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Gabriele Bonanno University of Messina

Some remarks on the variational methods

Some remarks on the classical Ambrosetti- Rabinowitz theorem are presented. In particular, it is

• bserved

that the geometry of the mountain pass, if the function is bounded from below, is equivalent to the existence of at least two local minima, while, when the function is unbounded from below, it is equivalent to the existence

• f

at least

• ne

local minimum.

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So, the Ambrosetti-Rabinowitz theorem actually ensures three or two distinct critical points, according to the function is bounded from below or not.

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HISTORICAL NOTES

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THE AMBROSETTI-RABINOWITZ THEOREM

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THE PUCCI-SERRIN THEOREM

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THE GHOUSSOUB-PREISS THEOREM

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In fact, in this case the mountain pass geometry implies the existence of two local minima and the Pucci-Serrin theorem ensures the third critical point. So, the Ambrosetti-Rabinowitz theorem, when the function is bounded from below actually ensures three distinct critical points.

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In

• rder

to apply the Ambrosetti- Rabinowitz theorem, it is important to establish the existence of a local minimum which is not a strict global minimum. In a similar way it is possible to see that, when the function is unbounded from below, the mountain pass geometry is equivalent to the existence of at least one local minimum.

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The existence of a global minimum can be

• btained
• wing

to the classical theorem

• f

direct methods in the variational calculus where the key assumptions are the sequential weak lower semicontinuity and the coercivity. THE TONELLI-WEIERSTRASS THEOREM

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Here, the version for differentiable functions is recalled.

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Our aim is to present a local minimum theorem for functions of the type: A LOCAL MINIMUM THEOREM

• Φ - Ψ

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An existence theorem

• f

a local minimum for continuously Gâteaux differentiable functions, possibly unbounded from below, is presented. The approach is based on Ekeland’s Variational Principle applied to a non- smooth variational framework by using also a novel type of Palais-Smale condition which is more general than the classical one.

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Let X be a real Banach space, we say that a Gâteaux differentiable function I :X → R verifies the Palais-Smale condition (in short (PS)-condition) if any sequence {un} such that PALAIS-SMALE CONDITION

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has a convergent subsequence. Let X be a real Banach space and let Φ : X → R, Ψ : X → R two Gâteaux differentiable functions. Put I = Φ − Ψ.

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Fix r1, r2 ∈ [−∞;+∞], with r1 < r2, we say that the function I verifies the Palais- Smale condition cut off lower at r1 and upper at r2 (in short

• condition) if

any sequence {un} such that

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has a convergent subsequence. Clearly, if r1 = −∞ and r2 = +∞ it coincides with the classical (PS)-condition. Moreover, if r1 = −∞ and r2 ∈ R we denote it by , while if r1 ∈ R and r2 = +∞ we denote it by .

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In particular,

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To prove the local minimum theorem we use the theory for locally Lipschitz functionals investigated by K.C. Chang, which is based on the Nonsmooth Analysis by F.H. Clarke, and generalizes the study

• n the variational inequalities as given by
• A. Szulkin .

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This theory is applied to study variational and variational-hemivariational inequalities . In particular, for instance, differential inclusions and equations with discontinuous nonlinearities are investigated.

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Here, by using the nonsmooth theory we

• btain results for smooth functions.

THE EKELAND VARIATIONAL PRINCIPLE

Arguing in a classical way of the smooth analysis (as, for instance, Ghossoub), but using the definitions and properties of the non-smooth analysis (as, for instance, Motreanu-Radulescu, the following consequence of the Ekeland variational Principle can be obtained.

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A CONSEQUENCE OF THE EKELAND VARIATIONAL PRINCIPLE IN THE NONSMOOTH ANALYSIS FRAMEWORK

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A LOCAL MINIMUM THEOREM

(1) (2)

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• Proof. Put

Clearly, J is locally Lipschitz and bounded from below. Hence, Lemma and a suitable computation ensure the conclusion.

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TWO SPECIAL CASES

(1)

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(2)

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A THREE CRITICAL POINTS THEOREM

From the preceding two variants

• f the local minimum theorem, a

three critical points theorem is

• btained. Here a special case is

pointed out.

