Do not worry about a priori estimates on critical points of unbounded (above + below) functionals: sometimes, it is possible to be happy.

Prof. Lucio Boccardo
Sapienza Università de Roma

Abstract: Let Ω be a bounded, open subset of R^N. In the talk, we will consider some integral functionals. The first two will be

J(v,φ) = ¹⁄_2 ∫Ω|Dv| − ¹⁄_2 ∫Ω|Dφ| + _∫Ωv[E(x)Dφ] − _∫Ωf(x)v.

I(v,φ) = ¹⁄_2 ∫Ω|Dv| - ¹⁄_2 ∫Ω|Dφ| + _∫Ωa(x)φg(v) − _∫Ωf(x)v.

where a, f ∈ L^m, E belong to some Lebesgue spaces and g(t) is an increasing and convex real function. I is related to a Schrödinger-Maxwell system, since its critical (saddle) points are solutions of the system; the existence (thanks to a regularizing effect) of saddle points (u, ψ) of I was presented in some brazilian talks.

We will see how some cancellation properties allow us to prove the existence of u, ψ even with very singular data a, f, E. In this case a very weak definition of minimum (maximum) is needed: the T-minima, introduced by the speaker some years ago and presented in the conference “70-Djairo”.