Positive solutions of Schrödinger equations with potentials G-symmetric and vanishing at infinity

We study the existence of a positive solution for a class of nonlinear Schrödinger equations aligned -Δ u+V(x)u=f(u), u ∈ D1,2(RN), \ N≥ 3. aligned - Δ u + V ( x ) u = f ( u ) , u ∈ D 1 , 2 ( R N ) , N ≥ 3 . Here the potential V is symmetric under a group action G ⊂ O(N) G ⊂ O ( N ) and decay to zero at infinity, and the nonlinearity f , under very mild hypotheses, is asymptotically linear or superlinear and subcritical at infinity, not satisfying any monotonicity condition.