functional not identically + ∞ which isbounded from below. Let ε > and u ∈ X such that f ( u ) ≤ inf x ∈ X f ( x ) + ε. Thenfor any given λ > , there exists v λ ∈ X such that f ( v λ ) ≤ f ( u ) , d ( u, v λ ) ≤ λ , and f ( w ) > f ( v λ ) − ελ d ( v λ , w ) , ∀ w = v λ . Ekeland’s variational principle has found numerous applications; in particular, wewould like to observe that prior to Ghoussoub-Preiss [10] it was used by Shi [24] toprove a Mountain Pass Lemma and general min-max theorems for locally Lipschitzfunctionals (K.C.Chang [5]). In this paper, we will use Ekeland’s variational principleto generalize the Ghoussoub-Preiss Theorem to the case of locally Lipschitz functionalof class C − satisfying the conditions ( CP S ) F,c ; δ or ( CP S ) F,c . Theorem 1.9.
Let X be a Banach space with norm || . || , C ([0 , X ) the space ofcontinuous mappings from [0 , to X , and Φ : X → R a locally Lipschitz functional.For z , z ∈ X , define Γ := { c ∈ C ([0 , X ) : c (0) = z , c (1) = z } , γ := inf c ∈ Γ max ≤ t ≤ Φ( c ( t )) , and set Φ γ := { x ∈ X : Φ( x ) ≥ γ } . If F ⊂ X is a closed subset such that F ∩ Φ γ separates z and z , then there exists a sequence { x n } ⊂ X such that dist δ ( x n , F ) → , Φ( x n ) → γ and (1 + k x n k ) min y ∗ ∈ ∂ Φ( x n ) k y ∗ k → . Theorem 1.10.
Under the assumptions of Theorem 1.9, if we add that the set F isnorm-bounded in the Banach space X , then there exists a sequence { x n } ⊂ X such that d ( x n , F ) → , Φ( x n ) → γ and (1 + k x n k ) min y ∗ ∈ ∂ Φ( x n ) k y ∗ k → . Theorem 1.11.
Under the assumptions of Theorem 1.9, if Φ satisfies ( CP S ) F,γ ; δ condition, then γ is a critical value for Φ : Φ(¯ x ) = γ, ∈ Φ ′ (¯ x ) . Theorem 1.12.
Under the assumptions of Theorem 1.9, if we add the condition thatthe set F is bounded in the norm of the Banach space X , then we can change the ( CP S ) F,γ ; δ condition to the ( CP S ) F,γ condition, and conclude there exists a criticalpoint ¯ x ∈ F for Φ on F with critical value γ : Φ(¯ x ) = γ, ∈ Φ ′ (¯ x ) .
6n 2009, Goga [11] studied a general Mountain Pass Theorem for local Lipschitzfunction. Let ( E, k · k ) be a Banach space, S a compact metric space and S a closedsubset of S . Let C ( S, E ) be the Banach space of all E − valued bounded continuousmapping on S with the norm k γ k := sup x ∈ S k γ ( x ) k . Let γ ∈ C ( E, S ) be a fixed elementand define Γ = { γ ∈ C ( S, E ) : γ ( s ) = γ ( s ) , ∀ s ∈ S } , c := inf γ ∈ Γ sup s ∈ S f ( γ ( s )) , where f is a real-valued function defined on E . Goga’s result is the following: Theorem 1.13.
