Some generalizations of minimax theorems for lower semicontinuous functionals and a new approach for logarithmic Schrödinger equations.

Abstract

The present work is built in two main directions: first,

new abstract theorems are developed for a class of semi-functional functionals continuous inferiorly as follows: given X a Banach space,

I = Φ + Ψ : 1 with a convex and inferiorly semicontinuous functional Ψ : X −→ (−∞, ∞] (Ψ ̸≡ ∞).

Our results refer to the Theory of Critical Points for n ̃ao-functionals. differentiables constructed by Szulkin in [81]; a generalization of the

Bartsch's source theorem [23] and also a theorem due to Heinz in [61] related to the notion of the genre of closed and symmetrical sets with respect to origin. A version of the symmetric mountain pass theorem is also proved. By applying the aforementioned abstract results, the existence of a plethora of solutions to a wide class of elliptical problems. The problems involve logarithmic nonlinearities, discontinuous nonlinearities and the operator 1-Laplacian. Later, as a natural consequence of our studies, we introduced a new approach to the study of logarithmic equations that allows us apply classical variational methods to class C functionals

1 in order to obtain solutions for different classes of logarithmic Schrödinger equations. That new idea is introduced using techniques explored in the study of spaces from Orlicz. The results obtained guarantee results of multiplicity of solutions to logarithmic Schrödinger equations involving the category of Lusternik-Schnirelmann, `the existence of positive solutions to a class of equations log ́ithmics over an exterior domain, considering different boundary conditions.