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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Review

Jean Mawhin’s contributions to critical point theory

Michel Willem

Author Affiliations

Institut de Recherche en Mathématique et Physique, L7.01.02, Chemin du Cyclotron 2, Louvain-la-Neuve, 1348, Belgium

Boundary Value Problems 2013, 2013:115  doi:10.1186/1687-2770-2013-115


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/115


Received:29 January 2013
Accepted:22 April 2013
Published:7 May 2013

© 2013 Willem; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this article is to describe some fundamental contributions of Jean Mawhin to critical point theory and its applications to boundary value problems.

Dedication

Dedicated to Jean Mawhin on the occasion of his seventieth birthday with friendship.

Content

The first paper by Jean Mawhin on critical point theory [1] was published in 1982 and was devoted to periodic solutions of a forced pendulum equation. One of the most recent papers in 2012 [2] concerns periodic solutions of difference systems with ϕ-Laplacian. It is impossible to describe all the contributions. We have selected 17 articles, 2 books and some fundamental topics:

– the forced pendulum equation,

– convex perturbations of indefinite quadratic functionals,

– construction of almost critical points,

– converse to the Lagrange-Dirichlet theorem, and

– Neumann problems for the ϕ-Laplacian.

1 From the classical to the relativistic pendulum

The forced pendulum equation is an important field of investigations of Jean Mawhin. We describe only some contributions (by variational methods) to the conservative forced pendulum, and we refer to the exhaustive survey [3] for other results.

Consider the classical second-order problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M1">View MathML</a>

(1)

where f is 2π-periodic. The solutions of (1) are the critical points of the action functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M2">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M3">View MathML</a>. The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M4">View MathML</a> is the space of absolutely continuous functions u which are T-periodic and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M5">View MathML</a>. Assuming that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M6">View MathML</a>

(2)

it is not difficult to prove that Ψ achieves its infimum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M4">View MathML</a> and, consequently, that (1) is solvable. This result, due essentially to Hamel in 1922, was rediscovered by Willem in 1981 and by Dancer in 1982.

Some sixty years after the first one, a second periodic solution was discovered in [4] under assumption (2).

Since, by assumption (2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M8">View MathML</a>

a natural space of definition for Ψ is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M9">View MathML</a>

The functional Ψ is bounded from below on X and, by a category argument, has at least two geometrically distinct critical points. A generalization to systems is contained in [5].

The argument in [4] was to use a refinement of the mountain pass theorem, observing that if v is a minimizer of Ψ, then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M10">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M11">View MathML</a>

Another proof, using a generalization of the Poincaré-Birkhoff theorem, was suggested by Franks [6]. However, this proof is not complete [7]. It seems that the variational proof is the only one until now. To find a proof using a fixed point theorem is an interesting challenge. Moreover, there is no exhaustive description of the set of h such that (1) is solvable assuming that f is 2π-periodic and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M12">View MathML</a> (see [3] and [8]).

Let us recall the general notion of ϕ-Laplacian. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M13">View MathML</a> (classical), or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M14">View MathML</a> (bounded), or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M15">View MathML</a> (singular) be an increasing homeomorphism such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M16">View MathML</a>. Canonical examples are, respectively, as follows:

The case of the p-Laplacian for the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M18">View MathML</a>

(3)

was recently solved by Jean Mawhin in [9]. The results are similar to the classical pendulum.

Consider now the forced relativistic pendulum and assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M19">View MathML</a>

(4)

On

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M20">View MathML</a>

we define the action

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M21">View MathML</a>

Let us describe the recent results (2010) of Mawhin and Brezis on the relativistic pendulum [10]. We sum up the simple and beautiful proof.

Theorem 1.1Under assumptions (2) and (4), problem (3) has a solution which minimizes Ψ onC.

Lemma 1.2The action Ψ has a minimizer onC.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M22">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M23">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M24">View MathML</a>, we can assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M25">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M26">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M27">View MathML</a>. By going if necessary to a subsequence, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M28">View MathML</a> uniformly on ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M29">View MathML</a>. It remains only to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M30">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M31">View MathML</a>, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M32">View MathML</a>

so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M33">View MathML</a>

We conclude by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M34">View MathML</a>. □

Let us recall the notion of critical point in the sense of Szulkin [11].

Definition 1.3 Let X be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M36">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M37">View MathML</a> is convex, proper (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M38">View MathML</a>) and lower semi-continuous (l.s.c. in short). A point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M39">View MathML</a> is a critical point of I if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M40">View MathML</a> and, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M41">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M42">View MathML</a>

The easy proof of the next lemma is given in [11].

Lemma 1.4Each local minimum of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35">View MathML</a>is a critical point ofI.

We conclude the proof by using an argument due to Bereanu, Jebelean and Mawhin [12].

Proof of Theorem 1.1 Let u be a minimizer of Ψ on C. We have only to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M44">View MathML</a> in order to verify the Euler equation.

Let us define on

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M45">View MathML</a>

the functionals

By an explicit computation, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M47">View MathML</a>

has exactly one solution w and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M48">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35">View MathML</a> is strictly convex, w is the only solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M50">View MathML</a>

But, by Lemma 1.4, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M51">View MathML</a>

We conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M52">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M53">View MathML</a>

 □

The case of Lagrangian systems of relativistic oscillators was recently treated by Mawhin and Brezis in [13].

An open problem from [10] is the extension in higher dimensions, for example,

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M55">View MathML</a>

2 Convex perturbations of indefinite quadratic functionals

The dual least action principle of Clarke (see [14,15]) was used in [16-19] and [20] to solve problems of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M56">View MathML</a>

(5)

in a closed subspace V of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M57">View MathML</a>, where Ω is a bounded domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M58">View MathML</a>. The linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M59">View MathML</a> is self-adjoint and the nonlinear potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M60">View MathML</a> is convex in its second variable.