Let f : E → R be a locally Lipschitz function and F a closed nonemptysubset of E . Assume that(a) γ ( S ) ∩ F ∩ f c = ∅ , ∀ γ ∈ Γ , where f c = { x ∈ E : f ( x ) ≥ c } ,(b) dist ( γ ( S ) , F ) > , where dist ( · , F ) is the distance function to F in E .Then for every ε > there exist x ε ∈ E such that(i) dist ( x ε , F ) < ε ,(ii) c ≤ f ( x ε ) < ε + ε ,(iii) dist (0 , ∂f ( x ε )) ≤ ε , where ∂f ( x ) is the Clark sub-differential of f at x . Thanks to a referee for pointing out the papers [13] - [14]. In [14], Livrea, R.and Marano S. A. considered more general compactness conditions: Let h : [0 , + ∞ [ → [0 , + ∞ [ be a continuous function enjoying the following property: Z + ∞
11 + h ( ξ ) dξ = + ∞ . (12)They call that f satisfies a weak Palais-Smale condition at the level c ∈ R when forsome h as above one has:( P S ) hc Every sequence { x n } ⊆ X such that lim n → + ∞ f ( x n ) = c and lim n → + ∞ (1 + h ( k x n k )) min y ∗ ∈ ∂f ( x n ) k y ∗ k = 0 (13) possesses a convergent subsequence. A weaker form of (
P S ) hc is the one below, where U denotes a nonempty closedsubset of X . For U := X it coincides with ( P S ) hc .7 P S ) hU,c Every sequence { x n } ⊆ X such that d ( x n , U ) → as n → + ∞ and (13) holdstrue possesses a convergent subsequence. Given x, z ∈ X , denoted by P ( x, z ) the family of all piecewise C paths p : [0 , → X such that p (0) = x and p (1) = z . Moreover, put l h ( p ) := Z k p ′ ( t ) k h ( k p ( t ) k ) dt, p ∈ P ( x, z ) , as well as δ h ( x, z ) := inf { l h ( p ) : p ∈ P ( x, z ) } . (14)Let B be a nonempty closed subset of X and let F be a class of nonempty compactsets in X . According to [9] Definition 1, they call that F is a homotopy-stable familywith extended boundary B when for every A ∈ F , η ∈ C ([0 , × X, X ) such that η ( t, x ) = x on ( { } × X ) ∪ ([0 , × B ) one has η ( { } × A ) ∈ F . Some meaningfulsituations are special cases of this notion. For instance, if Q denotes a compact set in X , Q is a non-empty closed subset of Q , γ ∈ C ( Q , X ),Γ := { γ ∈ C ( Q, X ) : γ | Q = γ } , and F := { γ ( Q ) : γ ∈ Γ } , then F enjoys the above-mentioned property with B := γ ( Q ). In particular, it holds true when Q indicates a compact topological manifoldin X having a nonempty boundary Q while γ := id | Q .In [14], Livrea and Marano made the following assumptions: ( a ) f : X → R is a locally Lipschitz continuous function.( a ) F denotes a homotopy-stable family with extended boundary B .( a ) There exists a nonempty closed subset F of X such that ( A ∩ F ) \ B = ∅ , ∀ A ∈ F and, moreover, sup x ∈ B f ( x ) ≤ inf x ∈ F f ( x ) . ( a ) h : [0 , + ∞ [ → [0 , + ∞ [ is a continuous function fulfilling (13), while δ h indicatesthe metric defined (12).Set, as usual, c := inf A ∈F max x ∈ A f ( x ) . Livrea and Marano [13]-[14] obtained the following theorems8 heorem 1.14.
Let ( a )-( a ) be satisfied. Then to every sequence { A n } ⊆ F suchthat lim n → + ∞ max x ∈ A n f ( x ) = c there corresponds a sequence x n ⊆ X \ B having the followingproperties:( i ) lim n → + ∞ f ( x n ) = c .( i ) (1 + h ( k x n k )) f ( x n ; z ) ≥ − ǫ n k z k for all n ∈ N , z ∈ X , where ǫ n → + .( i ) lim n → + ∞ δ h ( x n , F ) = 0 provided inf x ∈ F f ( x ) = c .( i ) lim n → + ∞ δ h ( x n , A n ) = 0 . Theorem 1.15.
Let ( a ) - ( a ) be satisfied. Suppose that either ( P S ) hc holds or F is bounded and ( P S ) hF,c holds, according to whether inf x ∈ F f ( x ) < c or inf x ∈ F f ( x ) = c .Then K c ( f ) = { x ∈ X : f ( x ) = c, x is a critical point of f } 6 = ∅ . If, moreover, inf x ∈ F f ( x ) = c , then K c ( f ) ∩ F = ∅ . Theorem 1.16.
