REMO GARATTINI On the existence of nontrivial solutions for a nonlinear equation relative to a measure-valued Lagrangian on homogeneous spaces

for every x ∈ X , where ν is a positive real number independent of r,R,R0. Such a triple (X, d,m) will be called a homogeneous space of dimension ν. We point out, however, that a given exponent ν occurring in (1.1) should be considered, more precisely, as an upper bound of the “homogeneous dimension”, hence we should better call (X, d,m) a homogeneous space of dimension less or equal than ν. Our setting is given by a couple (X,L), “a homogeneous space X with a Lagrangian L”, with the following properties (L1): L : C 7−→ M (X) is a map which associates with each function u from a given subspace C of C (X) a measure L [u] ∈ M (X), where C (X) denotes the space of all continuous functions on X and M (X) the space of all nonnegative Radon measures on X .