It is always assumed that F is dominated at infinity by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M61">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M62">View MathML</a> the first positive eigenvalue of L, and that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M63">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M64">View MathML</a>

(6)

Let us denote by K the inverse of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M65">View MathML</a>

The dual action is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M66">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M67">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M68">View MathML</a> is the Fenchel transform of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M69">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M70">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M71">View MathML</a>. The perturbed dual action is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M66">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M74">View MathML</a> is the Fenchel transform of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M75">View MathML</a>.

It is assumed that K is the sum of a compact and of a positive definite operator. Because of the non-resonance condition with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M62">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M71">View MathML</a> small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M78">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M79">View MathML</a> and has a minimizer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M80">View MathML</a>. It suffices then to use the interaction between F and the kernel of L given in (6) to prove a posteriori estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M80">View MathML</a>. Passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M82">View MathML</a>, we obtain a minimizer v of Ψ and, by duality, a solution u of (5).

Let Ω be a smooth bounded domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M58">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M84">View MathML</a> be a Caratheodory function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M85">View MathML</a>

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M86">View MathML</a> the first positive eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M87">View MathML</a>

We assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M88">View MathML</a>

satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M89">View MathML</a>

uniformly for, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M90">View MathML</a>.

We consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M91">View MathML</a>

(7)

Theorem 2.1[17]

Assume that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M92">View MathML</a>is nondecreasing for almost all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M90">View MathML</a>,

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M94">View MathML</a>on a subset of Ω of a positive measure.

Then problem (7) is solvable if and only if there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M95">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M96">View MathML</a>

A similar result for the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M97">View MathML</a>

is contained in [19].

The general results of [18] are applied to Dirichlet problems, Neumann problems and to periodic solutions of Hamiltonian systems and hyperbolic semilinear equations. In the latter case, the dimension of the kernel of L is infinite. See the survey [21] by Brezis.

It is important to note that the non-resonance assumptions are related to the potential <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M98">View MathML</a>, not to the gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M99">View MathML</a>. In particular, the Palais-Smale condition is not necessarily satisfied.

General non-resonance conditions are used in [22] in order to prove the existence and uniqueness for semilinear equations in a Hilbert space by variational or iterative methods. Applications are given to semilinear wave equations.

3 Two books

We describe some main features of two books by Jean Mawhin devoted to critical point theory.

The book Problèmes de Dirichlet variationnels non linéaires (1986) is a nice introduction to critical point theory. The main tools,

– minimization,

– dual least action principle,

– minimax methods, and

– Morse theory,

are applied to the simple model problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M100">View MathML</a>

A new methodology was used in the construction of Palais-Smale sequences.

Definition 3.1 Let X be a Banach space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M101">View MathML</a>. A Palais-Smale sequence (at level c) is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M102">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M103">View MathML</a>

The Palais-Smale condition (at level c) is satisfied if every Palais-Smale sequence (at level c) contains a convergent subsequence.

Let us also mention the recent survey [23] on the Palais-Smale condition.

As written in the introduction of [24], the usual minimax method

1. prove an a priori compactness condition, like the Palais-Smale condition,

2. prove a deformation lemma depending upon this condition, and

3. construct a critical value,

could be replaced by the following steps:

1. prove a quantitative deformation lemma,

2. construct a Palais-Smale sequence, and

3. verify a posteriori compactness conditions.

The book [16] contains the first application of this methodology, using the quantitative deformation lemma in [25]. (See [26] for another approach using Ekeland’s variational principle in the case of the mountain pass theorem).

The book Critical Point Theory and Hamiltonian Systems (1989) is motivated by the problems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M104">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M105">View MathML</a>

Among many other results, a new bifurcation theorem is given. Consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M106">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M107">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M108">View MathML</a>. If there is some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M109">View MathML</a> such that the critical groups satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M110">View MathML</a>

then there exists a bifurcation point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M111">View MathML</a> for (8).

4 Converse to the Lagrange-Dirichlet theorem

In 1971, Hagedorn proved that, for Lagrangian systems of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M112">View MathML</a>, the equilibrium is unstable if it corresponds to a strict local maximum of the potential energy. The proof, using the theory of geodesics on Finsler manifolds, was rather involved. A new proof is given by Hagedorn and Mawhin in [27].

The idea is to replace Jacobi’s principle of least action by a new variational principle due to van Groesen [28]. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M113">View MathML</a> be the kinetic energy and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M114">View MathML</a> be the potential energy. The functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M115">View MathML</a>

is minimized on the subset

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M116">View MathML</a>

of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M117">View MathML</a> for some suitable c.

5 Neumann problems for the singular ϕ-Laplacian

In this section, we describe some recent works motivated by the Neumann problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M118">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M119">View MathML</a>

The general problem treated in [12,29] and [30] is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M120">View MathML</a>

(9)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M121">View MathML</a> satisfies assumption (4). Let us define

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M123">View MathML</a>.

Then Szulkin’s critical point theory [11] is applicable to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M35">View MathML</a>, since K is a convex l.s.c. function and since J is a differentiable function. The strategy is to prove that a critical point of I in the sense of Definition 1.3 satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M125">View MathML</a> and hence is a solution of (9).

Assume, for example, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M126">View MathML</a>

Then (9) is solvable if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M127">View MathML</a>

or if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/115/mathml/M128">View MathML</a>

The first case corresponds to a ground state of I and the second case to a saddle point of I (see [12]). The case of mountain pass solutions is also treated. The generalization of those results to the non-radial case is a challenging open problem.

Competing interests

The author declares that he has no competing interests.

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