Let ( a ) and ( a ) be satisfied. Suppose that:( a ) There exists a closed subset F of X complying with ( γ ( Q ) ∩ F ) \ γ ( Q ) = ∅ for all γ ∈ Γ and, moreover, sup x ∈ Q f ( γ ( x )) ≤ inf x ∈ F f ( x ) .( a ) Setting c := inf γ ∈ Γ max x ∈ Q f ( γ ( x )) , either ( P S ) hc holds or F is bounded and ( P S ) hF,c holds, according to whether inf x ∈ F f ( x ) < c or inf x ∈ F f ( x ) = c . Then the conclusion ofTheorem (1.15) is true.Remark . Comparing our Theorem 1.9 with Livrea - Marano’s Theorem 1.14, wefound that although the conditions of Livrea - Marano’s theorems are more general, buttheir results are weaker, in fact, in Theorem 1.14 they can obtain lim n → + ∞ δ h ( x n , F ) = 0under the assumptions inf x ∈ F f ( x ) = c , without this condition, they can’t getlim n → + ∞ δ h ( x n , F ) = 0, generally, they can only obtain ( i ) lim n → + ∞ δ h ( x n , A n ) = 0. InTheorem 1.15 and Theorem 1.16, although the conditions ( a )-( a ) are more generalthan our Theorems 1.10-1.12, but they suppose two cases that ”either ( P S ) hc holds or F is bounded and ( P S ) hF,c holds, according to whether inf x ∈ F f ( x ) < c or inf x ∈ F f ( x ) = c ”,which are more difficulty to apply, in fact, in many applied examples, it’s very difficultto prove inf x ∈ F f ( x ) < c or inf x ∈ F f ( x ) = c . Remark . It should be poited out that the conditions (
CP S ) F,c and (
CP S ) F,c,δ generalize the Cerami condition (with a closed subset F ) in the smooth situation.For more on Cerami sequence in the smooth case, we refer to Schechter [23] and9tuart [26], Stuart’s paper proved the smooth case of our Theorem 1.9 and Theo-rem 1.10. The conclusions (i)-(iii) of Goga’s Theorem 1.13 and the condition ( P S ) c in Ribarska- Tsachev- Krastanov [22] are different from the conditions ( CP S ) F,c and(
CP S ) F,c ; δ stated here. Our results Theorem 1.9 and Theorem 1.10 are stronger since(1 + k x n k ) min y ∗∈ ∂ Φ( x n ) k y ∗ k → CP S ) F,γ ; δ and( CP S ) F,γ conditions are weaker than those used [11] and [22]; therefore, the argu-ments in our paper differ from [11] and [22] since they could utilize the Borwein-Preissvariational principle or a deformation lemma, whereas we use the classical Ekeland’svariational principle.
Remark . We should note the difference between our Generalized Mountain PassLemma (GMPL) and the following theorem of Struwe( [25]): Suppose M is a closedconvex subset of a Banach space V and E ∈ C ( V ) satisfies ( P. − S. ) M on M . Anysequence { u n } ⊂ M such that | E ( u n ) | ≤ c uniformly, while g ( u m ) = sup v ∈ M k um − v k < h u m − v, DE ( u m ) i → m → ∞ ), is relatively compact. Suppose further that E admits twodistinct relative minima u , u in M . Then either E ( u ) = E ( u ) = β and u , u can be connected in any neighborhood of the set of relative minima u ∈ M of E with E ( u ) = β , or there exists a critical point ¯ u of E in M which is not a relative minimizerof E .In Struwe’s Theorem, M is a closed convex subset of a Banach space, but in ourGMPL we don’t assume any convexity. We also don’t assume that E : M → R possesses an extension E ∈ C ( V ; R ) to V , but only that the functional is locallyLipschitz. Struwe’s Theorem assumes the existence of two local minimizers, but weonly require the existence of two valleys which may not be local minimizers. In theseways, we see the premise in our GMPL is weaker than the corresponding conditions inStruwe’s Theorem. Remark . The classical Mountain Pass Lemma and its many generalizations areprimarily concerned with saddle points, but we should note the saddle points encoun-tered in these various Mountain Pass Lemmas are different from those in the VonNeumann Minimax Theorem ( [28]). The Minimax Theorem of Neumann is essentiallyrelated with convexity and concavity, whereas the Mountain Pass Lemma is not related10ith convexity and concavity which is related with (
P S ) compactness condition andtwo valleys for functional. It seems interesting to use Ekeland’s variational principleto prove von Neumann Minimax Theorem.
Proof.
