Minimax Theorems

Frontmatter

Introduction

Abstract

Many boundary value problems are equivalent to

$$Au = 0$$

(1)

where A : X → Y is a Mapping between two banach spaces. When the problem is variational, there exists a differentiable functional φ : X → ℝ such that \(A = \varphi ' \), i.e.

$$\left\langle {Au,v} \right\rangle = \mathop{{\lim }}\limits_{{t \to 0}} \frac{{\varphi \left( {u + tv} \right) - \varphi \left( u \right)}}{t}. $$

Michel Willem

Chapter 1. Mountain pass theorem

Abstract

Let us recall some notions of differentiability.

Definition 1.1. Let φ : U → ℝ where U is an open subset of a Banach space X. The functional φ has a Gateaux derivative f ∈ X′ at u ∈ U if, for every h ∈ X,

$$\mathop{{\lim }}\limits_{{t \to 0}} \frac{1}{t}[\varphi (u + th) - \varphi (u) - \langle f,th\rangle ] = 0 $$

.

Michel Willem

Chapter 2. Linking theorem

Abstract

In order to extend the quantitative deformation lemma to continuously differentiate functions defined on a Banach space, we use the notion of pseudogradient defined by Palais in 1966.

Michel Willem

Chapter 3. Fountain theorem

Abstract

When a functional is invariant, we use a more precise version of the quantitative deformation lemma.

Michel Willem

Chapter 4. Nehari manifold

Abstract

Assume that φ ∈ C¹(X, ℝ) is such that φ′(0) = 0. A necessary condition for u ∈ X to be a critical point of φ is that \(\langle \varphi \prime(u),u\rangle \, = 0\). This condition defines the Nehari manifold

$$N\,: = \{ u \in X\,:\langle \varphi \prime(u),u\rangle \, = \,0,u \ne 0\}$$

.

Michel Willem

Chapter 5. Relative category

Abstract

We identify the two dimensional torus T² with the quotient space ℝ²/ℤ². We consider a function φ ∈ C¹(T², ℝ) having a maximum at M and a minimum at m.

Michel Willem

Chapter 6. Generalized linking theorem

Abstract

This section is devoted to the degree theory of Kryszewski and Szulkin.

Let (e _k ) be a total orthonormal sequence in a separable Hilbert space E and define

$$\left| {\left\| u \right\|} \right|: = \sum\limits_{{k = 0}}^{\infty } {\frac{1}{{{{2}^{{k + 1}}}}}\left| {u,{{e}_{k}}} \right|} . $$

Michel Willem

Chapter 7. Generalized Kadomtsev-Petviashvili equation

Abstract

This chapter is devoted to the existence of solitary waves of the generalized Kadomtsev-Petviashvili equation

$${{w}_{t}} + {{w}_{{xxx}}} + {{(f(w))}_{x}} = D_{x}^{{ - 1}}{{w}_{{yy}}} $$

(7.1)

, where

$$D_x^{ - 1} h(x,y): = \int_{ - \infty }^x {h(s,y)ds}$$

.

Michel Willem

Chapter 8. Representation of Palais-Smale sequences

Abstract

In this chapter, we describe losses of compactness in some variational problems. Minimizing sequences were considered by Pierre-Louis Lions in [50] and [51]. Palais-Smale sequences were studied by many authors (see the bibliography of [21]). Of course the Ekeland principle allows a reduction of minimizing sequences to Palais-Smale sequences.

Michel Willem

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