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A THREE CRITICAL POINTS THEOREM

(3)

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NONLINEAR DIFFERENTIAL PROBLEMS

Consider the following two point boundary value problem

A TWO-POINT BOUNDARY VALUE PROBLEM

where f : R → R is a continuous function and is λ a positive real parameter.

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Moreover, put

for all ξ ∈ R and assume, for clarity, that f is nonnegative.

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(1) (2)

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TWO-POINT BOUNDARY VALUE PROBLEMS

there exist two positive constants c,d, with c < d, such that

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there exist two positive constants c,d, with c < d, such that there exist two positive constants a,s, with s < p, such that

Fix p>1.

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NEUMANN BOUNDARY VALUE PROBLEMS MIXED BOUNDARY VALUE PROBLEMS

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STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS HAMILTONIAN SYSTEMS

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FOURTH-ORDER ELASTIC BEAM EQUATIONS BOUNDARY VALUE PROBLEMS ON THE HALF_LINE

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NONLINEAR DIFFERENCE PROBLEMS ELLIPTIC DIRICHLET PROBLEMS INVOLVING THE P-LAPLACIAN WITH P>N

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ELLIPTIC SYSTEMS ELLIPTIC NEUMANN PROBLEMS INVOLVING THE P-LAPLACIAN WITH P>N

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ELLIPTIC PROBLEMS INVOLVING THE p-LAPLACIAN WITH p≤N

(1)

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NONLINEAR EIGENVALUE PROBLEMS IN ORLICS-SOBOLEV SPACES NONLINEAR ELLIPTIC PROBLEMS ON THE SIERPI NSKI GASKET GENERALIZED YAMABE EQUATIONS ON RIEMANNIAN MANIFOLDS ELLIPTIC PROBLEMS INVOLVING THE p(x)-LAPLACIAN

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FURTHER APPLICATIONS OF THE LOCAL MINIMUM THEOREM

If we apply two times the first special case of the local minimum theorem and owing to a novel version of the mountain pass theorem where the (PS) cut off upper at r is assumed we can give a variant of the three critical

• theorem. In the applications it became

A VARIANT OF THREE CRITICAL POINT THEOREM FOR FUNCTIONS UNBOUNDED FROM BELOW

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(1) (2)

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If we apply iteratively the first special case of the local minimum theorem in a suitable way, we obtain an infinitely many critical points theorem. As an example of application, here, we present the following result.

INFINITELY MANY CRITICAL POINTS THEOREM

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(1)

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Previous results can be applied to perturbed problems,

• btaining for instance, results of the following type.

PERTURBED PROBLEMS

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Previous results can be applied to perturbed problems,

• btaining for instance, results of the following type.

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VARIATIONAL-HEMIVARIATIONAL INEQUALITIES

In a natural way the previous results have been also

• btained in the framework of the non-smooth Analysis.

As example, here, the following problem is considered.

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(1)

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The local minimum theorem can be directly applied to obtain the existence of at least one solution.

A DIRECT APPLICATION OF THE LOCAL MINIMUM THEOREM

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(1)

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Consider the following problem

(1’)

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(1) (AR)

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CO-AUTHORS and REFERENCES

These results have been obtained in the setting of PRIN 2007-ORDINARY DIFFERENTIAL EQUATIONS AND APPLICATIONS

(National Scientific Project Manager: Fabio Zanolin)

from the Unit of MESSINA.

Pina BARLETTA (RC) Pasquale CANDITO (RC) Roberto LIVREA (RC) Nuccio MARANO (CT) Giovanni MOLICA BISCI (RC)

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Others co-authors:

Giusy D’AGUI’ (ME) Beatrice DI BELLA (ME) Antonella CHINNI’ (ME) Pasquale PIZZIMENTI (ME) Angela SCIAMMETTA (ME) Diego AVERNA (PA) Nicola GIOVANNELLI (PA) Giusy RICCOBONO (PA) Elisa TORNATORE(PA) Stefania BUCCELLATO (PA)

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Donal O’REGAN (IRELAND) Vicentiu RADULESCU (ROMANIA) Dumitru MOTREANU (FRANCE) Patrick WINKERT (GERMANY)

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[3] [4] [5] [6] [7] [8] [1] [2]

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