Since the main ingredient is still Ekeland’s variational principle, we utilize somenotations and ideas from [8] and [10], but must deviate in a few key steps. Since theclosed set F γ := Φ γ T F separates z and z , we can write X \ F γ := Ω S Ω where z ∈ Ω , z ∈ Ω for open sets Ω and Ω with Ω ∩ Ω = ∅ .Choose ε which satisfies0 < ε <
12 min { , dist δ ( z , F γ ) , dist δ ( z , F γ ) } . (25)By the definition of Γ, we can find c ∈ Γ such thatmax ≤ t ≤ Φ( c ( t )) < γ + ε . (26)If we define t and t by t := sup { t ∈ [0 ,
1] : c ( t ) ∈ Ω , dist δ ( c ( t ) , F γ ) ≥ ε } ,t := inf { t ∈ [ t ,
1] : c ( t ) ∈ Ω , dist δ ( c ( t ) , F γ ) ≥ ε } , then since c (0) = z , we have by (25) and the continuity of c that t >
0; moreover,by c ( t ) ∈ ¯Ω and dist δ ( c ( t ) , F γ ) ≥ ε , we have c ( t ) ∈ Ω . Then Ω ∩ Ω = ∅ implies t > t . Again by (25) and the continuity of c we have t <
1. So altogether0 < t < t <
1. LetΓ( t , t ) := { f ∈ C ([ t , t ] , X ) : f ( t ) = c ( t ) , f ( t ) = c ( t ) } , (27)and consider the following distance in Γ( t , t ): ρ ( f , f ) := max t ≤ t ≤ t δ ( f ( t ) , f ( t )) , (28)where δ ( x , x ) := inf { l ( c ) : c ∈ C ([0 , , X ) , c (0) = x , c (1) = x } (29)with l ( c ) := Z k ˙ c ( t ) k k c ( t ) k dt. (210)11or x ∈ X , we define the functionΨ( x ) := max { , ε − εdist δ ( x, F γ ) } . (211)A map ϕ : Γ( t , t ) → R is defined by ϕ ( f ) := max t ≤ t ≤ t { Φ( f ( t )) + Ψ( f ( t )) } . (212)Since f ( t ) = c ( t ) ∈ Ω , f ( t ) = c ( t ) ∈ Ω , there exists t f ∈ ( t , t ) satisfying f ( t f ) ∈ ∂ Ω ⊂ F γ ; therefore, dist δ ( f ( t f ) , F γ ) = 0 , (213)and for any f ∈ Γ( t , t ), we have ϕ ( f ) ≥ Φ( f ( t f )) + Ψ( f ( t f )) ≥ γ + ε . (214)On the other hand, if we denote ˆ c = c | [ t ,t ] , then ϕ (ˆ c ) ≤ max ≤ t ≤ { Φ( c ( t )) + Ψ( c ( t )) } ≤ γ + 54 ε . (215)Notice that Γ( t , t ) is a complete metric space with respect to the metric ρ introducedin (28) [7],[8]. Since Φ and Ψ are lower semi-continuous, so is ϕ . Now (214) implies ϕ has a lower bound, and by (214) and (215) we have ϕ (ˆ c ) ≤ inf ϕ + ε . (216)In Ekeland’s variational principle, we use ε in place of ε , and take λ = ε , then thereexists ˆ f ∈ Γ( t , t ) such that ϕ ( ˆ f ) ≤ ϕ (ˆ c ) , ρ ( ˆ f , ˆ c ) ≤ ε , ϕ ( f ) ≥ ϕ ( ˆ f ) − ε ρ ( f, ˆ f ) , ∀ f ∈ Γ( t , t ) . (217)Let M := { t ∈ [ t , t ] : Φ( ˆ f ( t )) + Ψ( ˆ f ( t )) = ϕ ( ˆ f ) } . (218)Note that M is a non- empty set that is compact, since Φ and Ψ are lower- semi-continuous. We next show that t , t / ∈ M .By the definitions of t and t , we have dist δ ( c ( t i ) , F γ ) ≥ ε, i = 0 , , (219)12o by (211)we have Ψ(ˆ c ( t i )) = 0. By (26) and (214) and (217 ) we haveΦ( ˆ f ( t i )) + Ψ( ˆ f ( t i )) ≤ Φ(ˆ c ( t i )) + Ψ(ˆ c ( t i )) ≤ γ + ε < ϕ ( ˆ f ) , i = 0 , , (220)which implies t , t / ∈ M . Claim : There exists t ∈ M such thatmin x ∗ ∈ ∂ Φ( ˆ f ( t )) k x ∗ k (1 + k ˆ f ( t ) k ) ≤ ε . (221) Proof.
