An Introduction to Critical Points for Intergral Functionals

David Arcoya darcoya@ugr.es    Departamento de Análisis Matemático, Universidad de Granada, 18071-Granada, Spain

Lucio Boccardo boccardo@mat.uniromal.it    Dipartimento di Matematica, Università di Roma 1, Piazza A. Moro 2, 00185 Roma, Italy

1 Introduction

The study of minima of functionals defined in spaces of functions may be considered one of the keystones of the mathematical analysis. Remind the efforts by the great mathematicians of the last and present century to develop sufficient conditions on the functional for the existence of minimum. This theory is deeply related with the existence of solutions of boundary value problems. Indeed, this connection is estabilished by the well-known Euler-Lagrange equations associated to the functional.

However, there exist boundary value problems for which the associated functional is indefinite, i.e. it is unbounded from below and from above. This means that it has not global extrema and so we have to search the solutions of the problem among the critical points, i.e. the points for which the derivative of the functional vanishes.

From the abstract point of view there is a difference between the study of minima and of critical points. Indeed, for the existence of minima we need only assumptions on the functional. On the contrary, we point out that the results of existence of critical points involve additional hypotheses on the regularity of the functional to assure the existence of a derivative in some sense. This may explain why the theory of mimima handles classes of functionals with more general hypotheses of smoothness than the critical point theory.

In some papers [4], [5], [6], we overcame this difference by developing a critical point theory for nondifferentiable functionals. We observe explicitely that our model functionals does not involve similar functions to the modulus. In fact, the nondifferentiability of the considered functionals is due to the introduction of some smooth Carathéodory function A(x, u) (as smooth as you want). Specifically, we consider here functionals J defined in 01,2Ω (Ω ⊂ N open, N > 1) by

v=∫ΩAx,v|∇v|2dx−∫ΩFx,v+dx,v∈W01,2Ω,

with <α≤Ax,z≤β<∞,|A′zx,z|≤γ and f(x, z) ≡ F′(x, z)(derivative respect to z) a subcritical Carathéodory function. Observe that J is only differentiable along directions of 01,2Ω∩L∞Ω, even for smooth functions A (see [11]).

This note is devoted to present the critical point theory developed in [5] (see also [?, ?, ?, ?, ?, ?, ?]) for functionals which are not differentiable in all directions.

2 A mountain pass theorem for nondifferentiable functionals

Our abstract setting for the functionals J that we study is given by the following asumption:

(H)(X, || · ||X) is a Banach space and Y ⊂ X is a subspace which is a normed space endowed with a norm || · ||Y. Moreover, J : X → is a functional on X such that it is continuous in (Y, || · ||X + || · ||Y) and satisfies the following hypotheses:

a) J has a directional derivative 〈J′(u), v〉 at each u ∈ X through any direction v ∈ Y.

b) For fixed u ∈ X, the function 〈J′(u), v〉 is linear in v ∈ Y, and for fixed v ∈ Y, the function 〈J′(u), v〉 is continuous in u ∈ X.

Due to the lack of regularity of the functional, some words are needed in order to establish our definition of critical points.

In this framework a suitable version of the Ambrosetti-Rabinowitz Theorem has been proved in [5]. Specifically,

3 A simple model

The application of the abstract result quoted in the previous section to the study of the functional J defined in (1) is very technical. In particular, the verification of the condition (C). For this reason, we present here a simple but not natural functional which is not differentiable in 01,2Ω, but for which the verification of condition (C) has not technical difficulties like for functionals studied in [5], [6]. Specifically, we consider the functional J defined in 01,2Ω by setting

v=12fΩ|∇v|2dx+1qfΩax,v|∇v|qdx−1mfΩv+mdx,v∈W01,2Ω,}

where 1 ≤ q < 2 < m < 2* (2* = 2N/(N – 2), if 2 < N; 2* = ∞ if N ≤ 2) and a : Ω × → is a function which is measurable respect to x ∈ Ω, C1 with respect to z ≠ 0 and such that

(a1) There exist β > α > 0 such that

≤ax,z≤β,

for almost every x ∈ Ω and z ∈ .

(a2) There exists γ > 0 such that

a′zx,z|≤γ,foralmosteveryx∈Ω,∀z>0,

and a′(x, z) = 0, z < 0.

(a3)...