If not, for any t ∈ M , min x ∗ ∈ ∂ Φ( ˆ f ( t )) k x ∗ k (1 + k ˆ f ( t ) k ) > ε . (222)It is well known that k x ∗ k = sup v =0
0, ˆ f + hv ∈ Γ( t , t ); hence by ( 217), we have ϕ ( ˆ f + hv ) ≥ ϕ ( ˆ f ) − ε ρ ( ˆ f + hv, ˆ f ) . (226)By (212), we can choose t h ∈ [ t , t ] such that ϕ ( f ) = Φ( f ( t n )) + Ψ( f ( t n )) and then: ϕ ( ˆ f + hv ) = (Φ + Ψ)( ˆ f ( t h ) + hv ( t h )) . (227)Notice that here t h is defined for each h >
0. By the definition of ϕ , we know that forany h >
0, there holds ϕ ( ˆ f ) ≥ (Φ + Ψ)( ˆ f ( t h )). Combining with (226) and (227), wehave for every h > f ( t h ) + hv ( t h )) ≥ (Φ + Ψ)( ˆ f ( t h )) − ε ρ ( ˆ f + hv, ˆ f ); (228)that is,Φ( ˆ f ( t h ) + hv ( t h )) − Φ( ˆ f ( t h )) ≥ − Ψ( ˆ f ( t h ) + hv ( t h )) + Ψ( ˆ f ( t h )) − ε ρ ( ˆ f + hv, ˆ f )) . (229)If we recall the definition of Ψ, then Ψ is ε − Lipschitz with respect to the metric ρ , andso the above inequality impliesΦ( ˆ f ( t h ) + hv ( t h )) − Φ( ˆ f ( t h )) ≥ − ε ρ ( ˆ f + hv, ˆ f )) . (230)Notice that if h n → + , we can pass to a sequence { t h n } with t h n → τ ∈ M since M iscompact. Calculatinglim sup n → + ∞ Φ( ˆ f ( t h n ) + h n v ( t h n )) − Φ( ˆ f ( t h n )) h n ≥ − ε n → + ∞ ρ ( ˆ f + h n v, ˆ f )) h n (231)14nd further by Φ ∈ C − and the definitions of Clark’s generalized gradient and themetric ρ , we claimΦ ( ˆ f ( τ ) , v ( τ )) ≥ − ε t ≤ t ≤ t ( k v ( t ) k k ˆ f ( t ) k ) ≥ − ε . (232)In fact, by Φ ∈ C − and the continuity for v ( t ), we know thatΦ( ˆ f ( t h n ) + h n v ( t h n )) − Φ( ˆ f ( t h n ) + h n v ( τ )) h n ≤ L | v ( t h n ) − v ( τ ) | → , hence lim sup n → + ∞ Φ( ˆ f ( t h n ) + h n v ( t h n )) − Φ( ˆ f ( t h n )) h n ≤ lim sup n → + ∞ Φ( ˆ f ( t h n ) + h n v ( t h n )) − Φ( ˆ f ( t h n ) + h n v ( τ )) h n + lim sup n → + ∞ Φ( ˆ f ( t h n ) + h n v ( τ )) − Φ( ˆ f ( t h n ) h n = Φ ( ˆ f ( τ ) , v ( τ )) . Using the definition (28) of the metric ρ , we have ρ ( ˆ f + h n v, ˆ f )= max t ≤ t ≤ t inf { Z || ˙ c ( s ) || || c ( s ) || ds,c ( s ) ∈ C ([0 , , X ) , c (0) = ˆ f ( t ) , c (1) = ( ˆ f + h n v )( t ) } . Specifically, if we take the following loop connecting ˆ f and ˆ f + h n v for 0 ≤ s ≤ c t,n ( s ) = (1 − s ) ˆ f ( t ) + s ( ˆ f ( t ) + h n v ( t )) = ˆ f ( t ) + sh n v ( t ) , then we have c t,n ( s ) = h n v ( t ), and so we have the uniform convergence in t and s , c t,n ( s ) = ˆ f ( t ) + sh n v ( t ) → ˆ f ( t ) , n → + ∞ . So we have that R
10 11+ k c t,n ( s ) k ds → R
10 11+ k ˆ f ( t ) k ds = k ˆ f ( t ) k ,lim inf n → + ∞ ρ ( ˆ f + h n v, ˆ f )) h n ≤ max t ≤ t ≤ t ( k v ( t ) k k ˆ f ( t ) k ) , and (232) is proved, which violates (224) and shows that we cannot have the inequality(222) for every t ∈ M . Therefore, there is ¯ t ∈ M such thatmin x ∗ ∈ ∂ Φ( ˆ f (¯ t )) k x ∗ k (1 + k ˆ f (¯ t ) k ) ≤ ε , (233)this ends the proof of claim (221). 15y the definitions of t and t , we have that dist δ (ˆ c ( t ) , F γ ) ≤ ε for t < t < t ;furthermore, by continuity of ˆ c ( t ) and dist δ ( x, F γ ) on x , we have that dist δ (ˆ c ( t ) , F γ ) ≤ ε, ∀ t ∈ [ t , t ] . Here, we have used the notation dist δ (ˆ c ( t ) , F γ ) to denote the distance between ˆ c ( t ) and F γ deduced by δ in (29). We notice that ρ is the distance deduced by δ in (29), since ρ ( ˆ f , ˆ c ) ≤ ε , the triangle inequality implies that for all t ∈ [ t , t ] we have dist δ ( ˆ f ( t ) , F γ ) ≤ ε dist δ (ˆ c ( t ) , F γ )) ≤ ε ε = 3 ε . (234)Set x = ˆ f (¯ t ), we get dist δ ( x, F γ ) ≤ ε . Then by (217), ϕ ( ˆ f ) ≤ ϕ (ˆ c ); the fact that ¯ t ∈ M , (218), (214) and (215) yields γ + ε ≤ Φ( ˆ f (¯ t )) + Ψ( ˆ f (¯ t )) ≤ γ + 5 ε . (235)Hence Theorem 1.9 is proved.If F is bounded and a closed subset of X , then by the definition of δ , we knowthat ([7]) δ distance is equivalent to the norm distance, so there is c > dist δ ( x, F γ ) ≥ cd ( x, F γ ), where d ( x, F r ) is the distance between x and the set F r de-duced by the norm in the Banach space X .Then we get min x ∗ ∈ ∂ Φ( x ) k x ∗ k (1 + k x k ) ≤ ε , (236) d ( x, F γ ) ≤ c ε , (237) γ ≤ Φ( x ) ≤ γ + 5 ε . (238)If we let ε = n →
0, then we arrive at a sequence { x n } which satisfies the requirementsof Theorems 1.10, the Theorems 1.10 is proved. Theorems 1.11 and Theorem 1.12follow from Theorems 1.9 and Theorem 1.10. Let V ∈ C − ( R n , R ); that is, V is a locally Lipschitz potential function defined on R n . Let us consider the second order Hamiltonian systems − ¨ q ( t ) ∈ ∂V ( q ) (339)12 | ˙ q | + V ( q ) = h ∈ R (340)16 heorem 3.1. Suppose V ∈ C − ( R n , R ) and h ∈ R satisfy( V ) V ( − q ) = V ( q ); ( V ) ∃ µ > , µ ≥ , such that h y, q i ≥ µ V ( q ) − µ , ∀ y ∈ ∂V ( q ) , ∀ q ∈ R n ;( V ) there is an M > such that V ( q ) ≥ h , whenever | q | ≥ M .Then for any h > µ µ , the system (339) − (340) has at least one non-constant periodicsolution with the given energy h which can be obtained by Theorem 1.11. Corollary 3.2.
For a > , µ > , µ ≥ , let V ( q ) = a | q | µ + µ µ . Then for any h > µ µ , the system (339) − (340) has at least one non-constant periodic solution withthe given energy h which can be obtained by Theorem 1.11.Remark . If µ >
1, then V ( q ) = a | q | µ + µ µ ∈ C ( R n , R ); but if 0 < µ ≤ C ( R n , R ), but V ∈ C − ( R n , R ).In order to prove Theorem 3.1, we define the Sobolev space H := W , ( R /T Z , R n ) = { u : R → R n , u ∈ L , ˙ u ∈ L , u ( t + 1) = u ( t ) } . (341)Then the standard H norm is equivalent to k u k := k u k H = (cid:18)Z | ˙ u | dt (cid:19) / + | Z u ( t ) dt | . (342) Lemma 3.4. ([2])
Let f ( u ) := R | ˙ u | dt R ( h − V ( u )) dt and e u ∈ H be such that f ′ ( e u ) = 0 and f ( e u ) > . Set T := R ( h − V ( e u )) dt R | ˙ e u | dt . (343) Then e q ( t ) = e u ( t/T ) is a non-constant T -periodic solution for (339)-(340) . In a manner similar to Ambrosetti-Coti Zelati[2], from the symmetry condition ( V )we let E := { u ∈ H = W , ( R / Z , R n ) , u ( t + 1 /
2) = − u ( t ) } . A similar proof as in [2],we have
Lemma 3.5. If ¯ u ∈ E is a critical point of f ( u ) and f (¯ u ) > , then we have ¯ q ( t ) =¯ u ( t/T ) is a non-constant T -periodic solution of (339)-(340) .
17e define a weakly closed subset of H , F h := { u ∈ E : Z [ V ( u ) + 12 min y ∈ ∂V ( u ) h y, u i ] dt = h } . (344) Lemma 3.6. If ( V ) − ( V ) hold, then F h = ∅ ,for all h > µ µ .Proof. Take u ∈ E satisfying min t ∈ [0 , | u ( t ) | >
0. By condition ( V ), we know V (0) ≤ µ µ .We define g ( w ) = Z [ V ( w ) + 12 min y ∈ ∂V ( w ) h y, w i ] dt,g u ( a ) := g ( au ) = Z [ V ( au ) + 12 min x ∈ ∂V ( au ) h x, au i ] dt. Then we have g u (0) = g (0) = V (0) ≤ µ µ . We notice if a is large enough, min t ∈ [0 , | au | = a min ≤ t ≤ | u ( t ) | ≥ M. We use ( V ) − ( V )to get g u ( a ) = g ( au ) = Z [ V ( au ) + 12 min x ∈ ∂V ( au ) h x, au i ] dt (345) ≥ (1 + µ Z V ( au ) dt − µ ≥ (1 + µ ) h − µ h + 12 ( µ h − µ ) . (347)Hence any h > µ µ , when a large enough, g u ( a ) > h , we know there is a ( u ) > a ( u ) u ∈ F . Lemma 3.7. If ( V ) − ( V ) hold, then for any given c > , f ( u ) satisfies ( CP S ) F,c ; δ condition; that is, if { u n } ⊂ E satisfies dist δ ( u n , F ) → , f ( u n ) → c > , (1 + k u n k ) min y ∗ ∈ ∂f ( u n ) k y ∗ k → , (348) then { u n } has a strongly convergent subsequence.Proof. Notice that any u ∈ E, R u ( t ) dt = 0; hence, we know k u k E := ( R | ˙ u | dt ) / isan equivalent norm on E . By f ( u n ) → c , we have − k u n k E · Z V ( u n ) dt ≤ c − h k u n k E + ε, (349)where ε → n → + ∞ .By ( V ) we know that any y ∗ ∈ ∂f ( u n ), any x ∈ ∂V ( u n ),18 y ∗ , u n i = k u n k E · Z [ h − V ( u n ) − h x, u n i ] dt ≤ k u n k E Z [ h + µ − (1 + µ V ( u n )] dt. (350)By (349) and (350) we have h y ∗ , u n i ≤ ( h + µ k u n k E + (1 + µ c − h k u n k E ) + (2 + µ ) ε = ( − µ h + µ k u n k E + α, (351)where α = 2(1 + µ )( c + ε ).Since h > µ µ , then (348) and (351) imply k u n k E is bounded.The rest of the argument to show { u n } has a strongly convergent subsequence isstandard. Lemma 3.8.
Let G = { u ∈ E : Z [ V ( u ) + 12 min y ∈ ∂V ( u ) h y, u i ] dt < h } . (352) Then(i). F h is the boundary of G .(ii). If ( V ) holds, then F h is symmetric with respect to the origin .(iii). If V (0) < h holds, then ∈ G .Proof. (i). By the definitions of F and G .(ii). By ( V ), we have V ( − u ) = V ( u ), hence, ∂V ( − u ) = − ∂V ( u ) and V ( − u ) + 12 min y ∈ ∂V ( − u ) h y, − u i (353)= V ( u ) + 12 min y ∈− ∂V ( u ) h− y, u i = V ( u ) + 12 min − y ∈ ∂V ( u ) h− y, u i = V ( u ) + 12 min z ∈ ∂V ( u ) h z, u i By the definition of F and the above equations, we know that u ∈ F implies ( − u ) ∈ F .(iii). If we set W ( u ) = R [ V ( u ) + min y ∈ ∂V ( u ) h y, u i ] dt , then W (0) = Z [ V (0) + 0] dt = V (0) . Hence if V (0) < h , then W (0) < h and 0 ∈ G .19t’s not difficult to prove the following two Lemmas: Lemma 3.9. f ( u ) is weakly lower semi-continuous on F h . Lemma 3.10. F h is weakly closed subsets in H . Lemma 3.11. If h > µ µ , then the functional f ( u ) has positive lower bound on F h .Proof. By the definitions of f ( u ) and F h , we have f ( u ) = 14 Z | ˙ u | dt · Z min y ∈ ∂V ( u ) h y, u i dt, u ∈ F h . (354)For u ∈ F h and ( V ), we have12 Z min y ∈ ∂V ( u ) h y, u i dt = Z [ h − V ( u )] dt ≥ Z [ h − µ min y ∈ ∂V ( u ) h y, u i − µ µ ] dt, (355) Z min y ∈ ∂V ( u ) h y, u i dt ≥ h − µ µ + µ > . (356)So we have f ( u ) ≥ u ∈ F h . Furthermore, we claim that inf f ( u ) >
0, supposeotherwise, and note that u ( t ) = const attains the infimum of 0.If u ∈ F h , and u is constant, then by the symmetry u ( t + 1 /
2) = − u ( t ) or u ( − t ) = − u ( t ), we know u ( t ) = 0 , ∀ t . By ( V ) we have V (0) ≤ µ µ , by h > µ µ we get V (0) < h .From the definition of F h , 0 / ∈ F h . So inf F h f ( u ) > . Now by Lemmas 3.9-3.11, weknow f ( u ) attains the infimum on F h , and the minimizer is nonconstant. Lemma 3.12. ∃ z ∈ H such that z = 0 and f ( z ) ≤ . Proof.
We can choose y ( t ) ∈ C [0 ,
1] such that min ≤ t ≤ | y ( t ) | >
0. Let z ( t ) = Ry ( t ),then when R is large enough, by condition ( V ) we have Z ( h − V ( z )) dt ≤
0; (357)that is, f ( z ) ≤ . Lemma 3.13. f (0) = 0 . Lemma 3.14. F h separates z and .Proof. By V (0) < h , we have that 0 ∈ G . By ( V ) and ( V ) and h > µ µ , we can choose R large enough such that z = Ry ∈ { u ∈ H : Z [ V ( u ) + 12 min y ∈ ∂V ( u ) h y, u i ] dt > h } (358)20n fact, since we can choose y ∈ C [0 ,
1] such that min ≤ t ≤ | y ( t ) | >
0, hence, when R large enough, Z [ V ( Ry ) + 12 min y ∈ ∂V ( Ry ) h y, Ry i ] dt (359) ≥ (1 + µ Z V ( Ry ) dt − µ ≥ (1 + µ h − µ > h. (361)Hence, z = Ry ∈ { u ∈ H : R [ V ( u ) + min y ∈ ∂V ( u ) h y, u i ] dt > h } . So F h separates z and0. Theorem 3.1 now follows from Theorem 1.11 Since Ekeland’s variational principle imposes less restriction on the functional, wefound it very useful in proving our Generalized Mountain Pass Lemma with weakerassumptions. We were able to establish an immediate application for our GeneralizedMountain Pass Lemma to Hamiltonian systems with Lipschitz potential and a fixedenergy. It would be interesting to see what role it can play for other differentialequations.
Acknowledgements
This research was partially supported by NSF of China(No.11701463, No.11671278 andNo.12071316). The authors sincerely thank the referees and the editors for their manyvaluable comments and suggestions which help us improving the paper.